Lecture Notes in Earth Sciences Edited by Gerald M. Friedman and Adolf Seilacher
2 UIf Bayer
Pattern Recognition Probl...
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Lecture Notes in Earth Sciences Edited by Gerald M. Friedman and Adolf Seilacher
2 UIf Bayer
Pattern Recognition Problems in Geology and Paleontology
SpringerVerlag Berlin Heidelberg New York Tokyo
Author Dr. UIf Bayer Instltut f(3r Geologie und Pal~ontologie der Unlversit~t T0bingen S~gwartstr. 10, D7400 TfJbmgen, FRG
ISBN 3540139834 SpnngerVerlag Berlin Heidelberg New York Tokyo ISBN 0387139834 SprmgerVerlag New York Heidelberg Berlin Tokyo This work is subject to copyright All rights are reserved, whether the whole or part of the material ~s concerned, specifically those of translation, repnntmg, reuse of illustrations, broadcasting, reproduction by photocopying machme or similar means, and storage in data banks Under § 54 of the German Copyright Law where cop~es are made for other than prtvate use, a fee ~s payable to "Verwertungsgesellschaft Wort", Munich © by SprmgerVerlag Berlin Heidelberg 1985 Printed in Germany Printing and binding' Beltz Offsetdruck, Hemsbach/Bergstr. 2132/3140543210
To
Dorothee
Julia and Vincent
Preface
The
research
on
mathematical
methods
and
computer
applications in geology since 1977 was supported by the "Sonderforschungsbereich
53,
Seilacher.
the
volved:
During
PalOkologie" years,
several
"Konstruktionsmorphologie,
gesellschaftungen,
T{ibingen,
directed
"Teilprojekte"
Fossildiagenese,
FossilLagerstatten".
During
the
by
A.
were
in
Fossilverlast
period
of t h e "Sonderforscbungsbereich" a special project " Q u a n t i t a t i v e Methoden
der
serve
a
as
PalOkologie" final
report
was of
the
established:
Chapters
scientific
activities.
1 to
3
Further
i n f o r m a t i o n is available in the r e p o r t s of t h e "Sonderforschungsb e r e i c h 53".
The ideas on the seismic r e c o r d in c h a p t e r 4 arose during a c t i v i t i e s on Leg 71 of the DSDPprogram indebted
for valuable
in 1980, and I am
discussions to W. Gtlttinger, G. Dangel
mayr, D. A r m b r u s t e r , H. Eikenmeier of the "Institut far Inform a t i o n s v e r a r b e i t u n g " , Tt~bingen. During engaged
the
years,
somewhere
in
a considerable the
research
number activities.
of people was Here
I want
to express my special thanks to E. A l t h e i m e r and W, Deutschle, which
were
active
in
programming
problems
during
several
years. Ttlbingen
Ulf Bayer
CONTENT
1. INTRODUCTION 1.1 M a t h e m a t i c a l Geology and A l g o r i t h m i z a t i o n 1.2 Syntax and S e m a n t i c s 1.3 Stability
2. NOISY SYSTEMS AND FOLDED MAPS 2.1 R e c o n s t r u c t i o n of S e d i m e n t  A c c u m u l a t i o n
8
10
2.1.1 A c c u m u l a t i o n R a t e s and D e f o r m a t i o n s of the TimeScale
i1
2.1.2 E s t i m a t i o n of Original S e d i m e n t Thickness
12
2.1.3 Underconsolidation of S e d i m e n t s   a History E f f e c t
16
2.2 I n t r a s p e c i f i c Variability of Paleontological Species
19
2.2.1 A l l o m e t r i c Relationships
21
2.2.2 The ~Ontogenetic Morphospace ~
23
2.2.3 Discontinuities in the Observed Morphospace
26
2.3 Analysis of D i r e c t i o n a l D a t a
29
2.3.1 The Smoothing Error in Two Dimensions
30
2.3.2 Stability of Local E x t r e m a
35
2.3.3 Approximation and Averaging of D a t a
40
2.3.4 A Topological Excursus
45
2.3.5 Densities, Folds and the Gauss Map
46
2.4 R e c o n s t r u c t i o n of Surfaces from S c a t t e r e d D a t a
51
2.4.1 The Regular Grid
51
2.4.2 Global and Local E x t r a p o l a t i o n s
54
2.4.3 Linear Interpolation by Minimal Convex Polygons
56
2.4.4 Stability Problems with Minimal Convex Polygons
57
2.4.5 C o n t i n u a t i o n of a Local A p p r o x i m a t i o n A) A local continuous approximation B) C o n t i n u a t i o n of a local s u r f a c e approximation
62 62 65
3, NEARLY CHAOTIC BEHAVIOR ON FINITE POINT SETS 3.1 I t e r a t e d Maps
70 72
3,1,1 The Logistic D i f f e r e n c e Equation
73
3,1.2 The Numerical Approximation of a P a r t i a l D i f f e r e n t i a l Equation
76
3.1.3 Infinite Series of Caustics
79
3.2 Chi2Testing of D i r e c t i o n a l D a t a
82
Vt
3.3 P r o b l e m s with Sampling S t r a t e g i e s in Sedimentology
86
3.3.1 Markov Chains in Sedimentology A) D i s c r e t e Signals B) Equal Interval Sampling
86 88 89
3.3.2 Artificial P a t t e r n F o r m a t i o n in S t r a t i g r a p h i c PseudoTime Series A) Sampling of periodic functions B) The analysis of 'bed thickness' by equal d i s t a n c e samples
92 92 95
3.4 C e n t r o i d C l u s t e r S t r a t e g i e s   Chaos on F i n i t e Point Sets 3.4.1 Binary Trees
98 99
3.4.2 Image C o n c e p t s
t00
3.4.3 Stability Problems with C e n t r o i d Clustering Centroid cluster strategies Instabilities b e t w e e n c l u s t e r s Instabilities within clusters
103 104 t06 106
3.5 Tree P a t t e r n s b e t w e e n Chaos and Order
112
3.5.1 Topological P r o p e r t i e s of Open Network P a t t e r n s
115
3.5.2 P a t t e r n G e n e r a t o r s for Open Networks A) Algebraic models  prototypes of branching p a t t e r n s B) A m e t r i c model  the Honda t r e e
120 120 123
3.5.3 Morphology of Branches in Honda T r e e s A) Length of b r a n c h e s B) Branching angles  similarity and s e l f  s i m i l a r i t y C) Branches and b i f u r c a t i o n s   a quasicontinuous approximation
127 127 129 131
3.5.4 Evolution of Shape A) Trees, P e a n o and J o r d a n c u r v e s B) The outline of Honda t r e e s C) C h a n c e and d e t e r m i n i s m
134 135 138 141
4. STRUCTURAL STABLE PATTERNS AND ELEMENTARY CATASTROPHES 4.1 Image R e c o g n i t i o n of T h r e e  D i m e n s i o n a l O b j e c t s
144 146
4,1,1 The TwoDimensional Image of T h r e e  D i m e n s i o n a l Objects
147
4.1.2 The Skeleton of P l a n e Figures
151
4.1.3 T h e o r e t i c a l Morphology of WormLike Objects
153
4.1.4 Continuous T r a n s f o r m a t i o n s of Form
155
4.2 S u r f a c e Inversions in the Seismic Record  the Cusp and Swallowtail Catastrophes 4.2.1 C o m p u t e r Simulations of Rays, Wave Fronts and T r a v e l t i m e Records Linear rays Successive wave fronts T r a v e l t i m e record
157 160 160 161 163
4.2.2 Local Surface Approximation
165
4.2.3 Linear Rays, Caustics and the Cusp C a t a s t r o p h e
167
4.2.4 Wave Fronts and t h e Swallowtail C a t a s t r o p h e
172
4.2.5 Wave F r o n t Evolution and the T r a v e l t i m e Record
174
4.2.6 The T r a v e l t i m e Record as a Plane Map
176
VII
4.2.7 Singularities on the R e f l e c t o r Line
179
4.2.8 G e n e r a l i z e d R e f l e c t i o n P a t t e r n s in Two and T h r e e Dimensions A) The d e f o r m e d c i r c l e and the dual cusp B) T h r e e  d i m e n s i o n a l p a t t e r n s  t h e double cusp
183 183 189
4.2.9 D i s t r i b u t e d R e c e i v e r s
191
4.3 "Parallel Systems" in Geology
198
4.3.1 Some Examples of Parallel Systems
199
4.3.2 Similar and Parallel Folds
201
4.3.3 Bending a t Fold Hinges   t h e Hyperbolic Umbilic
206
4.3.4 N o t a t i o n of S t r a i n
210
4.3.5 G e n e r a l i z e d P l a n e Strain in Layered Media
211
4.4 SUMMARY
214
REFERENCES
217
INDEX
226
1.
INTRODUCTION
Theoretical modelling and the use of mathematical in importance
since
progress
methods are presently gaining
in both geology and mathematics offers new possibilities
to combine both fields. Most geological problems are inherently geometrical and morphological,
and,
therefore,
view". Geometrical their
essential
amenable
to a classification of forms from a "Gestalt point of
objects have to possess an inherent
quality under slight deformations.
stability
Otherwise,
in order to preserve
we could hardly conceive
of them or describe them, and today's observation would not reproduce yesterday's result ( D A N G E L M A Y R & GOTTINGER,
1982). This principle has become known as 'structural
stability' (THOM, 1975), i.e. the persistence of a phenomenon under all allowed perturbations. Stability is also, of course, an assumption of classical Newtonian physics, which is essentially the theory of various kinds of smooth behavior (POSTON &STEWART, 1978). However,
things sometimes "jump". A new species with a different morphology appears
suddenly
in the paleontological record (EI.DREDGE & GOULD, 1972), a fault develops,
a landslide moves, a computer program becomes unstable with a certain data configuration, etc. It is, surprisingly, the topological approach which permits the study of a broad range of such phenomena STEWART,
in a coherent manner (POSTON &STEWART, 1978; LU, 1976;
1982). The universal singularities and bifurcation processes derived from the
concept of structural stabiIity determine the spontaneous formation of qualitatively similar spatiotemporal
structures
in
systems
( DANGELMAYR & GI~TTINGER,
of
various
geneses
exhibiting
critical
TINGER & EIKEMEIER, t979; STEWART, 1981). In addition, this return to a tion of p h e n o m e n a "  
after
decades of a l g o r i t h m i z a t i o n  
geologist's intuitive geometric reasoning. examples,
how
the
behavior
1982; THOM, 1975; POSTON & STEWART, 1978; GI21T
It is the
qualitative geometrical
approach
'geometriza
comes much closer to the
aim of this study to elucidate,
by
allows one to classify forms and
to control the behavior of complex computer algorithms.
1.1 MATHEMATICAL GEOLOGY AND ALGORITHMIZATION The geometrical approach dominated the "mathematization" of geology until recently the computer
"changed the world". As VISTELIUS (1976)
summarized
in his discussion
of mathematical geology: "the restoration
(or
18th
of @eoloEical sciences . . . . are,
the more
of the problem.
century)
of
axiomatic
ideas
is due
to maturation
The more mature the geological ideas in the problem
the mathematical
tool
is determined
by the Ecological
meanin E
Less mature geological problems make it necessary to introduce
more routine mathematical means with restricted foundation to form geology".
Here,
two
types
of " m a t h e m a t i c a l
to "model" a specific lead
to
can
the
mostly
widely are
be
of
translated
viewed
treated
object
formulation
as
the
like
application" occur:
the classical
physical
as " s t a t i s t i c a l
"physical
laws",
mathematical
method of t h e o r e t i c a l
laws
"mathematical
The
and "routine
methods".
methods".
a situation
attempt
physics which can
mathematical
means" which
And, case studies by c o m p u t e r
Usually,
strongly
descriptive
criticized
statistical
by THOM
are
results
(1979).
He
gave the following reasons why the tool of m a t h e m a t i c s looses its s t r e n g t h as one goes down the scale of sciences: "...
the
first
is
that
at hand as physical
those sciences
laws
would like
which
do not
as fundamental
physics . . . .
as efficient
tools
to be like p h y s i c s and try to appear in
the eyes of other people as precise as physics. mathematized because it believes
have
Every science wants to become
that way it would be put on the same footin E
The second,
internal
reason now works in the re
verse sense: Inasmuch as a given science does not allow for precise mathematization
it
opens
practically
in that field, because they
indefinite
statistical
hypotheses,
possibility
of buildin E models
specific,
exact
and so
quantitative
workin E
possibilities
on,
and
there
is practically
in situations models . . . .
which actually
And
the
third
is the computer industry's lobby: Every laboratory wants puter workin E even in situations that
you
can
extract
any
to
scientists
make models of all kinds, with approximations,
can
kind
no limit
to the
do not allow for
reason,
of
course,
to have its own com
where a priori there is no reason to believe of
useful
information
out
of
the
things
you
have put into the computer. "
It was not Thom's aim to b l a m e those sciences which are not as precise as physics. R a t h e r
this was d i r e c t e d against the d e g r a d a t i o n of t h e m a t h e m a t i c a l
be t h e r e a s o n is t h a t
topologists like Thom "want q u a l i t i e s  
tool
may
though t h e s e s o m e t i m e s
acquire a f e a r s o m e l y algebraic, even numerical, expression" (POSTON & STEWART, 1978). Of
special
interest
is Thorn's second
argument,
the
indefinite working possibilities.
It
is always a very striking e x p e r i e n c e in applying " c o m p u t e r methods" t h a t some of these methods
allow
furthermore, data.
for various
that
some
and c o n t r a d i c t o r y
methods
Such observations were
interpretations
of the
same
data
and,
can even be influenced by t h e ordering of the input
the s t a r t i n g point to
analyze the q u a l i t a t i v e behavior of
p r o p a g a t e d algorithms in geology.
used
In
geology
as
strategies
sample,
a surface
profiles
are
and
paleontology
of p a t t e r n
statistical
is r e c o n s t r u c t e d
analyzed by
and
approximation
recognition. A density distribution
means
of
from
scattered
statistical
time
data
methods
are
is e s t i m a t e d
generally from
a
points, periodicity p a t t e r n s of
series
analysis,
etc.
Alternatively,
data are sorted, grouped and classified by using factor analysis, cluster or discriminant analysis, and so forth. These are the fields where "routine m a t h e m a t i c a l methods" dominate,
and it is the field where the computer allows one to analyze everything without
regard
to
any a priori scientific meaning and without the formulation of a scientific
hypothesis. During several years of work with the computer, and implementing c o m p u t e r programs
at
the
'Sonderforschungsbereich 53,
was a challenge to a c c e p t become r a t h e r
that
PalOkologie 
University
Tt~bingen',
it
r a t h e r sophisticated p a t t e r n  r e c o g n i t i o n programs may
unstable if some initial conditions, e.g.
the input data,
do not satisfy
the proper conditions, and that it is, in general, not known what the "proper conditions" are. On the other hand, such computer work allowed me to collect and to analyze e x a m ples of instable procedures and problems of i n t e r p r e t a t i o n . A collection of such examples is p r e s e n t e d h e r e t o g e t h e r with the 'qualitative' analysis of instabilities.
1.2 SYNTAX AND SEMANTICS
After
decades
of
algorithmization
in
science
the
computer
provides a
valuable
and indispensable tool. Much work has been invested in c o m p u t e r science to find rules for the verification of program c o r r e c t n e s s . The idea is to solve the programming problem "by
decomposing
subproblems
and
correctly,
and
fied
then
way,
the
overall
then if
the
the
problem
into
precisely
specified
that
each
subproblem
is
verifying solutions
original
if
are
problem
fitted will
together be
solved
in
solved
a speci
correctly"
(ALAGACIC & ARBIB, 1978}.
Thus, it s e e m s not very difficuIt to construct "correct" programs   as far as the syntax
is concerned
(WIRTH,
1972).
The
other
problem, however,
is a s e m a n t i c one:
The meaning of a c o m p u t e r output is not defined  no m a t t e r how c o r r e c t the syntax may be  until the meaning of the input is defined and until the input is consistent with the operations within the algorithm. In the same sense, the formulation of a program is usually not only a s y n t a c t i c problem, as in most cases semantics is initially involved to some e x t e n t . The problem, however, is not r e s t r i c t e d to c o m p u t e r applications in a narrow sense: It occurs whenever "formulas" are applied to data. In addition to the "correctness" of algorithms, t h e r e f o r e , the problem of the c o r r e c t application of algorithms arises furthermore,
and,
the question of how to "control" the computations. These are qualitative
problems because semantics itself is qualitative.
The problem and the necessity of algorithmcontrol in the field of geological applications will be elucidated by a collection of examples. The material is ordered in t h r e e chapters. These a t t e m p t to r e l a t e the observed instabilities with current areas of r e s e a r c h in topological, i.eo geometrical, areas. S o m e t i m e s the examples are only weakly c o n n e c t e d
with the t h e o r e t i c a l introduction to each chapter, as a t h e o r e t i c a l classification is not yet available for finite point sets from which most of the examples arose. However, it will be e l u c i d a t e d that it is commonly a question of the viewpoint   the question what we assume as variables and what as p a r a m e t e r s   if we classify a problem as a d i s c r e t e or
a d i f f e r e n t i a b l e system. Such systems are
commonly accounted whenever ~stability v
problems arise: Branching solutions, i.e. bifurcations, can be d e t e c t e d in many classical procedures: like Chi2testing of directional data, s u r f a c e r e c o n s t r u c t i o n from s c a t t e r e d data points and equal distance sampling in sedimentology. The widely used centroid clustering methods turn out to provide an excellent example of chaotic behavior on finite point sets, Their s t a t i s t i c a l value is strongly questioned because they lack structural stability. Smoothing of directional data on a sphere and the classical C h i 2  t e s t for orientation data further provide e x a m p l e s of a d e g e n e r a t e d bifurcation problem. A b r i e f discussion of i t e r a t e d maps gives a connection to present areas of research. The
application of the
c o n c e p t s of structural
stability and of c a t a s t r o p h e theory
to r e f l e c t i o n seismics provides a classification of structurally stable singularities in two dimensions. The analysis of image inversions in t e r m s of the local curvature of the ref l e c t o r and its depth produces a catalogue of images which allows a detailed, semiquantitative onsite
survey of the t r a v e l t i m e
record. For the geologist it can provide a f r a m e 
work for his qualitative s t r u c t u r a l i n t e r p r e t a t i o n . The and
concept
evolutionary
species or
the
of
structural
problems,
stability
e.g.
"bifurcation" of
narrow sense of this term
the
also provides new
analysis
species.
of
However~
the
insights in paleontological
"morphospace" of
such
models are
paleontological
qualitative
in the
and provide r a t h e r a framework for further analyses which
may t e r m i n a t e in models which can be t e s t e d e x p e r i m e n t a l l y or statistically. M a t h e m a t i c a l details are
ignored as far as possible: The object is to convey the
Vspirit~ of structural stability and r e l a t e d fields, and its application to geological p a t t e r n recognition problems. As far as m a t h e m a t i c s is required, it is kept to a minimal l e v e l examples of various fields are thought to be of more i n t e r e s t
than the m a t h e m a t i c a l
theory which has been summarized in various textbooks.
1.3 STABILITY ~Pattern recognition problems v as used here, cover a wide field of ~deformations t and
VinstabilitiesL Various
types of p a t t e r n
recognition problems 
which are
usually
solved by c o m p u t e r methods   are analyzed in t e r m s of ttopological stability v. The t e r m ttopological stability ~ or ~structural stability t means that the p a t t e r n does n o t drastically change under PRIGOGINE;
a small disturbance (ANDRONOV et 1977).
aL,
1966; THOM,
1974; NICOLtS &
However, p a t t e r n recognition problems may result even if the disturb
t 0
i m
m
0 0
a
~
migration
~
[..) ~
"~
Fig, 1.1: The phylogenetic history of horses: (a) the classical gradual phylogeny a f t e r SIMPSON {1951); (b) the same phylogeny redrawn along a modern time scale. In t e r m s of evolutionary velocities two quite d i f f e r e n t "modes of evolution" are r e p r e s e n t e d by the two figures. However, in t e r m s of p h y l o g e n y the relationship b e t w e e n s p e c i e s   the t r a n s f o r m a t i o n is s t r u c t u r a l l y stable as all pathways remain the same.
ance,
the t r a n s f o r m a t i o n , is s t r u c t u r a l l y stable. Such an example is given in Fig. 1.1. The
phylogeny  the evolutionary history  of horses is one of the most c e l e b r a t e d examples of gradual Darwinian evolution, and to some e x t e n t of directional selection {Fig. 1.1 a). However, if the time scale used by SIMPSON (1951 and others) is replaced by the absolute t i m e scale under c u r r e n t use, the phylogenetic p a t t e r n changes d r a m a t i c a l l y with r e s p e c t to mode and velocity of evolution (Fig,
1.1 b). All significant "evolutionary events" are
now c o n c e n t r a t e d within very narrow t i m e intervals. What does not chang% is the principal
structure
of
the
phytogenetic
lineages,
i.e.
are s t r u c t u r a l l y stable. The ttime~ axis in Fig.
the 1.1 b
ancestordescendant
relationships
is d e f o r m e d like a rubber strip
which is d i f f e r e n t i a l l y s t r e t c h e d without folding   a purely topological deformation. Although this d e f o r m a t i o n the
'semantic'
interpretation
slow
gradual
evolution under
is purely topological and s t r u c t u r a l l y stable, it changes
of
the evolutionary mode.
a longterm
While
Fig.
changing e n v i r o n m e n t , Fig.
1.1 a 1.1 b
indicates a indicates
periods of 'stasis' with little morphological evolution which are i n t e r r u p t e d by short t e r m
intervals of rapid evolutionary change and associated speciation events. A simple, however not trivial, t r a n s f o r m a t i o n of the scale, thus, may t r a n s f o r m a gradualistic picture into a p u n c t u a t e d one (cf. STANLEY, 1979). S t r u c t u r a l stability in a more precise sense can be r e l a t e d to the topological similarity of the " t r a j e c t o r i e s " of a process in this p h a s e  s p a c e (NICOLIS & PRIGOGINE, 1977; HAKEN,
1977). The "internal dynamic" of a process is usually described by d i f f e r e n t i a l
equations
which
usually
depend
on some
parameters.
In many physical
interpretations
these p a r a m e t e r s can be identified with some s t a t e of the e n v i r o n m e n t of the system, i.e. they depend on various kinds of d i s t u r b a n c e acting continuously on the system (HAKEN,
1977). As the system
and/or
its e n v i r o n m e n t evolves, some of these p a r a m e t e r s
can c h a n g e s m o o t h l y or suddenly, and during such
a change
t h e principal behavior of
the system c a n change.
homogeneous ~
~"
A
ed
B strain
stra/n
Fig. 1.2: Stressstrain diagrams of deepsea sediments: a) homogeneous sediments; b) s t r a t i f i e d sediments. Within each sequence c o m p a c t i o n and overload i n c r e a s e ( m o d i f i e d from BAYER, 1983). In physical systems not uncommonly a threshold occurs which, when passed, causes a sudden change of the behavior of the system. The most d r a m a t i c change in a dynamic system is t h a t its t r a j e c t o r i e s in the phasespace change t h e i r topological configuration. By a small p e r t u r b a t i o n of a p a r a m e t e r , the system then d e v i a t e s widely from the initial situation. Fig. t.2 i l l u s t r a t e s this situation roughly by the " t r a j e c t o r i e s " of a s t r e s s  s t r a i n diagram. The e x p e r i m e n t s were p e r f o r m e d with a r o t a t i n g vane (with the vanes i n s e r t e d parallel
to
the
bedding planes
of sediments;
cf. BOYCE,
1977; BAYER,
1983). In the
s t r e s s  s t r a i n diagram  the 'phase plane' of the process  two qualitatively very d i f f e r ent
patterns
of the
were
sediments.
observed
(Fig.
1.2; BAYER,
In homogeneous sediments
1983) depending on the " s t r a t i f i c a t i o n "
the s t r e s s  s t r a i n curves are smooth while
in s t r a t i f i e d two every
sediments
a sudden break
types
are
independent
set
the
trajectories
paction.
However,
sediments,
causes
of
the
evolve
the other
occurs.
Within the
range
compaction
(preloading)
of the
s m o o t h l y and s t r u c t u r a l l y
observed
parameter,
an essential change
in t h e
the
of m e a s u r e m e n t s
stable
lamination
mode of failure:
sediments, with
these
i.e. w i t h i n
increasing com
or s t r a t i f i c a t i o n A very distinct
of the point
of
f a i l u r e a p p e a r s in w e l l  s t r a t i f i e d s e d i m e n t s w i t h a " s u d d e n j u m p " in t h e s t r e s s v a l u e s . T h e r e is no c o m m o n
s e n s e w i t h r e g a r d to t h e t e r m
' s t a b i I i t y ' . As HOCH
S T A D T (1964} n o t e s : it.
"Often to
a
about by
not
problem, the
the
problem tion . . . . nition
is
but
solution
necessary it
is
...
In
feeling
that
should
result
The
seems
word to
be
a
to
important
to
many
physical
small
change
in
a
comparably
stability adequate
.
,A "~'~"
determine
is for
•
~'"
 .
4"t
very
all
.
.~.:...
a
".%
be
the able
problems in
the
small tricky
purposes."
0
.°.~
explicit
solution
to
say
something
one
is
motivated
conditions
of
the
change
in
the
solu
word.
No
one
defi
2.
NOISY
Many
SYSTEMS
geological
and
of some a n c i e n t s t a t e
AND
paleontological
FOLDED
problems
are
related
MAPS to
the
reconstruction
from the present remains. The present state, however, is usually
noisy as various f a c t o r s may have influenced the system during its history. This situation comes
very close
to
the r e c o n s t r u c t i o n
the theory of i n f o r m a t i o n important
role
(YOUNG,
of d e f o r m e d
signals in information theory.
the d i s t u r b a n c e of signals by random 1975).
The
stability
In
(white) noise pIays an
problems of such noisy systems
can be
NOISE
X
Fig. 2.1: D i s t u r b a n c e of signals by random, white noise. The initial signals are well s e p a r a t e d points or sufficiently small circles on the {x,y)plane at t i m e t=0. With increasing time, w h i t e noise is added, and the area increases w h e r e the signals are found with some probability. Where these areas i n t e r s e c t , two signals are in Competition for the r e c o n s t r u c t i o n process.
visualized in a t h r e e  d i m e n s i o n a l
model like Fig. 2.1. An initial signal is c h a r a c t e r i z e d
as a point (or as a sufficiently small circle) on a space plane (x,y). On its way to the r e c e i v e r w h i t e noise is added. As a result, t h e signal is driven out of its original position. When t h e random noise sums up during time, probability
within
a certain
A serious r e c o g n i t i o n problem signals s t a r t
t o overlap.
area
the signal will be found with a specific
surrounds
the
original position of the signal.
occurs when the probabilistic neighborhoods of d i f f e r e n t
Indeed,
r e c o n s t r u c t e d with c e r t a i n t y .
which
a signal found within an i n t e r s e c t i o n area c a n n o t be
The signals, discussed so far, disjunct areas. their
initial
are
isolated points which are originally located in
Now, if we cover parts of the (x,y)plane densely with signals so that
areas
of definition are c o n n e c t e d along boundary lines, then another way
to formulate the recognition problem is more appropriate. The evolution of the signalspace along the time axis can be described as a map
source
>
receiver.
The overlapping of the probability areas, in which a signal will be found {Fig. 2.1}, can then be described as local folding of the original definition space. Fig. 2.2 illustrates the local folding of the (x,y)plane. Within the folded areas it is not possible to solve
"
7
::.'..
a Fig. 2.2: A folded sheet as a model of a locally folded map (a). A continuous curve on a sheet may develop s e l f  i n t e r s e c t i o n s in the projective plane of a folded sheet (b).
uniquely the inverse problem, the reconstruction of the 'original signal'. Fig. 2.2b
illus 
t r a t e s the distortion which can be caused by a local fold. A regular curve on the original plane develops a s e l f i n t e r s e c t i o n on the projection of the local fold. However,
deforma
tions of this type will be discussed in more detail in the last chapter.
In an i n f o r m a t i o n  t h e o r e t i c approach the disturbance of signals is due to random forces, and the reconstruction process is mainly a s t o c h a s t i c problem. The possible drift of signals or particles is governed by a probability d i s t r i b u t i o n   the classical example is the Brownian m o v e m e n t of particles in a fluid. In geology and paleontology problems of
this
type arise
mainly if global properties of distributions are
r e c o n s t r u c t e d from
s t o c h a s t i c samples by local e s t i m a t i o n methods. In this first chapter, p a t t e r n recognition problems will be elucidated by examples which are related to the superposition of density functions and to double (multiple) valued local solutions of r e c o n s t r u c t i o n processes.
In the first example s t a t i s t i c a l problems are discussed in t e r m s of the r e c o n s t r u c t i o n of original sediment volumes and sediment accumulation rates. The system bears three
10
types of disturbances: U n c e r t a i n t y about the datum points; s t o c h a s t i c components, and a s y s t e m a t i c trend to underconsolidation of s e d i m e n t s with low overburden.
Then the analysis of ' i n t r a s p e c i f i c variability of paleontological species ~ is discussed in t e r m s of a probabilistic o n t o g e n e t i c morphospace. It will turn out that the covarianee s t r u c t u r e b e t w e e n (measurable) features within the o n t o g e n e t i c morphospace can be helpful for the taxonomist. But, we will see further t h a t one cannot e x p e c t linear relationships
between
the
features
a
nonlinear
theory
seems
appropriate
rather
than
just
an analysis by the s o  c a l l e d higher s t a t i s t i c a l methods.
In a qualitative discussion of the analysis of directional data the problem will be to find an optimal weighting function for the r e c o n s t r u c t i o n of a smooth density distribution from s c a t t e r e d data. It will b e c o m e c l e a r that the critical areas of the
reconstruc
tion are i n t e r s e c t i o n s of the areas~ on which the weighting function is defined,
In the next example the r e c o n s t r u c t i o n of s u r f a c e s from sparse point p a t t e r n s by c o m p u t e r methods is discussed. It will be shown t h a t the problems which arise in this c o n t e x t are mainly of a g e o m e t r i c a l nature. Therefore, the algorithmization of the reconstruction process is not trivial. The local e s t i m a t i o n methods in use provide no unique solution,
i.e.
they
are
very
sensitive to
example, t h e r e f o r e , leads over to
the
small
changes of the
next c h a p t e r
initial conditions. The
where this type of instabilities is
discussed in more detail.
2.1 RECONSTRUCTION OF SEDIMENTACCUMULATION
The problem to r e c o n s t r u c t accumulation and s e d i m e n t a t i o n r a t e s arises in sedimentology
in order
to
gather
information
about
sealevel changes, c l i m a t i c changes, and
the evolution of basins. F u r t h e r m o r e , accumulation
and s e d i m e n t a t i o n r a t e s clearly indi
c a t e hiatuses in the s e d i m e n t a r y sequence on the base of which local and global e v e n t s of the past are recognized (e.g. VAIL e t aI., t977). However, several processes and assumptions are involved in the r e c o n s t r u c t i o n of which the most important ones are ** the dating of the sediment sequence ** the e s t i m a t i o n of the original sediment thickness without compaction. Both r e c o n s t r u c t i o n s are biased and usually involve specific assumptions about the datum points, the initial porosities and the consolidation s t a t e of the sediments.
11
2.1.1 Accumulation R a t e s and Deformations of the TimeScale
The
computation
of
accumulation
rates
requires
e s t i m a t e s of
the
absolute
time
scale, i.e. a sufficient number of datum points along the sediment column. As soon as the datum points are given, the computation of the accumulation r a t e s is rather simple,
age
stages
accumulation rate cm/kyr 0
I
2
3
accumul, r a t e cm/kyr
4
t 14o [ K I M M E R I D GIAN
/
_1
stages
.J
r
KIMMERID GIAN
t
! L ........
OXFORDIAN 15o
I
OXFORD,
__J
I. . . . . . . . . . . . . .
.,, '
[]
m CALLOV0
CALLOVIAN
HI
m BATHON.
160
BATIqONIAN
l___ BAJ OCIAN I70
I
I
tl.,.
AALENIAN
AALENIAN TOARCIAN  180
BAJOCIAN
TOARCIAb
PLIENSBACHIAN
PLIENSBACHIAN
S1NEMURIAN

190
1
HETTANGIAN
~
t
SINEMUR.
a
HETTANG.
b
Cyclic accumulation rates of sediments in the South German Jurassic. on the time scale of VAN HINTE (1976); b) based on an a l t e r e d Jurassic time scale {see t e x t for explanation). On both scales a cyclieity is obvious, however, in (b) the cycles are much more regular (about 4 Ma), and a superimposed megatrend appears.
12
i.e. it is the quotient s e d i m e n t thickness time interval of d e p o s i t i o n . Fig.
2.3 gives such a c c u m u l a t i o n
rates
how
they
sequence
subdivide
the
Jurassic
for the South G e r m a n Jurassic and i l l u s t r a t e s into
generic
depositional
cycles.
However,
the p a t t e r n is not i n v a r i a n t against d e f o r m a t i o n s of the t i m e scale. Essentially the same situation arises as was discussed for the revolution of horses v if the t i m e scale is a l t e r e d (Fig. 2,a A,B). The
accumulation
rates
in Fig.
of VAN HINTE (1976). However,
2.3A have
been c a l c u l a t e d
using the
time
scale
for t h e t i m e  i n t e r p o l a t i o n within stages a more r e c e n t
b i o s t r a t i g r a p h i c subdivision (COPE e t al., 1980) was used (McGHEE & BAYER, in press}. A deformation
of
this t i m e
scale
2.aB) a l t e r s the cyclicity drastically, and the
(Fig.
p a t t e r n b e c o m e s much more regular. In addition, a well pronounced m e g a c y c l e b e c o m e s visible (Fig. 2.3 B). Van H i n t e ' s Jurassic t i m e scale is based on e s t i m a t e s of the upper and lower boundary of the Jurassic and of one additional ' c a l i b r a t i o n point v at the middle of the Jurassic (base of the Bathonian). Between t h e s e points he divided the t i m e scale linearly by the n u m b e r of a m m o n i t e Z o n e s  
with the result of an a v e r a g e duration t i m e of IMy for
e a c h a m m o n i t e Zone. Now, since he published his t i m e scale, the b i o s t r a t i g r a p h i c s c h e m e has been altered,
and t h e r e f o r e t h e duration t i m e varies from Zone to Zone. However,
by shifting the additional  middle Jurassic  c a l i b r a t i o n point into the Callovian the original
assumption
of
1My/ammonite Zone can be r e s t o r e d
for the Lower and Middle
Jurassic. This has been done in Fig. 2.3B, and this simple t r a n s f o r m a t i o n g e n e r a t e s the exceptional phase
cyclic
length
(cf.
pattern
which
agrees
EINSELE or McGHEE
with
otherwise
established
cycles
of
& BAYER in BAYER & SEILACHER,
similar eds.,
in
press}. As discussed earlier for the evolution of horses, the cyclicity per se is a s t r u c t u r al stable p a t t e r n which is preserved under topological t r a n s f o r m a t i o n s of the t i m e scale (even if o t h e r proposed scales are used, e.g. HARLAND e t al., 1978), while the regularity of the cycles, can
gather
t h e i r phase length
information
about
the
and
magnitude
velocity
of the
i.e. all p r o p e r t i e s process
change
from which we as the
scale
is
changed.
2.1.2 E s t i m a t i o n of Original S e d i m e n t Thickness One i m p o r t a n t process ~which a l t e r s the physical properties of sediments~ is
compac
13
Sediment Coml~sition
WaterContent(w) andPorosity(p)
Bulk Density
{%)
(%)
(g/cm3)
Computed Grain Density (g/cm3}
Sonic Velocity (m/s)
0 20 40 60 80 100 20 40 60 80 1.4 1.8 2.2 2.4 2.7 3.0 1.4 2.0 2.6 I I ¥ ] I I r"
c~ 4OO 5 0 0 ~ ~
fi
1
Fig. 2.4: Depthlogs for sediment composition and physical properties for DSDPsite 511 (adapted from BAYER, 1983). D a t a are m e a n values for cores (D: diatoms, N: nannofossils, c: clay c o n t e n t , O: o t h e r components).
tion under the overburden of l a t e r deposits. Especially in clays the physicochemical evolution is dominated by compaction.
Fig. 2.4 illustrates how the physical properties in a
s e d i m e n t column change with depth {i.e. overburden). In the example given, the s e d i m e n t column below 200 m depth is dominated by clay, parameters
porosity
and w a t e r  c o n t e n t
increasingly c o m p a c t e d .
decrease
and within this column the physical
continuousIy as
the s e d i m e n t
In the same course the density of the sediment
becomes
increases and
tends slowly towards the m e a n grain density of the sediment. tf one assumes
t h a t the void volume of the s e d i m e n t s is in equilibrium with t h e
overburden, then a first approximation
for the equilibrium c u r v e of c o m p a c t i o n can be
given by t h e equation dn dp
{where n: porosity
+
= r e l a t i v e void space;
cn
=
0
(2.1)
p: pressure or overburden). The c o n s t a n t
'c'
c a n be i n t e r p r e t e d as a c o e f f i c i e n t of volume change (TERZAGHI, 1943), which is specific for p a r t i c u l a r materials. I n t e g r a t i o n gives a simple declining exponential function n
=
no
exp
(cp).
(2.2)
14 n
1.o
0.8
0.5
0.,~
02 m
0.0 0
100
200
300
ZOO
500
500
200
800
Fig. 2.5: Porosity data from Fig. 2.4 with least square f i t t e d trend lines (see t e x t for explanation).
This model has been used in Fig. 2.5 to e s t i m a t e t h e
decline in porosity with r e s p e c t
to depth (i.e. t h e c h a n g e in bulk density has been neglected, cf. Fig. 2.4). Empirically the
data
are
well
approximated.
Besides the mean
trend
line some
more
trajectories
are given in Fig. 2.5~ which have been c o n s t r u c t e d under the assumption t h a t the coeffic i e n t of volume change is c o n s t a n t while the initial porosity of the s e d i m e n t s may have been variable and, thus, cause the s c a t t e r i n g of the d a t a points.
If we assume
that
the s e d i m e n t
it is no problem to e s t i m a t e without
compaction
(e.g.
is e v e r y w h e r e
in equilibrium with
the overload,
the thickness of the s e d i m e n t column which would result
MAGARA,
1968;
HAMILTON,
1976). Fig,
2.6
illustrates how
t h e two principal c o m p o n e n t s of a s e d i m e n t change under pure compaction: The volume
vn vs Fig, 2.6: The two c o m p o n e n t s of a s e d i m e n t   voids and s o l i d s   during compaction. Vn: volume of voids, Vs: volume of solids, p: pressure = overload.
15
of solids remains constant while the volume of voids decreases. The porosity is defined as the r e l a t i v e volume of the voids so that
V n = nV
where n: porosity,
and
Vs= ( 1  n ) V .
(2.3)
Vn: volume of voids, and Vs: Volume of solids. Because the volume
o f the solids is not changed by compaction, one has
Vs(t=O ) = Vs(t ) and, t h e r e f o r e , = (ln)V
(lno)Vo
from which we i m m e d i a t e l y have the compaction number C =
V
1no
Vo

Vo V

1,n
(2.4)
and the decompaction number D
The
decompaction
number
=
1
ln 1no

allows to compute
C
'
the
(2.5)
original
thickness of
any
sediment
layer if we know its original porosity no, i.e.
V o
=
Dr.
(2.6}
The thickness of the entire sediment column then can be c o m p u t e d by summing up all sediment layers or, if regression curves are used as in Fig. 2.5 by evaluation of the integral !n
i=l lno
V
i
or
f zln(z)
zo 1no
dz
(2.7)
Both techniques have been used in Fig. 2.7 whereby the original porosities of the samples have been e s t i m a t e d from the intersection of their associated t r a j e c t o r y (from Fig. 2.5) with
the
z e r o  d e p t h line.
In a s t a t i s t i c a l sense the t r a j e c t o r i e s of Fig. 2.5 are error
bounds to the mean regression line (probabilities can be a t t a c h e d to them by standard s t a t i s t i c a l techniques}, and so the curves in Fig. 2.7 can be interpreted. Thus, the reconstruction of the original sediment amount
is simply a s t a t i s t i c a l process. However, it
closely r e s e m b l e s the situation of Fig. 2.1. The data are biased by the sampling t e c h nique as well as by the laboratory technique. Now, if we add a small error to a data point, it will not a f f e c t the results much if the overburden (or depth) is small. However, as the overburden increases, the t r a j e c t o r i e s in Fig. 2.5 c o m e closer and closer. An error of the same magnitude, t h e r e f o r e , biases the results increasingly.
16
thickness
15
km t4
n= 09
085
/
1.2
0.8 1.0
075 0.8 0.6 0.4 02
depth
0.0 o
1oo
20o
3oo
4oo
500
6o0
zoo m
Fig. 2.7: D e c o m p a c t e d s e d i m e n t thickness based on the data of Fig. 2.5 (m: mean trend, n=0.9 etc.: integrals of the trend lines in Fig. 2.5.
While the r e c o n s t r u c t i o n of the initial s t a t e of the s e d i m e n t s depends on how far t h e " c o m p a c t i o n machine was run", we can, on the o t h e r hand, atways find the output if the " m a c h i n e would work until infinity". This stable limit, of course, volume of solids,
and the "dry s e d i m e n t a t i o n r a t e s " (of. S W I F T ,
is simply the
1977) are,
of course,
only biased by t h e t i m e scale and the laboratory technique.
2.1.3 Underconsolidation of S e d i m e n t s  a History E f f e c t Estimations that
of t h e original s e d i m e n t volume are usuatly based on the assumption
the consolidation s t a t e
of the s e d i m e n t
is in equilibrium with the overburden.
In
this case, as was pointed out, the e r r o r of the e s t i m a t i o n should increase with increasing overburden. However, if burial depth is small, then the t i m e  d e p e n d e n t flow of the porew a t e r c a n n o t be neglected; it bears on our understanding of the widespread underconsolidation of r e c e n t sediments, which is observed even under slow s e d i m e n t a t i o n r a t e s (MARSAL & PHILIPP, 1970; EINSELE, 1977). The consolidation of s e d i m e n t s is described by T e r z a g h i ' s model (e.g. TERZAGHI, 1943;
CHILINGARIAN
onedimensional
& WOLF,
sediment
column
eds., and
1975, under
1976; the
of consolidation T e r z a g h i ' s model takes the form
DESAI
assumption
& CHRISTIAN, of
a constant
1977). In a coefficient
17 3p =
a2p
 
m
(2.8)

at
3x 2 '
where p: the excess pore w a t e r pressure due to overload, m: the consolidation c o e f f i cient, and t: time. If this model is discretisized in space, i.e. if the sediment column is divided into small d i s c r e t e e l e m e n t s , then the partial
differential equation is trans
formed into a set of ordinary differential equations: dP x
d~
= m(PxAx
2Px + Px+fXx)"
(2.9)
Now, if one reduces the system to a single e l e m e n t  a situation which occurs in laboratory e x p e r i m e n t s   then we can rewrite equation {2.9) as dp +
cp
= i(t),
(2.10)
dt where the right side describes the " i n p u t "   i.e. the f l u x e s   at the boundaries of the e l e m e n t as a
function of
time,
and with
free boundary conditions (I(t)=O) a suddenly
imposed pressure declines exponentially with time.
The idea of Terzaghi's model is that a sudden imposed load increases initially the p o r e  w a t e r pressure (excess hydrostatic pressure) and that this pressure d e c r e a s e s a f t e r ward due to a loss of p o r e  w a t e r from the e l e m e n t whereby the excess hydrostatic pressure is t r a n s f o r m e d into a pressure at grain c o n t a c t s . Associated with the loss of porew a t e r is an increase in the number of grain c o n t a c t s . Therefore, the sediment approaches a new equilibrium
state
a f t e r compression which, of course, is usually not reversible.
The reduction of volume is r e s t r i c t e d to the volume of voids, and the change in pore volume is simply proportional to the decline in the excess hydrostatic pressure:
~n ap __dz = m   d z at ~t
.
(2.11)
Thus, we can solve equation (2.10) in terms of the pore volume, which in case of free boundary conditions takes the form: V(t)
= Ve+
(Vo V e) e
ct
(2.12)
for a load which is suddenly applied. The load is here r e p r e s e n t e d by the equilibrium volume Ve (cf. equation 2.2), and the excess hydrostatic pressure is proportional to the reducible void volume (VoVe). Now, if at time t=t 1 an additional load is applied, then equation (2.12} takes the form
V(t)
= Ve2+
(V(t I)  Ve2)
eC(ttl)
,
(2.13)
18
which c a n be r e w r i t t e n if V(tl) is i n s e r t e d from e q u a t i o n (2.t2): (2.14)
V(t) = Ve2+ (Vel Ve2)eCt2 eat+ (VoVel)e ct " As this equation
shows,
there
is some
remaining
reducible porevolume
from
the first
loading e v e n t , which has to be t a k e n into consideration. If f u r t h e r load is added in disc r e t e steps, we arrive finally at I"I
V(t)
= Vi +
( [
(Vi_lVi)eCti)e
(2.15)
ct
±=1 which i l l u s t r a t e s how t h e e a r l i e r loading s t a t e s c o n t r i b u t e to l a t e r s t a t e s . The equilibrium can only be approached if t h e t i m e intervals b e t w e e n Ioading are sufficiently long, o t h e r wise the s e d i m e n t layer will be underconsolidated. This history e f f e c t of loading is illust r a t e d in Fig. 2.8 for various t i m e intervals b e t w e e n loading events. The excess h y d r o s t a t ic pressure
{p in Fig.
3) develops clearly
a maximum
which d e g e n e r a t e s
to a simple
declining exponential function for a single loading e v e n t and to a sequence of such single events, as t h e t i m e intervals b e t w e e n loading b e c o m e large.
10ad. ~ t
i
i,
i
i
1
i
j iiiiliiiii
V
\ m,.. i
k
t t
1
t
Fig. 2.8: Responce of a single s e d i m e n t layer under stepwise loading when loads are apptied a t d i f f e r e n t t i m e intervals: t i m e i n t e r v a l d e c r e a s e from left to right; right: a single load. V: void volume; P: m o m e n t a r y reducible void volume which will vanish even if no additional load is applied; C:equilibrium void volume for e v e r y loading e v e n t . A t the top the loading i n t e r v a l s are marked, the t o t a l applied load is c o n s t a n t for all ' e x p e r i m e n t s ' .
With r e s p e c t to the previous discussion we have, t h e r e f o r e , to e x p e c t t h a t e s t i m a t e s of original s e d i m e n t volume are biased by the t i m e  d e l a y s in the consolidation process, the p a r a m e t e r s t i in equations (2.13) and (2.14) have, of course, the s t r u c t u r e of a t i m e delay.
Furthermore,
if the pressures
at
t h e boundaries of t h e s e d i m e n t layer are not
zero, i.e. if t h e s e d i m e n t layer is a s e g m e n t within a s e d i m e n t column, then the t i m e 
19
delay e f f e c t
increases further.
In case,
the permeability of the s e d i m e n t is low, the
excess h y d r o s t a t i c pressure will stay for r a t h e r long t i m e near the values of the o v e r burden, and the time lack b e t w e e n loading and equilibrium consolidation causes a continuation
of p o r e  w a t e r flow when sedimentation has stopped. On the other hand, if we
consider
a
two
or
threedimensional system
of strongly underconsolidated sediments,
any spatial disturbance like unequal loading can initialize an instable flow of porewater, which may lead to fluidization or Iiquidization of the upper sediment layers.
2.2
INTRASPECIFIC VARIABILITY OF PALEONTOLOGICAL SPECIES
~n 1966,
WESTERMANN observed that
in several a m m o n i t e stocks  a group of
cephalopods (Fig. 2 . 9 )   a specific intercorrelation of morphological f e a t u r e s occurs: ~Of
particular
tion, whorl ferent,
interest
section,
unrelated
explained"
and
is
ammonoid
(WESTERMANN,
the
coiling
intercorrelation which
stocks
has
and
between
been
cannot
observed he
costain
dif
satisfactorily
1966).
Fig. 2.9: E c t o c o c h l i a t e cephalopods, left r e c e n t Nautflus and two ammonites with well marked o n t o g e n e t i c changes in morphology.
Because BUCKMAN (1892) observed, probably for the first time, this particular type of covariation (intercorrelation) b e t w e e n the ornament and the whorl section in ammonites, Westermann a n c e'. "in
named
general
portion
This caused in this way, would
this
relationship
'B u c k m a n' s
1 a w
o f
c o v a r i 
In some cases, the 'covariance' extends to other features: the
complexity
to the d e c r e a s e
Westermann
the
sutureline
increases
in
pro
of o r n a m e n t "
to establish 'Buckman's second law of covariance'. Proceeding
any correlation between
lead to a new
of
features, which cannot be satisfactorily explained,
'law', and 'experiments' with other ammonite
disprove the specific correlation sufficiently 'to be a law'.
stocks would soon
20
,I
1 D 18
o
• e •
• •
e •
e
• •
•
• •
,
ee,~
•
•
•
e e e •
•
• •
• ,
• •
o
e •
•
•
e
1.0
•
Ib
'
50'
~~
~~~~
j
DSP. 100
mm
26o
Fig. 2.10: C o v a r i a t i o n of o r n a m e n t and c r o s s  s e c t i o n of Sonninia (Euhoploceras) adicra (Waagen), modified from WESTERMANN (1966). The s c a t t e r g r a m shows t h a t the morphotypes cover a continuous area in t h e p a r a m e t e r space; D: Raup's morphological p a r a m e t e r "ratio of whorl height to whorl width", DSP: end d i a m e t e r of the spinous stage. On the o t h e r hand, W e s t e r m a n n was able to show, by means of the e o v a r i a t i o n structure, t h a t 80 described species of the subgenus species
and
that
His b i o m e t r i c a l
Sonninia
the observed v a r i a b i l i t y must study
{cf. Fig. '2.10) shows
fill a continuous area
in t h e p a r a m e t e r
be viewed
that
the
space (DSP,
belong
(Euhoploceras)
as an i n t r a s p e c i f i c
specimens I/D)
to
a
single
property.
of this lumped species
and t h a t
the c o s t a t i o n
types
or ' f o r m a ' are regularly a r r a n g e d within this p a r a m e t e r space (of. Fig. 2.10 for explanation of p a r a m e t e r s ) . The c o v a r i a t i o n p a t t e r n described by WBuckman's law w is not unique within t h e a m m o nites, but
it is also not
universal.
Additional
studies
(e.g.
BAYER,
t977)
show t h a t
in
some cases Wage e f f e c t s w may play some role and t h a t t h e r e are some special conditions which m a k e ' B u c k m a n ' s law' easily visible. One of t h e s e conditions is t h a t t h e morphology changes
strongly
during
ontogeny
(cf.
Fig.
2.9
for
cases
of
rather
strong
ontogenetic
changes). The available i n f o r m a t i o n m a k e s it likely t h a t t h e observed c o r r e l a t i o n is due to oblique s e c t i o n s
through the o n t o g e n e t i c
morphospaee because t i m e is not accessible.
The problem t h a t age is not available in paleontology is well known; GOULD (1977} discusses in detail the problems, which arise, if equal sizes but d i f f e r e n t ages of specimens (and species) are c o m p a r e d by the a l l o m e t r i c relationship.
21
Evidence for an age control of 'Buckman's law' comes from additional f e a t u r e s of the s h e l l s  
the spacing of growth lines and s e p t a  
which both are
likely formed in
r a t h e r regular time intervals. Especially the spacing of septa (which is more easily lyzed) shows a
close
(BAYER,
1977).
1972,
relationship Fig. 2.11
to
crosssection
and
sculpture
in
certain
ana
ammonites
illustrates such a relationship b e t w e e n spacing of septa
and shell morphology.
:g
S
70°
5o
3o
to
r
0:2
l
oL5
"t
i
5
i
20 m m
lo
Fig. 2.11: Relationship b e t w e e n spacing of septa and morphology in a m m o n i t e s (modified from BAYER, 1972). s: angular distance of septa; r: radius of the shell.
2.2.1 Allometric Relationships If one a c c e p t s the hypothesis that 'Buckman's law' describes a phenomenon of intraspecific variation, we should be able to deduce it from more basic biological principles. Everyday e x p e r i e n c e on living organisms shows that with
age
and
that
most morphological features change
the relationship b e t w e e n two morphological f e a t u r e s
(which can be
quantified) leads usualiy to an allometric relationship, i.e. a relationship of the form: y = ax b
Actually, 'f i r s t
the
allometric
p r i n c i p 1e s
or
log(y)
relationship o f
can
=
be
g r o w t h'
log(a)
traced
+
{2.16)
bx.
further
down
to
the,
say,
(HUXLEY, 1932), The term 'first
22
principle' is here used in t h e sense t h a t it is very likely to observe such an a l l o m e t r i c relationship.
As HUXLEY
noticed,
two
measurements
(organs etc.) are in an a l l o m e t r i c
relationship when they both grow exponentially, i.e. let Yl' Y2 be the two m e a s u r e m e n t s , which grow exponentially Yl = a l e e l t ;
y2 = a 2 e C 2 t ,
(2.17}
then by e l i m i n a t i n g t i m e we find the a l l o m e t r i e relationship = Yl Now,
strictly
allometric
(Y2) cl/c2 ~'2
al
growth
results
also
"
in more
(2.18) sophisticated
growth
models
like
the " O o m p e r t z model". In this model one assumes t h a t the p a r a m e t e r ' c ' is not c o n s t a n t but d e c r e a s e s with age. Growth then can be described by a pair of d i f f e r e n t i a l equations dy d't + c ( t ) y
= 0
dc dt = c .
and an equation like
(2.t9)
The growth p a r a m e t e r ' c ' can be any function of t i m e , which goes to zero for large t i m e values {ideally as t i m e approaches infinity). Especially, any s t a b l e output of a linear control system
(e.g. homogeneous linear d i f f e r e n t i a l equations) provides a possible input for
the growth p a r a m e t e r .
A perfect
a l l o m e t r i c relationship results w h e n e v e r the two organs
under consideration are controlled by the s a m e mechanism, i.e. if t h e i r growth equations take the form: dy dt
c(t)*ay
dx d't c ( t ) * b y
= O;
= O;
(2.20)
by e l i m i n a t i n g t i m e one finds the p e r f e c t a l l o m e t r i c relationship
dy ay dx  b x
or
y = XoX a / b .
In both cases considered so far t h e a l l o m e t r i c relationship describes the relationship
be
t w e e n two growing organs in the phaseplane, i.e. the t r a j e c t o r i e s of growth without cons i d e r a t i o n of t h e velocity of growth. Indeed, we may still f u r t h e r generalize the r e l a t i o n ship to pairs of linear d i f f e r e n t i a l equations like dy f(t)~~ = ax + by;
dx f ( t ) ~ T = cx + dy,
(2.21)
and the relationship b e t w e e n the two m e a s u r e m e n t s takes the form dy dx
ax + by =
cx
+
dy
(2.22)
23
(a + d)
ters Fig.2.12: Relationship b e t w e e n type of equilibrium a n d c o e f f i c i e n t s of a pair of first order d i f f e r e n t i a l equations {equation 2.21}. The type of equilibrium depends on the eigenvalues of the c o e f f i c i e n t m a t r i x of equation 2.21. The eigenvatues are given by the root ),l,)t2 = { (a+d) +/((a+d) 2  4(adcb)) }/2 (e.g. HOCHSTADT, 1964; JACOI3S, 1974; HADELER, 1974).
which provides
allometric
relationships
for a wide
range
of p a r a m e t e r
values
(cf.
Fig.
2.12). Huxley's approximation
allometric of
growing organism.
the
relationship,
relationship
However,
there
therefore,
between
appears
growing organs
as
a
rather
likely
or m e a s u r e m e n t s
first
order
taken
on a
are numerous exceptions especially in ontogeny. Such
an example is given in Fig. 2.13  the nonlinear o n t o g e n e t i c trend in a Paleozoic a m m o nite which, however, c a n be approximated by a l l o m e t r i c relationships in d i f f e r e n t intervals.
2.2.2 The tOntogenetic Morphospace f If one picks individuals of a c e r t a i n age class from a species~ then the morphological
24
r
lo
I
2
i
i
4
t
I
i
6
t
I
8
whorl N °
Fig. 2.13: Nonlinear o n t o g e n e t i c relationships in a Paleozoic a m m o n i t e which can be stepwise approximated by simple a l l o m e t r i c relationships (modified from KANT & KULLMANN, 1980).
f e a t u r e s show usually a typical intraspecific variability, and in most c a s e s the d i f f e r e n t f e a t u r e s are c o r r e l a t e d within every age class, e.g. size and weight are c o r r e l a t e d and can be described by a twodimensional Gaussian distribution for every age class. In the most simple c a s e one needs two sources of variation to describe the o n t o g e n e t i c mophospace of a species:
a) for every age class a description of the variability of all f e a t u r e s under consideration
and
their
covariances.
As
a
first
approximation
time sections through the o n t o g e n e t i c morphospace are
one can
assume that
the
multidimensional Gaussian
distributions; b} a description how the
mean
of t h e s e distributions moves with
increasing age
through the morphospace. This gives a c h a r a c t e r i s t i c {mean) o n t o g e n e t i c t r a c e for the
entire
species 
for
measurements,
the
mean
{multidimensional)
allometric
relationship. Fig. 2.14 illustrates this description of the morphospace whereby the ' m e a n o n t o g e n e t ic t r a c e ' is approximated by a straight line {e.g. an ideal allometric relationship in
loga
rithmic coordinates), and the age sections are idealized as ellipsoids {ideal Gaussian distribution). It is obvious that this description cannot be used only for continuous o n t o g e n e t i c d e v e l o p m e n t {as in the a m m o n i t e example), it also holds for growth in finite steps like in c r u s t a c e a . Thus, this kinematic model provides a relatively general description of the
25
l m
,
•~~o m
•
•
•
•
•
O
t,: @@
;
'~:" 4 , • . • •
Oj
IU
O
N'," . 

$ Fig. 2.14: A linear modeI for the ' o n t o g e n e t i c morphospace' of a species. The variables u,v,s are p a r a m e t e r s or m e a s u r e m e n t s which c h a r a c t e r i z e the morphology. The ellipsoids are time sections, i.e. they are the probability distributions for a c e r t a i n age cIass. They are dislocated within the (u,v,s)space with t i m e t e i t h e r continuously or in d i s c r e t e steps. The hull of t h e s e ellipsoids {in the linear model a cone) is the probabilistie boundary of the o n t o g e n e t i c morphospace. Sections through this morphospace by another variable than time, e.g. size (s), are ellipses which contain various age classes which may appear to be strongly c o r r e l a t e d .
o n t o g e n e t i c development as well as a definition of a probabilistic morphospace for the whole ontogeny.
If this o n t o g e n e t i c morphospace is now sectioned by another variable than by age, e.g.
by
the
section
constant
size
contains
which parts
is an accessible c o n t r o l  p a r a m e t e r in p a l e o n t o l o g y   then
of the o n t o g e n e t i c trend. Thus,
even if
the
features
under
consideration are uneorrelated within an agesection, it is possible to find a strong c o r r e l a tion
within
the
sizesections
(Fig. 2.14).
l~low strong
this correlation will be, depends
on the specific o n t o g e n e t i c trace, on the correlation of f e a t u r e s in the a g e  s e c t i o n s and on the angle between' the principal axis of the age distribution and the o n t o g e n e t i c t r a c e .
26
'Buckmanfs law' was observed in those a m m o n i t e s which show specially strong morphological consists
through
ontogeny,
and
the
observed variability
for
a
constant
size
of morphotypes which are found as o n t o g e n e t i c growth s t a t e s in all specimens.
Therefore, the
changes
age
it
is likely t h a t
dependent
this
Vlaw' results simply from
morphospaces; whereby a
the oblique sections through
high c o r r e l a t i o n
between
f e a t u r e s on the
level of the age s e c t i o n s may inforce the strong c o r r e l a t i o n within the size sections.
2.2.3 Discontinuities in t h e Observed Morphospace
So far, the mean o n t o g e n e t i c t r a c e has been assumed to be a straight line or can be t r a n s f o r m e d into a straight line (i.e. if it is ideally allometric). However, even allometry is only an idealized first order approximation. Especially in ontogeny more complex relationships commonly occur,, which only allow an a l l o m e t r i c approximation through c e r t a i n intervals (Fig. 2.13, cf. KANT & KULLMANN; 1980). Nonlinear relationships are usually found if morphology is described by some index numbers  as it is the case in ttheoretical morphology ~ (e.g.
RAUP,
mean o n t o g e n e t i c t r a c e
1966).
Thus,
in the general case one has to e x p e c t that
the
is a t h r e e  or moredimensional curve. This causes complications
if t i m e is not available as the controlling variable; e.g. the size sections will show d e f o r mations as a function of age. A m a t h e m a t i c a l description of the morphospace without t i m e can, t h e r e f o r e , lead to r a t h e r c o m p l i c a t e d nonlinear equations.
In addition, one has to e x p e c t complications in any projection of the ndimensional o n t o g e n e t i c morphospace (all possible relationships) onto a subspace, say the twodimensional subspace of a point plot. Fig. 2.15 gives a sketch of such a curved o n t o g e n e t i c morphospace.
In the convex area
of its hull a singularity appears due to the projection into
...................:..:.:.:z::,:::::.::.::
Fig. 2.15: The hull of a curved o n t o g e n e t i c morphospace, a single o n t o g e n e t i c t r a c e and the probabilistic neighborhood of this trace. O t h e r trajectories~ which s t a r t close to the s k e t c h e d trace, will be within this probabilistic neighborhood. In the concave area of the hull a swallowtail singularity appears, which will be discussed in c h a p t e r 4.
27
the plane. Such structurally stable singularities will be discussed in detail in c h a p t e r 4, however, some
aspects of the
deformations in subspaces can be already discussed here
by the analysis of the o n t o g e n e t i c traces of single specimens.
If one picks a c e r t a i n set of o n t o g e n e t i c traces for single specimens from the probabilistic
o n t o g e n e t i c morphospace, then, by experience, one can
expect
that
they evolve
in a regular manner and that they do not depart too much from their original relative position within the age section: Experience not
turn
shows into
a
that
a juvenile
FleptosomeF
one
'p y k n i c ' h u m a n during
its
will,
in
general,
ontogeny.
Now, we can describe the evolution of the o n t o g e n e t i c morphospace as an i t e r a t e d (or continuous) map which describes the change of the age dependent probability
distribu
tion and the dislocation of its mean. And, one can assume that the map, which g e n e r a t e s the
probabilistic
ontogenetic
morphospace of
a
species
from
some
initial
distribution,
also describes the o n t o g e n e t i c traces of single specimens up to some error term. If one neglects the error rather
than
term, which causes the r e p r e s e n t a t i o n of o n t o g e n e t i c t r a c e s by tubes
by lines (Fig. 2.15),
then
a significant regular disturbance within a family
of o n t o g e n e t i c t r a c e s can result only from the projection of the multidimensional space onto
a
subspace.
What
then reasonably can be expected,
without
further
analysis, are
local folds of the map (Fig. 2.2).
A simple model of such a fold in two dimensions is the tangent space of a parabola (Fig. 2.16c) whereby the t a n g e n t s are local linear approximations of the o n t o g e n e t i c t r a jectories. The through
concave side of the
this area.
In contrary,
fold line,
on the
through every point of the plane. Naturally, local
model.
the parabola,
is empty, no t a n g e n t s pass
convex side of the fold line two tangents pass such a fold model can be valid only as a
In this sense Fig. 2.16 provides a paleontological example of a local fold
in the o n t o g e n e t i c morphospace.
The o n t o g e n e t i c traces of several individuals of the a m m o n i t e genus Hyperlioceras are drawn in a twodimensional parameter space (nonallometric) which includes size (=Dm). The specimens belong to different
species of this genus (BAYER,
1970)~ but this should
not be a serious problem because the idea is only to show that local folds can be expected in
paleontological
species size,
an
'growth'
may well be
data
lumped into
inversion of the
high relative whorl height
under
the
aspects
of
a single species. During
the
previous
late
discussion these
ontogeny,
measured by
morphological trend occurs (Fig. 2.16). Specimens with r a t h e r (N) turn into forms with m o d e r a t e values of this p a r a m e t e r
and vice versa. This p a t t e r n is very regular with r e s p e c t to the precision of the measure~ ments, and the inversion occurs within a relative small size interval. Thus, the local behav
28
50"
~0"
r....:......:;
/
/urn
i
16
2b
mm
Fig. 2.16: O n t o g e n e t i c t r a j e c t o r i e s of a m m o n i t e s of the genus Hyperlioceras (a: H. desori, b: H. subsectum, c: H. d ~ d t e s ) , modified from BAYER (1969). N: relative height of whorl, Dm: d i a m e t e r of the shell. During the late ontogeny, measured by size, an image inversion occurs, which can be i n t e r p r e t e d as a local fold. In the model t h e fold causes local i n t e r s e c t i o n s of the t r a j e c t o ries and an e m p t y area. If age (t) is used as an additional variable, one can e x p e c t that the t r a j e c t o r i e s are well separated, i.e. that the i n t e r s e c t i o n s are due to the projection onto the twodimensional p a r a m e t e r space.
ior
of
the
morphological t r a j e c t o r i e s can
well be c o m p a r e d with a local fold. If age
could be added as an independent variable, the t r a j e c t o r i e s would be lifted into the third dimension. However, if age is r e l a t e d to the e a r l i e r development in the p a r a m e t e r space (Dm,N), then
the t r a j e c t o r i e s will be arranged in a more or less regular manner within
the t h r e e  d i m e n s i o n a l space {Dm,N,t). The local singularity, where the t r a j e c t o r i e s inters e c t , may then appear like a piece of a ruled hyperbolic s u r f a c e (Fig. 2.16). The rulings model locally the o n t o g e n e t i c traces,
and their projection onto the (Dm,N)ptane is the
discussed t a n g e n t space of a parabola.
This is not the place to say that this is the way to study and to describe the patt e r n of Fig. 2.16. But it is a way to illustrate and perhaps to o v e r c o m e the difficulties which arise from singular s t r u c t u r e s like the regular i n t e r s e c t i o n of the t r a j e c t o r i e s . In c h a p t e r 4 it will be shown that singularity theory or, more specific, e l e m e n t a r y c a t a s t r o p h e theory provides a very elegant method to analyze such p a t t e r n s . Anyway, it b e c a m e clear t h a t the o n t o g e n e t i c development of morphology cannot always be considered to be linear, neither
on the
individual
probabilistic level of the o n t o g e n e t i c morphospace nor
ontogenetic
traces.
The
celebrated
analysis of
higher s t a t i s t i c a l methods {like f a c t o r analysis) has,
on the
morphology by
the
level of socalled
t h e r e f o r e , to be used with caution.
P a t t e r n s like in Fig. 2.16 cannot be linearized within the observed p a r a m e t e r space, and, therefor% they cannot
be analyzed with
linear models. On the
other hand,
the
earlier
29
discussion of the o n t o g e n e t i c morphospace shows that, even within the most simple linear model, the o n t o g e n e t i c trend cannot be ruled out for a linear f a c t o r analysis as is s o m e times
assumed
(BLACKITH
& REYMENT,
1971). If the c o v a r i a n c e s t r u c t u r e
is a l t e r e d
by age e f f e c t s within the size sections, then we c a n n o t r e c o n s t r u c t the original distribution from this sections without additional information  in paleontology qualitative information will then be p r e f e r a b l e against any q u a n t i t a t i v e measurement.On the o t h e r hand, the discussed models provide tools for the taxonomist. They give q u a l i t a t i v e a r g u m e n t s for the variability of species and, t h e r e f o r e ,
for the definition of a species. In addition, they allow to
f o r m u l a t e specific q u a n t i t a t i v e models.
2.3 ANALYSIS OF DIRECTIONAL DATA
The analysis of threedimensional directional data by means of the ' s t e r e o g r a p h i c projection'
(Fig. 2.17) is a standard procedure in t e c t o n i c s and sedimentology. The aim
of the procedure is usually to e s t i m a t e a density function of unknown form from data points on the sphere (el. MARSAL, requires a smoothing process,
1970). The r e c o n s t r u c t i o n of the density distribution
in general
a moving average.
The classical hand method
works with a counting a r e a (circle) of 1% of t h e s u r f a c e of the half sphere (or of its
C
Fig. 2.17: a) R e p r e s e n t a t i o n of a t a n g e n t plane in t h e unit sphere: by the ' c i r c l e of i n t e r s e c t i o n ' with the sphere, its unit normal and a point on the sphere (intersection of the normal with the sphere), b) a pair of idealized shear planes and a system of real shear plains in the s t e r e o g r a p h i c projection: r e p r e s e n t a t i o n by the 'circles of i n t e r s e c t i o n ' and the normals, c) Two s t e r e o g r a p h i c projections of the same set of joints; above: S c h m i d t ' s grid (equal area); below: Wulf's grid (equal angles).
30
projection into the plane). When the first c o m p u t e r programs for the analysis of directional data
appeared (e.g. SPENCER & CLABAUGH,
1967; ADLER et
al.,
1968; BONYUM
& STEPHENS, 1971; ADLER, 1970), they did not only simplify the analysis of directional data, but they added new ' d e g r e e s of f r e e d o m ' : to choose the size of the counting circle, to use various weighting functions or projections of the sphere (Fig. 2.17), and the c o m puter
allows to handle a r a t h e r
{KRAUSE, area
1970), was,
large number of data.
A question, which arose early
t h e r e f o r e , w h e t h e r t h e r e exists an optimaI size of the counting
with r e s p e c t to the number and to the distribution of data points on the sphere.
Alternatively,
new
'influence functions'
like
an
exponential
decay
function have been
introduced {BONYUM & STEVENS, 1971).
The problems associated with the smoothing process can be divided into more quant i t a t i v e and more qualitative ones. The variation of the influence area (either by changing the d i a m e t e r of the the total this
'counting c i r c l e '
or
by d i f f e r e n t 'weighting functions') alters
number of e x p e c t e d values at a grid point. The classical way to s t a n d a r d i z e
number
to
a p e r c e n t a g e of
all observed data
points causes d e f o r m a t i o n s of
distribution in the way that the maxima are s t r e t c h e d  is g r e a t e r than area',
the
the sum over all grid points
100%. The counted data need to be normalized into 'densities per unit
or the area
of influence has to be replaced by a weighting function for which
the integral over the area of influence equals one (BAYER,
1982), A more qualitative
aspect is that the smoothing process a f f e c t s the variance of the distribution (GEBELEIN, 1951). This d e f e c t is mainly a function of the size of the area of influence. These problems are
briefly discussed in the
first section. However, while they are important in
a s t a t i s t i c a l sense, they are less significant for the geological i n t e r p r e t a t i o n of orientation data.
In geology only the position of e x t r e m a may play a role for the structural
interpretation,
and
in this
case
the
described d e f o r m a t i o n s of
the
global distribution
do not a f f e c t
the
local i n t e r p r e t a t i o n . Therefore, most of the following discussion will
focus on the question w h e t h e r the local e x t r e m a are stably e s t i m a t e d by the methods currently
in use.
In the
final section we will return
to a more general problem and
analyze under which conditions we can suspect a density distribution at all.
2.3.1
The
estimation
projection onto
of
the
The Smoothing Error in Two Dimensions
a density
plane
function
involves a
from s c a t t e r e d data
moving average.
For
on the sphere or its
onedimensional histograms
the resulting e r r o r s and the d e f o r m a t i o n of the m o m e n t s have been discussed in detail by GEBELEIN (1951).
Fig. 2.18 illustrates how a onedimensional histogram is d e f o r m e d
if a moving average is used. Twodimensional data
and data on the sphere behave in
the same way (Fig. 2.19), and what we will do here is to e s t i m a t e the e r r o r of the smoothing
process,
i.e.
the
expected
difference between
the
true
and
the
computed
density distribution. Technically this requires Taylor expansions and integrations, however,
31
/'7= 3
J
1
f = ~Xf i
'
I
f = Xf i
1
d
~2.18: Smoothing a histogram by a moving average: a to c: normalized averages; d: not normalized histogram of a t h r e e point moving average.
the m a t h e m a t i c remains r a t h e r simple. The way to e s t i m a t e the error is to compare the observed densities with a t h e o r e t i c a l density function f(x,y) which is analytic (i.e. continuous and differentiable) with the values which result
from
averaging over a small interval. The error is the d i f f e r e n c e b e t w e e n the
true value of the density function and the average. In the plane we choose an interval {Ax,~y) in the way that
its c e n t e r   t h e
arithmetic m e a n  
has coordinates (0,0). We
can do this for any interval by simply shifting the coordinate system. To find the mean density within the interval we have to sum over all points within the interval and to divide by the area of the interval, i.e.
Ay 0, w(R)
= 0,
and w'(0)=w'(R)=0
are
only satisfied by
t h e polynomial 3
w(x)
= x
w(x)
= s(l
3 2 i  ~'x + ~
(2.40)
or + x2(2x
 3)).
43
R
R
Fig. 2.27: Interpolation by spline functions: a) the linear Euler spline; b) the cubic spline with vanishing d e r i v a t i v e s at the grid point and the boundary of the area of influence (R).
It turns out on
the
that
boundary
there
is a limited number of optimal weighting functions depending
conditions
(Fig.
2.27).
(2.39) allows a l t e r n a t i v e l y to adjust
Equation
the slope of the weighting function in an a r b i t r a r y way at the grid point and at the boundary of used
the area of influence. It is the cubic H e r m i t e
for cubic
spline interpolation
(De BOOR,
interpolation and can be
1978). The earlier
discussed triangular
weighting function is simply the linear Euler spline. The tinear and the cubic spline provide a simple i n t e r p r e t a t i o n of the radius of influence (R). Assume
the situation of two data points with d i s t a n c e R: Without loss
of generality we can locate them on the real line and s i t u a t e one of them a t the origin, the o t h e r
at
the point x=l
{i.e. x=r/R).
The weighting
functions for the two points
are w1(x ) =
w2(x)
(I
 x)
= (1

(1

Wl(X)
= (1
+ x2(2x
w2(x)
=
+
x))
(2.41)
= x
for the linear spline and
for the cubic spline.
(i

3);
(lx)2(2(lx)
In both cases
we
 3)
= x2(3
 2x)
(2.42)
find t h a t Wl+W2=l within the interval (0,R).
The approximated values within this interval are f(x)
= f(0)w 1 + f(1)w 2 =
f(O)(l
= f(O)
+
(2.4a)
 w2)
+
f(1)
w2
(f(1)
 f(O))w2
~
t h a t is a s t r a i g h t line in the case of the linear spline and a cubic function in the second case. However, if f{0}=f(R} (e.g. single m e a s u r e m e n t s at both points), then the c o n n e c t i o n b e t w e e n the two data points is simply the horizontal line f(0), cf. Fig. 2.27. The radius
44
of influence R, t h e r e f o r e , defines a threshold of resolution: D a t a points with distance less or equal R are subsumed in a single maximum larger
while data points with distances
than R are p r e s e r v e d as distinct maxima. As far as it is possible to r e l a t e R
to t h e number of d a t a (cf. KRAUSE, 1970), t h e spline functions provide optimal approxim a t i o n and smoothing capabilities. It is i n s t r u c t i v e to invert the above a r g u m e n t . If one defines the threshold property as the optimal solution~ then one can discuss any polynomial weighting function in t e r m s of t h e optimal solution, i.e. we h a v e to find c o e f f i c i e n t s so t h a t w(x)+w{1x) = c o n s t a n t . In t h e case of an a r b i t r a r y polynomial
wI =
a0
+
w2 =
a0 +
alx
+
a2 x2
+
...
+
an xn
...
+
an(lx)n
(2.44) al(lx
)
+
the condition can only be satisfied i f a(x) n = ax n, t h a t is, if the leading power t e r m is an odd number.
The
polynomials of equations
(2.33) and (2.36), t h e r e f o r e ,
are not
optimal. Now, we may analyze the special case of the cubic weighting function:
w I
=
ax 3
=
a(1x)
+
bx 2 +
cx
+
d
(2.45) w2
=ax
To satisfy
the
3 +
condition a l l
3 + b(1x) 2 + c(1x)
+ d
(3a+b)x 2
+,(a+b+c+d).
terms

(3a+2b+c)
which c o n t a i n powers of x have to vanish in the
sum Wl+W 2. This gives the following relations b e t w e e n the coefficients:
b +
(3a+b)
= 0
~
3a
= 2b
c
(3a+2b+c)
=
~
3a
=

0
2b,
(2.46)
and we have t w i c e the same relationship b e t w e e n ' a ' and 'b' which, of course, appeared already in e q u a t i o n (2.39}. The P a r a m e t e r ' c ' is a r b i t r a r y  if a=b=0 and c~0, the cubic function d e g e n e r a t e s to the linear Euler spline. The general solution for a cubic spline under the condition w(x,R)+w(Rx,R)=constant is lust
the
linear
Euler
linear c o m b i n a t i o n of the special spline.
We can
rewrite
equation
cubic spline of equation (2.39) and the {2.45) with
the p a r a m e t e r s
of equation
(2.46) as w(x)
= {a(x 3
3x2)
d + ~}
+
{ ~d + c x } ,
(2.47)
45
i.e.
as a linear c o m b i n a t i o n of the spline functions. To see what happens at the special
points x=0 and x=R we take the d e r i v a t i v e of equation (2.47) at these points w'
=
3ax 2

w'(O)
= + c
w'(1)
=
3ax
+
c
(2.48}
+ c
and find t h a t the weighting function has identical slopes at the points under consideration. Some weighting functions~ which satisfy the discussed conditions, are given in Fig. 2.28~ and it turns out t h a t
Fig.
these
functions are no proper weighting functions as they
2.28: S y m m e t r i c cubic weighting w(x)+w(1x)=constant.
have e x t r e m a
inside the
'counting circle'
functions which satisfy
and may even assume negative values. The
only remaining weighting function is the cubic spline of equation function
tends
to
produce
local
the condition
'platforms'
at
the
data
points
(2.39), (cf.
however, this
Fig.
2.27).
Thus,
the linear Euler spline r e m a i n s the optimal solution for a smoothing procedure.
2.3.4 A Topological Excursus The previous discussion focussed on the problem to find a proper weighting function which takes values g r e a t e r this
area.
lemma
A
similar
than zero inside a c e r t a i n area and the value zero outside
problem
for details see
occurs
J~NICH
in
topology
and
(1980). Formally
is
solved
there
by Urysohn's
we can f o r m u l a t e our problem
in
the following way (of. Fig. 2,29): What we are looking for is a function f: x* [0,1] on value 1 on U and value 0 on W. On V = W\U connection° The simple idea is (J~NICH,
U (V
(~W
which has
we are looking for a continuous
1980) to c o n s t r u c t a step function which
46
wbz,.
.J/" ...
,1
Fi~. 2.29: (a) The problem is to find a continuous connection b e t w e e n the areas U and W. (b) A possible solution is to c o n s t r u c t a s t e p function in V = W U. (c) However, if the boundaries of the subsets A,, A2, ... are Iocally in c o n t a c t , a further r e f i n e m e n t of the step function is not possible.
d e c r e a s e s from U to W (Fig. 2.29 B), i.e.
A =
AI(...
(An
(
one has to find a chain of sets
W\U.
The s t e p function takes the values 1 on U,
By
1  i/n on A.
1
stepwise
refinement
one
and 0 on W.
can
V and W. The only problem is that (Fig.
2.29
C).
connection,
i.e.
In this case, we
cannot
construct
the
continuous
connection
between
the boundaries of U i and Ui_ 1 do not m e e t
it would not be possible to c o n s t r u c t a continuous insert
an
additional
step
between
the
boundaries
(JNNICH, t980).
If one applies this to the discussion of weighting function G it b e c o m e s i m m e d i a t e l y clear
that
the
rectangular
weighting function is the e x t r e m e solution which does not
allow any f u r t h e r r e f i n e m e n t b e t w e e n the areas V and W because U = V\W = 0. On the o t h e r hand, the linear Euler splint is a possible connection, even when V shrinks to a point.
2.3.5 Densities, Folds and t h e Gauss Map
So
far,
the
r e c o n s t r u c t i o n of
density distributions was
discussed without
regard
to the question w h e t h e r t h e r e exists a density distribution for orientation data. In the case of c u r r e n t o r i e n t e d obstacles in sedimentology and of lineations and cleavages in
47
tectonics, we can use standard arguments. One can e x p e c t that in these c a s e s the orientation is controlled by a potential (BAYER, 1978). The systems stay at the minima of the potentials, i.e. in the
in the position of minimal drag in a current; cleavage will occur
direction of maximal
shear stress (minimal normal stress) e t c . To arrive at
a
probability distribution or a density distribution, one assumes that the objects are driven out of the minimum position by random forces. One way to find the density distribution is provided by the stationary solution of the FokkerPlanck equation (e.g. HAKEN, 1977}.
The situation is d i f f e r e n t when orientation data are taken from surfaces, e.g. from d e f o r m e d bedding planes in t e c t o n i c s . The orientation data are then the normals of the surface, and t h e r e f o r e the surface needs to be regular, that is, t h e r e exists a tangent plane at every point of the surface. If the surface is given as a map R2 ~
R 3, e.g.
as X(u,v)
= {x(u,v),
y(u,v),
z(u,v)},
(2.49)
then the unit normal v e c t o r at each point p of the surface is given by N(p)
= (X u A X v ) / ( I X u A X v l ) ;
(2.50) A is the vector Now, in differential g e o m e t r y the map
product.
N: S  
R 3 (S: the surface) is studied which
takes its values on the unit sphere S2 =
The map N:S ~ this map
interpretation
of
R3 [ x 2 + y 2 + z 2
= 1}.
(2.51)
S 2 is called the Gauss map of the surface S (DoCARMO, 1976), and
is equivalent
sphere. The Gauss
{(x,y,z)
to
the standard r e p r e s e n t a t i o n of orientation data on the unit
map has various properties, which can be interesting for geological directional data.
Detailed discussions are
given by DoCARMO (1976)
and DANGELMAYR & ARMBRUSTER {1983).
The first point to be discussed is how a surface e l e m e n t ~S' maps onto the unit sphere. If only local properties of a surface are considered, that is a small neighborhood of a s u r f a c e point, then typically t h r e e situations o c c u r  
the local surface s t r u c t u r e
is e i t h e r elliptic, parabolic or hyperbolic as illustrated in Fig. 2.30. With regard to the positive normals of the surface one finds that the Gauss map preserves orientation at an elliptic point and reverses it at a hyperbolic point (Fig. 2.30)   at a parabolic point a d e g e n e r a t e d situation arises, the normal v e c t o r s are all aligned in a plane which inters e c t s the sphere. In geological problems one has also to consider the inverse surfaces {synclines). In this case, the closed pathways around the surface points in Fig. 2.30 are inverted in the Gauss map.
48
elliptic
@@
parabolic
r. , ~
,,';
:.~
,¢.";..:...¢." ...
Fig. 2.30: The local s t r u c t u r e of a surface is e i t h e r of elliptic, parabolic or hyperbolic type. The o r i e n t a t i o n of a closed pathway (surrounding the c r i t i cal p o i n t ) i s p r e s e r v e d at an elliptic point and reversed at hyperbolic and parabolic points (a: positive normals, anticlines). For synclines (b) the e n t i r e patt e r n is reversed. A l t e r n a t i v e l y , (a,b) can be i n t e r p r e t e d as projections in the upper and lower h e m i s p h e r e   both methods are used in geology. Grids: inclined L a m b e r t ' s equal area projection; adapted from HOSCHEK, 1969).
A f t e r t h e s e g e o m e t r i c a l considerations we analyze how a surface e l e m e n t ' S ' onto
the
unit
sphere.
unit sphere,
and which value the area of t h e surface
If we choose
a very small surface
normals on t h e sphere is proportional
element
element
maps
takes on the
dB, t h e e x p e c t e d
density of
to dB/dS (where dS is the corresponding surface
e l e m e n t on the sphere). The area of a small surface e l e m e n t is
B =
/.flXu
^ x v I du d r ,
(2.52)
R
and t h e image of
B
on t h e unit sphere has a r e a
S =
f f i N u A NvJ R
du d v .
(2.53)
(R is the area in the (u,v)plane which corresponds to the surface e l e m e n t B). *S* can
49
be expressed by s
where K is the
=
;; R
K lXu A Xvl du
Gaussian curvature
of the
12.541
dv
surface. The
e l e m e n t s B and their image on the unit sphere
S
relation b e t w e e n the surface
is finally
lim B/S = (IX u A Xvl)/(KJX u A Xvl) B+ 0 For a detailed proof see DoCARMO (1966).
=
1/K.
{2.55)
The local density on the sphere is propor
tional to l/K, or for a larger surface area we e x p e c t the density at a point p to be
I Xu A Xvl that
is the inverse 'weighted average'
of the Gaussian curvatures. There is one point
one has to take care of. The Gaussian curvature valuessurface
(2.56)
du dv
K
can assume positive and negative
thus, it is necessary that the Gaussian curvature does not change sign on the element
under
consideration.
Otherwise we
cross a point where K=0,
and at
that point the density is not defined (equation 2.56).
However, if K=0 everywhere at
the surface, one has to distinguish two cases. If
the surface is planar~ then t h e r e exists no density distribution; all normals are mapped onto the identical point on the sphere. In the case of a parabolic surface e l e m e n t , e.g. a cylindrical or conical one,
we can
define densities if the
Gaussian curvature K is
replaced by the curvature k of the generating curve of the surface; the surface e l e m e n t s are replaced by arc length. The concept of a density distribution, t h e r e f o r e , is r a t h e r complex for orientation data
from surfaces. One has to distinguish various cases, and
one can handle only finite surface e l e m e n t s that satisfy certain conditions.
Anyway, trouble. which
it
is as usually,
In this case, the
Gaussian
the
most
interesting situations
are
those which cause
interesting situations arise when the s u r f a c e contains points at
curvature
changes sign.
Such situations arise
in the
most
simple
cases, e.g. if the surface is a sinusoidal cylinder   in this case, the Gaussian curvature K changes sign at the inflection points of the generating sine function. Somewhat more c o m p l i c a t e d surfaces are given in Fig. 2.31. The lines of parabolic points, which divide the surfaces, appear also on the Gauss map (stereographic projection) where they bound the
area
of normal vectors. The density of normals on the sphere is especially high
at (or near) the boundary of parabolic points which define a fold line of the Gauss map (cf. Fig. 2.2). The c o n c e p t of folded maps applies also to the s t e r e o g r a p h i c projection in the vicinity of parabolic points on a surface. The high (theoretically infinite) density
50
Fig. 2.31: Complex "sinusoidal" s u r f a c e s   t h e sets of parabolic points on the s u r f a c e s map onto ' c a u s t i c lines' {with high densities) bn the unit sphere. These ~caustics ~ are typical ~eatastrophe singularities ~.
of normals at t h e s e singularities has its analogy in caustics, and an i n t e r e s t i n g application to phonon focusing {cf. DANGELMAYER & ARMBRUSTER, 1983, for a thorough cal discussion). These with
minor
simulation {1977). to
The
the
patterns
modifications. of plastic
Further,
deformations
singularities,
singularities
likely apply
which
studied
in
to
the
focal m e c h a n i s m d a t a
such singularities o c c u r {Fig. 2.32)
bound
the
distribution
catastrophe
at
as they were
theory
of
normals
(DANGELMAYR
t983), and some of t h e m will be analyzed in more detail in t h e a c o m m o n problem
least
studied
theoreti
in seismology
in the t h e o r e t i c a l by LISTER e t are
closely
al.
related
& ARMBRUSTER,
4 th c h a p t e r . In geology
is to find the 'foldaxis' from a set of t a n g e n t planes. The linear
approach is to find a "regression circle" under the assumption t h a t t h e t e c t o n i c a l folds are cylindric {or conic). The approach by "caustic lines" probably allows to g a t h e r additional i n f o r m a t i o n as soon as the surfaces are more c o m p l i c a t e d .
Fig. 2.32: caxis pole figures for model q u a r t z i t e under plane s t r a i n {modified from LISTER e t al. 1978).
51
2.4 RECONSTRUCTION OF SURFACES FROM SCATTERED DATA Contouring by c o m p u t e r McCULLAGH,
became
a widely used m e t h o d in geology (e.g. DAVIS &
1975; KRUMBEIN & GRAYBILL,
MAN & PILRAU,
1980}. The use of computers,
1965; HARBAUGH et
al.,
1977; FREE
however, causes special problems which
do e i t h e r not occur within the classical hand method or are simply not recognized when contours are i n t e r p o l a t e d from a hand made triangulation net. Fig. 2.3a i l l u s t r a t e s with a ' c a t a s t r o p h i c ' example how d i f f e r e n t c o m p u t e r results may be dependent on t h e e s t i m a tion method. Fig. 2.34 gives a much earlier hand made e s t i m a t i o n for the identical data set (BAYER, 1975).
The computer
triangulation procedures
method because
(cf. it
CAVENDISH,
needs
too
1974)
much
is,
in
'intuition'
general,
to c r e a t e
not a
used
for
triangulation
net from s c a t t e r e d data points. On the o t h e r hand, it is relatively simple to i n t e r p o l a t e and draw c o n t o u r s from a regular grid {e.g. SCHUMAKER,
1976). Therefore,
contouring
procedures for c o m p u t e r s use mostly a two pass process: in a first step,
the surface
data
are e s t i m a t e d
for the points of a regular
grid, **
in the second step, the contour lines are i n t e r p o l a t e d from the regular
The final c o m p u t a t i o n
grid.
of the contour lines is a r a t h e r stable process. Useful methods
are well known from finite e l e m e n t s (ZIENKIWFI'Z, 1975) and from spline interpolation. Even more complex methods like global or semilocal surface fitting can be used to derive
contour
lines
from
regular
grids
{SCHUMAKER,
1976). Anyway,
on the
level of
the local grid cell the e s t i m a t i o n of contour lines is not uniquely d e t e r m i n e d , an aspect which wilt be discussed in some detail.
The
major
problems,
however,
derive
during
the
estimation
of
the
grid
values,
and this will be the major t h e m e of the following sections. The c e n t r a l example is an estimation
method by minimal convex polygons, which is capable to illustrate the prob
lems arising during the e s t i m a t i o n of the grid values.
2.4.1 The Regular Grid
The main problem with s c a t t e r e d regular grid. In geology, in particular,
data
is to e v a l u a t e
the surface
values
for the
this process can be c o m p l i c a t e d because the data
commonly show a natural ordering along outcrop lines such as river beds, t e c t o n i c zones etc. Therefore, very sophisticated methods have been developed for the gridding process. For
instance,
points;
weighting
also r a t h e r
functions
complicated
are
used
statistical
that methods
include directional
searching
have been developed
of data
for the prior
52
E
\ C
!
Fig. 2~Y3:'Catastrophic' contour maps from a data set with strongly fluctuating surface values, a,b: regular grids of different roughness; minimal polygon method (see text). c,d: the same estimations on an irregular spaced rectangular grid, every data point is located on a grid point, e: grid point estimation by Shepard's weighting function (see section 2.3.3). f: point distribution and bounding polygon.
Ca"H" u n d
+~'
l,,ig"H'_~jb e:t,schu~ r~val
2®° Fig. 2.34: A ~hand' estimation of contour lines for the data set of Fig. 2.33.
'~'~/~'~ I~ /
' < ~
53
i........ l
Fig. 2.35: Three possible solutions of isolines for the identical data set° a} Approximation by Shepard's method; b} minimal polygons; c) minimal polygons when the grid is g e n e r a t e d from the data points. Grid points are marked. Modified from ALTHEIMER et al., 1982.
and posterior analysis of the grid values (JOURNEL & HUIJBREYTS,
1978; HARBAUGH
et al., 1977; HUIJBREYTS, 1975).
In principle, the gridding technique is a map from one finite point set onto a n o t h e r finite point set, whereby no one to one correspondence exists. There may be more grid points than data points, or t h e r e could be less grid points than data points. This relationship may also change locally within the global gridding s t r u c t u r e ,
as illustrated in Fig.
2.35 in t e r m s of the grid points. Therefore, one may expect stability problems to occur, i.e. t h a t the map becomes locally folded. Fig. 2.33 shows a ' c a t a s t r o p h i c v result of computer
contouring,
due
to the
special,
very
inhomogeneous
data
configuration,
the grid
s t r u c t u r e and the gridding methods. This configuration was chosen to illustrate the problems
that
arise
from
computer
contouring.
As far
as stability
problems are involved,
they will be discussed below.
A minor problem in gridding is the special s t r u c t u r e of the regular grid (Fig. 2.35). In general,
the grid (or net} is chosen in such a way t h a t the distances b e t w e e n the
net points are constant; at least they are c o n s t a n t for every coordinate direction. Therefore, the resulting contours will depend on the initial decision about the grid s t r u c t u r e (Fig. 2.35}. If the grid is too rough, data points will be lost, or they are not c o r r e c t l y recorded.
On the o t h e r hand, if the grid is fine enough to provide locally the required
54
Fig. 2.36: Two isoline r e p r e s e n t a t i o n s of foraminiferal diversity (entropy) in Todos Santos Bay, California. a) Shepardts method; b) minimal polygons. Adapted from ALTHEIMER et al., 1982; data from WALTON, 1955.
resolution, the net
may be too fine in other areas where the data points are sparsely
s c a t t e r e d (cf. Fig. 2.35 c). For a c o m p u t e r program the l a t t e r case may be more tant than
lost
data
points
because
a very
fine grid can
cause
impor
an extensive need of
memory and unreasonably long computation times.
Furthermore,
the
original data
points may
not
be recorded correctly, even on a
very fine grid if they have i n t e r m e d i a t e position b e t w e e n the grid points. This can become important if the e s t i m a t i o n procedure has a smoothing e f f e c t like it is the case with Shepardts local method (Fig. 2.36 a, cf. section 2.3.3). This problem can be solved, to some e x t e n t ,
if it is possible to choose the grid points in such a way that every
data point b e c o m e s a grid point and that the grid is still a rectangular one {Fig. 2.35 c; ALTHEIMER et al.,
1982). A really satisfying solution of this problem would require
that the grid is formed over the data points. The classical way to do this is the triangulation
method, but
a good triangulation
net
is hard
to establish within the c o m p u t e r
{SCHUMAKER, 1976; CAVENDISH, 1974).
2.4.2 Global and Local Extrapolations
Another problem in order
to
is to bound the
avoid extrapolations into
e s t i m a t e d surface areas
values
where no data
to
a reasonable area
points are
available. Most
55
gridding techniques allow such extrapolations, and most c o m p u t e r not
programs
in use do
t e s t this simple problem. As can easily be derived from the classical triangulation
method (ALTHEIMER e t al.,
1982), the most extensive boundary giving reasonable esti
m a t e s is the convex polygon that surrounds all data points, though it does not ensure a
reasonable solution,
as
the
Shepard
estimation
in Fig.
2.33
illustrates. S o m e t i m e s
a b e t t e r r e s t r i c t i o n of the solution space can be forced by additional c o n s t r a i n t s like a limit
distance b e t w e e n data
points and grid points. But such restrictions a f f e c t all
estimations, not only the boundary   and a typical result are 'holes' within the working area.
There are several possible s t r a t e g i e s to establish a convex boundary for the data points (cf. ALTHEIMER et
al.,
1982).
However, we shall see in the next section that
a very simple method results from 'minimal polygons', which can be used to eliminate the interior points of the bounding polygon.
Indeed, the
problem
to
find a reasonable global boundary for the working area
has its local equivalent. One has to ensure that a grid point, for which a surface value is evaluated,
is located within a closed polygon of data points and that t h e s e points
are used to e s t i m a t e the surface value. Otherwise, local extrapolations may occur, which cause maxima or minima, which are not r e p r e s e n t e d in the data or ' p l a t f o r m s ' result if the local e s t i m a t i o n uses weighting functions.
The sectorial et
al.,
search method is an approach to solve this problem (HARBAUGH
1977)o The problem, of course, does not occur with the classical triangulation
method because,
in this special case,
the whole area
in question is densely covered
by triangles or by convex polygons with the data points at the corners. Again, most of the gridding techniques in use do not necessarily the
weighting average
methods do not.
For
the
test
these conditions, especially
usually complex geological data
•
•
///
/'//~ /'/
/
/ (1
b Fig. 2.37: a) the o c t a n t search o c t a n t for the e s t i m a t i o n of a tions, b) Surface r e c o n s t r u c t i o n dal contours even from a simple
\
method. A data point must be found in every grid point value. This avoids local extrapolaby an octand search method g e n e r a t e s sinusoicylindrical surface.
the
56
problem has been well recognized (HARBAUGH e t al., 1977), and s t r a t e g i e s like quadrant and o c t a n t These
search p a t t e r n s
for e v e r y grid point a set of data points must be found
2.37). Although the procedure secures t h a t
in a c e r t a i n n u m b e r of radial sectors (Fig. the
grid
2.37).
have been developed to o v e r c o m e the problem (Fig.
methods require t h a t
point is surrounded by a   n o t
it may produce o t h e r d e f e c t s .
necessarily c o n v e x  
polygon of data
A local gap will be produced
points,
if one s e c t o r is empty,
even if t h e r e exists a convex polygon which incIudes the grid point and which would secure a useful local e s t i m a t i o n . From this viewpoint, the method is not flexible enough because it requires locally a specific data configuration. On the o t h e r hand, the process tends to irregular c o n t o u r s b e c a u s e t h e s u r f a c e values e s t i m a t e d from the moving s e c t o r system
do not
change smoothly,
but
(Fig.
sector
a higher
is small.
Conversely,
they change r a t h e r
suddenly when a data
point
2.37), at least, if the required number of data per
e n t e r s or leaves the system
number
of
required
data
causes
an increasing
n u m b e r of gaps within the solution area.
2.4°3 Linear I n t e r p o l a t i o n by Minimal Convex Polygons The discussed problems caused us in 1980 to develop in the ' S o n d e r f o r s c h u n g s b e r e i c h 53, PalOkologie'
(ALTHEIMER
et
al.,
1982) a gridding
so far discussed conditions in a very simple w a y  
technique t h a t satisfies all the
the grid values are e s t i m a t e d from
t h e minimal bounding polygon of a grid point, by a triangle. The m e t h o d works r a t h e r well up to the point t h a t data
points are
available,
sometimes strange i.e.
the e x t r e m a
local e x t r e m a
A later
s t a b i l i t y analysis provided some r e m a r k a b l e
gridding
techniques,
especially
for
the
appear
in areas where no
c a n n o t be explained by the d a t a s t r u c t u r e .
discussed
results which also holds for o t h e r
sector
search
methods
and
for
the
classical triangulation method. It will turn out t h a t the problems are mainly g e o m e t r i c a l ones and t h a t our gridding t e c h n i q u e is not a really good way to e s t i m a t e surface points, but t h a t it illuminates very c l e a r l y t h e problems of contouring from s c a t t e r e d data. As was pointed out above, the locally s t a b l e e s t i m a t i o n of grid point values requires t h a t t h e grid point is l o c a t e d within a polygon spanned by a subset of the original d a t a points.
For
a plane
mapping
problem
the
smallest
possible polygon is a triangle with
d a t a points at its corners. As is well known from polygon theory and linear o p t i m i z a t i o n (COLLATZ & WETTERLING, 1971), this minimal polygon has t h e property t h a t the s u r f a c e values can be simply e s t i m a t e d by a linear interpolation for any point within the t r i a n g l e and
along
its
boundaries
(cf.
ZIENKIEWlTZ,
1975;
SCHUMAKER,
1976). G e o m e t r i c a l l y
t h e t h r e e points establish a plane s u r f a c e e l e m e n t . F u r t h e r m o r e , the principle of ' n e a r e s t neighborhood'
can
be
easily satisfied.
'Nearest
neighborhood' means
that
a grid point
value should be e s t i m a t e d from closest d a t a points (as far as this does not cause local
57
extrapolation). If t h e r e exists a polygon at all that encloses the grid point in question, then
t h e r e exists a convex polygon, especially a triangle, which both includes the grid
point and has the point of nearest neighborhood at one corner. The proof of this s t a t e ment can he outlined in the following way:
If the point of n e a r e s t neighborhood is c o n n e c t e d with every point on the convex boundary of a l l data points, then the whole area inside the global convex boundary is densely covered by nonoverlapping triangles, and the grid point must be e i t h e r an interior point or a boundary point of one of these triangles.
This secures that we find a bounding triangle. The minimal triangle without an interior point is found with the following strategy:
Start
at the point of nearest neighborhood and find the second and third nearest
points which
form,
t o g e t h e r with
the
point of nearest neighborhood, a bounding
triangle for the grid point.
We can call this triangle the minimal convex polygon  minimal because these are points of minimal distance with r e s p e c t
to
the
convexity condition. One should expect
that
such a triangle is a locally optimal form for the approximation of the grid point value. The approximation turns into a local interpolation. In addition, the local properties of the triangles project onto the global problem to find the bounding polygon of all data points. The local triangulations are bounded to the interior of the convex polygon boundary.
This
gridding technique has,
therefore,
the
additional
advantage
that
implemented in a very compressed way on a c o m p u t e r (ALTHEIMER et al.,
it
can
be
1982) and
t h a t it provides automatically control over the boundaries of the working area. On the other
hand, one can use the
boundary of minimal
the
data
set.
method to eliminate
all points inside the global convex
The points to be eliminated are simply those for which a
polygon exists which has data
points on its c o r n e r s  
or the global convex
polygon consists of the data points which are not interior points of a triangle with data points at its corners.
2.4.4 Stability Problems with Minimal Convex Polygons
Having for us that (cf.
Fig.
done
all
these
analyses of local
and global properties,
it was surprising
the i m p l e m e n t e d process b e c a m e unstable with c e r t a i n data configurations
2.33). The
main anomalies are local e x t r e m a which are not justified by the
data. They occur mostly within relatively large areas without data points, and they are especially strong
if
the
surface values of the nearby data
points are
very irregular.
58
Fig. 2.38,: The subdivision of a r e c t a n g l e by t h e local triangulation method {minimal polygons). The blank areas belong to the same triangulation (as the shadowed do).
In addition,
the
anomalies
are
very sensitive
to small changes of t h e
grid s t r u c t u r e .
Some of t h e s e p a t t e r n s r e s e m b l e very closely the problems which may arise from s e c t o rial search methods, To explain t h e s e anomalies it is n e c e s s a r y to study the local structure of the polygons, and to see how triangles may c o m e into c o m p e t i t i o n during the gridding process, Four points form the minimal polygon, which can be f u r t h e r divided into triangles. For
any
grid
point
located
within
this polygon
two possible
triangulations
exist
{Fig.
2.a8). Which t r i a n g l e will be chosen, depends on t h e d i s t a n c e functions b e t w e e n the grid points and t h e polygon c o r n e r s  condition
for
the
distance
i.e. on the n e a r e s t neighborhood rule. The minimizing
function
divides
The opposite r e c t a n g l e s belong, thereby,
the
polygon
into
four regions
(Fig.
2.38).
to the s a m e triangulation, and, t h e r e f o r e , have
s t a b l e and nearly smooth solutions in t h e i r interior. But the two possible triangulations give d i f f e r e n t grid point
moves over
interpolation
//
solutions which are surface
the
onto
not
triangulation the
connected
in a smooth way (Fig. 2,39). If the
boundaries,
alternative
solution.
then
the solution jumps from one
The
instabilities,
therefore,
/
Fig. 2,39: A ruled s u r f a c e over a r e c t a n g u l a r grid e l e m e n t and its approximation by the two possible t r i a n g u l a t i o n s which yield d i f f e r e n t solutions.
occur
59
. . . ~:~.. ~, . . . . . .
[
.!" "
."
5,:: ......
. :"
Fig. 2.40: The bifurcation surface which corresponds to the two possible triangulations of Fig. 2,a9 (cf. Fig. 2.38). The instability is independent of the density of the grid, it results only from the local triangulation method. On a sufficiently fine grid the contours r e f l e c t the bifurcation of the surface (b). in two ways:
1) Closely related grid points may deviate strongly in the e s t i m a t e d surface values because they belong to d i f f e r e n t local triangulations.
2) Even small changes in the structure of the regular grid can locally cause r a t h e r d r a m a t i c changes in the e s t i m a t e d surface v a l u e s  
e.g. for two independent con
touring processes over the identical data set.
These
effects
are
especially strong
within relatively
large areas
without data
points,
i.e. whenever the approximation involves data points which are far apart.
Fig. 2.40 illustrates in more detail how the local solution over a r e c t a n g l e bifurc a t e s into two disconnected surfaces with discontinuous contours. From which interpolation surface the e s t i m a t e d value will be taken, depends, as discussed, only on the position of the grid point(s). The situation becomes even worse if one considers higher polygons, i.e, a larger number of data points in competition. The possible number of local triangulations increases rapidly with the number of polygon corners {F{g. 2.41). The approximation process will b e c o m e more and more instable as the number of data points in c o m p e t i t i o n i n c r e a s e s   a situation favored by large e m p t y areas within the data space. A curious situation is that in such cases a r e f i n e m e n t of the grid increases the instability in the way that the resulting surface approximates the local discontinuities of the triangulation r a t h e r than a continuous surface (cf. Fig. 2.40 h). The situation is nearly the same with the o c t a n t search method where the triangles are replaced by open angular sectors.
60 Z
~
Fig. 2.41: The various possibilities for the triangulation and linear s u r f a c e approximation of a fivepoint polygon (1 to 5: the numbers z=l ... indicate the height of the corner points; r: the c e n t e r e d i n t e r p o l a t i o n   the central point is the a r i t h m e t i c mean of the corner points.
One
could
argue
that
the
discussed instabilities are
only problems of the
local
approximation method. But if we t r a n s f e r the results to the classical triangulation m e t h od, which covers the d a t a space densely with triangles, then it i s not hard to see that we have, in principle, the same problems. Let several people draw a map from the
iden
tical d a t a set with a f r e e choice of triangulation, then the results can diverge to a large extent.
The
d i f f e r e n t solutions for a
fivepoint polygon in Fig. 2.41 can be taken as
an example. What we find is that the instabilities appear now exclusively b e t w e e n d i f f e r ent maps.
We can go a s t e p further. The same problems e n c o u n t e r us again when we draw contour lines from the e s t i m a t e d regular grid. A simple way to do this is again a triangulation of the g r i d there
are
several
the contour lines are uniquely d e t e r m i n e d on a triangle. However, possibilities for
this
triangulation,
some of them
are
illustrated
in
Fig. 2.42. Every possible triangulation gives another solution, and we can only hope that these soIutions d e v i a t e not too widely because the surface is simple enough. Besides this triangulation from
the
method o t h e r s t r a t e g i e s are in use which draw the contour lines d i r e c t l y
rectangles, e.g. such a s t r a t e g y
II (SAMPSON,
1975).
is used in the
program
package SURFACE
Within this s t r a t e g y problems arise when two opposite corners of
a cell are higher and the two others are lower than the value for the contour line e n t e r 
61
\\\\\\\\ oo\\\\\\\\ N/'•/ \/\/ ~o/\/\ /\/\
//////// ////////o \/\/\/\
/\/~/\/
Fig. 2.42: Four a l t e r n a t i v e triangulations of a regular grid,
ing the cell. Fig, 2.43 illustrates this situation, and it becomes immediately clear that the cases (b) and (c) are just the two a l t e r n a t i v e triangulations of the grid cell, the decision problem is the same as discussed earlier. Case (a) is slightly different, it rep r e s e n t s a c e n t e r e d grid e l e m e n t
where
the c e n t r a l
value could have been e s t i m a t e d
as the a r i t h m e t i c mean of the corners (cf. Fig. 2.41). This approximation will be dis
+
4
/
/
a
+

,>
4
b
+

">
c 4
Fig. 2.43: Possible paths of contour lines through a grid c e l l   (+) corners higher and () corners lower than average of corner values. Modified from SAMPSON (1975).
cussed in the
next
section. In the SURFACE II program a decision is made b e t w e e n
the solutions (b) and (c) in Fig. 2.43:
(b) is chosen if the average of corner values is
higher than the entering contour line while (c) is chosen if the average is lower than the value of the contour line. This choice is arbitrary, however, it ensures that contour lines do not i n t e r s e c t within the grid e l e m e n t {Fig. 2.43
a)  this switch causes a jump
from the lower to the upper surfaces in Fig. 2.40 when the average height of the corner points is passed,
62
2.4.5 Continuation of a Local Approximation
It turned out that the method of minimal polygons or of a lbcal or global triangulation is instable with r e s p e c t to small changes of the initial conditions. In the case of
the
'hand m e t h o d ' , the initial condition is the choice of the triangulation, in the
' c o m p u t e r m e t h o d ' , it
is the choice of the grid. The same problem e x t e n d s to o t h e r
local gridding techniques, to the seetorial search methods and even to the approximations by weighting functions. They all are very sensitive to small changes of the initial conditions and
to changes of the p a r a m e t e r setting. A major problem arises if t h e r e are
large areas without data points. In this case, the interpolation process can be s o m e w h a t stabilized if one does not use the minimal convex polygons but tries to find the locally maximal
convex polygon. However, c o m p e t i t i o n b e t w e e n polygons can be only avoided
if the e n t i r e
interior of a locally bounded polygon is t r e a t e d as a local continuum,
and if all grid points inside the polygon are e s t i m a t e d from its corner points by some smooth process. The c o m p e t i t i o n during the formation of local polygons can be avoided if
the
local
polygons are
solution p r o j e c t s continuously into allowed until
they
have
the
neighborhood, i.e.
no overlapping
the s a m e solution inside the i n t e r s e c t i n g areas
and on the common boundaries. The problem has a formal analogy in the analytic continuation of a function in the complex plane. This analogy suggests that one could s t a r t from a local solution, a local contour line, and then c o n s t r u c t its continuation through the
data
analytic
s p a c e by use of some c o n v e r g e n c e c r i t e r i a . problem
The c o n v e r g e n c e circle of the
could t h e r e b y be replaced by convex polygons over the
finite data
set. The the
previous remarks lead to a g e o m e t r i c a l problem, which is hard to solve in
case of randomly s c a t t e r e d data.
Nevertheless, it s e e m s useful to discuss finally
how the linear interpolation over triangles can be generalized for any convex polygon and how a local solution over a regular grid e l e m e n t can be e x t e n d e d throughout the global data space,
A) A Local Continuous Approximation
In the case of a rectangle,
the simplest approach toward a stable continuous surface
approximation is to c o n s t r u c t a bilinear function over the corner points (SCHUMAKER, 1976)
f(x,y)
= a I + a2x
+ a3Y
+ a4xY '
(2.57)
The c o r n e r values of the grid e l e m e n t have to be used to d e t e r m i n e the c o e f f i c i e n t s . Now, any r e c t a n g l e can be standardized to a square of unit area by the map
68
C_
/"
/.._"
5 >< •
f
.0.
I
1
0
O.
Clearly, the bifurcation of the grid into two independent subsets causes two independent solutions 
one is c o n s t a n t because the initial conditions are c o n s t a n t on this subset
(the diagonal series of ones}, the o t h e r one resembles the solution of the first example, i.e. a spreading process from the left boundary into an e m p t y medium. It is not hard to see t h a t
the stable solution of the first example is not really stable, as s o m e t i m e s
is assumed
in t e x t s
on numerical
methods (MARSAL,
1976),
but
that
it also consists
of two independent solutions, which under the special conditions b e c o m e identical.
In case the critical p a r a m e t e r (3.8) takes values less than zero, the result fluctuates and assumes negative values on one of the bifurcation grids. In this case, the grids are again
connected,
but
the
negative
control
parameter
causes
alternating
signs of
the
U
values so t h a t one, in p r i n c i p l e , has still two d i f f e r e n t solutions of the discussed x,t type. In addition, we observe a close relationship to the discussion of regular and c e n t e r e d
grids
in sections 2.4.45.
Indeed, as
the
critical
value
of the p a r a m e t e r
(3.8)
is ap
proached, the stable regular grid {Fig. 3.61) evolves into two grids which r e s e m b l e the c e n t e r e d grids of the previous discussion. These grids are disconnected and provide two
79
independent component.
solutions. Another
The
stability
problem,
therefore,
has
a
strong
topological
analogy provides the discussion of linear systems in section 2.2.1,
where we observed a similar parabolic stability boundary. Of course, the partial
differen
tial equation ut+Uxx=0 can be approximated by a set of differential equations
Y'I
= allYl
+ a12Y2
+
"'"
+ alnYn
Y'2 = a 2 1 Y l etc.,
+ a22Y2
+
"'"
+ a2nYn
(3.1o)
which provide a discontinuous spatial but continuous temporal approximation.
To generalize this result,
we can briefly analyze the difference approximation of
first derivatives. There are three approximations in use (e.g. DESAI & CHRISTIAN, 1977), the forward difference
(Ui+l, jui, j) /Ax + O(Ax)
backward difference
(ui, ]Ui_l, j)/Ax + O(bx)
central difference
(Ui+l, jUi_l, j)/(gkx) + O((Ax)2).
Under numerical
aspects the central
difference should be the best one to approximate
a first partial derivative because its discretization error is only of order (Ax)2. But nearly all approximations using the central difference are instable (MARSAL, 1976). The previous discussion has shown that
this is not a numerical problem but a topological one. The
central difference causes a grid bifurcation as in the previous example, i.e. one computes two independent
solutions, and, therefore,
special initial conditions.
Actually,
the approximation can only be used for very
the problems,
which arise here,
are very close to
those discussed in the last chapter, especially the surface approximations from scattered data.
3.1.3
Infinite Series of Caustics
In refraction seismology one is sometimes interested in the socalled 'higher arrivals' and in the caustic formed by the rays. The caustic is the envelope of rays, which, in its totality, can be written as
F(x,y,p)
= O.
(3.11)
The equation of the envelope of this family of rays is obtained by eliminating the ray
80
parameter
p f r o m e q u a t i o n (3.11) a n d f r o m its p a r t i a l d e r i v a t i v e
3 F(x,y,p) 3p (e.g. B E N  M E N A H E M conditions, with
&
SINGH,
i.e. the situation
linear
aim
here
will
be
the
parabolic
velocity
is only
used,
(3.12)
= 0 1981).
where
increase,
rays
rays.
a general
the source
the
to demonstrate
parabolic
For
rays
circles
the
circular
can
(e.g. O F F I C E R ,
of iterated
If t h e c u r v a t u r e
approximate
one
choose
is located at the reflector.
are
the use
overview
maps,
of the
ones
to
For a m e d i u m
1974).
a still m o r e
Because simple
rays under consideration some
extent.
special
the
model
is l a r g e ,
If all p a r a b o l i c
rays
h a v e a c o m m o n s o u r c e point, e q u a t i o n (3.11) b e c o m e s
y 
(x 2  bx)
A specific ray with ray parameter At
x = b the
reaches
the
ray
is r e f l e c t e d ,
reflector
a
(3.13)
= 0.
b p a s s e s t h r o u g h t h e p o i n t s x = 0 and x = b if y = 0. and
second
because
time
at
the
source
x = 2b.
This
is l o c a t e d gives
the
at
the
general
reflector, iterated
it
map
for t h e r e f l e c t i o n p o i n t s
xi =
xi_ I +
x.
xo
(3.14)
b
or
=
+
ib.
1
The
ray
(b,2b),
itself
(2b,3b),
is d e f i n e d ....
Or
if
in
the
interval
a certain
(0,b)
interval
and
maps
is g i v e n ,
the
iteratively equation
into
the
for t h e
intervals
r a y s {3.13)
t a k e s t h e f o r m (by u s e o f e q u a t i o n (3.14)
y 
((xiib)2

b(xiib)
Fig. 3.8: C a u s t i c ( s ) o f a s i m p l e p a r a b o l i c ray system with reflection at the surface. T h e s o u r c e is l o c a t e d at t h e r e f l e c t o r .
) = 0.
(3.15)
81
By use of equations (3.11) and (3.12) one finds the equation of the caustics by eliminating the ray p a r a m e t e r b:
y  xi2(1(l+2i)2/(4i(i+l)))
= 0
or
(3.16) y + xi/(4i(l+i))
Thus,
the
caustics are
= O.
an infinite series of parabola with increasing curvature,
which
all pass through the point {0,0). Fig. 3.8 gives the first iteration as an example. More c o m p l i c a t e d ray systems like circular rays or arbitrary positions of the source can be t r e a t e d in the same way.
Of some i n t e r e s t is the case of a source located below the r e f l e c t o r . This situation causes
a
bifurcation,
which can be qualitatively described in the
source is now located at depth 'a'
(y+a)
all rays
and the r e f l e c t o r , as before, at depth zero. Then
(3.16) b e c o m e s
the ray equation
i.e.
following way. The
are

(x2bx)
passing through
= 0,
(3.17)
the source point (a,0). Now, the rays are r e f l e c t e d
at y = 0, and their horizontal position is then
x
To
find the
2
+ bx +a = 0 .
(3.18)
r e f l e c t i o n points, one has to solve the quadratic equation (3.18), and, in
general, this will give two reflection points because the ray propagates into the positive and
into
the
negative xdirection from
Fig. 3.9: Caustic of parabolic rays. The source is located below the r e f l e c t o r .
the
source point
(o,a). Because of s y m m e t r y
82
reasons, we have to consider the absolute values of the r e f l e c t i o n point; both solutions propagate into the positive and into the negative direction  we have to fold the solution space.
Only in this case,
the whole semiinfinite solution space
is covered with rays.
The c o n s e q u e n c e is t h a t the ray p a r a m e t e r 'b' is not further uniquely defined; it describes two rays, caustics
and both are
STEWART,
ray
connected 1978;
s y s t e m s are at
a
cusp
Fig. 3.9). The
capable
point,
to produce i t e r a t i v e l y caustics. The two
i.e.
one
bifurcation of the
finds cuspoid c a u s t i c s
(POSTON
&
solution arises also with other ray
systems, such as circular rays. It is not a property of a special ray system, but it only depends on the
position of the source; it is a structurally stable topological p r o p e r t y
of ray systems.
3.2
Within distribution
the has
CHI2TESTING OF DIRECTIONAL DATA
analysis of o r i e n t a t i o n pronounced e x t r e m a .
data
The
one
usual
has
way
to prove w h e t h e r the observed
is to
test
the contrary,
whether
the data differ significantly from a uniform distribution. The propagated method in t e x t books {e.g. MARSAL, 1979) is the Chi2test. BALLENTYNE & CORNISH {1979) observed that
the
Chi2value obtained
from
such a t e s t
depends on the
arbitrary
selection of
the o f f s e t point for the s e c t o r system, in which the directional data are grouped. By an e x t e n s i v e numerical analysis they showed that the Chi2values fluctuate as the s e c t o r p a t t e r n is r o t a t e d over the data. In their analysis, the Chi2values passed t h e r e b y several t i m e s the significance leveh T h e r e f o r e they concluded: "The
hitherto
considered
a
widespread valid
use
of
of
analysis,
mode
the
test
in
and
this
the
way cannot,
results
of
therefore,
previous
be
studies
employin E this methodolgy must be treated with extreme caution. "
While the numerical analysis of BALLENTYNE & CORNISH (1979) shows that this is a p r o b l e m a t i c t e s t , it does not elucidate why the uniform distribution as a zerohypothesis behaves in such an unpredicted way. Ballentyne & Cornish noticed that the e x p e c t e d frequency E is a c o n s t a n t for all s e c t o r s under the special condition of a uniform distribution:
E = N / k;
N: number
of data
k: number
(3.19)
of sectors.
The equation for the Chi2value can, t h e r e f o r e , be w r i t t e n as 1 k
X2= ~ i~=l((OiE)2
k
(3.20)
= i=l ~ (OiEi)2/Ei
if E. = constant; I 0.: number of o b s e r v e d i
data
in the sector
i.
83
But this simplified version contains still a constant value inside the sum. Further evaluation yields
X2=
1
} ] 0 i 2 _ 2 }]0 i
+ kE.
(3.21)
The indices of the sums are the same as in equation (3.20), they will not be r e p e a t e d in the following equations. By use of equation (3.19) one has the relations
O
1
= N
kE = N ,
and
and this gives finally N [ X = = ~
0i2

N.
(3.22)
Thus the obtained Chi2value depends only on the sum of squares of the observed values in the s e c t o r system. Because N and k are constants for a certain data set and a given sector pattern,
one can
define a modified t e s t s t a t i s t i c ,
which simplifies the f u r t h e r
analysis:
~k x % 1 =
X O.•2
(3.23)
•
The important point is that the t e s t s t a t i s t i c values Ei;
the only remaining variables are
does not really depend on the e x p e c t a t i o n the Oi's , and the Oi's can be altered by
a dislocation of the s e c t o r system or by another spacing of the sectors. The same situation arises in normal histograms and twodimensional data the
uniform distribution. The
behavior of the t e s t s t a t i s t i c
if they are
t e s t e d against
derived folmulae also hold in t h e s e cases. To study the it will be sufficient to alter the o f f s e t point of the s e c t o r
system.
Rotation of the s e c t o r p a t t e r n causes jumps of data points from one sector into the neighboring one when the s e c t o r boundary passes a data point. Such a jump modifies the sum
~Oi 2 locally:
(0iI)2 + (0i+I)2 = 0 i
2 +
Oi+ 1
2
+ 2(Oi+lOi)
+ 2.
(3.24)
For an arbitrary number of jumps b e t w e e n two classes one finds
(OIJ)2 + (02+J)2 = 012 + 022 + 2(0201 ) + 2J 2 J: number of jumps. Therefore, the t e s t s t a t i s t i c s is changed by a value
(3.25)
84
2(J(02_01) The
(3.26}
+ j2).
general equation for a dislocation of the s e c t o r p a t t e r n is found by summing up
all local changes, and the modified t e s t s t a t i s t i c
k Xa
where
the
+ 1 = I 0i
2
+ 2 ~Ji 2  2 [JiJi+l ,
+ 2 10i(JiJi+l)
s e c t o r k + 1 is identical with
(3.21) can be w r i t t e n as
the
sector
i.
(3.27)
The equation shows that
any
jump of a data point from one s e c t o r into another, under rotation of the s e c t o r p a t t e r n , will alter the t e s t value, as was numerically found by Ballentyne & Cornish. But equation
(3.27) allows a more detailed analysis. To have no change of ~ Oi 2 under a rotation of the s e c t o r s requires
o = X
o i ( a i  J i + 1) + XJi 2  X J i a i + l (3.28)
or a l t e r n a t i v e l y 0 =
XJi(Oi+l
 Oi) +
~ Ji
2

XmiJi+l"
These conditions can only be satisfied if J l = J2 . . . . all o t h e r cases,
= Jk' or if Ji = 0 for atl i. In
the original sum is altered, and one gets d i f f e r e n t t e s t values. Now,
the condition to have an equal number of jumps for all s e c t o r s is mainly a g e o m e t r i c a l problem.
The c o n s t r a i n t s given by equation (3.28) require that the data are equally spaced on the circle and t h a t
they have unique frequency. The unique frequency within every
s e c t o r is what one suspects to be proved by the t e s t method, but now spacing appears as
a new p a r a m e t e r ,
which a f f e c t s the t e s t value. In addition, on a sufficiently fine
scale the unique distribution does not play any role f u r t h e r m o r e . The d a t a are measured on a scale of real numbers, and,
t h e r e f o r e , any local c l u s t e r is due to the roughness
of the m e a s u r e m e n t ; as the scale b e c o m e s finer and finer, the cluster will be divided into single points. Therefore, on a sufficiently fine scale the data are single points on the
circle
{Fig. 3.10).
The
only remaining variable,
then,
is the spacing of the
data
points. One can i n t e r p r e t any t h e o r e t i c a l distribution in the way that the density over
Fig. 3.10: Equally spaced and uniform distributed points on a circle.
85
a c e r t a i n interval gives a m e a s u r e m e n t of the (infinitesimal} spacing of points on the real line. In the case of the uniform distribution, the density describes an (infinitesimal} uniform spacing. But what we e x p e c t , are not equally spaced data. Our opinion is that the
data
result
from a random process, which s e l e c t s the data with equal probability
from a c e r t a i n interval of the real numbers. Therefore, one cannot e x p e c t to find equally spaced data
in a finite size s a m p l e  
one has, of course, the combinational problem
to arrange N data on M points on the real line where M is given by the roughness of the m e a s u r e m e n t .
Now, one may ask how strong the a l t e r a t i o n s of the test value for d i f f e r e n t distribution p a t t e r n s are. The most simple system are two sectors, and 100 data give a likely sample size. In a first case, we may have nearly equal spacing, then every s e c t o r contains approximately 50 data points. The rotation of the s e c t o r system causes only jumps of a few data points at once, say, in the order 1 to 10. We find that ~ 0 i takes values like (equation (3.24)):
The change of the found if random second case
502
+
502
5000
492
+
512
5002
452
+
552
=
5050
402
+
602
=
5200.
test
value
is not very impressive. Changes of this magnitude are
numbers a r e , u s e d
that
the
to provide a numerical
test.
Now, assume in the
distribution has a single well pronounced maximum
so that
it
is possible to locate all data in one of the two sectors. On the other hand, there exists a rotation of the s e c t o r system, which divides the data into two sets with nearly equal frequency; t h e r e f o r e , under the rotation one will find sums in the range 02
+
1002
=
10 000
5()2
+
502
=
5 000,
and the e x t r e m a diverge quite clearly. For this data configuration the behavior of the system becomes r a t h e r chaotic with r e s p e c t to an arbitrary location of the s e c t o r system. In addition,
the
possible o u t c o m e s diverge
we have the striking situation that,
rapidly
with
increasing sample size. Thus,
in contradiction to the general s t a t i s t i c a l opinion,
an increase of sample size does not stabilize the result, but t h a t the c o n t r a r y is true: The
uncertainty
deviation certainty.
of
the
about sample
the
result
from
the
of
the
uniform
test
increases
distribution
as
the
approaches
86
These
observations allow
the t e s t . Take MARDIA,
a
final
discussion of
the
causes
for
the
instability of
a t w o  s e c t o r sample from the normal distribution over the circle (e.g.
1972)~ then
nearly all data,
there
are
two e x t r e m e cases: In the
first, one s e c t o r contains
in the second, both s e c t o r s contain an equal number of data. This is
again the situation considered above. But now, with the normal distribution, the e x p e c t a tion values are n e i t h e r independent of the data s t r u c t u r e  and variance of the d a t a  o f the
they depend on the mean
nor are t h e y independent of the choice of the o f f s e t point
s e c t o r system. As the
sector rotates,
the
observed frequency within a s e c t o r
is a l t e r e d in the same way as the e x p e c t a t i o n values for this s e c t o r and vice versa. To use a t e r m from s y n e r g e t i c s (HAKEN, 1977), both values are 'slaved' by the position of the s e c t o r system. In the case of a uniform distribution, the e x p e c t a t i o n values break out of this 'slaving', and this allows the system to fluctuate f r e e as described above. The 'revolt of the slaved p a r a m e t e r s ' b e c o m e s especially strong when the observed frequencies
are
strongly dependent on the
position of the
s e c t o r system,
i.e.
when the
distribution has a well pronounced maximum.
3.3
The by
PROBLEMS WITH SAMPLING STRATEGIES IN SEDIMENTOLOGY
analysis of p s e u d o  t i m e series (stratigraphic thickness against
means
of
correlation
classical
etc.
is
methods
well
like
known
polynomial curve
(FOX,
1975;
fitting,
some variable)
moving averages,
SCHWARZACHER,
1974).
Besides
cross these
methods, random models have been used, and here especially the concept of transitional probabilities and of Markov chains (KRUMBE1N, 1975}. In sedimentology and stratigraphy the general
problem with
these
techniques is that
the
'time
series' or the ' s e q u e n c e
of signals' is usually too short because the profiles are of limited length. Other problems result
from special, p r o p a g a t e d sampling techniques like equal d i s t a n c e sampling. These
problems
are
of special i n t e r e s t
in the p r e s e n t c o n t e x t
because
they cause problems
of convergence, and they are capable to g e n e r a t e artificial patterns: They are, in some way, the inverse problem of i t e r a t e d maps. It will turn out that the problems are again g e o m e t r i c a l ones, and that it is not advisable to replace geological (morphological} r e a s o n ing by a sampling formalism.
3.3.1
Markov C h a i n s in Sedimentology
There arise special problems if transitional probabilities are used to study periodicity p a t t e r n s of profiles. These problems are mainly of a classificatory nature; they r e s e m b l e closely the problem to define the g e o m e t r y of a bed or facies unit. F u r t h e r m o r e , they are r e l a t e d to the process of s e d i m e n t a t i o n
and from
this to the type of ' s i g n a l s '  
87
a
b
T°
Fig. ties find of a
c
7
m
e
3.11: Graphic r e p r e s e n t a t i o n of d i f f e r e n t scales for definition of probabilion profiles, a) The profile in classical r e p r e s e n t a t i o n , b) the probability to a c e r t a i n lithology measured in t e r m s of bed thickness, c) the probability lithotogy to be deposited during a t i m e interval.
w h e t h e r the sedimentation process consists of distinct e v e n t s or r e f l e c t s ,a continuously changing environment.
Transitional probabilities are just one possible definition of a large set of probability m e a s u r e m e n t s to be defined on. profiles. Some o t h e r types of probability m e a s u r e m e n t s on profiles may be briefly reviewed as a base for the later discussion of transitional probabilities.
From
a sedimentologicat viewpoint,
we
have
the
total
probability of
a
certain facies type, the likelihood to take a sample from the profile and to find a sandstone, carbonates, a claystone e t c .
In order to define the probability measurement, the
profile has to be classified into distinct facies units or beds, and the probabilities result from
bedthicknesses. Still
the
same
probability
structure
but
with
other
probability
values results if the depth scale of the profile is t r a n s f o r m e d into a time scale. So, any smooth d e f o r m a t i o n of the scale will give new probability values, but it will not disturb
the
somewhat
special different
probability algebraic
algebra rules
(Fig. 3.11).
(RENYI,
1977;
Another FISZ,
type
1976)
of
probabilities with
results
if two
objects
are analyzed simultaneously. In this case, one works with conditional probabilities, e.g, the probability to find a c e r t a i n fossil in a specified lithology. If one can define some ordering relation like before and a f t e r , then a series of d i s c r e t e signals can be analyzed in t e r m s of the conditional probabilities: to find a c e r t a i n signal before or a f t e r another one.
If the positional relationship is reduced to 'just before' or 'just a f t e r ' ,
then the
conditional probabilities b e c o m e the classical transitional probabilities of a Markov chain. The s t r u c t u r e of the conditional probability space depends not only on the scale used for
the
m e a s u r e m e n t s but
also on the definition of the relationship b e t w e e n the two
88
sets of objects. In sedimentological p r a c t i c e
the
main
problem
is to define distinct signals,
i.e.
distinct sedimentological units. In a very narrow sense, this is only possible if the sedim e n t a t i o n process is not continuous. But long series of s e d i m e n t a t i o n by events usually are r e s t r i c t e d to turbidites, in which the sedimentological s t r u c t u r e is very homogeneous
commonly without transitions b e t w e e n a larger number of lithologies. In all o t h e r
cases,
the
transitions b e t w e e n beds or
facies units
are
more or less continuous. But,
as sharp boundaries b e t w e e n the 'signals' disappear, the definition of a probability space b e c o m e s more and more subjective, and this c o n t r a d i c t s the aim of an objective s t a t i s t i cal analysis. For this reason, a modified sampling technique is frequently used to establish the empirical transition probabilities of a ' s e d i m e n t o l o g i c a l ' Markov chain (MIALL, 1973). Instead of d i s c r e t e signals, which have to be defined subjectively, one takes small samples in some regular distance. The reason is that the samples can be more easily classified. It s e e m s worthwhile to analyze how the d i f f e r e n t sampling methods may influence the probability structure.
A)
D i s c r e t e Signals First
some
additional
features
of
the
classical
approach
of
Markov
chains m a y
be discussed with r e s p e c t to the sedimentological questions. A s e d i m e n t a r y unit c h a r a c t e r ized by its lithological, sedimentological, paleontological etc. c o n t e n t needs to be defined as a signal, which is an event e i t h e r a p r i o r i  and
'just
above'
s e p a r a t e d from the events 'just below '
by distinct b o u n d a r i e s   or which can
become a 'distinct e v e n t ' by
a useful definition of its boundaries. Because the e v e n t is defined by its structure, the transition matrix is independent of bed thickness and profile depth. The transition matrix is of the form
(3.29)
nij: number of transitions from signal Sj to Si; Nj : total number of o c c u r r e n c e s of the signal Sj. Repetitions
of
identical
units
such
as
a
sequence sandstone

sandstone are
usual events in classical Markov chains. In sedimentology they are only possible if t h e r e exists some boundary which can
be recognized, a
f e a t u r e which usually is r e l a t e d to
banking. Now, the boundary b e t w e e n two beds, no m a t t e r how small, means some change in s e d i m e n t a t i o n , e.g.
a short
interval
of lowered s e d i m e n t a t i o n r a t e
that
caused the
physical boundary. Thus, if we do not identify the boundary b e t w e e n beds as a s e p a r a t e
89
Fig. 3.12: A sequence of three sedimentary units a, b, c and different possible transition graphs. Above: transitions if the sedimentary units are defined as 'distinct events t, middle: one possible transition graph for equal interval sampling; the number of singular loops depends on the spacing of the samples. Below: occurrence of repetitions, i.e. singular loops, due to lumping of lithologies; tat and 'b' are not distinguished.
event, this may be due to a high threshold that has disturbed the record. This view can be formalized in terms of a map from the transition matrix of degree n onto a transition matrix of degree n  k, e.g.
l0 P21
P12 0
P31
P32
0 / Pl3N~ P23]
$3
  ~ $2
.......................
QPll P21
A more
instructive representation
of
(3.30)
P12)
~
this
map can
P22
"
be given by a transition graph
{Fig. 3.12) which shows how singular loops, like a sandstone

sandstone sequence,
arise due to ignorance or to lumping of intermediate signals (e.g. sandy shales). As will be discussed below, the same structure with singular loops arises if the samples are taken in equal intervals. Within sedimentological problems, the occurrence of singular loops usually indicates that a transition state (such as nondeposition) is missing.
B)
Equal Interval Sampling If we now go on to the second sampling method, the sampling in discrete intervals,
90
we have to prove w h e t h e r it really t e r m i n a t e s into a Markov chain. MIALL (1973) writes t h a t this "method
can
relative
give
rise
frequencies
expense
of
to
a
of
accuracy
much
the
in
more
accurate
lithotypes
measuring
measure
present,
of
but
stepbystep
the
at
the
depositional
changes".
This r e m a r k
shows t h a t one has to take c a r e t h a t the probabilities are well defined,
i.e. t h a t one does not produce a m i x t u r e out of conditional and t o t a l probabilities (see above). Miall f u r t h e r e m p h a s i z e s
a sampling i n t e r v a l slightly less than the a v e r a g e bed
thickness. This additional r e c o m m e n d a t i o n will be studied in detail in the next section. In order
to analyze t h e i n t e r v a l  s a m p l e s t r a t e g y
one may assume to have a series of
well defined e v e n t s with t r a n s i t i o n a l probabilities
0
Pji
Pij
0
o . ,
.
.
.
.
.
. =
nij/N
.
lot of
transitions.
counts
of
is larger
As the
transitions
nj
/Nj (3.3I)
i
. .
The samples are t a k e n in equal t h e sampling d i s t a n c e
..
intervals from t h e identicaI universe. As long as
than the a v e r a g e bed thickness, one will miss quite a
sampling
stabilize,
0
but
distance the
becomes
sampling
less than
distance
can
the be
smallest further
bed,
the
reduced,
in
t h e e x t r e m e c a s e down to an infinitesimal small size. From t h e point, w h e r e t h e t r a n s i tion counts (not yet probabilities) b e c o m e stable, one will find new t r a n s i t i o n s only along the diagonal of the counting matrix.
The counts along the diagonal will increase with
d e c r e a s i n g sampling d i s t a n c e until the values equal the t o t a l thickness of e v e r y lithotype. The sum of the diagonal e l e m e n t s , t h e r e f o r e , gives t h e length of the profile: With every of
respect
the
total
measured of
the
to
the
content
intervalsampling
as
thickness
of
the
meterintervals,
sampling
of
process
the
diagonal
generates
different
lithotypes
cmintervals
elements,
nothing
or~
but within
therefore,
a
measurement a profile
generally,
in

units
distance.
The e l e m e n t s of the diagonal line, divided by t h e i r sum, provide the t o t a l probability to find a lithotype within the profile. If one c o m p u t e s transition probabilities from this c o u n t i n g matrix, then t h e probability m a t r i x reads
91
(:: ....
mi i / (mi i +Ni
(3.32) nj i/(mj
j+Nj)
.,o
m..: t h i c k n e s s o f t h e l i t h o t y p e 1; 11
n..: jl c o u n t s of t r a n s i t i o n s from l i t h o t y p e j to i; N.: t o t a l n u m b e r of t r a n s i t i o n s from l i t h o t y p e j to a n o t h e r s t a t e . l
This p r o b a b i l i t y
matrix
dimensionless numbers
First,
t h e p r o b a b i l i t i e s a r e not
because
t h e n..'s a r e c o u n t s while t h e m . . ' s have t h e d i m e n s i o n II 1J and this is independent of the roughness of the i n t e r v a l s  
of a length measurement, but, per definition,
has a suspicious s t r u c t u r e .
probabilities
are dimensionless numbers. On the other hand, if we
take a small sampling distance on a profile, where the bed thickness is not small relative to profile length, the diagonal elements approach one, and the transition graph (Fig. 3.12) is dominated by singular loops.
The suspicious s t r u c t u r e
o f this p r o b a b i l i t y m a t r i x
becomes clear
if one s e p a r a t e s
t h e c o u n t i n g m a t r i x into its i n d e p e n d e n t p a r t s
O
O
nil ...... .
.
.
nj.i .
.
= .
.
.
The
t.. a r e t h e real t r a n s i t i o n s U t h e individual l i t h o t y p e s .
From the a or
(3.33)
the
matrices
transitional lithotype
on t h e right
probabilities
m i i / ~mii.
sedimentological
a pseudoobjective
It
from
seems,
decision sampling
'what
j..0. I.
. . . 0
°
+
'
between
side one
lithotypes,
finds two
the
°
mii 0
m.. m e a s u r e 1l
0 ...
°
i]
the thickness of
independent probability
systems
the
t.. and t h e p r o b a b i l i t y or r e l a t i v e f r e q u e n c y of li t h e r e f o r e , r a t h e r d a n g e r o u s to t r a n s f e r t h e g e o l o g i c a l can
strategy.
be What
defined is t h e
as a d i s c r e t e meaning
of
lithological
the
signal'
'probability
to
matrix'
n..1j w h e n t h e t h i c k n e s s o f t h e beds v a r i e s strongly? In t h e c a s e t h a t t h e beds have n e a r l y t h e s a m e t h i c k n e s s , t h e additional rule t h a t t h e s a m p l i n g d i s t a n c e should be t a k e n n e a r t h e a v e r a g e bed t h i c k n e s s s e c u r e s t h a t
the estimated matrix approximates the transition
m a t r i x to s o m e e x t e n t . In t h e c a s e o f highly v a r i a b l e bed t h i c k n e s s , t h e m a t r i x n.. will q n o t m a k e m u c h sense. A t least, if one can c o m p u t e a m e a n bed t h i c k n e s s , one has s o m e idea w h a t a bed looks like in t h e p r o f i l e under study, and why should one t h e n go this doubtful way?
92
3.3.2
Artificial P a t t e r n Formation in Stratigraphie PseudoTime Series
It is known for a long time that several astronomic cycles in the order of 20 000 to 400 000 years
may cause c l i m a t i c changes, and t h e r e
has been a large number of
a t t e m p t s to r e l a t e geological phenomena to t h e s e astronomic cycles, e.go t h e r e are several good arguments in the case of m a r l  l i m e s t o n e rhythms, coming from the proposed c l i m a t i c changes (EINSELE & SEILACHER, eds. 1982). However, it is very hard to give a s t a t i s t i c a l proof of the correlation b e t w e e n bedding phenomena and astronomic cycles. To do this would require to have true time series of c a r b o n a t e production and of clay influx under controlled conditions. But already the relation b e t w e e n time and sediment accumulation is not known down to sufficiently small intervals in any profile. Therefore, this problem is usually ignored and a fairly c o n s t a n t r a t e of s e d i m e n t a t i o n assumed {SCHWARZACHER & FISCHER, series
1982).
analysis
An additional handicap is t h a t
require
equal
interval
r e c o m m e n d equal interval samples at
samples.
most c o m p u t e r procedures for t i m e  SCHWARZACHER
&
FISCHER
(t982)
a distance which is not smaller than the thinnest
beds e n c o u n t e r e d with some frequency. Surely, the bed is the {possible} unit for an analysis of cycles. It is the smallest visible fluctuation within the profile. If one has enough information about
the cycles, one will get a good picture of the periodic process if one
chooses the sampling distance exactly as half or as one wavelength. If, in addition, the sampling points are located at the e x t r e m a of the periods, one gets a very simple linearized approximation. For the s t r a t i g r a p h i c problem this would require to locate the sampling points at the e x t r e m a of a l i m e s t o n e  s h a l e sequence at the boundaries and the c e n t e r s of beds, in order
to
describe the
fluctuations of the
carbonate c o n t e n t properly. But,
if the data have to be analyzed with a c o m p u t e r algorithm, equal distant sampling points are necessary while bed thickness usually fluctuates. So, the c o m p u t e r obtrudes a c e r t a i n s t r a t e g y , w h e t h e r we like it or not. The question is what we have to do in order to get a physically or sedimentologically reasonable result and not only a meaningless product of the
c o m p u t e r . To elucidate some problems, two d i f f e r e n t a s p e c t s will be discussed,
the sampling of periodic functions and the analysis of bed thicknesses. In the first case, it will turn out that the propagated sampling distances can cause artificial p a t t e r n formation, in the second exampl% we will see that a not properly defined s t a t i s t i c a l hypothesis causes i n t e r p r e t a t i o n a l problems.
A) Sampling of Periodic Functions
SCHWARZACHER
& FISCHER
(1982)
r e c o m m e n d equal
interval
samples
at
a distance
which is not smaller than the thinnest beds e n c o u n t e r e d with some frequency, If we assume the bed to r e p r e s e n t approximately one wavelength of the smallest periods, we can study the influence of the sampling distance for an idealized periodic process. A simple cosine
93
J
..V.
TV_.V w,>v
v..
Fig. 3.13: Artificial p a t t e r n f o r m a t i o n due to intervaI sampling (intervaI width near ~r) on a cosine signal. The cosine signal and t h e sampling p a t t e r n s , which result from intervals of exactly the width ~r (but with d i f f e r e n t s t a r t i n g point), are drawn enlarged.
function provides a model for, e.g., the f l u c t u a t i o n of the c a r b o n a t e c o n t e n t (Fig. 3.13). Sampling at distances of exactly
~r will be successful if the s t a r t i n g point of the samples
does not coincide with the inflection point of t h e cosine function. In this special case, no periodicity then
the
will be d e t e c t e d  
anyway,
if the
sampling distance
is chosen as 2 ~ ,
periodicity will vanish for every s t a r t i n g point. For all o t h e r s t a r t i n g points,
one finds amplitude
fluctuations which c o r r e c t l y
represent
the wavelength, but the true
amplitude is only found if the s a m p l i n g  p o i n t s coincide with t h e m a x i m a of the cosine function. Now,
it
is unlikely t h a t
the sampling intervals have exactly
the length w ( o r
2Tr)
94
even in a c o m p u t e r simulation. Therefore, we disturb the sampling distance slightly, i.e. we take distances Jr +~
{or 2 Jr+ ~); What happens, d e m o n s t r a t e s Fig. 3.13. The original
cosine function is modulated, and new periodic p a t t e r n s of higher order occur which only depend on the disturbance p a r a m e t e r E . In the e x t r e m e case,
a new simple periodic
curve appears with several times the wavelength of the original signal. The error term causes the sampling points to the
error
parameter
this
Vmove' along
the periodic function, and depending on
shifting process adds an amplitude modulation
to the cosine
function. Already for small values of the error term we get an infinite number of possible higher periods and of pure chaotic b e h a v i o r   this depends on the quotient njr / { ~ + E ); if t h e r e exists a ~n' such that the quotient is a rational number, then we have a period of n ~ ;
otherwise, we have chaotic behavior. Now, in a numerical sense, the sampling
distance
near
Jr
is much
too
large
to
approximate the
by equally spaced samples would require
periodic function; an
analysis
sampling distances of only a fraction of J r
As the sampling distance becomes small in relation to phase length, we can safely use equal distance sampling as well as the m a t h e m a t i c a l ' m a c h i n e r i e s v of t i m e  s e r i e s analysis and control
theory.
In geology, however, we have
to
consider the sampling scale, and
this r e l a t e s the problem to i t e r a t e d maps, where similar problems with distances occur like
in the
example of the
finite logistic equation. The relation to i t e r a t e d maps can
be illustrated in more detail.
The cosine function is a well defined map of a rotating radius v e c t o r of unit onto
a Cartesian
identified with a
coordinate system
length
and vice versa. The rotating radius v e c t o r can be
mass point, which moves with c o n s t a n t velocity on the circle. Equal
distance sampling, then, is equiwatent to a uniform motion of the mass point inside the circle the
with regular elastic collisions with the circular wall. The points of collision are
sample points. This is a classical example of chaotic motion (GRENANDER,
If the first angle of collision is denoted by e
circle), then the arc length b e t w e e n successive impacts is Jr 2 e
Fig. 3.14: Elastic motions in a circle.
1978}:
{measured from the radius v e c t o r of the (from planimetry). The
95
resulting p a t t e r n depends on the ratio
2~r /(~T2@ }. If 2 @ and 7r are incommensurable,
t h e system tends to long t e r m c h a o t i c behavior {Fig. 3.14); however, the system depends also on the magnitude of @ . If
@ is large, we have a high frequency of collisions, and
the pathway of the mass point approximates the circle. If @ is small, a quite d i f f e r e n t pattern
appears,
like in Fig. 3.11b, where the pathway consists of quasi triangles, which
slowly r o t a t e . The lines of motion envelop again a circle, and in t e r m s of the approximation problem the relation of the radii of the outer and inner circle provides a m e a s u r e m e n t for the quality of approximation (the outer circle is the function which should be recorded).
B} The Analysis of 'Bed Thickness' by Equal D i s t a n c e Samples
In the previous example we had two variables, profile length and a second independent
variable,
variable
for
instance carbonate
is bed t h i c k n e s s  
content.
the sum
If rhythms
are studied,
of this variable being profile
usually the only length.
Therefore,
it is classically assumed t h a t the beds have been deposited in equal time intervals, i.e. the n u m b e r of beds is t a k e n
as a r e l a t i v e m e a s u r e m e n t
is a m e a s u r e m e n t of the varying s e d i m e n t a t i o n r a t e , independent FISCHER,
frame.
An
1982) t h a t
phenomenon
alternative
sedimentological
of time.
and t h e s e two variables form an model
is
the s e d i m e n t a t i o n r a t e was fairly c o n s t a n t ,
is due to
fluctuations
Then bed thickness
(SCHWARZACHER
&
and t h a t t h e bedding
in the lithological composition, as discussed in the
last example. To analyze this model in t e r m s of bed thickness by c o m p u t e r algorithms, S c h w a r z a c h e r & Fischer used a special sampling technique. They positioned t h e i r equally spaced
sampling points along the
profile
and then measured
the
thickness of the bed
which was hidden by a sampling point (Fig. 3.15a,b). Therefore, some beds are lost while others are r e c o r d e d several times. The method has the e f f e c t t h a t it appears more simply to decide which of several possible bedding planes has to be t a k e n as the boundary of the bed {there can be secondary f e a t u r e s due to solution). In addition, the method allows to hope t h a t e r r o r s will be s m o o t h e d out by the o b j e c t i v e sampling procedure.
In the
next step, they r e l a t e d bed thickness to profile length. The resulting graph (Fig. 3.15c) can well be again
analyzed
by methods
like a u t o c o r r e l a t i o n
and spectra.
Now, one can ask
what happens if the sampling distance becomes very small. For an 'infinitesimally'
small sampling distance, the graph resembles a step function (Fig. 3.15d), and 'bed thickness'
appears
profile.
twice
in this graph,
i.e. the beds are r e p r e s e n t e d
as squares along the
It is hard to identify this r e p r e s e n t a t i o n with a useful i n t e r p r e t a t i o n . The only
significant p a t t e r n s are the steps at the boundaries of the bed. Using only these i n t e r r u p tions {Fig. 3.1~e} one gets a series of phase modulated signals along t h e profile. These signals   t h e bedding planes  could be recorded directly along the profile by s e d i m e n tological reasoning, and then, of course, with higher precision than by an 'equal interval method',
where the distances are chosen by a rule like ' t a k e the thinnest beds which
96
Fig. 3.15: The analysis of 'bed thickness' along a profile {a). The points, at which bed thickness is measured, are t a k e n in equal intervals (b) and plotted against profile length (c). The t r a n s i t i o n to infinitesimally small sampling intervals gives a s e q u e n c e of 'squares' (d)   the remaining i n f o r m a t i o n are the i n t e r r u p t s b e t w e e n beds (e). The assumption of fairly c o n s t a n t s e d i m e n t a t i o n r a t e s allows to t r a n s f o r m the phase m o d u l a t e d signals (e) into amplitude modulated ones (f).
are
encountered
by some
frequency'.
Anyway,
the
decision
is necessary,
what
is and
what is not a bedding plane. The only point, which is c o n t r a r y to the geological method (the definition of bedding planes), is t h a t the signals are not equally spaced, analyzed can
with
standard
computer
programs
for
and t h a t they, t h e r e f o r e , c a n n o t be
time
series.
On
the o t h e r
be easily done if t h e equal d i s t a n c e sampling m e t h o d is u s e d  
a reason
to use
the
less suited method,
a simple
manipulation,
be brought into a form
the
phase
hand,
but can this be
are we really ' s l a v e d ' by t h e c o m p u t e r ?
modulated
system
that By
of bedding planes (Fig. 3.15e) can
which allows us to analyze the data by an 'equal step' a u t o 
c o r r e l a t i o n program. The approach, so far, was t h a t the s e d i m e n t a t i o n r a t e was considered fairly
constant,
i.e.
the
distance
between
the bedding planes provides a m e a s u r e m e n t
of t h e duration t i m e of a sedimentological 'signal'   of a bed. Therefore, one c a n give a plot
' n u m b e r of the initial signal' or ' n u m b e r of the i n t e r r u p t ' against the ' d u r a t i o n
97
time of the signal' (Fig. 3.15f). This transforms the phase modulated signal of bedding planes into an amplitude modulated one with equal distances b e t w e e n the data points. But now, it turns out that the two d i f f e r e n t sedimentological models b e c o m e identical. The assumption t h a t
the sedimentation r a t e
from the model that
the beds are deposited as e v e n t s in equal time intervals because
the
duration time,
is fairly c o n s t a n t cannot
be distinguished
as defined, is proportional to bed thickness. This, of course, holds
only for the ' s t a t i s t i c a l approach'; by geological reasoning it is usually possible to distinguish t h e s e two cases (EINSELE & SEILACHER, eds. 1982). Thus, one c o m e s out with the
result
that
the connection of the depositional models with a s t a t i s t i c a l
can cause a worse defined s t a t i s t i c a l problem, in the sense t h a t
formalism
the s t a t i s t i c s cannot
prove which of the sedimentological models is the c o r r e c t one while a successful s t a t i s t i cal t e s t always seems to prove the a priori sedimentological model. The problem an initial model is consistent with the statistical analysis
whether
is not a trivial one; in topology
the analogous problem is genericity.
"To
many
scientists
interesting. a
mapping
perturb
What f:U
we
~ R m,
f slightly
the
'genericity'
problem
are
looking
is
where
U
to o b t a i n
is
for an
a nicer
open and
has
the set
simpler
always
following: in
R n,
how
mapping?"
been given can
we
(LU,
1976).
The
idea
to
analyze
the
sedimentological problem
in t e r m s of a mapping leads
to the following diagram
~
s
2
It'
The set of observations (location and thickness of beds: P) maps under the sedimentological hypothesis s 1 (event sedimentation) onto the d i s c r e t e space D 1 (bed number and bed thickness). Under the hypothesis s 2 (constant s e d i m e n t a t i o n rate) the same set maps onto an alternative
discrete
space.
As
turned
map I which t r a n s f o r m s D 2 into Dt,
out
during the earlier discussion, t h e r e exists a
but only if D 2 is c o n s t r u c t e d from infinitesimally
small sampling intervals, i.e. if D 2 contains all bedding planes. Otherwise, if the sampling distance
becomes
sufficiently large,
D 1 cannot
be c o n s t r u c t e d in all details
from D 2.
On the o t h e r hand, from D 1 we can construct all possible o u t c o m e s of the model and sampling s t r a t e g y s2(n) (n for the number of sampling points). The reason is that P can be r e c o n s t r u c t e d in all details from D 1 but not from D 2. Thus, diagrammatically
98
s1 p
"., D I Sl 1
D2(o~) and it turns out t h a t only D I is a generic r e p r e s e n t a t i o n of t h e data, i.e. m a t h e m a t i c a l analysis justifies t h e way via geological reasoning. In r e t r o s p e c t i o n , the problems c o n c e r n i n g Markov chains a r e t h e same  geologically and m a t h e m a t i c a l l y .
3.4
CENTROID CLUSTER STRATEGIES  CHAOS ON FINITE POINT SETS
A
wide
problems.
field of
geological
A set of o b j e c t s  
a way t h a t
and paleontological
samples,
research
specimens e t c .  
focuses on c l a s s i f i c a t i o n
has to be classified
in such
the e l e m e n t s of a c l u s t e r show a maximum of ' s i m i l a r i t y ' while d i f f e r e n t
c l u s t e r s have a maximal dissimilarity. As the numbers of objects and variables b e c o m e large, it is c o n v e n i e n t to use c o m p u t e r procedures to solve the c l u s t e r p a t t e r n recognition problems. The classical approach into this direction b e c a m e known as ' n u m e r i c a l t a x o n o my'
{SOKAL & SNEATH,
which t r y
1964). Nowadays,
to solve the p a t t e r n
recognition problem
of various similarity and d i s t a n c e & LANGER,
1977; VOGEL, alternative
t h e r e exists a large n u m b e r of a l g o r i t h m s
1975). Clustering
distance
in a s t r a i g h t forward way by use
m e a s u r e m e n t s {e.g. HARTIGAN, strategies
measurement
or
even
produce,
1975; STEINHAUSEN in general,
solution;
an
a different
the data
can change the local and global s t r u c t u r e of the clusters
input
no unique
sequence
of
(VOGEL, 1975). In
addition, our opinion about t h e image of t h e s t a t i s t i c a l universe c a n fairly diverge from the
similarity
points and
structure
to be discussed:
stability
problems
a classification,
which is g e n e r a t e d
by a c l u s t e r i n g s t r a t e g y .
t h e r e p r e s e n t a t i o n of c l u s t e r s with d u s t e r
strategies.
in binary trees,
The binary t r e e s
There
are
three
image concepts,
imply t h a t
they
give
but t h e comparison with c l a s s i f i c a t i o n t r e e s shows t h a t this is not the
case. The image concept of most c l u s t e r s t r a t e g i e s is far from our g e o m e t r i c a l intuition.
99
It will turn out that this discrepancy explains much of the instable b e h a v i o r   because the clusters are not g e o m e t r i c a l objects, they cannot be used for a true classification, and any additional e l e m e n t destroys the local structure. Besides the instability b e t w e e n clusters,
there
appears
an
instability within clusters,
which
is more
striking
because
it does not depend on some metadefinition of a cluster and leads to true chaotic behavior on finite point sets.
3.4. I
The c l u s t e r s t r u c t u r e
Binary Trees
found by some algorithm is usually r e p r e s e n t e d as a binary
tree, which illustrates the similarities or distances b e t w e e n the hierarchies of e l e m e n t s and clusters. The binary t r e e s in cluster analysis are just a picture of one possible similarity s t r u c t u r e
(VOGEL,
sequence. Thus,
they do not allow to insert an additional object without changing the
structure,
at
least
1975);
locaiIy,
~hey are
not at all a classification by some ordering
nor do they allow to search an e l e m e n t by some decision
rule, as it is possible with binary t r e e s which r e p r e s e n t a relation or ordering b e t w e e n data.
Binary t r e e s are commonly used as classification s t r u c t u r e s that allow to search
and to insert e l e m e n t s very quickly in sophisticated c o m p u t e r programs (WIRTH, DENERT & FRANK,
1977),
for a m a t h e m a t i c a l discussion see
e.g.
1975;
SCHREIDER (1975).
The d i f f e r e n c e b e t w e e n binary t r e e s and cluster t r e e s can be illustrated in the following way: classification trees: (I)
B
1\
F
C
E
t
/"
/"
G
I
N
K
(II)
(L 0/\i j , o'\, /\, /\, I
)
)
I
I
I
~
000
001
010
OII
IO0
I0I
IIO
I
III
100
cluster tree: { A,B,C,D,E,F,G,H,I, J, K,L,M,N,O} !
{ A,B,C,D,E,F,G,H,I}
I
!
I
{ J,K,L,M,N,O}
{ E,F,G,H,I }
t
[
{ J,K,L}
{A,B,C,D}
i {E'F'G}
l
'i
{ M,N,O }
I
,
{A,B} { C,D}
r~
AB
The
{H,I}
{E,F}
r%
CD
G
EF
classification
trees
HI
{ JK
represent
A < B < C < ... < O. Therefore,
{
an
it
L
ordering
provides
an
MN
O
relationship optimal
like
strategy
'before' to
in
search
tree
(I}:
elements.
In the same way, o b j e c t s can be classified by a sequence of binary decisions ' t o have or to have not a c e r t a i n property'. Tree (II) gives an example for such a classification of the binary numbers. This t r e e allows to find e l e m e n t s by the decision rules at the nodes, and it allows to insert new e l e m e n t s at its proper position, e.g. it is no problem to
insert
a
number with
four digits. Tree
(I) provides a minimal classification while
t r e e (II) is an optimal dynamic classification because it allows to insert new e l e m e n t s without any a l t e r a t i o n of the p r e s e n t t r e e s t r u c t u r e .
In contrary,
the c l u s t e r t r e e
does not describe how to arrive at any one of the
e l e m e n t s if one s t a r t s at the highest hierarchical level. As far as some distance measu r e m e n t exists b e t w e e n A, B, ..., 0, the cluster t r e e simply s t a t e s that A is more similar to B than to C. But because the binary t r e e cannot describe a higher dimensional variable space, it does also not ensure t h a t the distance b e t w e e n A and t:3 may not be identical with the d i s t a n c e b e t w e e n A and C. This p r o p e r t y will later turn out to be the main source for instable clustering within clusters. The cluster t r e e only gives a possible structure of similarities and a possible grouping into s u b s e t s  
and not at
aI1 a rule how
to distinguish the subsets.
3.4.2
Image C o n c e p t s
"Cluster analysis is one of the P a t t e r n Recognition techniques and should be apprecia t e d as such" DIDAY & SIMON (1980) write, and
"the
general
tion
function
idea
of
all
(clustering)
these from
methods two
is
types
to of
build
an
identifica
information.
101
0 O
O Q
O
O
O
O O
Fig. 3.16: A point p a t t e r n in the plane (left} and the intuitive g e o m e t r i c a l interp r e t a t i o n as a bounded object (right).
on one hand, on
the
the e x p e r i m e n t a l
other,
the
a
result,
priori
representation
of
a
class
or
cluster."
Although the c o n c e p t s of clustering analysis can be well described in a formal m a t h e m a t i cal notation (DIDAY & SIMON, 1980; HARTIGAN, 1975), these c o n c e p t s have some 'weak w points. As Diday & Simon summarize:
"In
fact
the c o n c e p t specialist'
these
(the
cluster)
of p o t e n t i a l selects
the p r o p e r t i e s
or
this
rely
representations
inertial
functions.
representation
of his e x p e r i m e n t a l
from
The the
heavily
on
'classifier knowledge
of
universe."
Indeed, to work properly with such techniques does not only require to understand the mathematical
formulae,
but
also
to
understand
the
qualitative
behavior of
a cluster
s t r a t e g y and the concept of ' c l u s t e r s ' underlying a certain clustering process.
Given a point set in the plane (Fig. 3.16), our intention will be, in general, that t h e s e points are the image of a twodimensional object with a well defined boundary. In a more general sense, one e x p e c t s that the points are a sample from a ndimensional density distribution
which
is continuously defined
in R n,
and,
therefore,
one suspects
102
Fig. 3.17: A l t e r n a t i v e p a t t e r n c o n c e p t s for the point set of Fig. 3 . t 6  {left) and the binary fusion s t r u c t u r e of a weighted centroid method {right},
that
an increasing sample size gives a s t a t i s t i c a l
universe
a
line
which can be bounded by
smooth probabilistic surfaces. A consequence is that one identifies automatically ' c l u s t e r recognition' with classification. Indeed, if one has found a cluster and has bounded it by a closed surface,
then any additional data point is a c c e p t e d as a m e m b e r of the
point set if it is located inside the bounding hypersurface. A point outside of this ' o b j e c t ' will be a s s o c i a t e d with
it if its inclusion does not disturb our c o n c e p t of the image
too much. Thus, the g e o m e t r i c a l objects, of our opinion provide classification rules, and t h e s e are possibie because the g e o m e t r i c a l objects are stable.
Besides
this
intuitive
concept,
there
are
other
possibilities to
define
'clusters'.
The points of Fig. 3.16 can be c o n n e c t e d by a line; the resulting image approximates a twodimensional curve
(Fig. 3.17).
The
resulting ' o b j e c t ' can
be
accepted,
but it is
not the kind of things one e x p e c t s in a twodimensional space. The binary fusion p a t t e r n of a c e n t r o i d cluster these that
pattern
method (Fig. 3.17) diverges still
c o n c e p t s appear
they are not stable, and,
somewhat strange,
further
from our opinion. That
may have its reason in the
feeling
in fact, they are not stable in a structural sense. Any
additional point alters these patterns, at least locally, in an unpredictable way. F u r t h e r more, these objects g e n e r a t e no classification for additional points, t h e r e is no relation like 'inside' or 'outside', like 'belonging to' etc. Classificatory objects for points in R n
103
require t h a t they are bounded by a surface. Therefore, an object in R 3 must be t h r e e  dimensional itself. Points, lines and open surfaces in R 3 are d e g e n e r a t e d objects can
exist
neither
physically nor statistically,
they
are
objects
which c a n n o t
which
'contain'
something. The definition of a classificatory object in a d i s c r e t e ndimensional variable space
requires,
therefore,
at
describe just one object, R n, A c l u s t e r
least n+l data
points or samples. These n+t points then
a t r i a n g l e in R 2, a t e t r a h e d r o n in R 3, a h y p e r t e t r a h e d r o n in
and classification
concept
requires
that
the number of e l e m e n t s to be
classified exceeds clearly the number of variables defining the position of the e l e m e n t s in a hyperspace.
There
should not
be more
than
(n DIV m) o b j e c t s  
n: n u m b e r of
e l e m e n t s , m: number of variables, DIV: division of integers. In the
applications of c l u s t e r analysis it is c o m m o n l y the case t h a t
the n u m b e r
of variables equals, or even exceeds the number of e l e m e n t s or samples. The only structure one can find under such conditions is an ordering by some similarity rule, but one cannot expect sense, set
that
this provides a classification, as it is usually done. In a s t a t i s t i c a l
'classificatory
{EFRON,
objects'
are
closely
1965). In a topological
related
sense,
they
to the convex hull of a finite point are just higherdimensional analogues
of a smooth closed surface, a manifold. Because the point sets are finite, t h e continuous d i f f e r e n t i a b l e s t r u c t u r e s are replaced by simplices and t h e i r unions, polygons and polyhedra.
3.4.3 Within the various
Stability Problems with C e n t r o i d Clustering types
of cluster s t r a t e g i e s ,
hierarchical
methods are
the ones
most c o m m o n l y used, and within this subgroup the c e n t r o i d methods are most popular. The t e r m
' c e n t r o i d ' is here used in a very general sense. It simply expresses t h a t two
fused e l e m e n t s are f u r t h e r r e p r e s e n t e d by a single e l e m e n t , the e e n t r o i d of the original points. It is known from a r e s p e c t a b l e number of c l u s t e r s t r a t e g i e s t h a t the c o n s t r u c t e d d i s t a n c e t r e e depends, to some e x t e n t , on the initial ordering of the input data {VOGEL, 1975).
As
Vogel
found
by extensive
testing
with
one
special
data
set,
the
centroid
methods b e c o m e p a r t i c u l a r l y sensitive to the ordering of the d a t a if an e n t r o p y d i s t a n c e measurement
is used.
Cluster s t r a t e g i e s with o t h e r d i s t a n c e m e a s u r e m e n t s showed this
sensitivity only on the lower h i e r a r c h i c a l levels. In palecology one has very nice neous data s t r u c t u r e s  
homoge
the frequencies of species within a sample. In this special case,
the usual m e t r i c distance m e a s u r e m e n t s make not much sense while the entropy m e a s u r e ment
provides
a
'natural'
distance
measurement
for
these
frequency
data
of
species
(BAYER, 1982). This d i s t a n c e m e a s u r e m e n t has, in addition, t h e p r o p e r t y t h a t it approxim a t e s t h e C h i 2  t e s t for h o m o g e n e i t y of the samples (DIDAY & SIMON, I980); homogeneity of the samples being of special i n t e r e s t in palecological studies. Therefore, a program
104
~t
%. Fig. 3.18: C l a s s i f i c a t i o n and distribution of the f o r a m i n i f e r a fauna of Todos Santos Bay. Left: Classification by hand (WALTON, 1955); right: Ordering by a c l u s t e r program {KAESSLER, 1966).
was i m p l e m e n t e d t h a t allowed to analyze faunal data by the ' e n t r o p y m e t h o d ' (BAYER, 1982). The d a t a of a *hand m a d e ' ecological analysis (WALTON, been see
reevaluated how t h e
by c l u s t e r
strategy
analysis {KAESSLER,
works.
Fig. 3.18
gives the
1966), were spatial
1955), which had l a t e r rereevaluated
again
to
faunal distribution p a t t e r n
of
foraminifera, which had been found by the previous workers. Already the first few runs with
the
'entropy
analysis'
showed a strong d e p e n d e n c e of the result on the ordering
of the input data. The nice s t a t i s t i c a l p r o p e r t i e s of the e n t r o p y d i s t a n c e m e a s u r e m e n t , t h e r e f o r e , w e r e useless. But the use of o t h e r d i s t a n c e m e a s u r e m e n t s with r e p r e s e n t a t i o n of
the c l u s t e r s
VOGEL's
by t h e i r centroids
(1975) claim.
In fact,
did not e i t h e r stabilize the p a t t e r n ,
dependenc e from
in c o n t r a s t
to
t h e initial ordering of the data was
nearly the same. Fig. 3.19 gives some e x a m p l e s of various t r e e s t r u c t u r e s , which result from it was
slightly
is not the
different
the
initial
conditions and p a r a m e t e r
choice of a special
s t a r t i n g point
settings.
distance measurement,
for a more
detailed
The o b s e r v a t i o n
t h a t causes
that
the instabilities,
analysis of t h e s t a b i l i t y properties of the
c e n t r o i d c l u s t e r s t r a t e g i e s . This s t a b i l i t y analysis will now be outlined. The widespread use of c e n t r o i d c l u s t e r s t r a t e g i e s is, in part, due to t h e i r property to produce
nice binary h i e r a r c h i c a l
to belong to or
a cluster
unweighted.
are
'Weighted'
structures
replaced means
within a c l u s t e r while ' u n w e i g h t e d '
from
every
by t h e i r centroid,
that
the
means
centroid that
data
set.
The points
found
which can be e i t h e r weighted
is c o m p u t e d
from
all d a t a
points
a c o m p u t e d c e n t r o i d is f u r t h e r t r e a t e d
like a d a t a point, or t h a t it has equal weight as a d a t a point. In both eases, the progress of c l u s t e r i n g
causes a c o n c e n t r a t i o n of the original data
representatives,
points into fewer
and f e w e r
the centroids. Thus, e v e r y single fusion of two points e v a c u a t e s locally
the point space. As this c o n c e n t r a t i o n of the points proceeds in a c e r t a i n area, it forces
105
c//
t
°
! a
F
Fig, 3.19: Various classifications with centroid cluster s t r a t e g i e s of WALTONIs (1955) foraminifera data. Lower left and upper left: Two of a large number of possible cluster t r e e s which result only of an a l t e r e d input sequence of the data. Upper right: a l t e r e d metric, Lower right: d i f f e r e n t clustering algorithm,
106
the
process to branch into o t h e r areas which are still more densely covered by data
points, and the s a m e process is r e p e a t e d . This coupled process of local evacuation and subsequent branching into another area of the data s p a c e is the reason why every data set is mapped onto a nice binary tree. The t r e e s t r u c t u r e is due to the clustering process,
and it
that of
the these
is not a property of the data
distances b e t w e e n the
set. The same process causes, in addition,
clusters increase
s t r a t e g i e s (STEINHAUSER & LANGER,
monotonically, a c e l e b r a t e d p r o p e r t y 1977)
but
an
a l t e r n a t i v e viewpoint
is t h a t t h e s e s t r a t e g i e s impose a binary t r e e p a t t e r n onto every data set.
Instabilities b e t w e e n c l u s t e r s are
r e l a t e d to the problem
which images or which
point configurations can be a c c e p t e d as a cluster, i.e. one needs a metalanguage definition of a cluster. Although the p r o p e r t i e s of clusters and partitions, their homogeneity, are usually defined in t e r m s of the metric, the idea of a 'good' cluster is mainly a g e o m e t rical one. The question how
distant
two
allowed or not.
clusters
'what is a good c l u s t e r ' must
is commonly reduced to the problem
be to be separable
In clustering analysis the
terms
and w h e t h e r concave clusters are
convex and concave are
h o m o g e n e i t y and c h a i n  h o m o g e n e i t y (DIDAY & SIMON, use
the
earlier
discussed g e o m e t r i c a l
image
concept
1980). to
replaced by
At the moment, we may
define
'good'
clusters.
Then,
concavity means simply that every line connecting two points of a cluster is bound to the interior of an hyperpolyhedron, t h a t forms the boundary of the cluster. Consequently the
c e n t r o i d is not necessarily bound to the interior of a c o n c a v e cluster (Fig. 3.2O).
And because e v e r y fusion of two points increases locally the distances, the process can r e a c h a s t a t e w h e r e the image breaks into d i s c o n n e c t e d pieces, and where these pieces are c o n n e c t e d with data points which form s e p a r a t e clusters under the g e o m e t r i c a l image c o n c e p t (Fig. not
3.20). The striking point is that in such cases the centroid method does
follow the
'rule
of n e a r e s t
neighborhood'; not
the clusters, which appear
closest
because their boundaries are closest, are fused, but the more distant clusters are c o n n e c t ed (Fig. work
as
3.20). This behavior elucidates again that such a p a t t e r n r e p r e s e n t a t i o n cannot a
classification, at
least
not
as a 'natural
classification'; in some r e s p e c t s ,
this p a t t e r n c o n c e p t is even c o n t r a d i c t o r y to our understanding of similarity.
So far,
a m e t a d e s c r i p t i o n of clusters was necessary. One can also simply a c c e p t
t h a t a c l u s t e r formed by a c e n t r o i d process has nothing in common with our g e o m e t r i c a l imagination. If it can be uniquely c o n s t r u c t e d by an algorithm, it may be a totally abstract structure
but why should it not be as real as g e o m e t r y ? That clustering does
not produce such a well defined a b s t r a c t p a t t e r n , wiil be shown by a discussion of instabilities within clusters. To avoid any assumption about has to r e s t r i c t the a t t e n t i o n to local s t r u c t u r e s
the
s t r u c t u r e of clusters, one
as they are defined by the clustering
process itself. Then one can analyze how a local disturbance propagates into the e s t a b lished hierarchy.
Fig. 3.21 gives two c l u s t e r p a t t e r n s over a given point set (compare
107
121
........
Q
i.
E3
Fig. 3.20: Branching b e t w e e n clusters. The smaller cluster will be fused with the ' r e c t a n g u l a r points' because the formation of centroids changed the local topology of the point space. Consequently the larger cluster will be fused with the ~triangular points' in a later step.
Figs. 3.17 3.18). This time, an unweighted centroid method was used to find the similarity structure. This method causes a local decision problem because t h e r e are two choices for the same point to be fused, two neighbors have equal distance. Dependent on this choice,
two d i f f e r e n t local p a t t e r n s are
g e n e r a t e d {Fig. 3.21). These local a l t e r n a t i v e s
project onto the global t r e e s t r u c t u r e changing the distances or similarities all the way through the cluster t r e e (Fig. 3.21). The d i f f e r e n c e is strong enough to change the overall t r e e p a t t e r n . Thus, one of the two t r e e s appears more c o m p a c t , and, t h e r e f o r e , it will be more easily a c c e p t e d as a larger cluster if the t r e e s are embedded in a more complex cluster
tree.
On the
lower cluster
level
the
substructure appears also more c o m p a c t
in the right t r e e of Fig. 3.21 while in the left one two clusters can still be distinguished on the lower level. The decision, which one of the two clusters will be produced, depends only on the ordering of the data. The c o m p u t e r algorithm has either to take the first pair of data of the
points with smallest distance or the last one for f u s i o n   a consequence
binary t r e e
r e p r e s e n t a t i o n . Once
made the decision, the local topology of the
point p a t t e r n has changed, and, t h e r e f o r e , the local decision p r o j e c t s onto all l a t e r fusions contacting
this area.
Thus,
a true
bifurcation of the solution occurs,
and
it depends
108
/ %.
%.
Fig. 3.21: A single local decision problem (arrows) b e t w e e n two equally distant points a l t e r s the local t r e e s t r u c t u r e and the distances within the e n t i r e hierarchy.
only on the initial conditions (ordering of the data)
which branch of the solution will
be taken. The next question is w h e t h e r there exist situations that cause c a s c a d e s of bifurcations. In this case, the clustering process would b e c o m e chaotic, as far as this is possible on a finite point set. Such a sequence of bifurcations can be easily c o n s t r u c t e d by just
109
a
C Fig. 3.22: Centroid clustering as a decision game. The figure gives two possible cluster solutions for the identical, equally spaced point set. a) Every fusion causes the bifurcation into two identical subsystems; b) the associated fusion tree; c) the cluster trees.
choosing the most d e g e n e r a t e d case of input data  equally spaced data points. It does not m a t t e r w h e t h e r these points are arranged on a straight line, on a circle or on a regular (hyper)grid. Equally spaced data points on a straight line provide the most simple case;
Fig. 3.22
illustrates what
happens
in this special case
from various viewpoints.
First, the clustering process has a dynamical aspect   at the same time, only two points can
be
fused, and
the
fusion of
these
two points alters
the
local distance s t r u c t u r e
b e t w e e n the points. In the case of Fig. 3.22, the initial distance of izhe equally spaced points, ax, is a l t e r e d to (3ax)/2 b e t w e e n the centroid and its neighbors. Therefore, the data space is divided into two subsets of points which have still the original structure. But the two substructures are now independent; they can be t r e a t e d as parallel processes of
identical behavior. This bifurcation process proceeds until t h e r e are no data
points
left with the original spacing. Within the limit of a finite point space, any fusion on one of the subsets g e n e r a t e s new identical subsets, every bifurcation of the dynamical system g e n e r a t e s two dynamical s y s t e m s of the identical type. In addition, any fusion is a random choice b e t w e e n several possibilities, and the decision, which pair of data points is taken, depends only on the initial data configuration. One has a p e r f e c t l y c h a o t i c
110
system on a
finite point set.
finally. There
are,
Because
the
point set
is finite, the process t e r m i n a t e s
for e v e r y subsystem, two main possible o u t c o m e s which depend on
some initial choices of the fusion process {Fig. 3.22). One possibility is that during time all
points are
fused into
pairs.
Then the
resulting centroids are again equally spaced
(Fig. 3.22 left}, only with a l t e r e d distances. Thus, the process is cyclic. The a l t e r n a t i v e is that already the initial fusions d e t e r m i n e , to some e x t e n t , the further progress because t h e r e remain some points of smallest distance on every level which dominate the process {Fig. 3.22 right}. The c e n t r o i d clustering techniques, t h e r e f o r e , provide e x c e l l e n t examples of c h a o t i c behavior on finite point sets. In the case of equally spaced data, every slight change of the input data will cause another cluster tree, i.e. the system is e x t r e m e l y sensitive to the initial conditions. Because the number of data points is finite, the number of possible o u t c o m e s is finite as well, it is the number of disjunct permutations of the data. Therefore, the c h a o t i c aspect of the clustering process will increase with the number of (equally spaced} data, i.e. one has the same situation as with the C h i 2  t e s t against a uniform distribution or the t r a j e c t o r i e s of particles in Galton's machine.
Now, a p r a c t i t i o n e r could argue that
the c h a o t i c behavior of the c e n t r o i d c l u s t e r
s t r a t e g i e s requires a very special and d e g e n e r a t e d data s t r u c t u r e .
On the o t h e r hand,
it is well known that most cluster s t r a t e g i e s depend to some e x t e n t on the input sequence of the data {VOGEL,
1975}, and it s e e m s likely that this phenomenon is related to the
discussed instability. The remaining question, t h e r e f o r e , is how this instability can arise in a s t a t i s t i c a l sample. There are two s t r u c t u r e s which can e n f o r c e the chaotic behavior with any kind of data. The first one is that the data are measured with a c e r t a i n precision. Thus, with increasing number of data one will get an increasing number of equally distant data. Because multiple values at the same point do not change the local topology if they are fused under the centroid condition, the data p a t t e r n tends, at least locally, towards equally spaced data.
And once
more we have
the situation
that
an increase
of data does not stabilize the process, as one should e x p e c t from s t a t i s t i c a l reasoning, but
that
it
destabilizes the clustering process. In addition, once
more one finds that
a uniform or an equally spaced distribution leads to instability. This s t r u c t u r e probably is The
the
main
reason
frequencies are
structure,
which can
why
the
natural
palecological data
cause
particularly
numbers which e n f o r c e equal
e n f o r c e c h a o t i c behavior,
instable
clustering.
spacing locally. The
second
is a numerical one. Every c o m p u t a t i o n
with a finite number of digits causes numerical rounding or truncation errors. An excel10000 lent example due to WIRTH (1972) is the c o m p u t a t i o n of the sum i_~l/i, which takes d i f f e r e n t values if it is c o m p u t e d forward and b a c k w a r d  e.g., 3 of 12 tation
the numerical error a f f e c t s ,
valid digits of a 48bit machine. The same problem occurs for the compu
of distances b e t w e e n data
points (and centroids). If the number of variables is
high enough, one has to e x p e c t t h a t the d i s t a n c e b e t w e e n two points is not s y m m e t r i c , e.g. in the case of the Euclidian distance one has
111
}~ (XliX2i)2 ~ ~ (X2iXli)2 . Therefore,
this e r r o r will also depend on the input sequence of the data, and this t i m e
the error increases with the number of variables and with the complexity of the d i s t a n c e measurement.
This
can
explain
why
the
'entropy
distance
measurement'
is especially
instable. Logarithms are necessary to c o m p u t e this d i s t a n c e m e a s u r e m e n t ,
and this en
forces numerical errors.
Thus, the centroid cluster methods turn out to be highly instable due to t h e i r patt e r n recognition concept. They behave in a c h a o t i c m a n n e r with r e s p e c t to slight changes of the initial conditions, namely the ordering of the input data. The c h a o t i c behavior is triggered by the t e n d e n c y of measured data to form, at least locally, regular spacings, and
by the
numeric
error
computation
of order
the
distance
depends
measurement,
again on the
mainly by its a s y m m e t r y  
whereby
the
input sequence
of
the data.
in
totality,
it turned out during the last two c h a p t e r s t h a t most operations on finite point
sets with the c o m p u t e r have to be handled with special care. The local methods, which d o m i n a t e this field, are usually e x t r e m e l y instable due to small changes of the boundary conditions
and of
the
initial conditions.
What
we,
in general,
would need, are r a t h e r
more rigid topological methods than the highly sophisticated numerical procedures.
(3C
Fig.
3.23:
Classification
by probabilistic
neighborhoods.
Points
with overlapping
neighborhoods belong to a c l u s t e r with a well defined boundary (right).
112
Such a   p r o b a b l y m o r e   r i g i d
method was proposed by GRENANDER
(1981}.
The
idea is similar to the single linkage methods; however, it has a more g e o m e t r i c background, and
this returns to the discussion in section 3.4.2. A probabilistic neighborhood
is associated with every point: e.g., circles in the plane, spheres, hyperspheres or a l t e r n a tively polyhedrons (Fig. 3.23t. Every point has a c e r t a i n probability to be found elsewhere in this neighborhood, and a possible assumption about the probability is ' t h a t the curvature at the boundary is proportional to the density of the probability measure, ~ GRENANDER (1981),
or
the probability d e c r e a s e s , as
the d i a m e t e r of the
probabilistic neighborhood
increases. Two points are grouped in a similarity c l u s t e r if their probabilistic neighborhoods overlap. If the radius e of the neighborhood is fixed to a c e r t a i n value, then a unique
partition of the
data
set
arises, and
we get
partitions of d i f f e r e n t
roughness
for d i f f e r e n t values of e. The resulting s t r u c t u r e is not a t r e e because two or more e l e m e n t s may
fuse at
once for a c e r t a i n value of e. However, the resulting ' c l u s t e r s '
have now well defined boundaries as illustrated in Fig. 3.23. The point set of a ' c l u s t e r ' is bounded by a 'tubular neighborhood ~, and ' c l u s t e r s ~ are d i f f e r e n t on a c e r t a i n e  l e v e l if their boundaries do not overlap. Thus, given such a classification, we can later add samples and question w h e t h e r they belong to a ' c l u s t e r ~ for a c e r t a i n e  v a l u e . Clearly, the classification is not
stable in the s e n s e that
an additional object
may cause two
or more ' c l u s t e r s ' to fuse without changing the elevel. However, that is a quite more d i f f e r e n t instability than those discussed above. The classification certainly stabilizes as the number of data points increases and approaches the universe. This approach returns to the
exam
ples of the second c h a p t e r and is closely r e l a t e d to the continuation problem discussed there.
3.5 TREE PATTERNS BETWEEN CHAOS AND ORDER
The previous presentation focussed at practical problems which arise from unstable algorithms. Here, the viewpoint is much more theoretical, and the questions are mainly conceptual. The section serves as a kind of summary of the previous c h a p t e r s and conn e c t s them with the topic of this section, shape,
which may
following one. The morphology of t r e e  l i k e s t r u c t u r e s wilt be the especially the
arise
connection b e t w e e n branching p a t t e r n
under c e r t a i n
and overall
conditions: If one closely studies the crown of
a tree, the branching p a t t e r n appears r a t h e r irregular while an e n t i r e view of the crown, from
some
distance,
is c h a r a c t e r i s t i c
on
the
s p e c i e s  l e v e l (Fig.
3.24). On this basis
HONDA (1971) a t t e m p t e d to describe the multifarious form of t r e e s by a few p a r a m e t e r s of the branching p a t t e r n . This a t t e m p t , however, was not new: Already D'Arcy Thompson discussed the patterns (DVARCY
are
cymose
inflorescence of
'analogous
THOMPSON,
in
a
t952);
the
curious
and
and,
G.L.
as
botanists instructive Steucek
of t r e e s were already studied by Leonardo da Vinci.
and noticed that way
these botanical
to
the
equiangular
informed
me,
branching
spiral' patterns
113
Fig. 3.24~" Tree images.
R a t h e r similar branching p a t t e r n s occur in geomorphology, the network of drainage systems (Fig. 3.25). Trees and networks are equivalent on a c e r t a i n level, e.g. a t r e e g e n e r a t e s a network if two branches~ which come into c o n t a c t , lap
a
situation
fuse r a t h e r than over
which necessarily occurs if the branching process is bounded to
Fig. 3.25: The simplified drainage nets of the Amazon (left} and the Ganges delta (right).
a
114
surface
or
is
restricted
to
a
nearly
twodimensional object.
The
network
in leaves
provides an example for the l a t t e r case. As a concession to the literature,
the t e r m
' t r e e s ' will be equivocally used as the term 'open networks'.
The
botanical
attempts
toward
mainly d e t e r m i n i s t i c and have,
a
description of
branching p a t t e r n s
in t r e e s
t h e r e f o r e , be c r i t i c i z e d . Thus NIKLAS (1982)
are
writes in
a study on plant branching simulations: "Unless
the
processes ing
The
that
extent
can
be
of
apparent
determined,
a particular
structure
meaning of ' r a n d o m ' , however,
order
there
that
is
no
implies
is r a t h e r
can
arise
from
valid
basis
for
deterministic
vague.
random assert
causes."
If we a t t r i b u t e
a probability to
a growing system to branch or to branch not during a growth step, we may a l t e r n a t i v e l y formulate a d e t e r m i n i s t i c system which branches at the beginning of every growth step, but may be disturbed by some external cause which inhibits branching, and such e x t e r n a l e v e n t s may be truly s t o c h a s t i c . This aspect was clearly expressed by OSTER & GUCKENHEIMER (1976): "If
a
series
chaotic, one
. ..
exhibiting
of
(a)
of
three
the
cerises
no
are
collected
perceivable
. .. ,
regularities,
and
they
then
we
appear
conclude
things:
system
is
truly
stochasticdominated
by
random
influ
ences (b) ties (c) is
experimental are a
error
is
of
such
a
magnitude
that
all
regulari
obscured~ very
simple
obscured
"
deterministic
(by
the
mechanism
phenomenon
described
is
operating, here
in
but
section
3.1.1).
This is a viewpoint which now s e e m s to be common in physics (e.g. HAKEN, 1981; THOMPSON, sections,
and
1982).
Examples of
e x p e r i m e n t a l l y it
points (b) and
(c)
have
s e e m s more appropriate
been
given
in the
previous
to exclude points (b) and (c)
than to prove point (a); the failure of the C h i 2  t e s t in connection with a uniform distribution illustrates this point: There is still enough to be d e t e c t e d on the d e t e r m i n i s t i c level in which s t o c h a s t i c e l e m e n t s may e n t e r in t e r m s of disturbances.
Branching p a t t e r n s of t r e e  l i k e bodies are of some i n t e r e s t because they are not r e s t r i c t e d to trees, they occur commonly in biology and paleontology, on the level of organisms and on
the
level of organs. Besides this, they are even interesting o b j e c t s
in geology, geomorphology, and physics. The discussion of m e t r i c t r e e s is, to some e x t e n t , based on a study by Bayer
& McGhee (An Analytic Approach to
Branching Systems,
115
preprint
1984), a study which was initialized by a paper given by G.L. Steucek at Tfi
bingen in 1984o
3.5.1 Topological P r o p e r t i e s of Open Network P a t t e r n s In at
geomorphological
studies
the sources of streams,
open
networks
a viewpoint
and
tree patterns
naturally
originate
which is opposite to the b o t a n i s t ' s one where
the b r a n c h e s usually originate at the trunk. However, in t e r m s of m o r p h o m e t r y b o t a n i s t s take
a viewpoint
ordered
by
the
identical
to t h e
descriptive
geomorphological one.
"stream
number",
The s t r e a m s or b r a n c h e s are
which is due to S t r a h l e r
{e.g. BARKER
e t al., 1973; GRENANDER, 1976):
¢,./
, =",A ,. /
k
Fig. a.26: Trees with b r a n c h e s ordered by S t r a h l e r numbers. All t r e e s have equal numbers of end branches.
Definition
1:
The
S t r a h I e r
n u m b e r
's',
or
the
order
of
a
single branch, is given by the following rules: (1)
end branches have order s=l
(2)
two branches of order s produce a branch of order s+l
(3)
two b r a n c h e s of d i f f e r e n t order (Smax,Smi n) produce a b r a n c h of order Sma x,
The highest S t r a h l e r number S involved in an open network is called the order
of
the
network.
The S t r a h l e r n u m b e r {Fig. 3.26) of branching systems has special properties (e.g. BARKER et
al.,
1973;
GRENANDER,
in part
they
are
each order plotted the
deducible
are counted,
against
1976). from
In part
they
the definition.
are Thus,
based on empirical if t h e
observations,
numbers of b r a n c h e s of
then the logarithm of the number of b r a n c h e s in each order
the S t r a h l e r
n u m b e r gives a linear plot:
number of b r a n c h e s of order
s {BARKER et
al.,
i.e.
1973);
log n s = a+bs; where or, an observation
Us: from
116
geomorphology again
ns
be
(HORTON, the
1945),
number
of
the
bifurcation
branches
of
ratios
order
are
s,
approximately
then
the
constant.
Let
b i f u r c a t i o n
ratio ns_l/n s = R s
is a p p r o x i m a t e l y
constant,
ship. T h e s e ' e m p i r i c a l of
the
two
Strahler
branches
(3.34)
i.e
the
numbers
ns_ 1 a n d
ns a r e
laws', however, are not unexpected,
number.
of order
By d e f i n i t i o n sI,
and
the
every same
network
holds
for
in a g e o m e t r i c a l
relation
they result from the definition
of order any
s must
contain
substructure
in
the
at
least
tree
with
s > 1. T h e r e f o r e ,
ns  2 n s _ I > 0
or
ns_i/ns > I/2.
(3.35)
In the case ns_ l = I/2, the bifurcation tree is a perfect symmetric tree. The 'empirical laws' are expected whenever the deviation from the binary tree is random, i.e. whenever branches are randomly suppressed and added with a zeromean.
Another
measurement,
which
is s o m e t i m e s
o f a t r e e or a b r a n c h ( G R E N A N D E R ,
1
useful,
1
\1
1
1
2
1
Fig. 3.27: plexity'.
1
Definition
1
1
8
The
o f Fig.
3.26
end
1
branches
1
7
trees
2:
of
2 1
8
The
number
1/
1
, 3
is t h e
1976):
with
n u m b e r
2 1
branches enumerated
o f
1
by ' n u m b e r s o f c o m 
c o m p 1 e x i t y
N
c
of
a
branch
o f o r d e r s is t h e n u m b e r o f i t s e n d b r a n c h e s .
Fig.
3.27
illustrates
the
characterization
of
trees by
number of end branches and the order of the network ship b e t w e e n
their
numbers of complexity.
The
are not independent. The relation
t h e n u m b e r o f c o m p l e x i t y a n d t h e S t r a h l e r n u m b e r is s i m p l y
N
c
> 2S  l ,
(3.36)
II7
and the equality holds only if the t r e e is a p e r f e c t s y m m e t r i c tree. To prove this relationship it is helpful to introduce
Definition 3: The
s k e 1e t o n
of an open network is its largest s y m m e t r i c
substructure. The skeleton is c o n s t r u c t e d by eliminating the branches with S t r a h l e r number Smi n at nodes where SL,i_ 1 ~ SR,i_ 1, The remaining t r e e contains only nodes where the S t r a h l e r number increases and, t h e r e fore,
is p e r f e c t l y
symmetric.
Along a pathway
from
an end branch to the trunk the
Nc 4 s'321
SKELETONS S=1,...,5
, ,r V
Fig. 3.28: Trees p e r m u t a t i o n s of gically distinct S t r a h l e r numbers
of complexity Nc = 3,4, and 5 (s:Strahler numbers) with distinct branch p a t t e r n s . Note t h a t these p e r m u t a t i o n s are not all topoloas indicated by b r a c k e t s . Lower left: e l e m e n t a r y skeletons of 1 to 5.
118
Strahler
n u m b e r simply counts
the
n u m b e r of bifurcations.
If we invert this pathway,
the
n u m b e r of b r a n c h e s of equal S t r a h l e r n u m b e r increases
2 2,
. . . ., . .2.Ss,
monotonously like 20 , 21 , Arriving a t an end branch the n u m b e r of end b r a n c h e s is Nc= 2 S1 .
(a.a6) now is easily proved: If the t r e e is s y m m e t r i c , then the equality holds
Relation
as was shown, o t h e r w i s e b r a n c h e s have been e l i m i n a t e d during the c o n s t r u c t i o n of the skeleton, and, t h e r e f o r e , end b r a n c h e s have been e l i m i n a t e d , i.e. t h e inequality in equation
(3.36) holds. This discussion provides us f u r t h e r with a topological i n t e r p r e t a t i o n of the Strahler a
tree
number: or
It uniquely
its s u b s t r u c t u r e s
defines (Fig.
the
3.28),
e 1e m e n t a r y where
'elementary'
s k e 1 e t o n
means
that
the
of
length of
b r a n c h e s is not considered (empty nodes of t h e skeleton).
A n o t h e r question is how many d i f f e r e n t t r e e s with identical n u m b e r of end b r a n c h e s do exist, or generally: Given a c e r t a i n n u m b e r of end branches, how many d i s t i n c t n e t works c a n be g e n e r a t e d ? GRENANDER
(1976) gives the following solution, which is due
to SHREVE (1966): The n u m b e r of open networks N(n) for any branching system with n end branchings is N(n) = (2n2)! (n! (nl)!) 1,
and t h e n u m b e r N(nl,n2,...,nS_l,l} nS_l,t
n1
(3.37)
of topological distinct networks of order S with
b r a n c h e s of order 1 .. S is S1
N(nl,n2.... nS_l,l) =
11 2(ns2ns+l)((ns2)! ((ns'2ns+l)! (2ns+l2)!)l. (3.38) s=l
These two equations provide us with a probability m e a s u r e m e n t of finding an open network
with S t r a h l e r
numbers
nl,n2,
...
,ns_l,1;
when
the n u m b e r of end b r a n c h e s
is given, the d i s c r e t e probabilities are
(3.39)
P(nl,n2,...,ns_l,l) = N(nl,n 2 ..... ns_l,l) / N(n).
Fig. 3.28 i l l u s t r a t e s the networks which correspond to N(3)=2, N(4)=3, N(5)=14; i.e. the various c o m b i n a t i o n s of branches. However, t h e s e t r e e s are not all topologically distinct, some
of
them
are
simple
symmetric
inflections and c a n n o t
be distinguished:
t r e e s be t h e twodimensional projection of a t h r e e  d i m e n s i o n a l
image,
Let
the
then t h e inflec
tions mean simply t h a t an observer walked around t h e object by 180 ° and then classified it d i f f e r e n t l y b e c a u s e the s y m m e t r i e s have changed. The n u m b e r of topological distinct t r e e s is only
t19
N T = N{n) div 2
+
N(n) mod 2.
{3.40)
The d i f f e r e n c e of the two c o n c e p t s is not trivial, e.g. the probabilities will be d i f f e r e n t under the two hypotheses: The probabilities for a t r e e with four end b r a n c h e s {Fig.
3.28)
are p(3,2,1) = N(3,2,1)/N(4) = 1/3 and p(2,1) = 2/3. Based on topological similarity, however, we find the probabilities PT{3,2,1) = PT(2,1) = 1/2, and the g e o m e t r i c a l hypothesis influences the conclusions we may draw from the probabilities. Finally it seems worthwhile to emphasize
t h a t t h e probabilistic aspect,
mentioned
briefly, is based on the comparison of an observed p a t t e r n with a totally d e t e r m i n i s t i c pattern~
the
symmetric
tree
a possible question would be: How strong do we have
to disturb a binary t r e e to arrive at the observed t r e e p a t t e r n . The c o n c e p t of skeletons can be used to e m p h a s i z e this point. If the s y m m e t r i c t r e e s without r e p e t i t i o n s of e m p t y nodes
 the e l e m e n t a r y
skeletons
are defined as primitives of our t r e e s t r u c t u r e s ,
then any t r e e can be defined as ' p r o d u c t ' of e l e m e n t a r y skeletons: A node is the product {Si,S]}, w h e r e t h e first t e r m means left, t h e second right branching {for s y m m e t r y reasons this
notation
is
arbitrary)
and
Si
defines
an
elementary
skeleton
of
order
(Strahler
number) s=i. The t r e e s of Fig. 3.26 can be described as lists (Fig. 3.29)
",/ x/
V
v
N/
v
V,,/_v"9' V , , /
V
aL, then all p a t h 
w a y s ( m ' R , (n+i)*L) c o n v e r g e e v e n if 'i' is s o m e r a n d o m v a r i a b l e but s a t i s f i e s i _7000.
number of end branches increases, we expect increasing density of end
points and a trend towards a c o n t i n u o u s outline. Fig. 3.36 elucidates this point where only the end points of the t r e e s of Fig. 3.30 are marked. Clearly, the end points cluster in c e r t a i n areas and give the illusion of continuous lines which t o g e t h e r give the
i
Io
\
Fig. 3.36: Distribution of branch endings in the t r e e s of Fig. 3.30.
illusion of a continuous outline. One question to be discussed is w h e t h e r t h e s e points may define a continuous curve or even fill some space, as the number of bifurcation points goes to infinity. The o t h e r question is if t h e r e exists some defined envelope or hull of a c o n v e r g e n t Honda t r e e which can be considered as its limiting shape.
A) Trees, Peano and Jordan Curves
To illustrate the c o n c e p t of shape in more detail we consider a quite d i f f e r e n t system, continuous curves which are nowhere differentiable. A special example, which is useful in this context, is the Koch curve {cf, MANGOLDTKNOPP, 1968). The closed version of a Koch curve can be c o n s t r u c t e d in the following way (Fig.
3.37): A regular
triangle (equal angles) is i n f l e c t e d and superposed on itself so that the sides are divided into t h r e e s e g m e n t s of equal length, the resulting p a t t e r n is a regular star. The corners of this s t a r
are
again
regular
triangles, similar to the original ones, and with each
t36
vS
J i gd, 13tsaTrelC~nstTt~Otr:: ~hatteKrOCh cu
of these triangles the process is repeated, the resulting pattern
as illustrated
The Koch curve is the outer boundary of
in Fig, 3.37. If the process is repeated infinitely,
the triangles shrink to points and give rise to a continuous, nondifferentiable curve. If we now connect a tree pattern
the centers
of successive triangles by straight
lines, these form
with a triple point at each branching event. The branching angles and
the shortening ratios are constant,
i.e. if we delete one of the branch directions, the
resulting binary tree is a Honda tree.
.Fig, 3.38: Construction of a binary tree as sequence of similar triangles.
137
#
a
1I I
/
b
'4'
~
If
J(÷ Ji
C Fig. 3.39: Triangulation of a rectangle by i t e r a t i v e averaging (a; cf. Fig. 2.46) reconsidered as a space filling Peano curve (b). Connection of subsequent centroids g e n e r a t e s t r e e patterns, c: bifurcation and tripling with 'limit shape'; d: space filling t r e e p a t t e r n (see text).
The c e n t e r of every triangle
along the boundary is the termination point of a
end branch. As the iteration process goes to infinity, the triangle shrinks to its centroid which, by definition, is the t e r m i n a t i o n point of an end branch, and we e x p e c t a continuous curve as limit shape for the tree. In detail, this assumption is not quite c o r r e c t . If we only consider branches as indicated in Fig. 3.37,
then
there
are no branches
t e r m i n a t i n g in the concave intervals of the Koch curve while the original construction produces triangles in these areas. In the t r e e p a t t e r n an e m p t y interval remains b e t w e e n terminal points, and new ones are g e n e r a t e d with every iteration. The points belonging to the t r e e are only a subset of the Koch curve. If the points, however, are replaced by objects of
finite area,
the boundary becomes again
densely covered.
Numerically
this has been studied by HONDA & FISHER (1978, 1979) in t e r m s of the most equitable distribution of leaf clusters. Fig. 3.3a shows a similar construction of a binary bifurcation p a t t e r n with terminal triangular 'leaves' which tend toward a continuous outline, but again t h e r e are areas which are not covered, and in addition the leaves overlap.
The
relationship
to
the
discussion of cluster
trees
is obvious; however,
let
us
return for a m o m e n t to a problem of the first chapter, the r e c o n s t r u c t i o n of a continu
138
ous surface
by i t e r a t i v e averaging (section 2.4.5). If we draw the successive nets in
e i t h e r way illustrated in
Fig. a.ag, we get a nested sequence of squares with their
c e n t r o i d s as unique limit point. The sequence of centroids can be p a r a m e t r i c i z e d (e.g. MANGOLDKNOPP, 1968; GUGGENHEIMER, 1977}, and, as the process goes to infinity, we have
a space filling curve or a Peano curve. Again we can c o n n e c t a sequence
of centroids in the way indicated in Fig. 3.39 generating another special t r e e p a t t e r n with its t e r m i n a t i o n points a subset of the Peano curve. As the process goes to infinity, the t e r m i n a t i o n points of a binary t r e e cluster; however, they are always well separated. The t r e e b e c o m e s more dense if we allow triple points (Fig.
3.39),
and finally we can
turn the branching process into a space filling process by the condition that branches of any g e n e r a t i o n sprout into areas of least density (with maximal distance from neighboring branches).
B) The Outline of Honda Trees
The brief excursus to such strange s t r u c t u r e s as Jordan and Peano curves provides us with two c o n t r a s t i n g aspects:
It encourages the previous view t h a t
t h e r e are
tree
s t r u c t u r e s which g e n e r a t e an outline, and it discourages the a t t e m p t to search a description of this outline by inspecting the distribution of branch terminations. A conclusion could be t h a t we now need s t a t i s t i c a l methods to study the distribution of branch t e r m i nations more deeply; the situation is quite similar to the problem to find a closed boundary for a finite point set as discussed in t e r m s of cluster s t r a t e g i e s . However, in the special case of Honda t r e e s we have continuous d i f f e r e n t i a b l e functions which approximate the branches, and, what we ar looking for, is another continuous approximation of the most e x t e n d e d outline. T h e r e f o r e , we r e p l a c e the d i s c r e t e model by a continuous approximation hoping to r e p l a c e the multivarious outputs of a c o m p u t e r program by a t h e o r e m about the shapes, which possibly can be c o n s t r u c t e d from the model under consideration, similar to the
model's original a t t e m p t to describe the essential s t r u c t u r e s of complex
natural p a t t e r n s by a few p a r a m e t e r s .
Let us consider a leading branch and replace it by its spiral approximation. The spirals of the second generation are then most economically described in local coordinates of
the
leading spiral, i.e. in t e r m s of its local moving frame or F r e n e t
we consider straight branches of second order, we can
e.g.
trihedron. If
express them in t e r m s of
the t a n g e n t v e c t o r of the leading spiral and r o t a t e this v e c t o r into the proper position. The
local description of the d i s c r e t e system is analogous to equation (3.57), only the
v e c t o r ro is replaced by the r o t a t e d t a n g e n t v e c t o r bBt: Pi = (i~=O aiAi)(bBt)"
(3.63)
139
If we move this spiral of second order along the is
everywhere
preserved.
Consider
a
convergent
heading spiral, the branching angle
Honda
tree
with
infinite
number of
branching points, then by proposition {4) of the previous section the length of the second order spiral is always in an allometric relationship with the length of the leading spiral, and the same holds for the radii of the two spirals. We take the coiling c e n t e r of the leading spiral as the origin of the global coordinate system, and the global coordinates of a point fixed on the second generation branching p a t t e r n takes the form:
PB =Ps + a
lps Ib
(3.64}
(bBt)
where the index s denotes the leading spiral.
Before discussing this equation in more detail, we observe that b=l if we r e s t r i c t ourself to
leading spirals of the
we consider any
same
system because these are
Furthermore,
if
fixed point on a spiral of higher generation, this point is described
by a v e c t o r which originates at the
all similar.
the leading spiral. This holds for any point, even for
most distant one. Although it
is c u m b e r s o m e to evaluate the most distant point,
it will turn out that we can give a qualitative answer about the outline.
Remaining
in a
system
of leading spirals equation (3.64)
is
simplified
and can
be w r i t t e n in a more extensive form as
I: :1] I or
(3.65)
pB =
l
I + bB
sin~
cos~j
Lsin~,
and once more it turns out that this equation describes the original spiral which is rotated
around its coiling center.
by a similar s p i r a l  
Every
because a t r e e
family of leading spirals,
therefore, is bounded
consists of two systems of leading spirals, the
shape results from superposition of the two systems. On the other hand, within every system of leading spirals there only in size,
exists an infinity of identical subsystems which differ
and we can consider the maximal outline as result of the superposition
of smaller subsystems as illustrated in Fig. 3.40. Concerning the density of termination points we note that the distance of branching points d e c r e a s e s regularly along the leading spiral,
the
density of branching points is proportional to the curvature of the leading
140
Fig. 3.40: A Honda tree is selfsimilar on all levels. Superposition of images on a c e r t a i n level g e n e r a t e s the identical but enlarged image.
spiral, and the same holds for any secondary leading spiral, etc.. Thus, the density increases
towards
the
coiling c e n t e r s ,
and the density of coiling c e n t e r s
is proportional
to the c u r v a t u r e of the leading spiral. Density distributions of this type were already e n c o u n t e r e d with the ' c l u s t e r s t r a t e g i e s ' . Finally
we
consider
branches
of
finite
length.
The
propositions
of
the
previous
s e c t i o n still hold, t h e only d i f f e r e n c e is t h a t the leading b r a n c h e s do not coil to infinity. Equation 3,64, t h e r e f o r e , takes the form
PB =Ps + a(JPsJ IP 0 ])b(bBt), and
the
branches
terminate
for
Jpsl=JOOJ.
(3.66) Far
from
this singular point
and
for
b=l
the shape is quite close to t h e spiral of t h e leading branch. The system, t h e r e f o r e , is really s e l f  s i m i l a r on e a c h level.
141
C) C h a n c e and D e t e r m i n i s m
The previous discussions showed t h a t the Honda model g e n e r a t e s ideal selfsimilar and
deterministic
structures:
impresses its p a t t e r n
The morphology
of
branches
is (infinitely)
onto the shape of the e n t i r e s t r u c t u r e ,
repeated
and
and t h e shape is again
r e p e a t e d on the level of e v e r y generation of b r a n c h e s differing only in size, With t h e s e properties it is unlikely t h a t the Honda model r e f l e c t s r e a l i t y  
it is a strongly ideal
ized model, and in a l a t e r study HONDA & FISHER (1979) introduced additional sources of v a r i a t i o n to a p p r o x i m a t e real t r e e s more closely. However, s e l f  s i m i l a r i t y and f r a c t a l systems are c u r r e n t fields of i n t e r e s t (MANDELBROT, 1977), and the Honda t r e e provides a simple linear
model
although t h e basic
is some possibility t h a t
model
is r a t h e r
models "nearby" a p p r o x i m a t e
unrealistic.
reality,
However,
there
i.e. t h a t disturbed Honda
t r e e s provide b e t t e r approximations. There are several ways to disturb the original model, which involve s t o c h a s t i c and d e t e r m i n i s t i c aspects. A
simple
approach
is ' c h a n c e ' ,
added which a l t e r s branching ratios, However,
'chance'
i.e.
some e x t e r n a l
and perhaps internal
noise is
branching angles, and may even inhibit branching.
involves also events
as ' t o evolve at
a c e r t a i n place',
to 'evolve
within a c e r t a i n time interval', or to 'evolve under c e r t a i n and for the individual s t r u c t u r e not p r e d i c t a b l e e n v i r o n m e n t a l conditions', even if they are not random but exhibit only a complex been
that
spatiotemporal particular
distribution.
thunderstorm
at
The that
random
aspect
particular
time,
"if t h e r e then
would not
have
those b r a n c h e s would
not have been broken off and those buds would not have been inhibited"
is only a very
special viewpoint.
O t h e r aspects are e x t e r n a l and internal c o n s t r a i n t s , the system has to fulfill certain boundary conditions and c a n n o t grow freely. A possible result could be t h a t under c e r t a i n c o n s t r a i n t s only one or few of the topologically distinct t r e e s discussed in section 3.5.1
are stable.
regular,
they
Or, consider the drainage systems of Fig. 3.25; although they are ir
are
not
random;
the
geology d e t e r m i n e s
So the
influence of the Andes obviously d e t e r m i n e s
margin
of
the
Amazon
drainage
system,
and
much of the
stream
sealevel
drainage
directions
fluctuations
such systems
as random or partially formed by chance,
reconstruct
the historical
the w e s t e r n
through
periods d e t e r m i n e d much of the drainage p a t t e r n of the Ganges delta. cannot
at
system:
glaciation
If we consider
the reason is simply t h a t
we
evolution of boundary conditions to a sufficient pre
cision.
A third group of
factors,
which may be of interest,
are
internal constraints,
or
internal regulation systems, The Honda model is a special case of the family of linear maps Xi+l = a l x i + blYi
Yi+l = a2xi + b2Yi
(3,67)
142
which is equivalent to the d i s c r e t i z a t i o n of a pair of coupled linear d i f f e r e n t i a l equations
dx/dt = a l l x + a22Y; which g e n e r a t e Honda's
spiral p a t t e r n s
model
(cf.
(3.68)
dy/dt = a21 x ÷ a22Y
section
in the phase space
2.2.1).
Another
for p a r a m e t e r
possible extension
s e t t i n g equivalent
is t h a t
the
to
local evolu
tion depends not only on the m o t h e r b r a n c h (a "Markovian" situation) but on o t h e r branches nearby, e.g. t h a t the inhibition of a b r a n c h influences growth in some neighborhood. If t h e
resulting regulation
is linear,
then
the
map
(3.67) c o m e s close
to a t r a n s p o r t
equation as discussed in s e c t i o n 3.1.2. Any local d i s t u r b a n c e then would a l t e r the system in some neighborhood. Adding diffusion t e r m s coefficients
aij may
depend
to equation (3.68)
on (x,y) or I (x,y)],
leads
to the
and assuming t h a t the (kto)models which play
some role in spiral c h e m i c a l waves (e.g. DUFFY et al., 1980; VIDAL & PACAULT, 1982). Still more c o m p l i c a t e d s y s t e m s arise if nonlinear feedback mechanisms or a u t o c a t a l y t i c systems
are
introduced
s y s t e m is e.g.
(cf.
MEINHARD,
1984; NICOLIS
& PRIGOGINE,
1977). Such a
the H~non map (cf. THOMPSON, 1982)
Xi+l = Yi + l  a x i 2 ' which exhibits
a strange
Yi+l = bxi
attractor
(3.69)
and c h a o t i c
behavior. THOMPSON
(1982) concluded
from the random response of such a d e t e r m i n i s t i c modeh "Strange
attractors
modellin E that
a
For
with
Bogoliubov attractor
point
study c o m p u t e r
(as
averaging
a lon E period
may
a
of
their
must
any
sudden
apparent
profound
longer
nonlinear result
of
since
leap
in
it
be
one
may
seen
in
they must
conventional feature
our
now
essential
dynamics a
on
is
systems,
response
of
all mean
be
in
Krylova
occur
strange after
"
by this section:
models analytically to d e t e c t
effect
since
mechanical
quiescence.
is also e l u c i d a t e d
a
no
technique),
that of
have
behaviour,
deterministic
results care
is
thus random
modellin E
simple
computer
spected
Another
may
seemingly
stochastic
cases. that
of
Sometimes
it is worthwhile
t h e i r internal s t r u c t u r e .
to
We usually think
the o t h e r way, we f o r m u l a t e a problem analytically and t h e n use t h e c o m p u t e r to solve it.
Computer
mathematics
modelling
and
simulation
to some e x t e n t ,
of possible solutions.
In this r e s p e c t ,
are
but
not
summarized,
thus
became
decoupled
from classical
analytic
providing us usually with an enormous or infinite n u m b e r nature
is r a t h e r
closely approximated,
the
we can e s t a b l i s h an infinite c a t a l o g u e of forms without
d i s a d v a n t a g e to go to t h e field.
data the
143
A major point throughout this section was selfsimilarity which leads to the aspect that
it
is s o m e t i m e s worthwhile to
decompose complex s t r u c t u r e s
into
smaller
units
and to e x t r a c t the 'primitives' which allow to describe and to analyze the complex global pattern.
Fig. 3.41 illustrates this in the case of a m m o n i t e sutures which are close to
t r e e p a t t e r n s , and which are selfsimilar to some e x t e n t . By turning a into
a
local
one
the
global problem
analysis of a s t r u c t u r e commonly is s i m p l i f i e d   however,
the
inverse procedure is not always possible as was elucidated by various examples.
The analytic approach, however, usually condenses the possible p a t t e r n s and allows to discover u n e x p e c t e d s t r u c t u r e s which may evolve under c e r t a i n conditions. Thus, the previous discussion of shape was incomplete, as new p a t t e r n s may arise if the second order branches are d i r e c t e d to the concave side of the leading spiral. Straight branches e.g. then overlap and are bounded by their evolute, and this p a t t e r n does not require the
c o n v e r g e n c e of the Honda tree. Such p a t t e r n s  
evolutes and caustics   will be
the topic of the next chapter.
Fig. 3.41: A m m o n i t e suture lines are comparable to t r e e patterns. Commonly they are also selfsimilar (including reflections) on various levels. The enlarged 'primitives' can r e p e a t e d l y be found within the complex suture lines.
4.
STRUCTURAL
STABLE
ELEMENTARY
PATTERNS
AND
CATASTROPHES
In the preceding discussion it b e c a m e c l e a r that most of the observed instabilities are due to the f a c t that the p a t t e r n recognition process lacks an inherent stability property.
As LU (1976) states: "In any branch
of science,
objects under study.
out this classification. the stable objects . . . . appear. based
We all know on
a @rest
the
a challenge
to try to classify the
it is often extremely difficult
to carry
It becomes much easier if one tries to classify only stable objects have boundaries
that mathematics
differential
demand,
it is always
Unfortunately,
therefore,
calculus,
where discontinuities
used in almost all sciences so far i s which
presupposes
for a mathematical
continuity.
There is
theory to explain and predict
(if possible) the occurrence of discontinuous phenomena."
A very instructive e x a m p l e of s t r u c t u r a l stability and singularities we owe to CALLAHAN (1974): Take
a piece of fabric at one corner and put it onto a flat surface.
What forms can occur locally on the s h e e t {Fig. 4.1)? There are t h r e e possibilities: a) the s h e e t lies flatly and smoothly, these points are called the regular ones; b) a fold line appears on the sheet; c) a pleat forms at the end of a fold line.
'~Local
forms on a folded s h e e t (after CALLAHAN, 1974}; (a) a regular s h e e t lies flat; (b) a singular point on a fold line; (c) a singular point where the fold turns into a pleat; (d) a singular point which is not structurally stable.
The points (b) and (c) are
called the
singular points because of their special nature;
in particular, they are structurally stable singular points. To see this, take a point such as (d) in Fig. 4.1. This is an additional type of singular points, but it is not a structurally
145
s t a b l e one because any disturbance, slight as it may be, turns this point into one of type (a), (b} or (c). On the o t h e r hand, points of types (a), (b) or (c) cannot be made to disappear by a small perturbation. One can dislocate a fold or pleat by a small perturbation, but one c a n n o t a f f e c t
its presence. The discussed singularities yield in addition
a s t r u c t u r a l information. Put a stiff s h e e t of paper beside t h e fabric in the same way: It will only consist of regular and perhaps fold points. It is distinguished from the piece of fabric by the singular points which may appear under this e x p e r i m e n t . It is usually the set of s t a b l e singularities which allows us to classify objects.
It was THOM (1975) who pushed forward these c o n c e p t s in m a t h e m a t i c s and their applications. He has shown t h a t the concept of s t r u c t u r a l stability, i.e. the insensitivity to
small
perturbations,
is r e l a t e d
in one c o n t e x t
to stable singularities.
These
stable
singularities have then been classified by Thorn in his "seven e l e m e n t a r y c a t a s t r o p h e s " . Here,
examples are given
stand
patterns
generically
and to d e t e r m i n e
impresses
The examples
which d e m o n s t r a t e
are
on
a
derived
the
sensing from
geometric wavefield
Huygens'
detailed discussion of the m a t h e m a t i c a l can
be
found in LU (t976},
t h a t these c o n c e p t s are useful to undersingularities t h a t
(DANGELMAYR
an unknown surface
& GOTTINGER,
1982).
c o n s t r u c t i o n of wave fronts and envelopes. A
machinery and of
mainly physical applications
POSTON & STEWART (1978), GOTTINGER & EIKEMEIER
(eds. 1979), STEWART (1981, 1982). Fig.
4.1 r e p e a t s
Fig.
uniquely d e t e r m i n e t h e
2.1
with
the
addition of
the
local s t r u c t u r e of the s u r f a c e  
typical
singular points which
i.e. t h e set of singular points
on surfaces provides a skeleton of essential s t r u c t u r a l information. In the previous chapt e r s such sets of singular points were viewed as instabilities, in this c h a p t e r the inverse problem
will
be
studied,
the classification of "form" by the
intrinsic set
of singular
points.
In the
first
section
singularities are
discussed,
which occur
as discontinuities
in
the twodimensional images of threedimensional objects. The typical singularities provide useful i n f o r m a t i o n in picture processing. The concept of skeletons then r e l a t e s t h e analysis of
twodimensional
images
to
the
previous discussion of o p t i m i z a t i o n
problems on
one hand~ and to D'Arcy Thompson's classical Vtransformations of form ~ on the other. In the second section the theory of e l e m e n t a r y c a t a s t r o p h e s is applied to t h e linear ray model in r e f l e c t i o n seismology. Most of t h e discussion r e m a i n s r e s t r i c t e d to the twodimensional
track
line problem,
with basic m a t h e m a t i c s .
and the c o n c e p t s are derived from simple assumptions
Instead of dealing with waves directly,
the linear ray p a t t e r n s
and t h e i r caustics are studied. The wave fronts then are analyzed in t e r m s of a continuous plane the
map. Finally,
threedimensional
the t r a v e l t i m e r e c o r d is analyzed as a map which t r a n s f o r m s
spatiotemporal
system
into
the
traveltime
record.
It will
turn
146
out t h a t t h e t r a v e l t i m e record is locally equivalent to {oblique) sections through a swallowtail c a t a s t r o p h e
In the 'parallel
third section
surfaces'.
deformations
which is located on an oblique line in the d e p t h / t i m e coordinates,
some
aspects
This discussion
in t e c t o n i c s
takes
of folds and faults are discussed in t e r m s of up the
'precomputer'
and reviews this ' e a r l y '
analysis of
kinetic approach
large scale
in t e r m s of r e c e n t
d e v e l o p m e n t s in singularity theory, The l a t e r a l c o n t i n u a t i o n and the depth limit of folds will be discussed in d i f f e r e n t ways
and continues in some r e s p e c t the problems of chap
t e r 2.
4.1 IMAGE RECOGNITION OF THREEDIMENSIONAL OBJECTS A common
problem
in p a t t e r n
recognition
is the r e c o n s t r u c t i o n and c l a s s i f i c a t i o n
of t h r e e  d i m e n s i o n a l o b j e c t s from twodimensional pictures. Typically, this problem arises in transmission microscopy.
Fig.4.2 shows the larva of a medusa
in t r a n s m i t t i n g light.
Locally t r i a n g u l a r p a t t e r n s of increased density appear, which can be r e l a t e d to a bound
Fig. 4.2: The larva of a medusa in t r a n s m i t t i n g light. A swallowtail singularity occurs in the twodimensional image. The identical p a t t e r n is found in the twodimensional image of a t r a n s p a r e n t canal surface {modified a f t e r WUNDERLICH, 1966).
147
ary e f f e c t . of
locally
An identical p a t t e r n convex
surfaces,
is well known from twodimensional p e r s p e c t i v e views
especially
from
canal
surfaces
in c o n s t r u c t i o n a l
geometry
(WUNDERLtCH, 1966). Another
field,
where
similar
patterns
play
some
role,
are
tectonically
flattened
and d e f o r m e d images. The proper r e c o n s t r u c t i o n of such images yields useful i n f o r m a t i o n for the field.
paleontologist
The
normal
as well
approach
as
for the d e t e r m i n a t i o n of the local t e c t o n i c a l stress
in such
cases
are
affine
transformations.
t h e d e f o r m a t i o n s sufficientiy if the dimension of t h e object, altered:
if one has maps R 2 ~
These
describe
i.e. of its image,
is not
R 2 or R3 ~ R 3. In the case of a map R3 ~
new p a t t e r n s occur, discontinuities at the boundary lines of the image
R 2,
which are closely
r e l a t e d to t h e local surface s t r u c t u r e .
Fig. 4.3: Tectonically deformed and f l a t t e n e d bivalves show surface discontinuities similar to those in Fig. 4.2: In this case, the objects are not t r a n s p a r e n t , and, t h e r e f o r e , only parts of the surface discontinuities are visible {modified a f t e r ROLLIER, 1918}.
4.1.1 The TwoDimensional Image of ThreeDimensional Objects
Locally in two
dimensions
catastrophe pattern
hyperbolic (see
surfaces
(Fig. 4.4).
e.g.
POSTON
are
These
capable images
& STEWART,
is s t r u c t u r a l l y stable, and,
therefore,
to are
project
onto
closely r e l a t e d
swallowtaillike
images
to Thom's swallowtail
1978). This relationship secures it can be used to d e t e c t
that
the
locally c o n c a v e
surface e l e m e n t s or holes in the twodimensional images of t h r e e  d i m e n s i o n a l objects.
148 • .°°
...:.. ".'..... ......
...:..
°,°• •
::.'..::....:..'.......:...:.. :'......': ..
:, .. : ; . :
: ..:
.•
•
. . . . .
• . . . . .
,
.
•
. .
,
o••
: ..'..:.,::. • ..:...'.~.
,
. . . . .
.:.:';:!".':"
Fig• 4.4: Two projections of a hyperbolic s u r f a c e e l e m e n t • An oblique projection causes the o c c u r r e n c e of a surface discontinuity in the twodimensional image space. The t h r e e  d i m e n s i o n a l surface e l e m e n t can be identified with the c a t a s trophe manifold of the swallowtail c a t a s t r o p h e • The discontinuity is the t w o  dimensional image of t h e b i f u r c a t i o n set of t h e swallowtail (modified a f t e r POSTON & STEWART, 1978).
How such s u r f a c e
discontinuities develop in the twodimensional image space, can
be analyzed most simply by use of canal s u r f a c e s (WUNDERLICH, 1966)• A canal s u r f a c e is the e n v e l o p e HEIMER,
of a ( o n e  p a r a m e t e r )
1977). The canal surface,
family of spheres of c o n s t a n t
therefore,
radius (GUGGEN
can be g e n e r a t e d by moving the c e n t e r
of a s p h e r e along a (threedimensional) curve; t h e envelope s u r f a c e of the spheres then defines the canal surface• The most simple case of such a canal surface is the torus where the circular
leading
leading
c u r v e is a circle• Under an oblique projection i n t o the plane t h e c u r v e t r a n s f o r m s into an ellipse• This d e f o r m a t i o n of t h e
leading
c u r v e can be described by an affine t r a n s f o r m a t i o n which takes the circle into an ellipse• But the boundary lines of the torus in t h e twodimensionaI p r o j e c t i v e space b e h a v e differently. The g e n e r a t i n g spheres project onto R 2 as circles independent of a rigid r o t a t i o n of the torus• The result is t h a t the a p p a r e n t boundaries of the t r a n s p a r e n t torus develop into a c u r v e 'triangle'
with
a selfintersection
and with two cuspoid edges,
t h e earIier noticed
(Fig• 4•2)• Fig• 4•5 gives t h r e e d i f f e r e n t views of the torus and of the local
s u r f a c e discontinuities in the p r o j e c t i v e plane. The identical p a t t e r n appears if one keeps the
leading
curve c o n s t a n t and
alters
the d i a m e t e r of the canal s u r f a c e (Fig. 4.6)• In this case, classical c o n s t r u c t i o n a l g e o m e try
(WUNDERLICH,
1966)
tells
us
that
the
locally discontinuous
the c i r c u l a r projections of t h e spheres e n t e r the e v o l u t e of t h e
image leading
appears
when
curve, along
which the c e n t e r s of the spheres are located• A family of such surfaces with v a r i a b l e
149
I>cI
Fig. 4.5: Three p e r s p e c t i v e views of a (transparent) torus and of its apparent surface discontinuities. The l a t t e r develop at the interior hyperbolic surface points, as the projection becomes oblique.
canal d i a m e t e r maps, t h e r e f o r e , onto a family of involutes or of parallel curves, which unfold inside the evolute of the
leading
curve into swallowtaiIlike curves {Fig. 4o6}.
The l a t t e r observations allow to relate
the evolution of the image boundaries to
Huygens ~ principle (Fig, 4.7). If the image of the canal surface is studied in two dimensions~ then the envelope of the spheres reduces to the envelope of circles with a c o n s t a n t radius. A change of the d i a m e t e r of the canal surface corresponds to a change of the radii of these circles. In two dimensions, this is identical with a continuous transport of the
boundary
lines along the
normals of the
leading
curve. The image changes
locally when the area is r e a c h e d where the normals i n t e r s e c t {Fig. 4.7). This constructional principle allows to study all possible images of canal surfaces that can occur in two
, ::%
:
..:.,
" ~ ~,:~!..'..:~"
Fig. 4.6: Canal surfaces with identical generating curve but of d i f f e r e n t diam e t e r . The boundaries of the surface map onto a family of parallel carves in the twodimensional image space.
150
Fi~. 4.7: Construction of the twodimensional images of canal surfaces by use of Huygens v principle. The rays indicate the dislocation lines for the surface boundary in the projective plane.
dimensions (Fig. 4.8). The only discontinuities, which appear, are the swallowtaillike singular curves. The same holds for the p e r s p e c t i v e views of the torus; the only d i f f e r e n c e is that,
in this case, not the d i a m e t e r of the canal surface is a l t e r e d but the local
c u r v a t u r e o f the the
surface
leading
~moves into
curve. The result is the same: As the curvature decreases, the
evolute v of the
leading
curve, and the boundary lines
deform in the identical way discussed above.
The critical points of a canal s u r f a c e are the hyperbolic points. They behave under the
projection onto a
the
~swallowtait' is typical
4.2.
The
relation
to
twodimensional image in the discussed way. The o c c u r r e n c e of for
locally convex structures,
hyperbolic s u r f a c e
e l e m e n t s allows
as will be shown in section for a simple construction of
\
\ Fig. 4.8: A family of boundary lines for canal surfaces parabolic generating curve.
which have a common
151
Fig. 4.9: C o n s t r u c t i o n of a folded polygon.
the
swallowtail
singularity
as a ruled surface
over
the swallowtail. A hyperbolic surface, which locally takes the form z = xy, can be generated
as a ruled surface.
If t h e s e rulings are p r o j e c t e d
then the envelope of the s t r a i g h t
into two dimensions (Fig. 4.9),
lines gives again the various possible images of the
surface in two dimensions including the swallowtail discontinuity.
The normals,
concept
of
parallel
can be r e l a t e d
to
curves,
which
are
generated
by
a t r a n s p o r t along t h e i r
the c o n c e p t of p o t e n t i a l s and to e l e m e n t a r y c a t a s t r o p h e s .
tn t h e l a t t e r case, t h e original t h r e e  d i m e n s i o n a l obiect resembles the c a t a s t r o p h e m a n i fold. If the
leading
curve is continuous,
it can be locally described
as an implicit
function f(x,y) = 0. In addition, one can suspect t h a t the family of parallel curves, which is g e n e r a t e d
by
the
transport
along
the
normals,
can
be
given
in implicit
form
as
c = fix,y). But this l a t t e r formulation can be i n t e r p r e t e d as a potential. The directional derivatives, t h e gradient of this local potential, define the dislocation field. C a t a s t r o p h e theory One of
t h e n gives a c l a s s i f i c a t i o n of t h e s t a b l e singularities of such 'local' potentials. the
catastrophes,
which appear
under
the
map R3 ~
R 2, is the swallowtail.
In the p r e s e n t c o n t e x t it is a stable discontinuity, which allows to identify local hyperbolic s u r f a c e e l e m e n t s in the twodimensional image. A more detailed discussion of this singularity will follow in the next sections.
4.1.2 The Skeleton of P l a n e Figures In
Picture
the c o m p u t e r ,
Recognition
the
problem
arises
to
represent
objects
economically
in
and to classify objects as equivalent even if they are disturbed to some
e x t e n t . Especially the l a t t e r problem leads to the c o n c e p t of skeletons or 'medial axes' (BLUM, 1973; BLUM & NAGEL,
1977). BOOKSTEIN (1978) gives the following definition
152
of the skeleton of a plane figure: The skeleton of a plane figure is a c e r t a i n graph inside the figure, t o g e t h e r with a function on the
graph.
The skeleton
is the locus of all points which do not
have a unique n e a r e s t boundary point upon the shape; the function is the d i s t a n c e to any of the set of equally distant n e a r e s t points.
Thus,
the
skeleton
arose
from
is lust the set of singular points discussed in section 2.4.5, which
the problem
to
find the n e a r e s t
boundary point. The most n a t u r a l way of
finding the skeleton of a given object is by shrinking it until it reduces to its skeleton (ROSENFELD wave
front,
& WESZKA, which s t a r t s
with c o n s t a n t v e l o c i t y  
1980). This
shrinking c a n  
theoretically
be done by a
i n s t a n t a n e o u s l y at e v e r y point along the boundary and moves i.e. by Huygens' principle, as discussed in the previous section
(for t e c h n i c a l details of c o m p u t a t i o n see e.g. ROSENFELD & WESZKA, 1980). Still more i l l u s t r a t i v e is t h e "grassfire" model (BOOKSTEIN, 1978): "We imagine and fired
its starting ously burn
a shape
boundary
(simultaneously).
from twice.
locus until two
loci
'drawn'
it encounters
directions,
Such
to be
the dry grass of a prairie,
on
The fire will burn evenly in all directions
whereupon
comprise
the
from
points at which it arrives simultaneit
quenches
skeleton,
itself~
and
the
as
function
grass we
cannot seek
is
the time it takes the fire to arrive there and Eo out. "
The
critical
set
of
singular
points
we
encountered
in the
optimization
problem
{section 2.4.5) has now a totally d i f f e r e n t quality: T o g e t h e r with a v e c t o r valued function (which defines the d i r e c t i o n to the n e a r e s t boundary points) and a d i s t a n c e m e a s u r e m e n t {which defines t h e location of the original boundary line) t h e set of singular points or the
Vmedial axis'
generically
defines
our object,
Fig.
4.10 shows various examples
plane figures and t h e i r 'medial axes'.
Fig. 4.10: Skeletons (medial axes) of various twodimensional objects,
of
153
4.1.3 T h e o r e t i c a l Morphology of W o r m  L i k e O b j e c t s
The c o n c e p t of s k e l e t o n s b e c o m e s e s p e c i a l l y simple in t h e c a s e of cylindrical objects
when
parts
of the boundary line and the s k e l e t o n are
parallel
curves. Fig,
4.11
i l l u s t r a t e s t h e c a s e of a cylinder with spherical caps t e r m i n a t i n g its ends. We m a y think about such a cylindrical object in t e r m s of a wiggly worm or a c h r o m o s o m e if we allow
qg7 Fig. 4.11: Morphology of a wiggly worm, which p r e s e r v e s length, width, c i r c u m f e r e n c e , and a r e a when it bends (a,b). The c u r v a t u r e of coiling, however, is limited by the width of t h e worm (c).
X and Ylike p a t t e r n s . Now, if our wiggly object s t r e t c h e s or c h a n g e s its form in s o m e way,
"our intuition e x p e c t s
that
the width of its
form will s t a y p r a c t i c a l l y
the
same
and we e x p e c t in addition its length to be invariant" (BOOKSTEIN, 1978). What we then need are precise definitions of width and length and a map which p r e s e r v e s t h e s e properties if the ' w o r m ' bends.
Let the cylindrical object be s t r a i g h t , then we always can find a c o o r d i n a t e s y s t e m so t h a t the boundaries are described by t h e map
x = s y = X
if
ISl
(F(x),G(x))
+ X ~ ( F ( x ) , G ( x ) )
(4.3)
or u = f(s)

Xg(s)
v = g(s)
+
x~(s)
,
where N is t h e normal to t h e medial line. The spherical caps (4.1b) are still located at t h e endpoints of t h e medial line and are not d e f o r m e d {cf. Fig. 4.11). Finally, we are i n t e r e s t e d how the c i r c u m f e r e n c e altered.
of the object and its area are
From equation (4.3) we find the length i n c r e m e n t ds of the parallel pieces of
the boundary to be
ds B = (I  Xk(s))
•
~/(f(s) "2 + g(s)
2
) ds
, (4.4)
where
k(s)
is the
curvature
of the m e d i a l
• 2 / ( ~ ( s ) 2 + g(s) ) = I because c i z e d by arc l e n g t h ,
the m e d i a l
line
line
and
is p a r a m e t r i 
and t h e length of the boundary is LB=
f(lXk)ds = 2 fds
+ f(l
+%k)ds
+ 2Lspherical
+ 2 L s p h e r i c a I caps caps
;
k: c u r v a t u r e
of m e d i a l
axis.
(4.5)
That is, the length of the boundary is simply twice the length of the medial axis, independent of the deformation of the medial line! By definition, the length of the medial line does not
change under arbitrary deformations, and the length of the circumference
is an invariant property of the m a p (4.3). The area of the object is given by
155
A = ff(lXk)dsdl dsdX
= If
+ ff(l +Xk)dsdl
+ 2A
+ 2A
spherical
caps
spherical caps,
and again the d e f o r m a t i o n does not a f f e c t the area which, t h e r e f o r e , is a n o t h e r invariant
for any fixed value of t h e p a r a m e t e r
tions
bending and c o i l i n g  
)~. The map (4.2) describes ideal d e f o r m a 
of wormlike objects, which p r e s e r v e t h e length of the
medial axis, the length (surface) of t h e boundary, and the area (volume) of the o b j e c t properties which  by intuition  can be e x p e c t e d for biological objects. However, there
are
if we r e t u r n to the discussion of the preceding section, it is c l e a r t h a t
limits
for the
deformation
of such wormlike
objects.
As the c u r v a t u r e
or
the width of the object increases (Fig. 4.6), the surface would develop s e l f  i n t e r s e c t i o n s , which c a n n o t be realized. wiggly worm
(Fig. 4.11),
These singularities, which occur at are
'standard catastrophes',
the convex areas of the
which will be discussed in more
detail in the next sections. For our wormlike object, however, we find a strong correlation b e t w e e n width and the c u r v a t u r e of coiling.
4.1.4 Continuous T r a n s f o r m a t i o n s of Form
D'Arcy Thompson introduced the c o n c e p t of shape t r a n s f o r m a t i o n s to describe the morphological
evolution
in
phylogeny
and
ontogeny
(THOMPSON,
1952).
BOOKSTEIN
(1978) summarizes: "The formal of map
one
theme of D'Arcy Thompson's
shape
of
one
into
shape
another onto
by
the
the
other,
method is this:
single and
mathematical
then
to
to represent object
visualize
a change
which
this
is
the
mathematical
object. "
This program find the ideal
'ideal
has been followed with much emphasis, and a c o m m o n goal was to
family of t r a n s f o r m a t i o n s ' ,
hydrodynamic
attempts,
problem
is
solved
by
which solves the biological problem, the
conformal
mappings.
Many
as the
published
of course, r e l a t e D ' A r c y Thompson's problems to classical physical m e t h o d s  
conformal
mappings
(BOOKSTEIN,
t978),
(Fig.
4.12;
RICHARDS
& KAVANAGH,
1947),
biorthogonal
grids
and to the NavierStoke equation as a general solution (GRENAN
DER, 1976). The and
methods,
morphological
based states.
a continuous d e f o r m a t i o n
on
D'Arcy
Thompson's
They describe
a map
as it is usually
program,
from
compare
one s t a t e
obvious in ontogeny
different
objects
to the o t h e r but not and appears likely for
156
,V

'
F";" t
',
~:I
; J
L
'.
~__,
4
..
J
.... 4
, ,
j
"'/
t
s ~
,>';
A
"
i
B
Fig. 4.12: Transformation of form in a bivalve (A) and a brachiopod (B}. Both mappings are produced by the conformaI map w=a(z + l/z) + b log z. (Adapted from BAYER, 1978).
phylogeny if one does not believe in m a c r o  m u t a t i o n s . Further, grids'
implies
a
continuous,
a
topological
deformation.
However,
the use of ' C a r t e s i a n in the
real
systems
we find s o m e t i m e s r a t h e r sudden changes although the d e f o r m a t i o n is continuous.
Equation capable
(4.3)
provides a ' p r o t o t y p e ' of such continuous deformations, which are
of sudden morphological changes as soon a s the surface e n t e r s the
'caustic'
of normals. Of course, it is not hard to argue that real biological t r a n s f o r m a t i o n s are much more c o m p l i c a t e d than the map (4.3). However, if we do not a t t e m p t to describe the e n t i r e morphology at once
but r e s t r i c t the study to interesting local patterns, then
Fig. 4.13: The early ontogeny of Sepia (after NAEF, 1928 and BLIND, 1976) and the c i c a t r i x of Nautilus (after BLIND, 1976) compared with a swallowtail surface (after THOM, 1970).
157
the map (4°3) provides a first order approximation for the evolution of a locally cylindrical surface
element
what
may happen, is the evolution of folds on the surface
(cf.
Figs. 4.14.9). Fig. 4.13 illustrates folding processes in the early ontogeny of cephalopods, which c a n
be
related
to the evolution of a wave
front
the s u r f a c e  
which folds
and g e n e r a t e s singular lines.
The map (4.3) describes a simple case of a wave front and is equivalent to THOM's (1970,1975}
approach
singular sets
are
to
morphogenesis,
emphasized.
as soon as not
the
morphology
In the simple example of equations
itself but
the
(4.3) the i n t e r e s t i n g
singularities are the cusp and swallowtail c a t a s t r o p h e s . How these e l e m e n t a r y c a t a s t r o p h e s are r e l a t e d to the map
(4.3), will be discussed in the next section.
4.2 SURFACE INVERSIONS IN THE SEISMIC RECORD 
THE CUSP AND SWALLOWTAIL CATASTROPHES In a wide field of applications geologists are c o n c e r n e d with the problem to i n t e r pret the records of r e f l e c t i o n seismology in a q u a l i t a t i v e way. The morphological {geometrical} i n t e r p r e t a t i o n is here usually much more i m p o r t a n t than the r e c o n s t r u c t i o n of true depth relationshipsproblem
a major problem for t h e seismologist. To solve t h e i n t e r p r e t a t i o n
q u a n t i t a t i v e l y requires to solve the full dynamics of the wave equation, which
c a p t u r e s t h e process of r e m o t e sensing. In the most general sense, waves are spreading processes 1968). The
which satisfy a t o t a l hyperbolic d i f f e r e n t i a l equation (COURANT & HILBERT, trouble
is t h a t one does not only need the initial conditions but also the
f
Fig. 4.14: Sketches of t r a v e l t i m e records, which have been i n t e r p r e t e d as salt domes with s e l f  i n t e r s e c t i n g r e f l e c t o r s (above: a f t e r DRIVER & PARDO, 1974; below: a f t e r BIJUDUVAL et al., 1974).
158
entire
boundary conditions to solve the
general
equation.
hyperbolic equation one can pick a visible s t r u c t u r e
Instead of solving the total
from the wave field, say a c r e s t
or trough line or, in a general notation, a wave front (WRIGHT,
1979)o To solve the
spreading process of the wave front one can use Huygens ~ principle (COURANT & HILBERT,
1968; OFFICER,
1974). If a wave front is known at a c e r t a i n time, one studies
the evolution of the w a v e l e t s t h a t spread from every point of the generating wave front. The successive wave fronts are the envelopes of the wavelets. In the threedimensional case the w a v e l e t s are spheres, and the successive wave fronts are surfaces F(x,y,z) = constant. If the propagation of the wavelets is fairly c o n s t a n t along the generating wave front,
the successive wave
fronts can be approximated by a transport of the original
s u r f a c e along its no,reals with c o n s t a n t velocity. In two dimensions the spherical w a v e l e t s reduce
to circular
ones,
and the
problem is m o d e r a t e l y simplified. In a m a t h e m a t i c a l
sense this construction of the propagating wave fronts can be described as a continuous map, and, at least locally, one can suspect that this map can be summarized in a p o t e n tial equation V = w(x,y,z).
The
interesting structures
geometrical
singularities can
& GUTTINGER
(1982)
of
these
potentials
are
their
stable
singularities.
The
welt be studied by topological methods. DANGELMAYER
discussed the physical a s p e c t s of r e m o t e s e n s i n g  
topographies, d i f f r a c t i o n p a t t e r n s e t c .
Fresnelzone
  carefully in t e r m s of c a t a s t r o p h e theory. They
showed that the topological approach yields reasonable results for the inverse s c a t t e r i n g problem as well as for the o n  s i t e survey: "Since, in practice, mologist  because to
classify
his
the analytic analysis is of little interest to the seishe needs an overall
forms
tackling
the
picture, inverse
a
'Gestalt' point of view
problem
at
its
topological
roots comes much closer to the interpreter's intuitive geometric9 i.e. qualitative, approach" (DANGELMAYER & GUTTINGEB, 1982).
This "qualitative the
approach"
is,
indeed, still
seismologist. Interpreting
reasonable within are
question
the
whether
seismogram.
intersecting
Shadow
zones,
well known p a t t e r n s of the
such
'anomalies'
show up
more important
s e i s m o g r a m s from
record.
in the
areas
reflectors double
are
traveltime
salt
and
and 4.15
record.
domes
realistic
reflections
Figs. 4.14
for the geologist than
with
Much
or
it
may
singular
be
for a
patterns
'hyperbolic r e f l e c t i o n s '
give some examples how more
impressive s t r u c t u r e s
of t h e s e types have been described from 3.5 to 12 kHz records (echograms) {JOHNSON & DAMUTH,
1979;
of
and
the
echograms last
decade,
DAMUTH, of
when
patterns
1980;FLOOD, like
1980;
EMBLEY,
'hyperbolic r e f l e c t i o n s '
sedimentologists s t a r t e d
to
interpret
t980). became the
The of
interpretation interest
deep sea
during
morphology
for their purposes. In the
case of high frequency echograms, the wave front evolution can be well
159
Fi,g. 4.15: T r a v e l t i m e records with i n t e r s e c t i n g reflections, 'extensions of s e d i m e n t a r y layers into the b a s e m e n t s ' and high energy zones within an isotropic s e d i m e n t cover,
approximated
by linear rays (FLOOD,
1980). tn the case t h a t the model is reduced to
the 1seismic track', the problem is further simplified. The relation b e t w e e n a (cylindrical) surface and t h e t r a v e l t i m e r e c o r d along the t r a c k line can be viewed as a map R 2 R 2 or ( x , y ) ~
....
(u,2t), where (x,y) are the spatial coordinates of a section through the
s u r f a c e and (u,2t) are t h e s p a c e  t r a v e l t i m e c o o r d i n a t e s of the record. In this notation, the i n t e r p r e t a t i o n of the seismogram
becomes
identical with the problem to d e t e r m i n e
t h e map tsL This approach was r e c e n t l y stressed by FLOOD (1980), who analyzed periodic wavefields. He found t h a t 'hyperbolic r e f l e c t i o n s ' depend on the wavelength of sinusoidal surfaces and w a t e r depth. But the approach by any global s u r f a c e approximation is much
160
too general: T h e r e is no real c h a n c e to establish a sufficient c a t a l o g u e of global morphologies. One problem, t h e r e f o r e , is w h e t h e r one can classify s u r f a c e points in such a way that
t h e i r image on t h e seismic r e c o r d can be uniquely identified. This, of course,
a topological traveltime
problem.
images
for
It is t h e the
purpose
of
twodimensional
this section (seismic
is
to derive
a c l a s s i f i c a t i o n of
problem.
This c l a s s i f i c a t i o n
track}
will be one which r e l a t e s the t r a v e l t i m e p a t t e r n s to local topological properties of the reflector.
There
the traveltime
are image,
two ways
to discuss
versus t h e wave
the
relation between surface properties
and
fronts or versus t h e plane map approach. Both
methods will be discussed to d e m o n s t r a t e d i f f e r e n t a s p e c t s of c a t a s t r o p h e theory.
4.2.1
C o m p u t e r Simulations of Rays, Wave F r o n t s and T r a v e l t i m e Records
Ray theory b e c o m e s drastically simplified if one assumes t h a t the rays are s t r a i g h t lines, and t h a t t h e source and the r e c e i v e r are located in a single point. This situation is nearly r e a l i z e d a first
for deep sea e c h o g r a m s
approximation
(FLOOD,
1980), but it can also be used as
for o t h e r r e f l e c t i o n seismograms.
the s u r f a c e of the r e f l e c t o r
Under t h e s e special conditions,
can be viewed as the envelope of the wavelets,
it forms
a ' w a v e front', and the normals of the r e f l e c t o r are the rays along which the r e f l e c t e d wave front p r o p a g a t e s (e.g. GRANT & WEST, 1965). If the section through the r e f l e c t o r along the t r a c k
line is given analytically, one can i m m e d i a t e l y w r i t e down the (linear)
ray equations. tf t h e r e f l e c t i n g s u r f a c e is given in explicit form as y = f(x), then the linear rays a r e given in p a r a m e t r i c i z e d form as
where
u
=
Xo
w
=
f(xo)

Tf'(xo) +
(4.6)
T
(Xo, f(Xo)) defines a point on the r e f l e c t o r ,
(u,w) are the spatial c o o r d i n a t e s of
t h e ray which passes through t h e point (Xo, f(Xo)), and
T is a p a r a m e t e r which g e n e r a t e s
the ray. It turns out t h a t the p a t t e r n s , which can be found, are not a property of t h e r a y s , but t h a t 1979).
The
they need to be formed by the morphology of t h e r e f l e c t o r
reflector
impresses
its
pattern
generically
onto
the
sensing
(WRIGHT,
wave
system
(DANGELMAYER & GOTTINGER, t982), in this case, onto the family of rays. Equation (4.6) allows to draw the linear ray p a t t e r n for any d i f f e r e n t i a b l e function. This was done for a sinusoidal function in Fig. 4.16, and it b e c o m e s c l e a r t h a t the seismic r e c o r d will be disturbed in the areas of ray overlap. In these areas one finds double and triple reflections. The two p i c t u r e s of Fig. 4.16 show the farfield p a t t e r n and, en
161
Fig. 4.16: Computer simulations of the linear ray system of a sinusoidal r e f l e c tor. Left: farfield p a t t e r n , right: nearfield pattern.
larged, the nearfield p a t t e r n (close to the surface). Fig. 4.17 illustrates that the r a t h e r c o m p l i c a t e d farfield p a t t e r n results from superpositions of various nearfield patterns.
If the rays, which pass through the surface e l e m e n t of a wave front, are known, then the evolution of the wave front along the ray family can be simulated, a f t e r the ray equations have been normalized by arc length, tn the case of linear rays one finds the successive wave fronts from the continuous map u
=
x

Tf'(x)//(l+f'(x)
2
(4.7)
)
w = f(x)
+ "r//(l+f'(x)2).
(u,w} are the new spatial coordinates of a point {x,f(x)) on the original wave front, and T
is the distance b e t w e e n these two points { T = vt, v: velocity, t: time). The equations
(4.7)) define a continuous map from the plane into the plane
{(u,~)~(x,y)]S(~):(~,v)},
(4.8)
where (Xo,Yo) are the points on the (cylindrical) r e f l e c t o r . Fig. 4.17 illustrates how the wave fronts evolve from sinusoidal surfaces, and how they are folded within the areas in which the rays i n t e r s e c t . As was noted earlier, the r e f l e c t o r impresses its s t r u c t u r e onto the family of rays and onto the wave fronts. This is illustrated in Fig. 4.18 where the r e f l e c t o r was simulated by a cycloid. When the generalized cycloid develops a cusp, a
local singular point,
Now, it
then
the
is not hard to see that
ray and the wave
front p a t t e r n change dramatically.
this is not a structurally stable situation.
First one
162
.
.
.
.
.

_
=

=

=

=

=

:
~~¢~,~,~~
•
Fig. 4.17: C o m p u t e r s i m u l a t i o n s from sinusoidaI r e f l e c t o r s .
~
of
the
evolution
of
rays
Fig. 4.18: Linear ray s y s t e m and w a v e f r o n t s of a cycloid. c a s e o c c u r s when t h e cycloid develops a cusp.
and
wave
fronts
The d e g e n e r a t e d
163
can
argue
that
into
a pattern
the cusp is morphologically not stable. Any small disturbance turns it like Fig. 4.17.
In addition,
in this d e g e n e r a t e d case,
the ray approach
does not fit Huygens' principle. Fig. 4.19 illustrates how the envelope of the wavelets evolves near
the singular surface point.
It
turns
out
that
the successive wave fronts
'ignore' the singularity of the r e f l e c t o r . They are again continuous functions, which can be approximated by a surface model like the generalized cycloid of Fig. 4.18. This observation allows to ignore such singularities for most of the following discussion.
Fig. 4.19: Huygens' wave front construction by wavelets near a singular surface point.
The last point to be discussed is how the surface maps onto the t r a v e l t i m e record. To
find the
distance of dinate 'u'
map
(Xo,Yo)
{u,vt/2)
one has to
section the
ray p a t t e r n in a certain
the r e f l e c t o r , i.e. along the survey track line. The modified horizontal coor(Fig. 4.20) is a function of the inclination of the rays. If the track
line is
taken as the zero level, y = 0, then the horizontal dislocation is
U X
0
Fig. 4.20: The p a r a m e t e r system for the linear ray model: (x,y,): surface coordinates of the r e f l e c t o r , u: horizontal coordinate of s o u r c e and r e c e i v e r at y = 0, r: distance b e t w e e n source (receiver) and surface.
164
AX = ( x  u )
(4.9)
= f'(x)f(x).
The horizontal dislocations can be easily derived from equations (4.6}} and the condition y = w = 0 if the p a r a m e t e r T is eliminated. Now, the t r a v e l t i m e is proportional to the distance b e t w e e n the shot point and the r e f l e c t o r point {Fig. 4.20) or proportional to
r
=¢'((ux) 2 + f(x)2)
•
(4.10)
Thus, if all variables are expressed in t e r m s of x and f(x), one finds the map (FLOOD, 1980)
The
{x,f(x)) ~
equations
surface
trace.
{u,w)
allow
as
u
=
x
r
=
f(x)/(lf'(x)
to
Fig. 4.21

f'(x)f(x)
simulate
{4.11) 2)
traveltime
=
vt/2.
record
numerically
for
any
differentiable
gives some examples of such simulations. The figures include
the ray s y s t e m s and the simulated t r a v e l t i m e records. The surface models are all convex, and this gives 'hyperbolic r e f l e c t i o n s ' on the t r a v e l t i m e record. The mapping equations, which have been discussed so far, allow to simulate the linear ray p a t t e r n , the evolution of wave fronts and the t r a v e l t i m e record for any d i f f e r e n t i a b l e surface e l e m e n t . Indeed,
Fig. 4.21: C o m p u t e r simulations of rays and of the t r a v e l t i m e record for various s u r f a c e models {fourth order polynomials}. The saddle points of the t r a v e l t i m e record have been set to zero depth.
165
:~c."..:';:iI'~':'.':..~..x.
. .....~?.:.'.'..... ".::,a. . . . . .
. . '
:',::,>
L.' . .....t"
S, 7 •

' "~'bx"
 ,,:
Fig. 4.22: Simulations of the t r a v e l t i m e record like in Fig. 4,21 plots for 'layered media'.
but as point
these approximations can be extended to threedimensional surface structures. But they are still much too general. There is an infinite number of possible functions~ which can be used to approximate a r e f l e c t o r surface, and these functions may depend on a large number
of p a r a m e t e r s
so that
we are
not
able
to catalogue
the
traveltime patterns
within the p a r a m e t e r space. The major aim of the next section is, t h e r e f o r e , to analyze the local properties of r e f l e c t o r s , and to show how these local properties impress their s t r u c t u r e generically on a sensing wavefield and, t h e r e f o r e , on the t r a v e l t i m e record.
The be
computer
simulated.
simulation
Thus,
one can
has
the
transform
advantage the
that
more
geometrical
c o m p l i c a t e d systems can
simulation
into
a point
plot
(Fig. 4.22) which resembles the received energies to some e x t e n t , i.e. the observed traveltime
record.
The
comparison of such a plot
for
a multilayersystem (Fig. 4.22)
with
t r a v e l t i m e records {e.g. Fig. 4.15) shows that not only 'hyperbolic r e f l e c t o r s ' may arise from
local
concave
surface elements. High energy zones, which are
s o m e t i m e s found
in otherwise nearly isotropic areas, may well be r e l a t e d to the s u r f a c e morphology r a t h e r than to a property of the sediment cover.
4,2.2
Local Surface Approximation
tn the preceding discussion it b e c a m e clear that
the r e f l e c t i n g surface impresses
its s t r u c t u r e onto the rays, onto the wave fronts and, t h e r e f o r e , finally onto the t r a v e l time record. It is this generic situation which makes r e m o t e sensing a structurally stable process
and structural stability alone secures t h a t one has a chance to r e c o n s t r u c t
the s u r f a c e properties. On the other hand, it turned out that it is unreasonable to work with global s u r f a c e structures. Therefore, at first one needs a classification of the criti
166
cal
points
on
the
& GUTTINGER will
be r e s t r i c t e d
along
the
the
surfaces
we
surface,
This
1982) in d e t a i l
to p l a n e
track
describes
reflecting
{1980,
line.
curves,
can even
was
done
i.e. to a f i r s t a p p r o x i m a t i o n
In t h i s c a s e ,
surface,
classification
the usual situation
is o f b o u n d e d suppose that
by D A N G E L M A Y E R
for t h r e e  d i m e n s i o n a l p r o b l e m s . H e r e t h e d i s c u s s i o n
variation
a surface
will be
of cylindrical that
(GUGGENHEIMER,
the
reflectors
function,
1977).
line is o f s m a l l v a r i a t i o n ,
For
which
most
real
and, t h e r e f o r e ,
t h e u s u a l s i t u a t i o n will be t h a t t h e c u r v e c a n be l o c a l l y a p p r o x i m a t e d by a T a y l o r s e r i e s f(x)
= a 0 + alx
+ a2x2
+
...
+ higher
terms.
(4.12)
Fig. 4.23: R a y s y s t e m s (x,z) and ' t r a v e i t i m e r e c o r d ' (x,t) o f l i n e a r s u r f a c e e l e m e n t s . T h e s t r a i g h t lines m a p o n t o s t r a i g h t lines, b u t t h e h o r i z o n t a l d i s l o e a tion c a n c a u s e o n l a p p i n g f e a t u r e s a n d s h a d o w z o n e s . Now, t h e r e a r e t w o i n t e r e s t i n g c a s e s : If t h e c o e f f i c i e n t s a 2 = 0 a n d a 1 ~ 0, t h e r e f l e c t o r equals
locally
a straight (4.11). line
(near
The
only
is d i s l o c a t e d
shadow ance
zones of
x = 0)
a straight
line on t h e t r a v e l t i m e spectacular along
patterns
the
record
are
are
horizontal
and onlapping
the
line,
r e c o r d (Fig.
patterns the
coordinate.
u n d e r t h e m a p , i.e. t h e r e is no s p e c t a c u l a r
more
interesting
if t h e
expansion
system
at x = a ° by u s e o f t h e m a p x   ~ x
system
in s u c h a w a y
and that
element
in Fig.
4.23.
in t h e
the transformed
maps
onto
This horizontal
between
record, sediment
An i n c l i n e d s t r a i g h t dislocation
can
cause
E x a m p l e s for t h i s d i s t u r b cover
and
basement
in
p r o p e r t y ' t o be a s t r a i g h t line' is p r e s e r v e d
d e f o r m a t i o n on t h e t r a v e l t i m e
Taylor
that
reflector
as c a n be p r o v e d by u s e of e q u a t i o n
in t h e t r a v e l t i m e
intersections
can be simplified
straight
4.23),
summarized
Fig. 4.15. On t h e o t h e r h a n d , t h e g e o m e t r i c a l
Things become
This
parameter
following
a 2 is n o n  z e r o .
way:
One
locates
 a o. T h e n o n e r o t a t e s xaxis coincides with
record.
In t h i s c a s e ,
the
a new coordinate the new coordinate
the tangent
a t x = 0,
t h e y  a x i s c o i n c i d e s w i t h t h e n o n  o r i e n t e d n o r m a l at t h i s p o i n t . T h e t r a n s f o r m a 
tion r e d u c e s t h e T a y l o r s e r i e s to t h e f o r m
f(x)
The parameter
1 = ~ kx 2 +
...
+ higher
terms.
k is t h e local c u r v a t u r e o f t h e c u r v e a t t h e p o i n t x = 0
(4.13)
167
k = f"(0)/(1f'(0)2)
3/2
•
(4.14)
Dependent on the sign of k the point is either a maximum or a minimum in the local coordinates. The
discussed
transformation
relates
the
local
structure
of the r e f l e c t o r
to its curvature at the critical point, a well known procedure from differential g e o m e t r y (GUGGENHEIMER, 1977; DO CARMO, 1976). If the Taylor series s t a r t s with higher 2 t e r m s than x , one has a d e g e n e r a t e d situation, and one will find p a t t e r n s like in the case of the cycloid of Fig. 4.18.
Such situations will be avoided during most of the
following discussion.
In the case
0 0, t h e n t h e c u b i c is m o n o t o n o u s l y
increasing
(or d e c r e a s i n g ) ,
f a m i l y of n o n  i n t e r s e c t i n g
and
the
and t h e r e
e x i s t s an a r e a w h e r e
intersect.
The
case
map x ~u
is o n e to one, i.e. t h e r e e x i s t s only o n e
r a y s . If s < 0 , t h e n t h e c u b i c h a s a m a x i m u m the map x ~u
s = 0 defines
the
and a m i n i m u m ,
is n o t u n i q u e l y d e t e r m i n e d ,
transition
state
between
these
i.e. t h e r a y s
two
possibilities,
it d e f i n e s t h e loci w h e r e t h e c a u s t i c i n t e r s e c t s t h e t r a c k line.
Now, equation.
the
critical
area
These extrema
in
the
(x,s}space
is g i v e n
d e f i n e t h e loci on a t r a c k
i n t e r s e c t s t h e t r a c k line. T h e e x t r e m a U
by
the
extrema
of
the
cubic
line, w = c o n s t a n t , w h e r e t h e c a u s t i c
a r e g i v e n by
= 0 = s + 3x 2 X
or
(4.21)
s = 3x 2 •
F r o m e q u a t i o n (4.20) and (4.21) o n e finds t h e c r i t i c a l
line
in t h e
(u,s)space
by e l i m i n a t i o n
o f t h e v a r i a b l e x:
U2/4 i.e. up
to
change cubic
a
of
proper
the
parameter
parameter
parabola
already
= s3/27
that
the
setting
just the meaning
captures
patterns
the
the
possible One
spatial
can
use
threedimensional
equation
space
discussed
is
local
much
curvature
equation
to
This
height
as
On
general
of t h e r e f l e c t o r most draw
folded
general a picture
surface
before.
If one
(b = constant),
caustic.
more
in t h e i r
(4.20)
{u,x,s).
same
of track
earlier now
i n c l u d e s also c h a n g e s o f t h e
parameters.
the
's' to a change
describes
seen
(4.22)
is
the
then
other
because
hand,
the
relates
we
have
parameter
line. T h u s , e q u a t i o n
the
shown
critical in
surface
Fig. 4.24.
's'
(4.20)
s e n s e by a m i n i m a l s e t of
a
the semi
of in
Every
s e c t i o n s = c o n s t a n t t h r o u g h t h i s folded s u r f a c e d e s c r i b e s t h e d i s l o c a t i o n of t h e h o r i z o n t a l r e f l e c t o r c o o r d i n a t e a l o n g a p o s s i b l e t r a c k line (for a f i x e d local c u r v a t u r e ) .
To a r r i v e once in
more.
terms
rays
may
at
The
of
the
the cubic
final c a t a s t r o p h e equation
number
intersect
at
a
of
its
point
(4.20) roots. in
the
representation,
t h e v i e w p o i n t h a s to be c h a n g e d
c a n be d i s c u s s e d in t e r m s This gives spatial
the
additional
coordinates.
The
of i t s d i s c r i m i n a n t information discriminant
how
or
many
takes
the
form
D = u2/4
 s3/27.
{4.23)
170
K
Fig, 4.24: The c a t a s t r o p h e set of the cusp c a t a s t r o p h e ,
For D = 0 one has the points which s e p a r a t e the p a r a m e t e r s e t t i n g s leading to a single root
(D > 0) from
those t h a t cause triple roots (D < 0). Now, the question how many
roots a cubic e q u a t i o n has is identical with t h e question how many e x t r e m a a quartic e q u a t i o n m a y have. The cubic can, t h e r e f o r e , be e m b e d d e d into the c a t a s t r o p h e p o t e n t i a l
V = x4/4
+ ux2/2
(4.24)
+ sx,
which has been published as t h e cusp c a t a s t r o p h e (RiemanHugoniot c a t a s t r o p h e in THOM, 1975).
This
catastrophe
potential
captures
the
discussed
twodimensional
ray
patterns
in t h e i r most general topological behavior, The
previous discussion of the cusp c a t a s t r o p h e
allows to classify t h e t r a v e l t i m e
record in t e r m s of depth and local c u r v a t u r e of the r e f l e c t o r line. To do this explicitly one c a n w r i t e
the local parabolic approximation as y =  a
' a ~ indicates depth, The t r a c k
line is then located
+ bx 2, where the p a r a m e t e r
at d e p t h
zero.
From t h e equations
(4.11) one finds t h e local t r a v e l t i m e image:
u = (l2ab)x
+ 2b2x 3
r = (a+bx2)(l4b2x
(4.25)
2) 1/2
As turned out from the analysis of the cusp c a t a s t r o p h e , the c r i t i c a l set is given by
s = ( l  2 a b ) / ( 2 b 2) = O. The p a r a m e t e r
identification a =w
relates
(4.26)
this r e p r e s e n t a t i o n to equation (4.20). Now
the c r i t i c a l set can be r e w r i t t e n in t e r m s of the p a r a m e t e r s (a,b) as
171
b
i O _ _
W
OK~
~?
A w
/k
e%7
"T
m
•
Y
Fig. 4.25: The morphology of local elements on the reflector line (upper graph) and their image on the traveltime record. The surface elements and their images are located in the parameter system depth (a) and local curvature of the reflector line (b). Inside the hyperbolic boundary the reflector image is inverted, i.e. it is the domain of hyperbolic reflections.
172
1 
2ab
= 0.
{4.27)
Equation {4.27) describes a hyperbolic boundary line in the (a,b)space, and equation (4.25) allows to c o m p u t e the image of various parabolas dependent on the choice of the p a r a m e ters 'a' and tbL Fig. 4.25 illustrates the relationship b e t w e e n the local morphology of the r e f l e c t o r and its image on the t r a v e l t i m e record within the p a r a m e t e r space (a,b). From
the
discussion of the cusp c a t a s t r o p h e
we know t h a t
the p a r a m e t e r 's' a f f e c t s
the i n t e r s e c t i o n of the c a u s t i c with the track line. In analogy to the p a r a m e t e r 's' one can vary equation {4.27): c This
defines a
family
of
2ab
hyperbolae
= 0. in the
(4.28) (a,b)space of identical
travettime
image
(different depth location), which result from d i f f e r e n t conditions (Fig. 4.25).
Thus,
the previous discussion provides us with some practical results, at least for
the i n t e r p r e t a t i o n of echograms. The analysis of the t r a v e l t i m e record, in t e r m s of local properties of the r e f l e c t o r line and of caustics, allows to classify the t r a v e l t i m e images by a minimal set of p a r a m e t e r s , and
local
curvature
are
not
and it b e c o m e s c l e a r that independent with r e s p e c t
these p a r a m e t e r s  
to
depth
the e f f e c t s they produce
on the t r a v e l t i m e record. In addition, it b e c o m e s c t e a r that the e x t r e m a of a r e f l e c t i n g s u r f a c e are stabte points. In this case, one can approximate the r e f l e c t o r line locally by a parabola without any r o t a t i o n of the local c o o r d i n a t e system, and the point (0,f(x)) is r e c o r d e d at
its c o r r e c t l y horizontal position as well as with the c o r r e c t t r a v e t t i m e .
This allows to e s t i m a t e t h e wavelength of sand waves and similar s t r u c t u r e s along the track line from the original t r a v e l t i m e record. F u r t h e r m o r e , the amplitude can he estim a t e d as well in t e r m s of t r a v e l t i m e , and the relation to the cusp c a t a s t r o p h e allows to draw charts, from which the local c u r v a t u r e can be e s t i m a t e d .
4.2.4
Wave F r o n t s and the Swallowtail C a t a s t r o p h e
The next point of i n t e r e s t is how the wave fronts evolve near a cuspoid caustic. Within the
linear ray model, a wave front is given as a set of points {on the family
of rays} which have equal d i s t a n c e from the r e f l e c t o r
{(u,w) E This
(x,y)
twodimensional
1 ((xu) 2 + (YW)2) I/2 = r = const.}.
relationship
( D A N G E L M A Y R & GOTTINGER, continuous map
for the
wave
can
be
1982).
fronts,
extended
to
the
threedimensional
(4.29)
case
Here it is more appropriate to return to the
as it was derived in equation (4.7). From
these
173
equations one finds a wave front by s e t t i n g • = c o n s t a n t . The only c r i t i c a l set for the rays is the cuspoid caustic. Therefore, the c r i t i c a l
as WRIGHT (1979) points out, we should "expect
value graph of t h e cusp c a t a s t r o p h e ,
which is t h e b i f u r c a t i o n set of the
swallowtail". Indeed, if one draws t h e wave fronts for several values of the p a r a m e t e r T
to s i m u l a t e
their evolution
from
a locally parabolic
reflector,
then
they
take
the
form of sections through the swallowtail c a t a s t r o p h e (Fig. 4.26) as far as they are located inside the caustic.
To see in detail how the swallowtail is r e l a t e d to wave fronts one
/~il'l//llll/l/ltlllllllltlllltllfllll'llllllllllllll~ IIIIIIIIIIItltlIItlfltt'tIII'tlIIItlIIHIIlItt~ / /ltfft/////llltlltflliltttt![II!{llllIlllllil;
Fig. 4.26: Evolution of parabolic wave fronts into swallowtails. The unfolding of the wave fronts is caused by the folded ray system, which is due to the cusp c a t a s t r o p h e . The numbers indicate values for t h e p a r a m e t e r of evolution vt. 
=
c a n develop the equations (4.7) in Taylor series. If the local parabolic surface approximation formula is inserted into equations (4.7), then the Taylor expansion of these equations up to order 4 gives the approximations u = (l2bT)x
+ 4b3~x 3
(4.30)
w = T + b ( l  2 b I ) x 2 + 6 b 4 x 4.
If b ~ 0 and
T~ O, then t h e s e equations can be s t a n d a r d i z e d to t h e form W = sx 2 + 3x 4 U = 2sx
where
(4.31)
+ 4x 3
W = (wT)/(2b4T),
U = u/(b3T),
174
s =
(l2bT)/(b3T).
On the o t h e r hand, the c a t a s t r o p h e potential of the swallowtail is defined as V = x5/5
Its critical
value
graph
+ ax3/3
in the
+ bx2/2
parameter
+ cx.
space
{4.32)
(a,b,c)
is defined by
its derivatives
(THOM, t975} V
=
x4 +
ax 2 +
=
4x 3 +
=
12x 2 +
bx
+
c
= 0
(4.33)
x
V V
xx xxx
2ax
+
b
2a
=
0
=
0.
If one uses the first two derivatives to solve for the p a r a m e t e r s b and c in t e r m s of a and x, one finds the map b
=
4x 3

c
=
3x 4
+
By a proper choice of
2ax
(4.34)
ax.
the signs (take
x
x) this
map b e c o m e s equivalent to the
local Taylor expansion {4.31) if one takes the foIlowing p a r a m e t e r identifications IJ At
least
locally
are equivalent tail. Again,
W
~ b,
approximation
a up
to sections a = s = constant
one
approximation
(by an
c,
finds that
a
standard

s.
to order
4), one
through
catastrophe
finds that
the
wave
fronts
the catastrophe set of the swallowon T H O M ' s
(1975)
list gives a good
to the ray model. In this case, the swallowtail catastrophe describes rather
pretty the evolution of w a v e
4.2.5
fronts.
Wave Front Evolution and t h e T r a v e l t i m e Record
An examination of the original Taylor approximations for the wave fronts (equations (4.30)) shows that the s e c t i o n s s = const, through the swallowtail are located on a line w = T.
The
projections onto
the
{u,w)plane
the modified swallowtail (4.30) give {Fig. 4.26}.
The
sections through
the
{ {u,w)~(x,y) ) of
these
sections
through
the typical evolution p a t t e r n for the wave fronts swallowtail
are
sitting one behind the o t h e r
in
the caustic. Now, the p a r a m e t e r T can be w r i t t e n as ~r = vt (v: velocity, t: time}, and we find that
the
way,
in which the
swallowtail
is sitting above the cuspoid caustic,
depends on the sonic velocity of the medium or on the velocity of the wave front dislocation. On the other hand, the velocity cannot a f f e c t the spatial p a t t e r n  the caustic as turned out during the discussion of the cusp c a t a s t r o p h e . The caustic is a structurally
175
stable
spatial
pattern,
w h i c h only d e p e n d s on t h e
local
surface
structure,
i.e.
t h e local
curvature.
Now,
the
sional spatial
a n a l y s i s of w a v e coordinates.
fronts
The map
adds a t h i r d d i m e n s i o n , t i m e ,
from
the reflector
to
)f
the
to t h e t w o  d i m e n 
traveltime
record,
x
Fig. 4,27: T w o v i e w s o f t h e m o d i f i e d s w a l l o w t a i I c a t a s t r o p h e . T h e s w a l l o w t a i l s i t s on a line y = vt, T h e s e c t i o n s y = c o n s t a n t t h r o u g h t h i s c a t a s t r o p h e set are the recorded reflections near a concave surface element.
there
176
fore,
turns
out
to be a map
R 3 ~ R 2 or
(x,y,t)~
(u,t).
From
the caustic we know
t h a t it is a stable p a t t e r n in the (x,y)plane. In addition, we know t h a t the images of the sections
through
the swallowtail
need
to have s t a b l e positions inside the c a u s t i c s
(Fig. 4.27}. A c h a n g e of the sonic velocity, t h e r e f o r e , c a n n o t a l t e r these spatial p a t t e r n s , it can only a f f e c t the r e c o r d e d t r a v e l t i m e , i.e. t h e spreading velocity of the wave front. For
the
In the
traveltime spacetime
record
this means
coordinates
that
t h e sonic
the
timeaxis
is s t r e t c h e d
velocity can only a f f e c t
the
or compressed. timeaxis.
The
only allowed t r a n s f o r m a t i o n of t h e c a t a s t r o p h e set {Fig. 4.27) by a change of the v e l o c i t y is,
therefore,
pure
does not a l t e r
(4.30)),
does
the
local
one has to section
(y,t)plane
with equation
t = ay. This t r a n s f o r m a t i o n
spacetime
modified
reflector
which
approach
area
sections
pattern
map
onto
modified swallowtail
coordinates,
swallowtail,
catastrophe
surface the
the
(4.30)
traveltime
record?
To study
by a plane w = y = c o n s t a n t
(4.30) 'w' means depth). This gives the image of the local surface
(in the equations in the
plane
in the
i.e. t h e i r projections onto the spatial (x,y)plane.
How this,
shear
the spatial c o o r d i n a t e s of the sections through the modified swallowtail
in the (u,vt/2)plane. Fig. 4.27 gives two views of this
have
summarizes
been the
in a very condensed through
the
sectioned
patterns, way.
by a plane
which c a n
By comparison of
threedimensional
catastrophe
w = constant.
arise set,
from
Again
the
a local c o n c a v e
the observed r e c o r d one can
get
with
reasonable
q u a l i t a t i v e i n f o r m a t i o n about the local surface s t r u c t u r e . Especially t h e 'hyperbolic r e f l e c tions'
turn
out
to
represent
local surface
inversions, which are
related
to
the wave
the local r e f l e c t o r
geometry
fronts, which have e n t e r e d the local c a u s t i c .
4.2.6 The Traveltime Record as a Plane Map A second
approach
to
analyze the r e l a t i o n b e t w e e n
and the t r a v e l t i m e r e c o r d is versus the plane map (x,y}  ~ ( u , v t / 2 ) , which has been defined by equations (4.11}. This method
is very close to FLOOD's (1980) study of 'hyperbolic
r e f l e c t i o n s ' in deep sea echograms. Again the c a t a s t r o p h e approach versus local properties of t h e r e f l e c t o r will provide general results. First, one has to specify t h e mapping equations (4.11). To introduce d e p t h explicitly, t h e r e f l e c t o r line is locally a p p r o x i m a t e d by a p a r a b o l a f(x) = a + bx 2 like in equations {4.25). The Taylor expansion of ' r ' (equation (4.25)) up to order 4 gives the local map
u =
(l+2ab)x
+ 2b2x 3
(4.35) r = a + b ( l + 2 a b ) x 2 + 2 b 3 ( l  a b ) x4.
177
Although this
map is very similar
to
the
evolution equation of the wave fronts
(4,30), it is not possible to transform it into the standard form of the swallowtail (4,34) by
means
of simple transformations, which preserve the
as
follows from
the
previous
discussion, we
topological structure,
should e x p e c t
arbitrary
sections
Indeed, through
the swallowtail r a t h e r than its standard form,
Now, instead of r we can use r 2 = v2t2/4 as the distance m e a s u r e m e n t b e t w e e n 2 the source and the reflection point. The square r ms a monotonous function of r because r > 0 (Fig. 4.20). This t r a n s f o r m a t i o n is not unusual to a seismologist (e.g. KERTZ, 1969}, and it allows to formulate the distance r as r2 = =
(f(x)2 2 + a
+
( n  x )2
(l2ab)
(4.36)
x2 +
2 u
2ux
+
b 2 x 4.
This equation can be r e w r i t t e n as a ' c a t a s t r o p h e potential' if b ¢ 0, V  (r 2  a 2 ) / b 2 = x4 +
(l2ab)x2 b2

2Ux
+ U2
(4.37)
or V = x4 +
2vx 2  2Ux
+ U2
with obvious p a r a m e t e r identifications.
The first derivative of this ' c a t a s t r o p h e potential' defines U:
Vx
= 0 = 4x 3 + 4 v x
i.e. the original cusp catastrophe

(4.38)
2U,
(eq. 4.20).
This c a t a s t r o p h e potential does not appear in Thorn's list of e l e m e n t a r y c a t a s t r o phes, but he discusses it of
the
as a selfreproducing singularity or as the stopping potential
cusp c a t a s t r o p h e (THOM,
1975),
In t e r m s of c a t a s t r o p h e theory this potential
is the universal unfolding o f the cusp c a t a s t r o p h e , and we can e m b e d it into a local potential V1 =
x5/5 +
vx3/3
+ ux2/2
+ u2x
(4.39)
by a proper choice of the p a r a m e t e r s . This is a swallowtail with a d e g e n e r a t e d p a r a m e t e r space. The p a r a m e t e r s ' c ' and 'b' from equations (4.33) are now r e l a t e d by b = c 2. The critical set appears in the (V,U,v)space (Fig. 4.28), and the t r a v e l t i m e record is r e l a t e d to s e c t i o n s v = c o n s t a n t through the critical set. Thereby one has to keep in mind that 2 V means r , not r. The sections v = c o n s t a n t have locally a swallowtaillike appearance, but, in addition, they have two maxima sentation of Fig. 4.28).
where the curves bend down again (in the repre
178
V
v
i
U
Fig. 4.28: The stopping potential of the cusp c a t a s t r o p h e (a} in the p a r a m e t e r space (V,U,v). The positive Vaxis is drawn downward for the convenience in comparing it with the standard swallowtail (b) and the hyperbolic r e f l e c t i o n of the t r a v e l t i m e record.
The appearance of the two additional maxima above the point of s e l f i n t e r s e c t i o n in Fig. 4.28 needs an explanation because we cannot e x p e c t this p a t t e r n from the simple parabolic approximation of the r e f l e c t o r . Similar p a t t e r n s can be found in the simulated record of Fig. 4.21,
but
it will turn out that
these p a t t e r n s are of a very d i f f e r e n t
type because they are really r e l a t e d to the s u r f a c e structure.
What happens with the
stopping potential, illustrates Fig. 4.29. There, the parabolic r e f l e c t o r line extends over the track
line (S). Now, one can c o n s t r u c t the image of this a b s t r a c t s u r f a c e on the
t r a v e l t i m e record in a very simple way. One has just to project the length of the rays, which c o n n e c t the r e c e i v e r with the r e f l e c t i o n points s t r a i g h t downward from the point where they i n t e r s e c t the track line. This gives the curve (r), i.e. the image on the t r a v e l time
record. This c o n s t r u c t i o n can also be done for those parts of the r e f l e c t o r line
which e x t e n d above the track line. Because t r a v e l t i m e is measured without a directional component, i.e. it can only assume positive values, the curve (r) bends down again, as one moves away from the i n t e r s e c t i o n point of (S} and (r). Therefore, one has to choose
ssA
Fig. 4.29: The a b s t r a c t situation that the track line (S) i n t e r s e c t s the r e f l e c t o r line. In this case, the t r a v e l t i m e record (r) r e a c h e s the track line at the i n t e r s e c t i o n point and bends then down again because r can assume only positive, values.
179
carefully the c o r r e c t interval if the stopping potential is used as a model for the travelt i m e record. The c o r r e c t interval is, in any case, located b e t w e e n the two maxima of the sections v = c o n s t a n t of Fig. 4.28.
If one
analyzes
the
critical
mind, then it turns out that
surface
of Fig. 4.28
with the
noted r e s t r i c t i o n s in
the typical 'hyperbolic r e f l e c t i o n s ' with a swallowtaillike
appearance are r e s t r i c t e d to a limited range of the p a r a m e t e r v. If v is positive, one has a convex r e f l e c t o r , which in a topological sense is recorded correctly. As v assumes sufficiently large negative values, the 'hyperbolic reflections' turn smoothly into a more parabolic appearance, which, like the 'hyperbolic r e f l e c t i o n s ' , is an inversion of the local r e f l e c t o r topology   a concave surface e l e m e n t turns into a convex image. Those 'parabolic r e f l e c t i o n s ' are also well known from echograms (FLOOD, 1980), but, more c o m m o n ly,
they are found within b a s e m e n t r e f l e c t i o n s (Figs. 4.14, 4.15).
As was shown in the last section, the approach versus wave fronts provides another f r a m e to summarize the images on the t r a v e l t i m e record. The advantage of the plane map approach is that the 'stopping potential' r e p r e s e n t s the images in a still more condensed way.
4.2.7
Singularities on the Reflector Line
So far, a very simple r e f l e c t o r model was used. In the case of faults, folds and flexures the situation may b e c o m e more c o m p l i c a t e d
although a local parabolic approxi
mation with rotation of the coordinate axes may be still possible. The most simple case, w h e r e one can find such a critical situation, are flexures and folds. A first impression
);ii f
Fig. 4.30: First flexures.
order
approximation
of
ray
systems
and
wave
fronts
near
180
of
what
may
happen
(Fig. 4.23). What
near
a
fault
gives
the
simple
linear
model
of
section
4.2.2
is actually new in this linear approximation, is the appearance of a 3 + ay which also
shadow zone. A flexure can be simulated by a cubic equation x = y includes simple folds. Fig. 4.30
gives a rough approximation of rays
and wave
fronts
which arise from the cubic r e f l e c t o r line model with a > 0, a = 0 and a < 0. For a < 0, the caustic p a t t e r n s can be approximated by a parabolic approximation at the e x t r e m a of the cubic equation, but only parts of the wave fronts s c a t t e r back to the t r a c k line, i.e.
only
one branch of the caustic
i n t e r s e c t s with the
track
line. Fig. 4.31
trys
to
capture the behavior of the caustic over a family of cubic r e f l e c t o r lines. For compari
Fig. 4.31: The caustics of a family of cubic r e f l e c t o r lines. Left: The family of caustics of only one e x t r e m u m {the lower one). Right: Separation of the c a u s t i c s into their relevant parts, i.e. the branches which reach the track line.
son, the family of caustics for only one e x t r e m u m is also shown. These graphical methods only give a very rough idea of what happens near such s t r u c t u r e s , but it is not the scope h e r e to analyze t h e s e problems in detail. In this c o n t e x t it b e c o m e s at least n e c e s sary to study d i f f r a c t i o n patterns. For this approach see
DANGELMAYR & GUTTtNGER
(1982). Similar
problems
arise
if
the
reflector
has
singular
points
like
the
cycloid of
Fig. 4.18 in section 4.2.1. The cycloid can be described by the map x
=
t

sin(t)
y
=
I

cos(t).
(4.40) By
taking a Taylor expansion near the cusp point, one finds
181
x =
t3
(4.41)
Y  t 2
where t
the
constants
is e l i m i n a t e d ,
The
main
isolated the
one
point,
point
critical
have
been
finds that
however,
of t h e
point
absorbed the
is n o t
reflector
cusp point
that
we
line a t
is g i v e n as t h e
in x a n d y for c o n v e n i e n c e . equals
have
the
a cusp,
If t h e p a r a m e t e r 3 2 parabola y = x.
semicubic
but
that
the
singularity
is an
w h i c h d x / d t = 0 and d y / d t = 0. T h e c a u s t i c n e a r
loci of t h e radii
of c u r v a t u r e
on t h e n o r m a l s o f t h e
semieubic parabola: x Yc Thus,
not even
too close
(4.42)
 4t 3 + o4t .3
C
 t2.
  ~~ t4
to the isolated
singular
point (t = 0) the caustic behaves
like
the m a p
i.e.
it is a fold c a t a s t r o p h e
t 3 is m u c h s m a l l e r r a y s a r e only l o c a t e d zone. a
A
detailed
topological
than
u =
t2
V
t,
=
(Fig. 4.32; LU,
t and that
(4.43)
1976). T h e t e r m
t 4 is m u c h s m a l l e r
'not
too c l o s e '
t h a n t 2. A t a fold c a u s t i c
on o n e side o f t h e c a u s t i c s and c a u s e , t h e r e f o r e ,
analysis
classification,
of
such and
it
singular would
points be
on
the
necessary
means that
to
reflector study
line the
would
require
wavefield
than the ray system.
m
F i g . 4.32: T h e fold c a t a s t r o p h e causes a shadow zone.
the
locally a s h a d o w
m
( c a u s t i c ) n e a r a s i n g u l a r p o i n t on t h e r e f l e c t o r
rather
182
Table 41: S u m m a r y of t h e Ray Model The t r a v e l , l i n e r e c o r d in its most c r i t i c a l case corresponds to sections y = c o n s t a n t (y:depth) through a swallowtail c a t a s t r o p h e which is located on a line y = vt in the t h r e e  d i m e n s i o n a l space {x,y,t). The various types of specialized d e f o r m a t i o n s depend on t h e l o c a I c u r v a t u r e of t h e r e f l e c t o r line, on t h e d i s t a n c e b e t w e e n t h e t r a c k line and t h e c r i t i c a l point on t h e r e f l e c t o r line, and on t h e sonic v e l o c i t y of the medium. In t h e p a r a m e t e r s p a c e d e p t h of t h e c r i t i c a l point (a) and local c u r v a t u r e (b), t h e c r i t i cal boundary line for an image inversion, i.e. for t h e o c c u r r e n c e of ~hyperbolic r e f l e c tions', is given by t h e hyperbola 1  2ab = 0. This hyperbolic equation simply c o m p a r e s t h e local c u r v a t u r e of t h e r e f l e c t o r with a c i r c u l a r wave f r o n t a t d e p t h ' a ' . In detail, one finds t h a t t h e s e p a r a m e t e r s a f f e c t t h e t r a v e l t i m e r e c o r d in the following way:
I) Spatial p a t t e r n s , t h e cusp c a t a s t r o p h e 1) The local c u r v a t u r e of the r e f l e c t o r line: Only convex a r e a s of t h e r e f l e c t o r line are s p e c t a c u l a r (cause trouble within the record) b e c a u s e a cuspoid c a u s t i c evolves. Two special situations occur: a) The local approximation of the r e f l e c t o r line requires a r o t a t i o n of the local coordin a t e system with r e s p e c t to t h e global one. The sections through t h e c a t a s t r o p h e set b e c o m e s oblique. This p a t t e r n c a n be d e t e c t e d on t h e t r a v e l t i m e r e c o r d because t h e 'hyperbolic r e f l e c t i o n s ' are a s y m m e t r i c . b) D i f f e r e n t local c u r v a t u r e s (b = k/2) or the r e f l e c t o r c a u s e a dislocation and s t r e t c h i n g (compression) of the c a u s t i c in t h e spatial coordinates. This d e f o r m a t i o n c a n only be distinguished from (2) if t h e t r u e d e p t h position of t h e s p e c t a c u l a r point on the r e f l e c t o r line is known. 2) The hei_~.h_t_of ,_he t r a c k line above t h e r e f l e c t o r line: Because t h e r e f l e c t i o n p a t t e r n depends on t h e r e l a t i o n b e t w e e n t h e c u r v a t u r e of t h e i n c i d e n t wave front and t h e c u r v a t u r e a t t h e s p e c t a c u l a r point on t h e r e f l e c t o r line, this c a s e c a n n o t be distinguished from a c h a n g e of t h e local c u r v a t u r e of t h e r e f l e c t o r w i t h o u t additional i n f o r m a t i o n (e.g. a m e a s u r e m e n t of t r u e depth). This p a r a m e t e r chooses a special line through t h e c a t a s t r o p h e set of the cusp which is stably located in t h e space coordinates. Because t h e cusp c a t a s t r o p h e is t h e b i f u r c a t i o n s e t for the swallowtail and t h e discussed stopping potential, this p a r a m e t e r also appears in the other catastrophes. 3) E x t r e m a of c u r v a t u r e : In case t h e r e f l e c t o r has a local minimum of c u r v a t u r e , it can be a p p r o x i m a t e d by a parabola, and the discussion of sections 4.2.17 holds: Typical p a t t e r n s inside t h e c a u s t i c are 'hyperbolic r e f l e c t i o n s ' . However~ if t h e r e f l e c t o r has a local maximum of c u r v a t u r e , t h e c a u s t i c p a t t e r n is inversed, as discussed in section 4.2.8. Anyway, t h e previous discussion r e m a i n s valid if t h e propagation of wave fronts is inversed. A f t e r ~ t h e wave fronts have passed through t h e c a u s t i c , a parabolic r e f l e c t i o n p a t t e r n r e s u l t s which allows to distinguish this c a s e from t h e ' s t a n d a r d s i t u a t i o n ' . I!) S p a c e  t i m e p a t t e r n s : t h e swallowtail c a t a s t r o p h e The sonic v e l o c i t y of t h e medium does only a f f e c t t h e t r a v e l t i m e . This p a r a m e t e r can, t h e r e f o r e , c a u s e only those t r a n s f o r m a t i o n s which l e t t h e s p a c e p a t t e r n i n v a r i a n t  pure s h e a r in t h e ( y , t )   plane. The c a t a s t r o p h e set, which describes t h e evolution of t h e wave fronts, is a modified swallowtail which is l o c a t e d on a line y = vt. The t r a v e l , l i n e images a r e plane s e c t i o n s through this c a t a s t r o p h e set. A l t e r n a t i v e l y , t h e t r a v e l t i m e image c a n be described by t h e unfolding of the cusp c a t a s t r o p h e , i.e. by its stopping potential. The l a t t e r approach gives a description in t h e c o o r d i n a t e s (x,y,v2t2).
183
Table 42: S u m m a r y of s t r a t e g i e s in the analysis of t r a v e l t i m e records
"wave f r o n t approach"
"plane mapping method"
C o n s t r u c t i o n of the ray syste m (normals of t h e local r e f l e c t o r e l e m e n t )
The c a t a s t r o p h e map along the t r a c k line, the cusp c a t a s t r o p h e
The caustic or the envelope of the rays c e n t e r s of curvature) Evolution of the w a v e f r o n t s along t h e rays, t h e swallowtail c a t a s t r o p h e
t r a v e l t i m e s e c t i o n s through the c a t a s t r o p h e set of t h e wave f r o n t s   t h e modified swallowtail
Unfolding of the cusp c a t a s t r o p h e , t h e 'stopping p o t e n t i a l '
i¢ The local image of the t r a v e l t i m e record
4.2.8 G e n e r a l i z e d R e f l e c t o r P a t t e r n s in Two and T h r e e Dimensions In case
the r e f l e c t o r
discussion provides modeI is sufficient
can be described by an explicit function y=f(x), the previous
a finite classification of r e f l e c t o r and c a t a s t r o p h e
theory
patterns
provides a f r a m e
as long as a linear ray for this classification,
as
s u m m a r i z e d in tables 41 and 42. However, the application of c a t a s t r o p h e theory requires local coordinate
changes,
which s o m e t i m e s
may be assumed i n a d e q u a t e for the problem.
tn the previous discussion it turned out t h a t the t r a v e l t i m e record depends on a p a r a m e t e r s= 12ab which appears in alI equations  for t h e caustic, the wave fronts and the t r a v e l t i m e record.
The p a r a m e t e r
'a'
is equivalent
to the depth of the r e f l e c t o r ,
and '2b=k'
is its local c u r v a t u r e (cf. equation 4.13). The p a r a m e t e r 's', t h e r e f o r e , provides a simple interpretation,
it m e a s u r e s
t h e r e l a t i o n b e t w e e n an incident wave front with radius ' a '
(depth} and the radius of c u r v a t u r e of the r e f l e c t o r . Image inversion occurs for a > l/(2b), i.e. if the radius of the incident 'wave front' is larger than the radius of curvature, multiple r e f l e c t i o n s
arise
locally because
the c u r v a t u r e
of
the r e f l e c t o r
increases,
as one
departs from the c r i t i c a l minimum. Fig. 4.33 iIlustrates this viewpoint.
A) The D e f o r m e d Circle and the Dual Cusps A n a t u r a l question is
what happens if the r e f l e c t o r has a d i f f e r e n t s t r u c t u r e , i.e.
184
~
J tI
• ~
t
Fig. 4.33: The c o n t a c t b e t w e e n the incident wave front and the circle of curvature d e t e r m i n e s the possible number of r e c e i v e d reflections: In the c a s e of a parabolic r e f l e c t o r , multiple r e f l e c t i o n s result only if the curvature of the incident wave front is larger than the local curvature of the r e f l e c t o r , i.e. if the shotpoint is located inside the ' c a u s t i c ' of normals. The usual situation is a fold point on the c a u s t i c (b); a cusp point appears only at a local e x t r e m u m of curvature.
if the c u r v a t u r e d e c r e a s e s , as one d e p a r t s from the minimum. This causes a d i f f e r e n t type of c o n t a c t b e t w e e n the circle of c u r v a t u r e and the r e f l e c t o r : The r e f l e c t o r is totally bound to the convex side of the circle of curvature, a situation which cannot arise in the case of a locally 'parabolic r e f l e c t o r ' . An appropriate way to study both situations simultaneously is to consider a p e r f e c t circular
arc,
and to transform it by a simple
affine t r a n s f o r m a t i o n
{ ,)= r[10
co ,1
which takes the circle into an ellipse. Fig. 4.34 illustrates how the r e f l e c t o r e l e m e n t , its c o n t a c t with the circle of curvature and the caustic are a l t e r e d by a smooth change of the p a r a m e t e r 'e': In t h e c a s e 0 < e < 1 the ellipse has a local minimum of curvature. The circle of curvature
is bounded to the c o n c a v e side of the r e f l e c t o r , which, t h e r e f o r e , can
be approximated by a parabola, and the previous discussion can be applied. For
e=0,
point
the a
r e f l e c t o r is a p e r f e c t
singularity
with
circular
arc. All rays pass through a single
indefinite codimensions. This situation
is structurally
unstable, as any small disturbance t r a n s f o r m s the singular point into a caustic. If e >1,
the
circle of curvature
and a new p a t t e r n
arises.
caustic of a cycloid (Fig.
is located on the convex side of the r e f l e c t o r ,
However, 4.18);
but
the caustic is again cuspoid, similar to the the cusp points into the opposite direction
than in the case of a parabolic r e f l e c t o r .
I85
Fig. 4.34: R a y s a n d w a v e f r o n t s f r o m an e l l i p t i c r e f l e c t o r , a: 0 < e < t , b: e=0, c: e > 1. S e e t e x t for d i s c u s s i o n .
T h e t y p e o f c a u s t i c t h u s d e p e n d s on t h e t y p e o f c o n t a c t and
the
viewpoint
reflector.
The
ellipse
still
provides
a rather
between
special
and c l a s s i f i c a t i o n c a n be d e r i v e d if t h e a r g u m e n t s
t h e c i r c l e of c u r v a t u r e
example.
A more
general
o f s e c t i o n 4.2.2 a r e a p p l i e d
to m o r e g e n e r a l c u r v e s .
Locally, dimensional
the
circle
curve.
of curvature
Choosing
provides
its c e n t e r
as
the
a rather origin
of
good a p p r o x i m a t i o n a polar
of a two
coordinate
system
we
c a n d e s c r i b e t h e r e f l e c t o r by an e q u a t i o n r
where
R is t h e
from
the perfect
local
= R + f(e),
r a d i u s of c u r v a t u r e
circular
arc
(of. Fig.
(4.45}
and f(0} d e s c r i b e s
the deviation
of the curve
4.35). T h e q u e s t i o n is w h a t we c a n i n f e r a b o u t
t h e f u n c t i o n f(0). T h e r a d i u s o f c u r v a t u r e in p o l a r c o o r d i n a t e s is g i v e n by
186
d Fig. 4.35: The t h r e e possible and its circle of curvature.
contacts
between
a
twodimensional r e f l e c t o r
R = (r2 + r'2)3/2 r 2+2r'2rr"
At O =0 the r e f l e c t o r has curvature
(4.46)
R, and this is the case if f{0) satisfies the t h r e e
conditions f(0)=0, f'(0)=0, and f"(0)=0 as can easily be verified from the standard equation (4.46).
If we use a power series to approximate fie), then this series cannot
involve
powers less than three, i.e. we need at least a function frO)= fla+...+higher terms. Such functions, of course, are really flat at the origin, their curvature vanishes at
However,
8 =0.
f(fl)= ( 3 is an odd function, and if we insert it into equation (4.45), it
b e c o m e s clear that the circle of curvature i n t e r s e c t s the r e f l e c t o r in some neighborhood of
e =0; the local r e f l e c t o r model is a 'spiral arc' with monotonously increasing (decrea
sing) curvature in a sufficiently small neighborhood of of the leading t e r m
(f(e)=+e a)
e =0 (Fig.
4.35a).
A sign change
simply r e f l e c t s the intersection p a t t e r n at the ray
fl =0;
the p a t t e r n , however, does not change. The term.
situation
Then
the
becomes different
if
the
power
series
r e f l e c t o r deviates s y m m e t r i c a l l y from
the
starts
with
a
fourth order
circle of curvature,
and a
sign change of the leading fourth order term changes the type of c o n t a c t : For +8 4 the c i r c l e of curvature
is e n t i r e l y on
the concave side of the r e f l e c t o r while for 0 4 it
is on the convex side (Fig. 4.35). The
two a l t e r n a t i v e power series with leading t e r m s of order t h r e e or four are
really distinct and exclude one another, as now will be shown. A local r e f l e c t o r approximation involving both t e r m s could always be brought to the form f(8)
=
O3 +
ae 4 +
...
+
higher
terms.
(4.47)
187 However, by a redefinition of the zero angle ( 8 
[
8  ~  a ) , e q u a t i o n (4.47) can be trans
formed into
04/(4a) tn
3 2 ~0 + 2a20 + (a4a3).

(4.48)
equation (4.48) the radius of curvature is given by (R+c), and f~) has again to satisfy
f(0)=f'(0)=f"(0)=0, i.e. 1 3 a0 
302
3e2
6(? . 0 .

+ 2a 2 = 0 (4.49)
a
These two equations, however, are usually not zero, and the function f(O) is dominated by the first and second order t e r m s with nonvanishing first and second order derivatives and, t h e r e f o r e , does not satisfy the requested approximation.
Therefore,
our
problem
is,
locally,
strongly equivalent
to
a power
series
which
s t a r t s e i t h e r with a third or a fourth order term, and c a t a s t r o p h e theory implies a fold or cusp c a t a s t r o p h e . The critical point in our problem is the point r=R, the c e n t e r of the
circle of curvature
which, of course, is a point on the evolute of the rays,
i.e.
a point on the caustic. Sufficiently close to 0 =0, the radii of our polar coordinate system coincide with the rays. The t r a n s f o r m a t i o n p =rR maps the r e f l e c t o r {the wave front} to the critical point. Near this point, we take the r e f l e c t o r as f{0)=04 or more conveniently, we use the unfolding
0 =
+04/4
+
u02/2
We cannot choose u and v freely because to
+
v@.
(4.50)
f(O) has to satisfy f'(O)=f"(O)=O. This leads
the s e t o f equations v = ¥0 3 
uO
and
(4.51) u = $3@ 2
If we solve for u and v in t e r m s of
fl and insert this in equation (4.50), this equation
simplifies to a fourth order t e r m as required. However, if we use u and v as local o r t h o gonal coordinates, then we can eliminate fl u (g)
3
v = ~(~)
2
and find one of the dual cusps
(4.52)
188
i.e.
the c a u s t i c we e x p e c t .
In a spatial i n t e r p r e t a t i o n ,
s e c o n d d e r i v a t i v e of t h e f u n c t i o n p .
u is t h e (negative) first, v t h e
I n t e r p r e t e d as v e c t o r s , t h e y provide a local o r t h o 
gonal f r a m e and c a p t u r e q u a l i t a t i v e l y t h e dislocation of r a y s close to t h e c r i t i c a l point r=R.
Similar
arguments can
be
applied
to
the
case
f(9)= 0 3, the
critical
points
are
fold points.
If we cusp
restrict
points on
our a t t e n t i o n
a caustic.
Their
to local occurrence
structures,
there
is not m o r e t h a n fold and
is a f u n c t i o n of t h e c o n t a c t
between
the
r e f l e c t o r and its local c i r c l e of c u r v a t u r e , as i l l u s t r a t e d in Fig. 4.35. In t e r m s of r e f l e c tion p a t t e r n s , however, 'local t is r a t h e r r e l a t i v e . In this c o n t e x t , a fold point is a point where
two
rays
intersect;
however,
this is only the c a s e on t h e c a u s t i c itself. In the
4.34),
which is not r e l a t e d to a singular point on t h e r e 
i n t e r i o r of a c a u s t i c (cf. Fig.
4.32),
f l e c t o r {e.g. Fig.
we find t h a t t h r e e r a y s i n t e r s e c t at e v e r y point. Thus, fold p o i n t s
a r e not s u c h i m p o r t a n t f r o m a less tocal viewpoint. I m p o r t a n t , h o w e v e r , is t h e d i f f e r e n c e b e t w e e n t h e dual cusps b e c a u s e t h e y provide an e s s e n t i a l s o u r c e for t h e s e i s m i c i n t e r pretation.
In at are
the all
one
sense
the
x  a x i s which related
a reflector
dual
result
patterns.
cusps are from
it
only
shows t h a t
the the
wave propagation
different,
they
the
are
simply dual
leading power
terms,
reflections and t h u s
This is obvious b e c a u s e any w a v e front c a n be c o n s i d e r e d as
and vice v e r s a  
here
r a y s p r e s e r v i n g angles. The w a v e identical,
not
a sign c h a n g e of
direction
of
a wave front
front
p a t t e r n s of t h e
propagation
and t h e
two
along t h e
dual cusps, t h e r e f o r e ,
are
is inversed. This is a nice r e s u l t b e c a u s e
previous discussion holds also is inversed;
is a m a p of t h e r e f l e c t o r
earlier
for t h e dual c u s p if t h e d i r e c t i o n of
discussion provides really
a c a t a l o g u e of
t h e e s s e n t i a l r e f l e c t i o n p a t t e r n s as far as a linear a p p r o a c h is s u f f i c i e n t .
On t h e o t h e r band, t h e r e r e m a i n s a d i f f e r e n c e b e t w e e n the dual cusps. In the c a s e of
a
locally
parabolic
reflector, t h e
wave
fronts
are
sections
through the
swallowtail
w i t h its c u s p s and s e l f i n t e r s e c t i o n s , and t h e t r a v e l t i m e r e c o r d s in t h e c r i t i c a l c a s e are
F i ~ 4.36: W a v e f r o n t s of t h e dual cusps.
189
'hyperbolic r e f l e c t i o n s ' , cusp,
the r e f l e c t o r
caustic;
again with cusps and s e l f i n t e r s e c t i o n s .
is an elliptic arc
In the case of the dual
which is bounded to the interior of the cuspoid
as soon as the image passes through the cusp point, the wave
'parabolic' appearance,
fronts have a
and thus has t h e t r a v e l t i m e record; cusp points and s e l f i n t e r s e c 
tions then are missing. A typical p a t t e r n , which commonly arises, is a series of parabolae which a l t e r n a t i v e l y
correspond
traveltime
but
record,
to synclines and anticlines,
without
cusp
points.
In the
case
and which i n t e r s e c t the
track
on t h e
line sections
the
caustic, swallowtail p a t t e r n s may arise, but they are i n v e r t e d with respect to the p a t t e r n s arising from a 'parabolic r e f l e c t o r ' (Fig. 4.36).
In summary, the various r e f l e c t i o n p a t t e r n s , in t e r m s of t h e c o n t a c t b e t w e e n t h e r e f l e c t o r , wave
front
available
(distance
for
from
the
source).
a qualitatively c o r r e c t
which may arise, are well classifiable its circle of c u r v a t u r e and the incident
Usually
there
interpretaion.
should
be
enough
The linear ray model,
information of course,
is
only a first approximation, but the principal relationships remain stable even if the sonic velocity of the medium is not a c o n s t a n t .
B) T h r e e  D i m e n s i o n a l P a t t e r n s  The Double Cusp
At least,
a few r e m a r k s shall be made
in what r e s p e c t the simplified model of
linear rays and especially of a twodimensional r e f l e c t o r line gives insight into a larger class of images which may result from c o m p l i c a t e d r e f l e c t o r topologies. The twodimensional
approach
generally,
to
extends
parabolic
without surface
difficulties points.
Fig.
to 4.37
cylindrical gives
surface
elements
two e x a m p l e s  
or,
more
a cylindrical
and a conical surface  t h a t show how the caustic and a single wave front are located over the surface. In such cases, the t r a v e l t i m e record will depend on the r e l a t i o n
be
t w e e n t h e axis of the syncline and the t r a c k line  one may find 'hyperbolic r e f l e c t i o n s ' , onlapping p a t t e r n s ,
doubted or
tripled r e f l e c t i o n s
(Fig. 4.38). Thus, an irregular topo
graphy, which impresses its s t r u c t u r e onto the wavefield, can cause nice multiple r e f l e c tion p a t t e r n s
which
look like p e r f e c t l y
s t r a t i f i e d sediments;
and,
therefore,
one may
ask how much onlapping f e a t u r e s in Fig. 4.15 are real, and which ones are due to the rough
topography of the
considers
farfield
effects
basement. or
more
The complexity complicated
of these e f f e c t s
surface
elements
increases if one
like
hyperbolic
and
elliptic surface points. R e p r e s e n t i n g the surface near (Xo,Yo,Zo) by z=f(x,y) the evolution equation for the wave fronts becomes
{(U,V,w) E (x,y,z) I ((xu)2 + (YV)2 + (wf(x'y))2)I/2 = r = const. }.
(4.53)
In the case of a parabolic or hyperbolic surface point, the family of rays is given by the (vector) e q u a t i o n
190
Fig. 4.37: The caustic and a single wave front over a cylindrical (above) and conical (below) s u r f a c e .
r = (x, y, x 2
±
ay 2) + k(2x, +2ay, i),
(4.54)
and a point on the track line may be given as (Xo,Yo,Zo). To see, which surface points map onto the track line, one has to solve the equation
(Xo' YO' Zo) = (x, y, x 2
+
ay 2) + ),(2x, +2ay, i).
(4.55)
Let the track line be located at Zo, then by elimination of the parameter ), , one finds the relationship
2 = x 2 + ay
 z0
x 0 = (12Zo)X + 2x 3 + 2ay2x YO = (1 ~ 2z 0 + 2ay 3 + 2x2y,
(4.56)
191
Fig. 4.38: Sketch of the t r a v e l t i m e record of a cylindric syncline with track line sections parallel and oblique to the syncline axis.
a map which is a special d e g e n e r a t e d case of the double cusp c a t a s t r o p h e the standard umbilic c a t a s t r o p h e s . The caustic patterns, which result
including
from the double
cusp, are rather complicated. A full discussion of threedimensional phenomena is above the scope of this discussion; however, a detailed study in t e r m s of standard c a t a s t r o p h e s was given by DANGELMAYER & GUTTINGER
4.2.9
(1983).
Distributed R e c e i v e r s
Seismic shooting rarely resembles the idealized situation that source and r e c e i v e r are at the same place. However, as turned out during the previous discussion, the results found
from
rather
idealized
assumptions hold
for
a
much wider class of 'disturbed'
problems. It will be shown here that the principal results still hold if source and receiver are at d i f f e r e n t places, or if a chain of receivers is used. In the l a t t e r case, not a single r e f l e c t i o n
but the r e f l e c t e d and d e f o r m e d wavelet is recorded. What we
shall do here is, t h e r e f o r e , to study how the r e f l e c t e d wavelet deforms.
The r e f l e c t i o n of a wavelet is governed by Snell's law of equal angles, i.e. the angle an incident ray forms with the normal of the r e f l e c t i n g surface is the same as
192
the angle the r e f l e c t e d ray forms with t h e s a m e normal. A convenient way, t h e r e f o r e , is to view the incident and the r e f l e c t e d rays in t e r m s of the r e f l e c t o r . Let the r e f l e c tot be given in t e r m s of its local curvature, i.e. with the c e n t e r of the global coordin a t e system at the c e n t e r of its local circle of curvature (cf. equ. 4.45): Locally the r e f l e c t o r can be w r i t t e n
(4.57)
and the normal rays are
(4.58)
Yn
=
r (sin 0 +
X
r
Lsin
+
r ( cos
Now, in a plane problem we can express the incident rays in local c o o r d i n a t e s by means of the t a n g e n t (t) and normal (n) v e c t o r s at the surface:
r.
= r
+ X(an
+ gt),
(4.59)
1
and the r e f l e c t e d rays are simply the r e f l e c t i o n s of incident rays at the normals
r
The
coefficients 'a'
source,
e.g.
and
'b'
= r
r
can
be
+ X(ccn  B t ) .
determined
to
(4.60)
satisfy special conditions of the
in the c a s e of a point source, equation (4.59) leads to a pair of linear
equations from which the c o e f f i c i e n t s can be d e t e r m i n e d . A very simple system arises if the r e f l e c t o r is locally a p e r f e c t circular arc. The equations for the incident and r e f l e c t e d rays then simplify to the pair of equations
cos
[sin (4.61)
[~]
The condition that

[c°s
o001.
Ab [sin
the incident rays originate from a point source requires that t h e s e
rays pass through the source point for some value of I . we can choose the value
k=l,
d e t e r m i n e d from the linear equations
(ra)cose  bsinO = x 0 (ra)sin@ + bcose = YO to be
Without loss of generality,
and the values for the p a r a m e t e r s 'a' and 'b' can be
193
a = r  (YosinO b YoCOSO
Because
of
+ xncosO)  x~sinO
(4.62)
the s y m m e t r y of the circular arc a simple r o t a t i o n allows to locate
the
source formally at (Xo,0) so t h a t the previous equations simplify further. If one inserts 'a'
and 'b' from equation {4.62) into equations {4.61), one finds a simplified equation
for the incident rays
,xi] =
r
+
(1X
)`
(4.63}
sin and the r e f l e c t e d rays are
(Xrl [ fc°81 II )`x r cos2e
=
Yr
As previously,
( 1  ) , ) [sin @
r
the c a u s t i c
+
of
the
reflected
ray
(4.64)
0 sin20
system
is of special
interest.
If we
consider e q u a t i o n (4.61) as a map, the caustic is equivalent to its singular set, which can be d e t e r m i n e d from the condition t h a t the Jacobian of the map vanishes, i.e. t h a t
J
From
this condition
=
I
xe
xk I
YO
YX
and equations
=
xey ~  xky ~ = O.
(4.61) and (4.64) we d e t e r m i n e t h e c r i t i c a l set
in
t e r m s of )`:
=
a 2(a2+b2)_a
If we insert these values for
= r
2 ....... x° 
Yr
which
xocose
complicated.
c o m p u t e the values ( l  X ) and
(I~)
if y o = O .
(4.65)
X into equation (4.64), we find an equation for the c a u s t i c
l+2x23x
looks r a t h e r
I  xocose l+2x2_3xocos 8
=
xo 
cose{sineJ
However,
XoCOSe2
l+2x2axocosO
a simple observation
)`Xoat e=0:
2  2 Xo  xo i+2xo2
;
kxo
2 Xo  x~ =  I +2x 2
(4.66) ~.sin2ejJ
is important.
Let
us
194
F_ig. 4.39: The cardioid caustics of a circular r e f l e c t o r and their relation to point sources.
Locally, near
0=0,
we have the simple relationship 2(1%)= kxo, and this means that
near this special point the caustic behaves like a cardioid independent of the complexity of our original equation. The cardioid, however, has a cusp point at
O =0, and this
is a standard cusp point, as can easily be shown by developing the equations x = r(2cos8
 cos2e);
in Taylor series near the critical point x n, 1 +
82;
y = r(2sin9
 sin2e)
>
= (.~. y)2
e =0:
y ~ nl~0e3
(x_l)3
What we now can do with the source point, is to dislocate it along the x  c o o r d i n a t e
{Fig. 4.39).
Clearly,
a critical
situation
c e n t e r of the c i r c l e of curvature.
arises if the
source is located at
{0,0), the
In this case, all rays pass through the origin, the
caustic d e g e n e r a t e s to a singular point, and we would not r e c e i v e any r e f l e c t i o n s at points besides this d e g e n e r a t e d singularity.
If 0 R, we find t h a t the caustic has formally two cusp points if we consider not simply a circular arc and
but a full circle. These cusp points are given by
0=0
8 =~r, tn addition, we observe t h a t t h e s e cusp points are simple inversions of the
corresponding source locations x o ~  x o .
A somewhat striking point is that
we always
have the same type of a cusp (what we called the dual cusp of the r e f l e c t i o n problem) independent of the radius of the incident wavelet. The caustic p a t t e r n , t h e r e f o r e , does not depend on the c o n t a c t b e t w e e n the r e f l e c t o r and the (circular) incident wavelet. Another special situation occurs if point,
the
other
one d e g e n e r a t e s into a
tXoi =R. In this case, we have only one cusp fold point with
its tangent
coincident with
195
t h e t a n g e n t of t h e c i r c u l a r r e f l e c t o r  
t h e c a u s t i c b e c o m e s a p e r f e c t cardioid, in the
c a s e the s o u r c e point is l o c a t e d outside the circle,
t h e r e r e m a i n s only one cusp point,
but in addition we find two c r i t i c a l fold points where the d e f o r m e d cardioid has t a n g e n tial
contact
with
b i f u r c a t i o n point.
the
circular
However,
reflector.
if X o ~ m ,
In s o m e sense,
the
s i t u a t i o n Xo=l d e f i n e s a
we find again a s y m m e t r i c solution, t h e c a u s t i c
is now a nephroid (cf. POSTON & STEWART, 1978} w h e r e b y t h e s y m m e t r y
refers
to
the two s o u r c e s Xo=+~.
The c a u s t i c p h e n o m e n a a s s o c i a t e d with a point s o u r c e and a p e r f e c t c i r c u l a r r e f l e c tor,
therefore,
c a n be s u m m a r i z e d as continuous d e f o r m a t i o n s of a cardioid. The s t a b l e
p a t t e r n is t h e cusp point of the cardioid, which locally r e m a i n s the identical cusp c a u s tic and
i n d e p e n d e n t of t h e location of the source. Now, t h e c i r c u l a r r e f l e c t o r the
question a r i s e s
what
happens if it
is d e f o r m e d .
is u n s t a b l e ,
Before going in details,
we
Fig. 4.40: The virtual s o u r c e s of a planar and c i r c u l a r r e f l e c t o r . The wave f r o n t s provide v i r t u a l r e f l e c t o r s . first o b s e r v e t h a t wavelet
are
there
normals.
e x i s t s a virtual
In t h e
case
of
surface,
a plane
for which the r e f l e c t e d rays of t h e
reflector,
this virtual
s u r f a c e is again
a point source, a s t a n d a r d e x a m p l e in s e i s m o l o g y (Fig. 4.40). tn a m o r e g e n e r a l s e n s e , e v e r y wave front is a p o t e n t i a l l y virtuaI r e f l e c t o r s u r f a c e b e c a u s e t h e wave f r o n t s i n t e r s e c t t h e rays orthogonatty. In the c a s e of linear rays, t h e wave fronts are found from the n o r m a l i z e d e q u a t i o n (4.61), i.e. from
196
[Xr} IcosO Yr
The
= r [sinej
traveltime
(a2+b2) I/2
is e q u i v a l e n t
rays. If t h e r e c e i v e r s a r e
[afc°sO r s,n011 [sinOj
to t h e s u m of t h e
b [ c o s O J .
length of t h e
(4.67)
incident and r e f l e c t e d
on t h e s a m e xlevel as t h e s o u r c e , , t h e t r a v e l t i m e is given
by
2t = (a2+b2)I/2(l 
and t h e
identical
t r a v e l t i m e record
r e c e i v e r coincide, e i t h e r
from
x__~o_ r cos 8 ~ bsinO )'
would be r e c e i v e d
a virtual
(4.68)
acos
in a s y s t e m w h e r e s o u r c e
and
s o u r c e or a virtual r e f l e c t o r which, of course,
is simply a w a v e front (Fig. 4.40).Now, we can use t h e discussion of t h e last s e c t i o n . A critical deformation the
s i t u a t i o n a r i s e s if the v i r t u a l r e f l e c t i n g s u r f a c e b e c o m e s a circle. Any small then
deforms
i t , and
the
singular c a u s t i c
dual cusps. We c o n s i d e r this d e g e n e r a t e d
point
e v o l v e s in e i t h e r
s i t u a t i o n and d i s t u r b t h e
one of
reflected
rays
(the normals) by a not n e c e s s a r i l y c o n s t a n t r o t a t i o n
[XrJ = riO@sO} Yr
It°s01 +
[ainOJ
[sinSJ
[ coseJ
(4.69)
where
f C O S e (0) A = [sins(e)
sins (8)] cose(O)J.
(4.70)
The c r i t i c a l s e t c a n again be found from the J a c o b i a n to be
X = (p2+p'2)e°se
{l+c~, } p2 {2+e,) p, 2_pp,,
(4.70)
The d e f o r m a t i o n of t h e original ray s y s t e m , t h e r e f o r e , c o n s i s t s of a r o t a t i o n as defined by the
matrix
t h e c a s e cos e
' A ' and a dislocation along t h e rays which is proportional
to cos ~ .
In
v a n i s h e s , t h e r e f l e c t o r b e c o m e s identical with its c a u s t i c , and t h e i m a g e
is i n v e r s e d , as cos e a s s u m e s n e g a t i v e values. However, this would r e q u i r e r a t h e r s t r o n g d e f o r m a t i o n s . We c o n c l u d e ,
therefore,
that
the caustic pattern
f o r m e d by t h e n o r m a l s
r e m a i n s s t a b l e as long as t h e d e f o r m a t i o n s a r e of r e a s o n a b l e order.
We n o t e finally t h a t we c a n r e f o r m u l a t e cos c~ A as
1 [l+cos2e (cose)A = ~_ [sin2e
sin2e 1 l+cos2~J .
The caustic formed by the rotated norrnals can now be written
(4.71)
197
Fig. 4.41: Normal and r e f l e c t e d ray s y s t e m and c a u s t i c s at a parabolic and h y p e r bolic r e f l e c t o r . A point source does not c h a n g e t h e c a u s t i c p a t t e r n .
Fi G. 4.42: R o t a t e d n o r m a l s of a circular r e f l e c t o r . The singular point is t r a n s f o r m e d into a c a u s t i c . The r o t a t e d rays can be c o n s t r u c t e d as a v e r a g e of t h e n o r m a l s and rays r o t a t e d twice t h e original angle (but which still h a v e the length of the normals).
198
rr
=
r
The resulting p a t t e r n Fig.
4.42
reflector. circle

f(r,r')
~
[sin2u
cos2~jn
(4.72)
is the a v e r a g e of two v e c t o r fields which differ by a rotation.
elucidates
this point and i l l u s t r a t e s once
Even a c o n s t a n t
into a c i r c u l a r
more
the instability of a c i r c u l a r
r o t a t i o n of the normals deforms the singular point of the
fold line. Fig. 4.42 elucidates
in addition t h a t only a r o t a t i o n
with angles larger than ~r/2 can really change the c a u s t i c p a t t e r n , as can also be i n f e r r e d from
equation
{4.73). Thus,
normals of a s u r f a c e the
rotation
we can
remains stable
of the normal at
this r e f l e c t o r .
finally conclude
that
the c a u s t i c
pattern
even if r e f l e c t e d w a v e l e t s are r e c e i v e d
a circular reflector
is equivalent
of the because
to a d e f o r m a t i o n of
This point is especially elucidated by equation (4.73), which s t a t e s t h a t
a monotonous d e f o r m a t i o n of t h e r e f l e c t i o n angle within reasonable bounds c a n n o t really change
the
original
caustic
and
the
related
patterns.
Therefore,
the c a u s t i c
formed
by the normal rays must be a c c e p t e d as a s t r u c t u r a l l y s t a b l e p a t t e r n , even under reasonable disturbances.
4.3 "PARALLEL SYSTEMS" IN GEOLOGY
Structural
geology describes and analyzes the "geometry"
of deformed
rocks. The
procedure is mainly geometrical, and the relations to the physical processes are established by "classification procedures" ( G Z O V S K Y
et al., 1973). The base for these relation
ships is developed from various physical, e x p e r i m e n t a l and n u m e r i c a l methods for which a wide v a r i e t y al.,
of m a t h e m a t i c a l
1971; JOHNSON
1971; DIETRICH,
methods has been used (BAYLY,
& POLLARD,
1970; FLETCHER,
1974; MATTHEWS e t
1973; BEHZADI & DUBEY, 1980; COBBOLD e t al., 1979; SMITH,
1975 to give a few examples). Most
of t h e geological s t r u c t u r e s are the result of complex s t r a i n fields. These s t r a i n fields, in general, are not the result of similar complex global fields of forces, but the complex and
inhomogeneous
behavior of rocks,
strain i.e.
field
from
results
their
from
primary
the
v a r i a b l e elastic,
plastic,
and viscous
inhomogeneities. The d e f o r m a t i o n s of rocks
can be very large, and then they are outside of the scope of classical d i f f e r e n t i a l calculus. This is especially occur,
true
if transitions
from
elastic
if the boundary conditions are not k n o w n  
dislocations of m a t e r i a l
by solution and r e c r y s t a l l i z a t i o n
t h e d e f o r m a t i o n process (STEPHANSON,
to plastic and viscous behavior
in general
they are n o t  
and if
play an i m p o r t a n t role during
1974; TRURNIT & AMSTUTZ, 1979). A classical
approach to study d e f o r m a t i o n s of rocks, t h e r e f o r e , is the g e o m e t r i c a l analysis. A c o m m o n way is to apply the
methods of finite strain analysis {RAMSAY, 1967; JAEGER,
1969;
HOBBS, 1971) to regions for which a nearly homogeneous s t r a i n c a n be assumed. Basically, this type of analysis is the study of some special mappings, and some of them will be briefly discussed here.
199
4.3.1 Some Examples of Parallel Systems
Much of the previous discussion focussed on systems of quasiparallel layers, which posses a formal g e o m e t r i c a l similarity with d e f o r m e d s t r u c t u r e s in geology. Considering a t h r e e  d i m e n s i o n a l space such a parallel system can be w r i t t e n F(u,v;t) = x(u,v) + tN(u,v)
(4.73)
where N(u,v) = (Nx,Ny,Nz) , the unit normal v e c t o r at x(u,v).
If F(u,v;0) is e v e r y w h e r e differentiable, then the Jakobian d e t e r m i n a n t of such a system is nowhere zero (DoCARMO, 1976):
det J(F)=
= I ( F u)
(F v)
(F t) l = ]XuA x v ]
~ 0
(4.74)
where F u etc. are column vectors of the Jacobian m a t r i x (see section 4.3.4 for details). Equation (4.74) shows t h a t t h e r e exists a tubular neighborhood to t h e surface x(u,v) which is uniquely defined. Given a solution for a surface x(u,v) under c e r t a i n conditions, we can e x t e n d this solution into a small but finite neighborhood of x(u,v). In a conceptual sense
this secures
that
the solutions can be applied to a real physical system
where
a surface is always of finite thickness. Assume equation (4.73) is applicable as a linear first
order
approximation,
then we i m m e d i a t e l y
get an e s t i m a t e
of the maximal
local
e x t e n t of the tubular neighborhood, i.e. the area into which we may extend the solution. This area is hounded by the 'focal s u r f a c e s '
Xl(U,V) = x(u,v) + p "IN(U,V)
(4.75)
x2(u,v) = x(u,v) + 92N(u,v) where
0 1' P2
are
the
principal
curvatures
of the s u r f a c e
(cf. DoCARMO,
1976; GUGGENHEIMER, 1977).
The
assumption
of
parallel
folds in t e c t o n i c s (e.g. HILLS, surface point,
measurements obvious
from
depth~ as several
layers
Fig. 4.43,
a
long tradition
in the r e c o n s t r u c t i o n
of
1963 for an overview). The r e c o n s t r u c t i o n of folds from
is i l l u s t r a t e d
segments
has
in Fig.
is t h a t
the
vanish along the
4.43
after
fold c a n n o t
an example
of GILL (1953). A
be e x t e n d e d
continuously into
~caustic of the normal rays' as discussed
in the previous section; and it has been assumed (e.g. BUSK, 1956) t h a t t h e s e ' l i n e s (or sufaces) evolve into faults. Of however,
rather
course, equation (4.74) is only a first order approximationl
similar a r g u m e n t s
hold for t h e
which can be w r i t t e n (DoCARMO, 1976)
'normal
v a r i a t i o n ' of a s u r f a c e x{u,v)
200
~
Reconstruction
of parallel
folds from s u r f a c e data.
Modified a f t e r GILL
(4.76)
F(u,v;t) = x(u,v) + th(u,v)N(u,v) w h e r e h(u,v) is some s c a l a r variable.
Such a f o r m u l a t i o n provides us with t h e possibility to a d a p t some conditions which have to be satisfied by h(u,v), and equation (4.76) can be considered as a variational problem, or we may consider equation (4.76) as t h e disturbed linear problem described by equation (4.73)• P a r a l l e l s y s t e m s are e n c o u n t e r e d in various areas. With r e s p e c t to geology~ an import a n t one is the c o n c e p t of sliplines in the theory of p e r f e c t plasticity, which is closely related
to e v o l u t e s and involutes as s t a t e d
by Hencky~s and P r a n d t l t s t h e o r e m s (LING,
1973): HENCKY's
theorem:
The
angle
formed
by t h e t a n g e n t s of two fixed shear lines
of one family at t h e i r points of i n t e r s e c t i o n with a shear line of the second family does not depend on t h e c h o i c e of t h e i n t e r s e c t i n g shear line of t h e second family. PRANDTL~s t h e o r e m : Along a fixed s h e a r line of one family, t h e c e n t e r s of curvature of t h e s h e a r lines of t h e o t h e r
family form an involufe of t h e fixed shear
line. Given one nonlinear shear a
first
approximation
intersecting straight
for
line, the tlinear system t of normals and involutes provides the
slipline
field.
The most
simple cases
are
orthogonally
lines and t c e n t e r e d fans ' of c i r c u l a r arcs~ which provide reasonable
first order approximations of plastic d e f o r m a t i o n (e.g. LING, 1973). Fig• 4.44 i l l u s t r a t e s P r a n d t P s solution for sliplines below a s t r i p load. A more general soIution consists of
20I
Fig. 4.44: P r a n d t l ' s solution for sliplines below a strip load above a homogeneous hal fspace.
centered
arcs
of
logarithmic
as discussed in section spiral x(u,v),
then
spirals:
Consider
x(u,v)
a generalized
logarithmic
spiral,
3.5.3, and h(u,v) to be proportional to the c u r v a t u r e of the leading
equation
(4.76} describes a family of possible solutions,
from which
we have to choose the locally valid one which then can be e x t e n d e d to neighboring areas by c o n n e c t i n g local solutions along the s t r a i g h t c h a r a c t e r i s t i c s .
ODE
(1960) applied
the
slipline
theory
to
the
f o r m a t i o n of faults in sand and
clay under t h e conditions of plane strain. By a similar a t t e m p t , also more geometrically, FREUND analysis
(1974) studied the
curva'ture
the of
t e r m i n a t i o n of t r a n s c u r r e n t
transcurrent
faults
can
faults by splaying;
be r e l a t e d
to t h e
from
formation
his
of an
evolute of a fan of faults. Evolutes, as lines {or surfaces) of discontinuity, occur f u r t h e r under unidirectional glide in solid crystals (e.g. KLEMANN,
1983).
4.3.2 Similar and Parallel Folds Concerning geologically 'shallow' d e f o r m a t i o n s (without phase transitions} HOEPPENER (1978) found from e x p e r i m e n t s t h a t most folds c a n be t r a c e d back to t h e following types: 1) similar folds 2) parallel folds a) c o n c e n t r i c folds b) box folds. Parallel folds occur usually near t h e free surface or near shear planes while elsewhere
the
more
energy consuming similar
folds develop.
The
differences between
the
202
=
A Fig. 4.45: A) Ideal (chevron folds},
parallel
folds (kinks or box folds) and B) ideal
similar
folds
two types of folds are s c h e m a t i c a l l y illustrated in Fig. 4.45. Parallel folds are of finite depth range,
i.e. they resemble the parallel wave fields discussed in the last section.
Similar folds in c o n t r a r y continue (ideally) infinitely. The strain in folded layered s y s t e m s has e x t e n s i v e l y be studied by HOBBS (1971), h e r e we consider only volume preserving systems. Similar folds with c o n s t a n t divergence are described by maps of the form X = ax
+
f(y,z)
Y = by
+
h(z)
{4.77)
Z = cz w h e r e a,b,c: constants; f,g,h: arbitrary functions. The Jacobian d e t e r m i n a n t
Net J=
3 fi
axTI =abc
is c o n s t a n t and by choosing a,b,c in ratios such that abc=l, the d e f o r m a t i o n described by equation (4.77) is volume conserving, locally and globally. If we consider cylindrical folds, equation (4.77) reduces to a twodimensional system and describes a twodimensional dislocation field as illustrated in Fig. 4.46. A special p r o p e r t y of this case is that the
principal
strains are
identical along every s t r a i g h t
'shear line' of
the dislocation
field. As t h e s e parallel dislocation lines never i n t e r s e c t , the fold e x t e n d s ideatly into infinite depth, and laterally the local fold p a t t e r n can easily be continued if we c o n n e c t local solutions along a straight dislocation line (cf. Fig. 4.46c). Of course, along such lines the solution is discontinuous with r e s p e c t to
the curvature,
a discontinuity which
occurs in sinusoidal s y s t e m s at the inflection points. JOHNSON & ELLEN out
that
such
(1973) pointed
lines of discontinuities may be of some value in the analysis of folds
203
I
a
b 1
Fig. 4.46: D e f o r m a t i o n of a homogeneous halfspace (a) into similar folds (b,c). A local solution (c) can be e x t e n d e d laterally by c o n t i n u a t i o n along a slipline.
and c o m p a r e d them with ' c h a r a c t e r i s t i c s ' as they occur in the slipline theory of plasticity. The possibility to continue local solutions laterally is c o m m o n for both fold types. In the case *normal
of parallei
cylindrical
ray v as i l l u s t r a t e d
no solution
with c o n s t a n t
folds, a local solution can be continued along any
by Fig. '4.47. Jacobian,
However,
for a parallel
and thus it c a n n o t
describe
system
there
exists
a deformation
which
preserves volume locally. On the o t h e r hand, we have already seen in section 4.1.3 t h a t it is possible to connect deformed pieces in such a way t h a t the e n t i r e volume of the systems is not a l t e r e d (Fig. 4.47}. Concerning global volume changes, similar and parallel folds provide c o m p a r a b l e solutions. Parallel folds are best considered as l a m i n a t e d systems which allow the laminae to glide one above the o t h e r as illustrated in Fig. 4.47. Within
Fig. 4.47: Ideal (concentric) folds lateralIy continued along 'rays' or 'lines of discontinuity'. Heavy lines i n d i c a t e intervals of equal length for the layers, Black grid elements: d e f o r m e d 'volume e l e m e n t s ~ of originally r e c t a n g u l a r grid elements.
204
Fig, 4.48: Buckling of a card deck under lateral stress c o n f i r m e d by v e r t i c a l plates. Right: details of kink formation.
205
laminae the ideal model allows only for m e m b r a n e stresses, a situation s o m e t i m e s applicable
to
deformations
in
liquid
crystals
{KLEMAN,
1983). The d e f o r m a t i o n s b e t w e e n
subsequent layers then are proportional to the change of surface e l e m e n t s {rather than volume elements): let
F(u;t) = x(u) + zN(u)~
(4,78)
then ds/dt = (1zk) where k: the local c u r v a t u r e of the, leading curve,
and
the
We
find
deformation that
the
is simply proportional surface
to the
curvature
of
the
surface
element,
e l e m e n t s vanish along the evolute of the normals or a t
the focal surface, which w% t h e r e f o r e , can e x p e c t to be part of the shear surface s e p a r a t ing successive parallel folds.
The two fold types are both idealized systems, and e x p e r i m e n t a l l y transitions occur between
the
experiments the
'ray
two that
method'
types. the
JOHNSON
common
is of
folding under twoaxial
& HONEA
assumption
limited value.
Fig.
that 4.48
{1975) concluded one can e s t i m a t e illustrates some
stress. The transition from parallel
from
their
depth
of
multilayer folding by
phases of multilayer
to similar folds is again a
c o n t i n u a t i o n problem. Assume t h a t the solution is known a t the free surface of a half space and t h a t this solution is give n by a box fold: The range of the parallel fold solution is of limited depth, and it is bounded by a cuspoid focal line (Fig. 4.49). We are i n t e r e s t ed to e x t e n d the d i s t u r b a n c e into depth and require t h a t the fold lines are continuous along the focal line. To continue the disturbance we project the focal line into depth, i.e. assume the focal line is given by a function g(x,y) = 0, then we consider the family
Fig. 4.49: Continuation of parallel folds into similar folds by p r o p a g a t ing the cusp discontinuity into depth.
206
of functions g(x,y) = c.
(4.79)
The fold lines of the parallel system i n t e r s e c t the focal line perpendicular as was discussed in t e r m s of wave fronts. The e x t e n d e d fold lines, t h e r e f o r e , have also to i n t e r s e c t the original focal line by right angles. A possible continuation, t h e r e f o r e , are the orthogonal t r a j e c t o r i e s of the family g(x,y) = c which are found by solving the d i f f e r e n t i a l equation
gy
(4,80)
+ gxy' = O.
In the c a s e of a cuspoid focal line, x
2
(yc) 3 = 0 or y  x 2/3 = c, the orthogonal
t r a j e c t o r i e s are the family of functions y = (8/9)x 4/3 +c
(4.8t)
which clearly provide a set of similar folds (Fig. 4.49). Usually the solution will be bounded to a strip of finite length, however, the strip can be continued laterally as discussed previously, and if we consider a layer of finite thickness, this continuation can be adjusted to p r e s e r v e volume globally. Clearly, the discussed models are only first order approximations which, however, allow graphical analysis of even c o m p l i c a t e d large scale d e f o r m a tions and which c a p t u r e some essential qualitative p r o p e r t i e s of e x p e r i m e n t s .
4.3.3 Bending at Fold Hinges
The previous discussion focussed on systems composed of layers of vanishing thickness, or of negligible thickness with r e s p e c t to the e n t i r e system. Concerning a c o m p a c t layer of finite thickness one has to consider the deformations near the fold hinge, as schematicaIly illustrated in Fig. 4.50. The previously discussed linear approach of paraI
Fig. 4.50: Idealized parallel folds (kinks) composed of layers of d i f f e r e n t finite thickness.
207
lel
layer
state.
reveals
Bernoulli's
theorem,
i.e.
undeformed c r o s s  s e c t i o n s in the
deformed
A more realistic model provides St. Vernant~s solution for bending of a bar by
couples. The d e f o r m e d s t a t e is described by the map c X = x(1 + ~ z) Y = y(1  ~
(4.82)
z)
z + ~ (co ( y 2  z
Z = where
c:
2)  x 2)
strength
of
couples;
E: Young~s modulus,
o:
Poisson~s ratio
and
c / E = RI; R: radius of curvature (see e.g. BUDO, 1974; LOVE, 1944).
In engineering the usual procedure is to study the deformation of an object under specific stress configuration. In geology we usually know little about the original stress field. Therefore, it is worthwhile to work with models, and the question is not mainly how the object d e f o r m s within a certain stress field but how far the model is applicable. One question, which can be pushed forward by m a t h e m a t i c a l analysis, is how the various p a r a m e t e r s i n t e r a c t and whether the solution is bounded to some region, i.e. concerning the bending model we are i n t e r e s t e d if the thickness of the bar is unlimited.
The limits of the solution are given by the condition that the Jacobian of St. Vernant's
map vanishes; however, in this case we can
simplify the
analysis by reducing
the map to a standard c a t a s t r o p h e on Thomas list. If we slide the bar along the line y=0, i.e. by a vertical plane along the long axis (z), equation (4.82) simplifies to
Z = ~ E ( o z 2 + x 2) + z c X =Exz
(4.83)
+ x,
and by means of the t r a n s f o r m a t i o n o z 2 ~ z 2 this equation simplifies to the standard form
Z
*
X
*
= z
2
+ x
= 2xy
2
2E +c'o¢z
2E +x. c
The only assumptions involved are that d e f o r m e d states) and that by use of the
(4.84)
standard
section (E,c, =constant),
the couples ~c' do not vanish {we consider only
o ¢0. The singular set of this map is illustrated in Fig. 4.51 form of the hyperbolic umbilic.
Fig. 4.52 illustrates a single
and it becomes clear how the solution space is limited by a
cusp and a fold line, i.e. even for small deformations of this type the bar cannot exceed a c e r t a i n thickness. Fig. 4.52a ° illustrates the shape of the undeformed area,
i.e. the
boundaries defined by J=0 {J: Jacobian determinant). By s e t t i n g J=c one finds lines of
208
. ..
Fig. 4.51: The s t a n d a r d form i n d i c a t e the local tpotential~.
a
;i !.
•. ,'£ •
of Thomas hyperbolic umbilic.
~"
b
Isolines inside circles
i ........."
Fig. 4.52: a) The nonlocal section through a bent b a r along its long axis (see text). The c r i t i c a l s e t ( s e l f  i n t e r s e c t i o n s of parabolas) corresponds with the section through t h e hyperbolic umbilic. P a r a b o l a s indicate lines which are parallel in the u n d e f o r m e d s t a t e , a °) a s s o c i a t e d u n d e f o r m e d state: Only the blank a r e a can be d e f o r m e d to the image i n d i c a t e d in (a). b) The same bar with lines of c o n s t a n t values of the J a c o b i a n d e t e r m i n a n t , b °) the associated u n d e f o r m e d image.
209
Fig. 4.53: The 'hyperbolic umbilic' as a sheet of paper folded in its plane.
210
"equal
volume
change"
in
the
undeformed
state
(Fig.
4.52b °)
and
by
means
of
the
mapping (4.83 or 4.84) t h e i r image in the d e f o r m e d s t a t e (Fig. 4.52b). F u r t h e r p r o p e r t i e s will be analyzed
in a more general sense in the next sections. The reduction of the
original m a p to a twodimensional problem, clearly, system
reacts~
to r e I a t e
however,
t h e solution is c o r r e c t
gives only an idea how t h e e n t i r e
for
the
plane selected,
and it allows
the d e f o r m a t i o n to a r a t h e r simple e x p e r i m e n t {Fig. 4.53): A s h e e t of paper
' b e n d e d ' in its plane i l l u s t r a t e s in a r a t h e r simple way how t h e limiting fold line evolves. To c o n n e c t
this section with the
f u r t h e r analysis of parallel
systems we observe
t h a t equation (4.83) can a l t e r n a t i v e l y be w r i t t e n (using v e c t o r notation):
(4.85) =
y
e
where
the
term
first
term
+
~z
2
2 (oY x
is just
the
(~y

2E z
E/c
description of t h e
" n e u t r a l surface" and the second
is t h e n o n  n o r m a l i z e d normal of the surface e l e m e n t s . Thus, t h e first two t e r m s
on the right side describe t h e 'normal v a r i a t i o n ' of the surface, i.e. h(u,v) = I Xu A Xv{ in e q u a t i o n (4.76). The final t e r m on t h e right side c a n be t a k e n as a nonlinear disturbance
of the
quasiparallel
system.
This nonlinear t e r m
depends only on z such
that
the bending equation is properly a p p r o x i m a t e d by the quasiparallel system if z is sufficiently small.
4.3.4 N o t a t i o n of S t r a i n Whenever
elastic
or
plastic
deformations
are
considered,
the
problem
is usually
f o r m u l a t e d in t e r m s of s t r e s s e s and strains. The procedure is to solve a given problem in t e r m s of dislocations {e.g. LOVE,
Fll
1944). The d e f o r m e d s t a t e then is given in t h e form
discussed with similar folds, i.e. by a map
X2
=
x2
+
•'
~i = f ( x l , x
2 'x3)
3 deformed
undeformed
dislocations
The elements of strain are related to the Jacobian matrix of the dislocations (e.g. LOVE, 1944).
211
"agl agl. ag~axl ax 2 ax 3
(4.87)
= [ I)=
<Xv,Xv>+ 2z + z2< Nv'Nv> I)
(4.97)
213
1 xy = ~{2