Principles of the Gravitational Method, Volume 41 (Methods in Geochemistry and Geophysics) (Methods in Geochemistry and Geophysics)

PRINCIPLES OF THE GRAVITATIONAL METHOD

METHODS IN GEOCHEMISTRY AND GEOPHYSICS (Volumes 1–28 are out of print) 29. V.P. Dimri – Deconvolution and Inverse Theory – Application to Geophysical Problems 30. K.-M Strack – Exploration with Deep Transient Electromagnetics 31. M.S. Zhdanov and G.V. Keller – The Geoelectrical Methods in Geophysical Exploration 32. A.A. Kaufman and A.L. Levshin – Acoustic and Elastic Wave Fields in Geophysics, I 33. A.A. Kaufman and P.A. Eaton – The Theory of Inductive Prospecting 34. A.A. Kaufman and P. Hoekstra – Electromagnetic Soundings 35. M.S. Zhdanov and P.E. Wannamaker – Three-Dimensional Electromagnetics 36. M.S. Zhdanov – Geophysical Inverse Theory and Regularization Problems 37. A.A. Kaufman, A.L. Levshin and K.L. Larner – Acoustic and Elastic Wave Fields in Geophysics, II 38. A.A. Kaufman and Yu. A. Dashevsky – Principles of Induction Logging 39. A.A. Kaufman and A.L. Levshin – Acoustic and Elastic Wave Fields in Geophysics, III 40. V.V. Spichak – Electromagnetic Sounding of the Earth’s Interior 41. A.A. Kaufman and R.O. Hansen – Methods in Geochemistry and Geophysics

Methods in Geochemistry and Geophysics, 41

PRINCIPLES OF THE GRAVITATIONAL METHOD

A.A. Kaufman 158 Del Mesa Carmel, Carmel, CA 93923, USA and

R.O. Hansen EDCON-PRJ, Inc., 171 S. Van Gordon St., Suite E, Lakewood, CO 80228, USA

Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2008 Copyright r 2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-52993-0 ISSN: 0076-6895 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in The Netherlands 08 09 10 11 12 10 9 8 7 6 5 4 3 2 1

Contents

Introduction

ix

List of Symbols

xi

Chapter 1 Principles of Theory of Attraction 1.1. Newton’s Law of Attraction 1.1.1. Interaction between a particle and an arbitrary body 1.1.2. The gravitational constant, k 1.2. The Field of Attraction and Solution of the Forward Problem 1.2.1. Example 1.3. Different Types of Masses and their Densities 1.4. Two Fundamental Features of theRAttraction Field 1.4.1. Independence of the integral l gdl on the path of integration 1.4.2. Relation between the flux of the attraction field and its sources, (masses) 1.5. System of Equations of the Field of Attraction 1.6. Laplace’s and Poisson’s Equations 1.7. The Potential and its Relation to Masses 1.8. Fundamental Solution of Poisson’s and Laplace’s Equations 1.9. Theorem of Uniqueness and Solution of the Forward Problem 1.9.1. The first boundary value problem 1.9.2. The second boundary value problem 1.9.3. The third boundary value problem 1.9.4. The fourth boundary value problem 1.10. Green’s Formula and the Relationship Between Potential and Boundary Conditions 1.11. Analytical Upward Continuation of the Field 1.12. Poisson’s Integral 1.13. Behavior of the Attraction Field 1.13.1. The attraction field of a spherical mass 1.13.2. The attraction field of a thin spherical shell, Fig. 1.12b 1.13.3. Evaluation of a mass of the arbitrary body 1.13.4. The normal component of the attraction field due to planar surface masses 1.13.4.1. Case one: A planar surface of an infinite extent 1.13.4.2. Case two: A plane of finite extension, (Fig.1.13d) 1.13.4.3. Case three: A plane surface has a form of a disk with radius a 1.13.5. Field caused by a volume distribution of masses in a layer with thickness h and density d 1.13.6. Determination of layer density 1.14. Legendre’s Functions and a Solution of Laplace’s Equation 1.14.1. Expansion of the function 1/Lqp in the power series 1.14.2. Laplace’s equation and Legendre’s functions

1 3 4 6 7 9 10 11 13 14 18 20 21 25 28 29 30 32 33 37 40 42 42 46 47 47 49 50 50 51 53 54 55 57

vi

Contents

Chapter 2 Gravitational Field of the Earth 2.1.

Forces Acting on an Elementary Volume of the Rotating Earth and the Gravitational Field 2.1.1. Equation of motion of an elementary volume 2.1.2. The field gs of the surface forces in a fluid 2.1.3. The gravitational field g 2.1.4. The centrifugal force 2.1.5. Frame of reference rotating with a constant angular velocity (two-dimensional case) 2.1.6. Frame of reference rotating with the constant angular velocity (three-dimensional case) 2.2. Gravitational Field of the Earth 2.2.1. General features of the field g on the earth’s surface 2.3. Potential of the Gravitational Field of the Earth 2.3.1. Level surfaces and plumb lines 2.3.2. Curvature of level surfaces and vector lines 2.3.3. Potential and distribution of density of mass 2.3.4. Poincare theorem 2.4. Potential and the Gravitational Field due to an Ellipsoid of Rotation 2.4.1. Formulation of the boundary value problem for the potential U a 2.4.2. The system of coordinates of oblate spheroid 2.4.3. The system of coordinates of an oblate spheroid 2.4.4. Solution of Equation (2.129) by the method of separation of variables 2.4.5. Expressions for the potential 2.4.6. The gravitational field due to the rotating ellipsoid 2.4.7. The gravitational field on the surface of the ellipsoid, e ¼ e0 2.4.8. Relation between the reduced and geographical latitudes 2.5. Clairaut’s Theorem 2.5.1. The linear approximation 2.6. Potential of the Gravitational Field in Terms of Spherical Harmonics 2.7. Geoid and Leveling 2.7.1. Geoid and quasi-geoid 2.7.2. A height and an elevation 2.7.3. Leveling 2.8. Stokes’s Formula 2.8.1. Bruns’s formula 2.8.2. Boundary condition on the geoid surface 2.8.3. Boundary value problem 2.8.4. Spherical approximation of the boundary condition 2.9. Molodensky’s Boundary Problem 2.9.1. Molodensky’s problem and Bruns’s formula 2.9.2. Boundary condition for the disturbing potential T 2.9.3. The boundary value problem for the function T 2.9.4. Solution of the boundary value problem

59 59 61 64 65 66 70 72 74 75 77 78 82 82 84 85 85 87 90 91 96 97 98 100 102 106 114 116 118 119 120 121 122 123 124 128 129 132 132 134

Contents

2.10. Attraction Field of the Spheroid 2.10.1. Integration of Equation (2.318) 2.10.2. Potential caused by masses of a homogeneous spheroid 2.11. Spheroid and Equilibrium of a Rotating Fluid 2.11.1. Equation of equilibrium and level surfaces 2.11.2. Relationship between density, angular velocity, and spheroid eccentricity 2.11.3. Solution of Equation (2.344) 2.11.4. Oblate spheroid with very small eccentricity 2.11.5. About a stability of a figure of equilibrium 2.12. Development of the Theory of the Figure of the Earth (Brief Historic Review) 2.12.1. I. Newton, 1643–1727 2.12.2. Ch. Huygens, 1629–1695 2.12.3. C. MacLaurin, 1698–1746

vii 135 139 142 143 143 145 145 147 148 149 149 153 153

Chapter 3 Principles of Measurements of the Gravitational Field 3.1. History of Measurement of the Gravitational Field 3.2. Principles of Ballistic Gravimeter 3.2.1. Equations of motion 3.2.2. Two methods of field measurements of the field 3.2.3. Non-symmetrical motion 3.2.4. Symmetrical motion 3.3. Pendulum Devices 3.3.1. Small oscillations 3.3.2. Oscillations with an arbitrary amplitude 3.3.3. Physical pendulum 3.3.4. Equation of a motion 3.3.5. Reversing pendulum 3.4. Influence of Coriolis Force on Particle Motion 3.4.1. Equation of motion 3.4.2. Motion of a free particle 3.4.3. Foucault’s pendulum 3.5. Vertical Spring–mass System 3.5.1. Vertical spring balance and lever spring balance 3.5.2. Vertical spring balance 3.5.3. Mechanical sensitivity of the system 3.5.4. Equation of free vibrations of a mass 3.5.5. Limiting case, n ¼ 0 3.5.6. Three types of attenuation of free vibrations 3.5.6.1. Weak attenuation, nox0 3.5.6.2. Critical attenuation, n ¼ x0 3.5.6.3. Strong attenuation 3.5.7. Forced vibrations 3.5.8. Mechanical sensitivity and stability of vertical spring–mass system 3.5.9. Stability of the vertical spring–mass equilibrium

161 163 163 165 166 167 169 170 172 175 176 178 180 180 183 184 187 188 188 189 190 192 192 193 193 193 193 196 197

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3.6. Spring with an Initial Compression and Hooke’s Law 3.7. Torsion Spring–mass System 3.7.1. Three types of points of equilibrium 3.7.2. Equation of mass rotation 3.7.2.1. Unstable equilibrium, @s=@b40 3.7.2.2. Stable equilibrium, @s=@b40 3.7.2.3. Indifferent equilibrium, @s=@b ¼ 0 3.7.3. Mechanical sensitivity of the torsion balance 3.8. Lever Spring–mass System 3.8.1. Zero lever spring system 3.8.2. About measurements in the presence of a high-frequency noise 3.9. Measurement of Second Derivatives of the Potential of Gravitational Field 3.9.1. Resultant force and moment 3.9.2. Equation of equilibrium and motion 3.9.3. Second derivatives of the gravitational potential 3.9.4. Equation of equilibrium in the system of coordinates x, Z, z 3.9.4.1. Case one 3.9.4.2. Case two

197 201 202 203 204 204 205 205 206 207 208 210 210 211 213 214 215 215

Chapter 4 Uniqueness and the Solution of the Inverse Problem in Gravity 4.1. Concept of Uniqueness and the Solution of the Inverse Problem 4.1.1. Uniqueness and its application 4.1.2. Example 4.2. Solution of the Inverse Problem and the Influence of Noise 4.3. Solution of the Forward Problem (A Calculation of the Field of Attraction) 4.3.1. Two-dimensional model 4.3.1.1. Thin two-dimensional layer 4.3.2. Three-dimensional body 4.3.2.1. The first approach 4.3.2.2. The second approach

217 221 223 225 229 230 233 234 235 235

Bibliography

237

Appendix

239

Subject Index

243

Introduction The subject of this monograph is physical and mathematical principles of gravitational method. The first chapter is devoted entirely to the field of attraction caused by masses. Proceeding from Newton’s law of attraction we introduce the concept of this field and describe its fundamental features. Special attention is paid to the system of equations of the attraction field at regular points and interfaces where a density is discontinuous function. Then after introduction of the potential we perform transition from the system of equations to Poisson’s and Laplace’s equations and discuss their fundamental solutions. Also the theorem of uniqueness and different boundary value problems are studied in detail. As illustration, we consider two examples, which play an important role in the exploration and global geophysics, namely, analytical upward continuation of the field and Poisson’s integral. At the end of this chapter there are several examples, which characterize a field behavior inside and outside masses. The second chapter considers the gravitational field of the earth. At the beginning, we study forces acting on elementary volume of the rotating Earth in an inertial frame. Then, considering the second Newton’s law in non-inertial system, we introduce the centrifugal force and gravitational field, as well as its potential. Special attention is paid to the gravitational field and potential, caused by an ellipsoid of rotation. This study allows us to describe Clairaut’s theorem. Also, taking into account results derived in Chapter 1, we represent the potential of the gravitational field in terms of spherical harmonics. This chapter also discusses such concepts as geoid, Stokes formula, and Molodensky’s boundary problem which play an important role in studying Earth’s figure. In conclusion, we briefly consider equilibrium of rotating fluid and history of development of the theory of the earth’s figure. Principles of measurement of the gravitational field are discussed in Chapter 3. We describe the theory of ballistic gravimeter, pendulum, and different types of static gravimeters, including such questions as stable and unstable equilibrium, ‘‘zero-length’’ spring and mechanical sensitivity. Also we consider the principles of measuring of the second derivatives of the potential. Finally, in Chapter 4 we focus on a solution of the inverse problems in the gravitational method and discuss uniqueness and non-uniqueness, ill and well-posed problems, stable and unstable parameters, regularization, as well as different methods of a solution of the forward problem.

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Methods in Geochemistry and Geophysics

ACKNOWLEDGMENTS We express our thanks to Dr. M. Brodsky, Dr. I. Fuks, Dr. T. LaFehr, Dr. A. Levshin, Dr. K. Naugolnykh, Dr. T. Niebauer, Dr. L. Ostrovsky, Dr. W. Torge, Dr. J. Wahr for very useful comments and suggestions. We also want to thank Marianna Borukaeva for her technical assistance. Finally, we thank our wives, Irene and Kathleen for their patience and support.

List of Symbols a a A b C d,d0,d* dF do dl dH E e F1 Fr F f f* G G g ga gc gR, gy, gf gx, gy, gz gi, ge ga gt, gn gU gN gs H Hor h h1, h2, h3 I i, j, k iR , iy , if J20 J j ¼ (1)1/2 k Lqp L

major semi-axis of spheroid acceleration moment of inertia minor semi-axis of spheroid moment of inertia functions elementary force elementary solid angle elementary displacement elementary height Young modulus, harmonic function outside a geoid deformation Coriolis force restoring force force flattening of ellipsoid, frequency parameter characterizing a change of gravitational field, (gravity flattening) Green’s function normal field gravitational field attraction field centripetal field components of the field in spherical system of coordinates components of the field in Cartesian system of coordinates fields caused by masses inside and outside of small sphere secondary field tangential and normal components of the field useful signal noise field of surface forces height, mechanical ellipticity orthometric height elevation metric coefficients moment of inertia, mean moment of inertia unit vectors in Cartesian system of coordinates unit vectors in spherical system of coordinates parameter mean curvature imaginary unit gravitational constant distance between points q and p rod length

xii l l0 M MN m N n P p Pn(m) Pn(e), Q(e) Pn(je), Q(je) R R, R1 r0 S S* S(c) t T T(p) t t0 U(p) Uc U0 Uav v W(p) a a0 g l m j e,Z,j s o

Methods in Geochemistry and Geophysics string length reduced length mass, mass of the earth function describing system of coordinates of oblate spheroid ratio of centrifugal force to gravitational one at the equator function describing system of coordinates of oblate spheroid, distance unit vector pressure observation point, pressure Legendre’s function of first kind Legendre’s functions of the first and second kind with real argument Legendre’s functions with imaginary argument radius of sphere radius vectors radius vector of the center of mass surface, weight of particle small spherical surface function in Stokes formula time period potential caused by irregular part of masses moment, angle pre-tension moment potential of gravitational field, potential of normal field potential of centrifugal field potential of level surface average value of potential linear velocity potential parameter, angle pre-tension angle magnitude of normal gravitational field latitude, linear density elastic parameter angle of twist, coordinate in the spherical system of coordinates coordinates in spheroidal system of coordinates surface density, normal stress solid angle, angular frequency

Chapter 1 Principles of Theory of Attraction 1.1. NEWTON’S LAW OF ATTRACTION The phenomenon of attraction of masses is one of the most amazing features of nature, and it plays a fundamental role in the gravitational method. Everything that we are going to derive is based on the fact that each body attracts other. Clearly this indicates that a body generates a force, and this attraction is observed for extremely small particles, as well as very large ones, like planets. It is a universal phenomenon. At the same time, the Newtonian theory of attraction does not attempt to explain the mechanism of transmission of a force from one body to another. In the 17th century Newton discovered this phenomenon, and, moreover, he was able to describe the role of masses and distance between them that allows us to calculate the force of interaction of two particles. To formulate this law of attraction we suppose that particles occupy elementary volumes DV(q) and DV(p), and their position is characterized by points q and p, respectively, see Fig. 1.1a. It is important to emphasize that dimensions of these volumes are much smaller than the distance Lqp between points q and p. This is the most essential feature of elementary volumes or particles, and it explains why the points q and p can be chosen anywhere inside these bodies. Then, in accordance with Newton’s law of attraction the particle around point q acts on the particle around point p with the force dF(p) equal to dFðpÞ ¼ k

DmðqÞDmðpÞ Lqp L3qp

ð1:1Þ

where k is a coefficient of proportionality, called the gravitational constant. In the International System of Units (SI) its value is k ¼ 6:674  1011 m3 =kg  s2 Dm(q) and Dm(p) are masses which may have arbitrary values, and they are measured in kilograms. As follows from Newton’s second law, mass is a quantitative measure of inertia, since with an increase of mass the rate of a change of the particle velocity for a fixed force becomes smaller. Also Lqp is the vector: Lqp ¼ Lqp L0qp

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Methods in Geochemistry and Geophysics

(a)

(b) L qp • dm(p)

dm(q)

dm(p)

M

F

dm(q)

•

(c)

(d)

M

m Lqp q

•p

m M

Fig. 1.1. (a) Newton’s law of attraction, (b) illustration of Equation (1.1), (c) field caused by an arbitrary mass, (d) Cavendish experiment.

here L0qp is the unit vector directed from the point q to the point p. As was mentioned above, DV(q) and DV(p) are elementary volumes. It is clear that only in this case the force of interaction of the particles does not depend on the position of the points p and q within the volume, and with an increase of the distance Lqp Equation (1.1) gives a more accurate value of the force. Note that dimensions of elementary volumes can change in different problems from very small to extremely large ones. Thus, in accordance with Equation (1.1) the mass Dm(p) is subjected to the force dF(p) which is directly proportional to the product of both masses and inversely proportional to the square of the distance between them, and it has a direction opposite to Lqp, (the presence of minus at the right hand side of Equation (1.1) illustrates this fact). This extremely simple formula describing the basic physical law of the gravimetry may need some comments. 1. Values of masses can be different. 2. Newton’s law of attraction states that the force of interaction of particles is inversely proportional to the square of the distance between them. However, in a general case of arbitrary bodies the behavior of the force as a function of a distance can be completely different. 3. In the SI system of units the distance is measured in meters, mass in kilograms, and the force in Newtons. 4. Equation (1.1) does not contain the physical parameters of the medium where the masses are located, and this means that the force of interaction between two masses is independent of the presence of other masses. For instance, if we place a mass M between masses Dm(q) and Dm(p), Fig. 1.1b, the force caused

Principles of Theory of Attraction

5.

3

by Dm(q) remains the same. Certainly, it is very interesting that the mass M does not influence the transmission of the force from the particle q to that near point p. Thus, the medium surrounding particles does not have any influence on the force of interaction between them. By analogy with Equation (1.1) the force acting on mass Dm(q) caused by the mass Dm(p) is dFðqÞ ¼ k

DmðqÞDmðpÞ Lpq L3qp

Inasmuch as L3qp ¼ L3pq and Lpq ¼ Lqp, we see the validity of the Newton’s third law: dFðqÞ ¼ dFðpÞ Note that the notation dF indicates that we deal with the force caused by elementary masses. Since the gravitational constant is extremely small, it is natural to expect that the force of interaction is also very small too. For illustration, consider two spheres with radius 1 m and mass 31.4  103 kg made from galena, with the distance between their centers 10 m. Then, the force of interaction, Equation (1.1), is F ¼ 6:6  104 N Indeed, it is a very weak force. 1.1.1. Interaction between a particle and an arbitrary body In order to determine the force with which an arbitrary body acts on a particle located around the point p, we mentally divide the volume of the body into many elementary volumes, so their dimensions are much smaller than the corresponding distance from the particle p. It is clear that the magnitude and direction of each force depends on the position of the point q inside a body. Now, applying the principle of superposition, we can find the total force acting on the particle p. Summation of elementary forces gives: FðpÞ ¼ k

X DmðqÞ n¼1

L3qp

Lqp DmðpÞ

In the limit when elementary volumes tend to zero we have Z dðqÞdV Lqp FðpÞ ¼ kDmðpÞ L3qp V

ð1:2Þ

ð1:3Þ

Here d(q) is the volume density of mass: dðqÞ ¼ lim

DmðqÞ DV

ð1:4Þ

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Methods in Geochemistry and Geophysics

if DV ! 0 or dm ¼ ddV

ð1:5Þ

3

The dimension of the density is kg/m . It is appropriate to note that to carry out numerical integration the elementary volumes have to satisfy two conditions, namely, a. Their extensions are much smaller than the distance to the observation point p. b. The masses are practically uniformly distributed within each elementary volume. Thus, integration over an arbitrary volume allows us to find the force caused by any distribution of masses. It is essential that the particle p can be located either outside or inside of a body and at any distance from its surface. Equation (1.3) describes the total force that is a result of a superposition of the elementary forces, vectors, at the same point. Correspondingly, this force can cause a translation of the particle only. It is also instructive to consider the force F generated by the particle and acting on an arbitrary body. Each elementary volume is subjected to the force dFðqÞ ¼ k

DmðqÞDmðpÞ Lpq L3qp

and, therefore, the total force has the same magnitude as F(p), but the opposite direction. In other words, we again observe the Newton’s third law to be valid. At the same time, it may be appropriate to comment: inasmuch as the elementary forces dF(q) are applied at different points of a body, their action usually causes both a translation and a rotation. Of course, the motion of a particle and a body can drastically differ from each other, and one vivid example is the system consisting the earth and a particle. Applying the same approach it is a simple to find an expression for the force of interaction between two arbitrary bodies. It is obvious that the force acting on any elementary volume of a body is the sum of the forces due to other body and the force caused by different elements of the same body. In particular, the resulting force due to body 1 acting on body 2 is Z Z dm2 dm1 F ¼ k L12 L312 V2 V1 Thus, we have derived a generalization of Newton’s law of attraction. 1.1.2. The gravitational constant, k The coefficient k is one of the fundamental constants of physics and astronomy. From Kepler’s laws it is possible to express the masses of all planets in terms of the mass of the sun. However, in order to find the mass of the sun and,

Principles of Theory of Attraction

5

correspondingly, masses of other planets we have to know the gravitational constant. In general, its knowledge allows us to calculate the force of interaction between two arbitrary bodies. The numerical value of the gravitational constant depends on the system of units in which force, mass, and distance are defined. Its magnitude is found experimentally by measuring the force of the gravitational attraction between two bodies of known masses, located at a given distance. For bodies of a moderate size this force is extremely small, and for this reason the value of this constant remained unknown for more than two hundred years. Finally, this task was solved by Cavendish in 1792 using a torsion balance. Earlier the same approach was applied by Coulomb to study forces of electrical attraction and repulsion. In principle, the Cavendish balance consists of two small spheres of the same mass m, Fig. 1.1d, mounted at opposite ends of a light horizontal rod, which is suspended at its center by a thin vertical fiber (quartz thread). A small mirror mounted on the fiber reflects a beam of light onto a scale. In order to use the balance two relatively large spheres of mass M, usually made of lead, are brought close to masses m. The forces of gravitational attraction between the large and small spheres form a couple, which twists the fiber, as well as mirror. Correspondingly, the light beam moves along a scale. At equilibrium the elastic force of the fiber compensates the external one, and we have FL ¼ mj Here F is the force of interaction between masses m and M, L the rod length, m the elastic parameter of the fiber, and j the angle of the twist. By using a very fine fiber the deflection of the light beam may be sufficiently large so that the gravitational forces, depending on the unknown constant, k, can be measured quite accurately. Of course, this requires the reduction of different types of noise, such as the influence of charges on the surfaces of the masses. Note, that the expression for F implies that the spherical masses interact as particles and this assumption will be proved later. To appreciate this experiment we compute the force of the gravitational attraction between the large and small spheres in the Cavendish balance when m ¼ 103 kg and M ¼ 0.5 kg and the distance between their centers is 0.05 m. Then F¼

6:67  1011  103  0:5 ¼ 0:13  1010 N 25  104

Certainly, this is an extremely small value and is the reason why the determination of the gravitational constant with very high accuracy is a rather complicated experiment. During the last two hundred years there were many measurements of this constant, but still only three digits after decimal point are reliable. One can say that due to Cavendish’s measurements it became possible to develop the theory of gravity and evaluate mass of the earth. In fact, determination of this mass was the main goal of this experiment.

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Methods in Geochemistry and Geophysics

1.2. THE FIELD OF ATTRACTION AND SOLUTION OF THE FORWARD PROBLEM In accordance with Equation (1.3) the force acting on the particle around the point p is directly proportional to the mass, Dm(p). Now let us imagine that this mass decreases. Then, the force dF(p) decreases too, but the ratio dFðpÞ dm remains the same and it characterizes a quantity g, which is called the field of attraction at the point p: Z dðqÞLqp gðpÞ ¼ k dV ð1:6Þ L3qp V Assuming that the distribution of masses inside the volume V is given, this vector function g(p) depends only on the coordinates of the observation point p, and by definition it is a field. It is appropriate to treat the masses in the volume V as sources of the field g(p). In other words, these masses generate the field at any point of the space, and this field may be supposed to exist whether a mass is present or absent at this point. When we place an elementary mass at some point p, it becomes subject to a force equal to FðpÞ ¼ DmðpÞgðpÞ

ð1:7Þ

that causes motion of the mass. In essence, all measurements in the gravitational method are based on the use of Equation (1.7). One can say that at each point the attraction field is ‘‘waiting’’ for a mass to create a force and cause a motion. The function g(p) can be also interpreted in a completely different way. As follows from Newton’s second law, g(p) represents the acceleration of an elementary mass, caused by the force F(p). We will mainly apply a concept of a field, but sometimes both approaches will be used. Inasmuch as the interaction of masses usually does not influence their density, it can be specified. In other words, the density of each material is a known parameter, and this is a very important fact, because it means that Equation (1.6) allows us to solve the forward problem by integration. However, in general it is not true and, for instance, a rotation of a compressible fluid causes a change of its density. In other geophysical methods generators of the field cannot be usually determined before the calculation of the field, and for this reason solution of the forward problem requires solution of a boundary value problem. Certainly, the remarkable simplicity of the solution of the forward problem in the field of attraction is exceptional. Equation (1.6) can be applied at any point of space, including a volume where masses are located. In such a case the distance Lqp may tend to zero, and this fact naturally causes a suspicion that the use of Equation (1.6) is invalid in these places. However, as was already pointed out and will be proved later, the singularity of the integrand is removable and Equation (1.6) gives the correct value of the attraction field inside a mass. It is clear that with an increase of separation between the observation point

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p and a body, the vector Lqp becomes practically independent of the point q. Correspondingly, in the limit when Lqp tends to infinity, it can be taken out of the integral in Equation (1.6) and we obtain R dðqÞdV m gðpÞ ¼ k V 3 Lqp ¼ k 3 Lqp ð1:8Þ Lqp Lqp where q is an arbitrary point of a body and m its mass. Thus, regardless of the dimensions and shape of a body, it generates at relatively large distances practically the same attraction field as that of an elementary particle. 1.2.1. Example To illustrate Equation (1.8), consider a solution of the forward and inverse problems in the simplest possible case, when the field is caused by an elementary mass. Suppose that a particle with mass m(q) is situated at the origin of a Cartesian system of coordinates, Fig. 1.2a, and the field is observed on the plane z ¼ h. Then, as follows from Equation (1.8), the components of the attraction field at the point p(x,y,h) are gx ðx; y; hÞ ¼ k

mx my mh gy ðx; y; hÞ ¼ k 3 gz ðx; y; hÞ ¼ k 3 3 Lqp Lqp Lqp

ð1:9Þ

because Lqp ¼ xi þ yj þ hk (a)

(b)

z gx x h 0 m

gz

x (c) gx

x

Fig. 1.2. (a) Elementary mass beneath an observation plane, (b) behavior of the horizontal and vertical components of the field, (c) influence of mass position.

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Methods in Geochemistry and Geophysics

Here i, j, k are unit vectors along the coordinate axes. Thus, the solution of the forward problem is trivial. In particular, along the line where y ¼ 0 and z ¼ h we have: gx ðx; 0; hÞ ¼ k

mx ðx2 þ h2 Þ3=2

gy ¼ 0 gz ðx; 0; hÞ ¼ k

mh ðx2 þ h2 Þ3=2

ð1:10Þ

The behavior of both components as functions of x is shown in Fig. 1.2b, and these can be easily explained. First of all, the horizontal component vanishes above the body, since the field is directed toward the body. Also gx(x) has two extremes, and their position is defined from the condition: @gx ¼0 @x Taking a derivative we find: h xe ¼ pffiffiffi 2

ð1:11Þ

and with decrease of the depth h of the particle an extreme is observed at smaller values of x, but its magnitude increases, Fig. 1.2c. For negative values of x the horizontal component, gx, is positive, and this is understandable because the field is directed toward the mass and it forms an angle with the x-axis which is smaller than 901. On the contrary, if x40 this angle exceeds 901 and the component is negative. It is clear that the vertical component jgz j has a maximum at x ¼ 0 and gradually decreases with the distance x. At relatively large distances we have: gx4gz, which follows from the fact that the field becomes almost horizontal. Note, that at such distances the vertical component decreases more rapidly than L2 qp . The solution of the inverse problem is very simple too. For instance, if we know only the component gx then its zero value determines the position of the body (x-coordinate). At the same time, the depth h is defined from Equation (1.11). It can be calculated in different ways; taking the ratio of this component at two points, we obtain the equation with respect to h: gx ðx1 Þ ðx22 þ h2 Þ3=2 x1 ¼ gx ðx2 Þ ðx21 þ h2 Þx2

ð1:12Þ

Repeating these calculations with different pairs of gx(x) we may increase the accuracy of the evaluation of h. Next, making use of the value of this component at any point, the mass m is evaluated. In the case when only the vertical component is known, the determination of the position of mass and its value is similar. Here it is appropriate to notice the following. Inasmuch as an arbitrary body, located at a large distance from an observation point p, creates a field, known always with some error, often it cannot be practically distinguished from that of an elementary particle, and for this reason we are able to determine only the product of volume and density, mass, but each of them remains unknown. It is the first illustration of the fact that the solution of the inverse problem in gravity, as well as in other geophysical methods, is an ill-posed one, because some parameters of a body

Principles of Theory of Attraction

9

cannot be defined. In Chapter 4 we will discuss this subject in some details, including such questions as uniqueness and non-uniqueness, the well- and ill-posed problems and regularization.

1.3. DIFFERENT TYPES OF MASSES AND THEIR DENSITIES As was pointed out earlier, Equation (1.6) allows us to find the attraction field everywhere, but it requires a volume integration, that in general is a rather cumbersome procedure. Fortunately, in many cases the calculation of the field g(p) can be greatly simplified. First, consider an elementary mass with density d(q), located in the volume DV. Now let us start to increase the density and decrease the volume in such a way that the mass remains the same. By definition, these changes do not make a noticeable influence on the field because the observation point p is far away. In the limit, when DV ! 0

and

d!1

we arrive at the mathematical concept of the point mass, m(q). Correspondingly, its attraction field is defined by Equation (1.8). Of course, it is impossible to place a mass into a point, and a point mass is only an approximate representation of a real body. Suppose that an elementary volume is a cube with sides of length h. Therefore, the transition to the point mass leads to an increase of the volume density as h3. Next, we assume that the volume V(q) has the shape of a rod, Fig. 1.3a, with cross section S ¼ h1h2 and distance to the observation points satisfying conditions: Lqp  h1

and

Lqp  h2

Then, we can reduce each elementary volume dV ¼ h1h2dl and increase the volume density so that the elementary mass: dm ¼ dh1 h2 dl does not change. Here dl is an elementary displacement along the rod. In the limit, when the cross section S becomes a point of the line l, we obtain a linear (a)

(b)

dS q dl h(q) h2

h1

Fig. 1.3. (a) Linear distribution of masses, (b) surface distribution of masses.

10

Methods in Geochemistry and Geophysics

distribution of masses with linear density l: lðqÞ ¼ dðqÞh1 h2 where dðqÞ ! 1

and

h1 ! 0; h2 ! 0 2

It is clear that d(q) tends to infinity as h , and we deal with the second mathematical concept of a mass. Respectively, the expression for the attraction field becomes Z lðqÞ gðpÞ ¼ k Lqp dl ð1:13Þ 3 l Lqp and the volume integration is reduced to a linear one that is much simpler. Finally, suppose that masses are situated in a relatively thin layer, so that its thickness h(q) is much smaller than the distance Lqp, Fig. 1.3b. Let us consider an elementary volume dV ¼ h(q)dS. Its mass is dm ¼ dðqÞhðqÞdS. Now, reducing the layer thickness and preserving the mass dm, we obtain in the limit a surface mass with the density s(q): sðqÞ ¼ dðqÞhðqÞ

ð1:14Þ

In this case the volume density d(q) tends to infinity as h1, when h(q) approaches zero. Correspondingly, the expression for the field is Z sðqÞ gðpÞ ¼ k Lqp dS 3 S Lqp and the volume integration is replaced by the surface one. Thus, the field of attraction caused by volume, surface, linear, and point masses is "Z # Z Z X mðqÞ dðqÞ sðqÞ lðqÞ gðpÞ ¼ k Lqp dV þ Lqp dS þ Lqp dl þ Lqp ð1:15Þ 3 3 3 L3qp V Lqp S Lqp l Lqp We have found that the volume density tends to infinity differently near point, linear, and surface masses, and this fact influences the field behavior in the vicinity of such places. Of course, Equation (1.6) always allows us to calculate the field of attraction g. At the same time, in many cases the use of Equation (1.15) greatly simplifies this procedure.

1.4. TWO FUNDAMENTAL FEATURES OF THE ATTRACTION FIELD Inasmuch as Equation (1.6) allows us to solve the forward problem for any distribution of masses, we may say in this sense that the theory of the gravitational method is completely developed. However, in order to understand better the behavior of the field of the earth and sometimes to improve the quality of the

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Principles of Theory of Attraction

solution of the inverse problem, it is useful to study general features of the attraction field and introduce also the concept of potential. 1.4.1. Independence of the integral

R l

gdl on the path of integration

First, consider the field of an elementary mass and evaluate the integral along the straight line shown in Fig. 1.4a. This gives   Z p2 Z p2 Lqp dl dLqp 1 1 km ¼ km ¼ km  ð1:16Þ 2 Lqp2 Lqp1 L3qp p1 p1 Lqp since Lqp dl ¼ Lqp dl

and

dl ¼ dLqp

It is clear that the integral along an arc of the circle with radius Lqp is equal to zero, because the field g(p) is perpendicular to the path of integration. For this reason, as before, the integral along the path shown in Fig. 1.4b is defined by the interval along the straight line. Next consider a more complicated path, shown in Fig. 1.4c. Integration between points p1and p3 gives:     Z p3 1 1 1 1 gdl ¼ km   þ Lqp3 Lqp2 Lqp2 Lqp1 p1 or Z

p3 p1



1 1 gdl ¼ km  Lqp3 Lqp1

 ð1:17Þ

Comparison of Equation (1.16) to Equation (1.17) shows that in both cases the value of the integrals is defined by the distance from the particle to terminal points (a)

(b)

Lqp dl p1

q

p2

p3

dl p2

m(q) p1

q

p3

(d)

(c)

(e) p2

p2

d ln

p1 p1

q Fig. 1.4. (a–e) Different paths of integration of field g.

d l2n

d l1n

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Methods in Geochemistry and Geophysics

of the path. Now we generalize this result for an arbitrary path, Fig. 1.4d. Each elementary displacement can be represented as a sum of two perpendicular vectors, Fig. 1.4e: dln ¼ dl1n þ dl2n Here dl1n is the displacement along the radius vector, drawn from the point q, but dl2n is the element of the arc of the radius Lqp. Then we have: gðpÞdln ¼ gðpÞdl1n and the integral along the elementary displacement of the path dln can be written as: Z



pnþ1

gdln ¼ km pn

1

Lqpnþ1



1 Lqpn

 ð1:18Þ

Taking into account the fact that the interval dl is very small, the integral is equal to gdl1n and the right hand side of Equation (1.18) describes this product. Performing a summation of these integrals along the path with terminal points p1 and p2, we obtain: Z

p2

p1



1 1 gðpÞdl ¼ km  Lqp2 Lqp1

 ð1:19Þ

As in the first case, the integral is defined by the position of the terminal points alone, while the shape and length of the path do not have any influence on its value. We have demonstrated that in the case of a single elementary mass, the integral in Equation (1.19) is path independent. Now suppose that there is an arbitrary distribution of masses. Applying the principle of superposition we conclude that the same behavior holds in the general case when the field is caused by any distribution of masses. This result can be formulated differently, namely, that the work performed by the attraction field between two points does not depend on the path of integration. Certainly, this is an amazing fact which would be difficult to predict. Indeed, changing the path connecting two given points we deal with different values of the field, different orientation of the displacement, length of the path, and the dot products of vectors g and dl, but the value of the integral remains the same. It seems as if there is communication between the field g at different points of the path, so that they control the behavior of each other and produce the same value of the sum of dot products. In order to emphasize this feature of the field, let us imagine the following experiment. Suppose that the distance between points a and b is 1 m. The first path is a straight line connecting these points. Performing an integration of the dot product gdl, we obtain some value of the integral. The second path is completely different, and it goes through all mountains of the earth, as well as oceans and finally returns to the point b. During this journey we measure the attraction field at each point of such path and form the product gdl, but its summation gives exactly the same result as integration along the short straight line!

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Principles of Theory of Attraction

(a) p

(b) dS1

dS

g(p)

dS2

Lqp

Lqp

dS3

q m(q) z (c)

gθ

(d) gθ gθ

gR

θ p

gθ a gR

0

(e)

R Fig. 1.5. (a) Flux through elementary surface, (b) flux as a sum of elementary fluxes, (c) field components due to spherical mass, (d) symmetry of field, (e) the field inside and outside spherical mass.

1.4.2. Relation between the flux of the attraction field and its sources, (masses) Now we establish the second remarkable feature of the attraction field. As before, at the beginning consider the field of a point mass, m(q). By definition, the flux of the field through an elementary surface dS, Fig. 1.5a, is gðpÞdS ¼ k

m Lqp dS L3qp

ð1:20Þ

As is known, Lqp dS ¼ do L3qp

ð1:21Þ

is the solid angle, which the surface dS subtends at the point q. Thus, in this case the flux can be represented as: gðpÞdS ¼ kmðqÞdoðqÞ

ð1:22Þ

Note, that the flux of the attraction field is a purely mathematical concept; it does not have any physical meaning. Next, we evaluate the flux through the surface S,

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Methods in Geochemistry and Geophysics

Fig. 1.5b. Performing the integration, we obtain Z gðpÞdS ¼ kmðqÞoðqÞ

ð1:23Þ

where o(q) is the solid angle which the surface S subtends at the point q. It can be positive, negative, or zero. Now consider the most interesting case, when the surface S is closed. Then, as is well known, the solid angle is equal to 4p if the point q is located inside the volume V, enclosed by the surface S, and it is zero, when q is outside the volume. Correspondingly, Equation (1.23) can be written as: I gðpÞdS ¼ 0 ð1:24Þ S

if the mass is located outside the volume V, and I gðpÞdS ¼ 4pkmðqÞ

ð1:25Þ

if it is somewhere inside V, (p is a point of the closed surface S). In accordance with Equation (1.24) the flux caused by the mass located outside V is zero regardless of a position of the mass m(q). Certainly, this is a remarkable fact. Indeed, at each point of the surface this mass creates a field and the dot product gdS in general differs from zero. However, the sum of these elementary fluxes over a closed surface is zero. Moreover, a change of the surface leads to a change of elementary fluxes, but their sum, integral, vanishes as long as the closed surface does not intersect the mass. Thus, the flux is insensitive to mass located outside the volume V. At the same time, as follows from Equation (1.25), the flux through a closed surface defines the amount of mass inside V. It is essential that the flux remains the same, regardless of the location of this elementary mass and a shape and position of the closed surface surrounding it. Applying now again the principle of superposition, we obtain for an arbitrary distribution of masses: I gdS ¼ 4pkM ð1:26Þ S

where M is the total mass situated inside the volume V that is enclosed by the surface S. This equation describes the second fundamental feature of the attraction field, and it can be also treated as ‘‘the bridge’’ between sources (masses) and the field. It may be appropriate to notice that the similar relationship holds for any field and is often called Gauss’s formula.

1.5. SYSTEM OF EQUATIONS OF THE FIELD OF ATTRACTION In the previous section we established two fundamental features of the attraction field, and both of them follow from Newton’s law of attraction and the principle of

15

Principles of Theory of Attraction (b)

(a) g b

L1

g

a

dl

g L g

L2

g

(c)

(d) g2

n

g2

n2

n

t

n1

g1

g1

Fig. 1.6. (a, b) Illustration of Equation (1.28), (c) tangential component of the field at interface, (d) normal component of the field at the interface.

superposition. The first one, independence of the work of the attraction field on a path between two points, is written as Z Z gdl ¼ gdl ð1:27Þ L1

L2

Here L1 and L2 are two arbitrary paths, connecting the same terminal points, Fig. 1.6a. Changing the direction of one of the paths, for example, L2, we have Z Z Z Z gdl ¼  gdl or gdl þ gdl ¼ 0 ð1:28Þ L1

L2

L1

L2

Since these paths form a closed path L, Equation (1.27) can be rewritten as I g  dl ¼ 0 ð1:29Þ that is, the circulation of the attraction field is always equal to zero. This means that vector lines of this field are always open, and their terminal points are either located inside the masses or at infinity. The second fundamental feature of the field g is described by Equation (1.26), and it states that the flux of this field through any closed surface characterizes amount of mass in the volume V enclosed by this surface. Both Equations (1.26 and 1.29) represent the system of equations of the attraction field in the integral form: I I gdl ¼ 0 and gdS ¼ 4pkm ð1:30Þ l

S

Here m is the total mass inside V. Inasmuch as this system does not contain derivatives of the field, it is valid at any point, including boundaries of media with

16

Methods in Geochemistry and Geophysics

different densities. These equations can be treated as relationships between values of the field g at different points, located at any distance from each other, and, correspondingly, they are integral equations with respect to the field g. Next, we obtain the system of field equations in differential form, that is, relationships between values of the field and the density of masses in the vicinity of the same point. First, assume that this point is regular, and, therefore, the field has first derivatives in that point. Taking into account the definition of curl and divergence, we arrive at the system of equations at regular points: curl g ¼ 0 and

div g ¼ 4pkd

ð1:31Þ

Here d is the volume density at a point. For instance, at points where masses are absent div g ¼ 0. Let us discuss the physical and mathematical content of these equations. The first one clearly shows that the attraction field does not have vortices and, correspondingly, the work done by this field is path independent. In other words, the circulation of the field is equal to zero. At the same time, the second equation demonstrates that the field g is caused by sources (masses) only. As illustration, consider the set of these equations in the Cartesian system of coordinates:    i j k   @gx @gy @gz @ @ @  þ þ ¼ 4pkd ð1:32Þ  @x @y @z  ¼ 0;   @x @y @z  gx gy g z  Respectively, in place of the first equation we have @gz @gy ¼ ; @y @z

@gx @gz ¼ ; @z @x

@gx @gy ¼ @y @x

ð1:33Þ

Thus, we deal with a system of four differential equations of the first order and, in general, there are four unknown functions: gx, gy, gz, and d. If the distribution of masses is known, then we do not need to use the set (1.31) to find the field. In a fact, this task is solved by integration, using Equation (1.6): Z Lqp gðpÞ ¼ k dðqÞ 3 dV ð1:34Þ Lqp As follows from Equation (1.33) there are relationships between different components of the field, and they indicate that each component of the field contains the same information about a distribution of masses. Note, that Equation (1.33) directly follows from Equation (1.34). By definition, we have: Z dðqÞðxp  xq Þ gx ðpÞ ¼ k dV L3qp V Z

dðqÞðyp  yq Þ

gy ðpÞ ¼ k V

L3qp

dV

Principles of Theory of Attraction

Z gz ðpÞ ¼ k

17

dðqÞðzp  zq Þ dV L3qp V

Taking the corresponding derivatives, we again arrive at Equation (1.33). The system (1.31) is written for points where the density d is defined. However, there are exceptions; for instance, an interface between media with different densities, Fig. 1.6c, since in such places the density of masses is a discontinuous function. Now, making use of Equation (1.30), it is easy to derive a surface analogy of Equation (1.31). Let us calculate the circulation along the path shown in Fig. 1.6c. From Equation (1.29) it follows: g2 dl2 þ g1 dl1 ¼ 0

ð1:35Þ

since displacements dl are small, the integrals can be replaced by dot product of the field and the displacement, while the integrals along the path h, perpendicular to the surface, vanish when h tends to zero. Taking into account that dl2 ¼ dl1 we obtain g2t dt  g1t dt ¼ 0 or g1t ¼ g2t

ð1:36Þ

where gt is the tangential component of the field. Equation (1.36) is the surface analogy of the first equation of the attraction field, and it shows that the tangential component of g is a continuous function at the boundary between media with different densities. Next, imagine an elementary cylinder around some point q of this surface, Fig. 1.6d. Then, applying Equation (1.26), we have Z g2 dS2 þ g1 dS1 þ gdS ¼ 4pkm Sl

where dS2 ¼ dSn

and

dS1 ¼ dSn

but S1 is the lateral surface of the cylinder, n the unit vector directed from the back to front side of the surface. In the limit, when the cylinder height tends to zero, mass m also vanishes, and it gives g2n ¼ g1n

ð1:37Þ

The latter is the surface analogy of the second equation of the field. Thus, both components are continuous functions at an interface, Fig. 1.6d. As a summary, it is useful to illustrate a transition from Newton’s law of attraction to the system of equations in the form of diagram. The arrows show that all equations are derived from Newton’s law of attraction, and they do not contain more information than this law. At the same time, the system (1.38) allows us to

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Methods in Geochemistry and Geophysics

understand better the general features of the field and often leads to some simplification in solving of the forward and inverse problems of the gravitational method.

1.6. LAPLACE’S AND POISSON’S EQUATIONS As was pointed out, the set (1.31) is a rather complicated system of differential equations, and it is natural to attempt to replace them by much simpler equations. This is the first motivation for introducing the potential of the attraction field. Taking into account the equality curl grad U ¼ 0 Newton’s law of attraction

g( p) = − k ∫

(q)

V

3

L qp

Lqp dV

∫ gdl = 0

ð1:38Þ ∫ g d S = − 4π km

l

S

g1t = g2t

g1n = g2n

the first equation of the set (1.31) gives gðpÞ ¼ grad U

ð1:39Þ

Thus, we have expressed the vector field g in terms of a scalar function, U(p), by a relatively simple operation, gðpÞ ¼

1 @U 1 @U 1 @U i1 þ i2 þ i3 h1 @x1 h2 @x2 h3 @x3

ð1:40Þ

where h1, h2, and h3 are metric coefficients; x1, x2, and x3 are coordinates of the observation point; and i1, i2, and i3 are unit vectors of the coordinate system. For instance, in the Cartesian system of coordinates h1 ¼ h2 ¼ h3 ¼ 1, and the unit vectors are i, j, k. Correspondingly, grad U ¼

@U @U @U iþ jþ k @x @y @z

It is clear that Equation (1.39) defines the potential U up to a constant, since grad C ¼ 0, that is, an infinite number of potentials describe the same field g. For this reason, it is appropriate to consider the potential as an auxiliary function introduced with mainly one purpose, namely, to simplify the analysis of the more complicated field g. Our next step is obvious: we have to find an equation,

Principles of Theory of Attraction

19

describing the behavior of U. In order to introduce the potential we already used the first equation, curl g ¼ 0. Now, substituting Equation (1.39) into the second equation of the system (1.31), we obtain: div grad U ¼ 4pkd

or

r2 U ¼ 4pkd

ð1:41Þ

Thus, we have obtained Poisson’s equation, and, as is well known, Equation (1.41) can be represented in the form:        1 @ h2 h3 @U @ h1 h3 @U @ h1 h2 @U þ þ ¼ 4pkd h1 h2 h3 @x1 h1 @x1 @x2 h2 @x2 @x3 h3 @x3 At the same time, in the vicinity of points where masses are absent, (d ¼ 0), in place of Equation (1.41) we have Laplace’s equation: r2 U ¼ 0

ð1:42Þ

Both Poisson’s and Laplace’s equations describe the behavior of the potential at regular points where the first derivatives of the field exist. To characterize the behavior of the potential at the boundary of media with different densities, let us make use of Equation (1.39) according to which a component of the field along some direction l is equal to the derivative of the potential in this direction: @U @l Thus, instead of the surface analogy of the field equations we obtain: gl ¼

ð1:43Þ

@U 2 @U 1 @U 2 @U 1 ¼ and ¼ ð1:44Þ @t @t @n @n where U1 and U2 are values of the potential at the back and front sides of the surface, respectively. It is obvious that the continuity of tangential derivatives of the potential follows from the continuity of the potential itself, and therefore the first equality of the set (1.44) can be replaced by much simpler one: U1 ¼ U2

ð1:45Þ

Note, the equality of the normal derivatives does not follow from the continuity of the potential, because the normal derivatives also depend on values of the potential above and beneath of the interface. Thus, the behavior of the potential is described by the system given below r2 U ¼ 4pkd and @U 2 @U 1 ¼ on S ð1:46Þ @n @n Certainly, the system of equations for the attraction field is much more complicated than that for the potential. Before we continue it may be appropriate to make the following comment. In all geophysical methods the fields, such as the particle displacement caused by elastic waves, the constant and time-varying electric U2 ¼ U1

20

Methods in Geochemistry and Geophysics

and magnetic fields can be expressed either in terms of the scalar or vector potentials or both of them. Respectively, all methods may be equally called the potential methods, and in this sense the gravitational method is not an exception.

1.7. THE POTENTIAL AND ITS RELATION TO MASSES Now we establish the relation between the potential U and masses. First, consider an elementary mass dm ¼ ddV. In accordance with Equation (1.8) we have gðpÞ ¼ k

dm 1 Lqp ¼ kdm gradp 3 Lqp Lqp

ð1:47Þ

since Lqp 1 ¼ gradp Lqp L3qp where the index p means that the gradient is considered in the vicinity of the point p. Comparing Equations (1.39 and 1.47) we can conclude that the function U, corresponding to the field of the elementary mass located at the point q, is UðpÞ ¼ k

dm þC Lqp

ð1:48Þ

because when the gradients of two functions are equal, then the functions may differ by at most a constant. Taking into account the fact that the field g caused by mass the dm tends to zero at infinity, it is natural to assume that its potential also vanishes if Lqp ! 1. Then from Equation (1.48) it follows that C ¼ 0 and we have: UðpÞ ¼ k

dm Lqp

ð1:49Þ

Next, applying the principle of superposition, we arrive at an expression for the potential caused by a volume distribution of masses: Z dðqÞdV UðpÞ ¼ k ð1:50Þ Lqp V Equations (1.6 and 1.50) clearly show that the potential is related to masses in a much simpler way than the field g, and this is the second reason for its introduction. Along with volume masses, it is possible to introduce other types of masses, and we have Z Z Z X mðqÞ dðqÞ sðqÞ lðqÞ UðpÞ ¼ k dV þ dS þ dl þ ð1:51Þ Lqp V Lqp S Lqp l Lqp where mi is an elementary mass and s and l surface and linear density, respectively. It is useful to discuss one more application of the potential, and with this purpose in mind we consider the change of this function, dU in the vicinity of some

Principles of Theory of Attraction

21

point. As is well known, dU ¼

@U @U @U dl 1 þ dl 2 þ dl 3 @l 1 @l 2 @l 3

ð1:52Þ

where dl1, dl2, and dl3 are elementary displacements along coordinate lines x1, x2, and x3. It is easy to see that the right hand side of this expression can be written as a dot product of two vectors, namely, dl ¼ dl 1 i1 þ dl 2 i2 þ dl 3 i3 and grad U ¼

@U @U @U i1 þ i2 þ i3 @l 1 @l 2 @l 3

ð1:53Þ

Here dl 1 ¼ h1 dx1 ;

dl 2 ¼ h2 dx2 ;

dl 3 ¼ h3 dx3

Therefore, dU ¼ dlgrad U ¼ g dl

ð1:54Þ

After integration of this equation along an arbitrary path with terminal points a and b, we obtain Z b UðbÞ  UðaÞ ¼ gdl ð1:55Þ a

Thus, the integral of the field g along a path is expressed by the difference of potentials at terminal points of this path. Certainly, it is much simpler to take a difference of the scalar at these points: U(b)U(a), than to perform an integration, and this fact demonstrates another advantage of using the potential.

1.8. FUNDAMENTAL SOLUTION OF POISSON’S AND LAPLACE’S EQUATIONS Now we return to Poisson’s and Laplace’s equations, which describe the behavior of the potential inside and outside masses, respectively. Earlier we have already derived an expression for the potential: Z dðqÞdV UðpÞ ¼ k Lqp V that is valid at any point. Therefore, this function is a solution of Poisson’s equation inside the masses, and it satisfies Laplace’s equation outside them. If an integration is performed over all masses then U(p) represents the fundamental solution of these equations. It is instructive to show in a different way that U(p) obeys these

22

Methods in Geochemistry and Geophysics

equations. For instance, in the case when the observation point is located outside the masses, that is, p6¼q, we have: Z 1 r2 UðpÞ ¼ k dðqÞr2p dV L qp V where r2p UðpÞ ¼

@2 1 @2 1 @2 1 þ þ @x2p Lqp @y2p Lqp @z2p Lqp

and 2 2 2 1=2 L1 qp ¼ ½ðxp  xq Þ þ ðyp  yq Þ þ ðzp  zq Þ 

Taking the first and then the second derivatives, we find: @2 1 2 5 ¼ L3 qp þ 3ðxp  xq Þ Lqp @x2p Lqp @2 1 2 5 ¼ L3 qp þ 3ðyp  yq Þ Lqp @y2p Lqp @2 1 2 3 ¼ L3 qp þ 3ðzp  zq Þ Lqp @z2p Lqp that is, r2

1 ¼0 Lqp

and correspondingly the Laplacian of the potential is equal to zero, if p6¼q. In a similar manner, but taking into the account an influence of masses in the vicinity of the point q, we will demonstrate later that U(p) obeys the Poisson’s equation inside the masses. By definition, the Laplacian of U represents the divergence of the attraction field, and, correspondingly, its value characterizes the density of masses at same point. Now the following question arises. What does the Laplacian tells us about the behavior of the potential? To answer this question we first consider the simplest case, when U depends on one argument, x, Fig. 1.7a. Then, we can represent the derivatives as:      dUðxÞ 1 Dx Dx ¼ U xþ U x dx Dx 2 2 and     dU x  Dx d 2 UðxÞ 1 dU x þ Dx 2 2  ¼ dx2 Dx dx dx

ð1:56Þ

23

Principles of Theory of Attraction

(a)

z

(b)

y

U(x)

x h = Δx x- Δx x x +Δx

x

Fig. 1.7. (a) Illustration of Equation (1.59), (b) elementary cube around point x, y, z.

Inasmuch as

 dU x þ Dx 1 2 ½Uðx þ DxÞ  UðxÞ ¼ Dx dx  dU x  Dx 1 2 ½UðxÞ  Uðx  DxÞ ¼ Dx dx

we have d 2 UðxÞ 1 ¼ ½Uðx þ DxÞ þ Uðx  DxÞ  2UðxÞ 2 dx ðDxÞ2

ð1:57Þ

d 2 UðxÞ 2 ¼ ½U av ðxÞ  UðxÞ 2 dx ðDxÞ2

ð1:58Þ

or

Here Uav(x) is the average value of the potential at the point p: Uðx þ DxÞ þ Uðx  DxÞ ð1:59Þ 2 Thus, the first derivative defines the rate of change of the function U(x), while the second derivative shows how its average value differs from the value of the function at the same point. For instance, if U 00 ðxÞo0, we have UðxÞ4U av ðxÞ and there is a maximum. Next, suppose that within some interval of x the average value of the function coincides at each point with the value of this function: U av ðpÞ ¼

U av ðxÞ ¼ UðxÞ

ð1:60Þ

and therefore d 2 UðxÞ ¼0 ð1:61Þ dx2 The last equation represents the simplest class of functions in the one-dimensional case, namely, the linear functions, for which the condition (1.61) is met. Correspondingly, we can say that the second derivative is a measure of how the behavior

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Methods in Geochemistry and Geophysics

of the function in the vicinity of some point differs from that of a linear function. In other words, the second derivative characterizes the curvature of a line describing the function. In order to generalize this study for three-dimensional case, imagine an elementary cube around some point p(x,y,z), so that its sides are directed along the coordinate lines, Fig. 1.7b. The length of each side is 2h. In accordance with Equation (1.58) the second derivatives along the coordinate axes are @2 U 1 ¼ ½Uðx þ Dx; y; zÞ þ Uðx  Dx; y; zÞ  2Uðx; y; zÞ @x2 ðDxÞ2 @2 U 1 ¼ ½Uðx; y þ Dy; zÞ þ Uðx; y  Dy; zÞ  2Uðx; y; zÞ 2 @y ðDyÞ2

ð1:62Þ

@2 U 1 ¼ ½Uðx; y; z þ DzÞ þ Uðx; y; z  DzÞ  2Uðx; y; zÞ 2 @z ðDzÞ2 Taking into account the fact that Dx ¼ Dy ¼ Dz ¼ h and substituting Equation (1.62) into the expression for the Laplacian: @2 U @2 U @2 U þ 2 þ 2 ¼ r2 U @x2 @y @z we obtain

" # 6 1 X U i  6UðpÞ r U¼ 2 h i¼1 2

or

2 r2 U ¼

6 P

3

Ui 7 66 6i¼1 7  UðpÞ 6 7 5 h2 4 6

ð1:63Þ

Here Ui is the value of the potential on ith face of the cube, while U(p) is its value at the center of the cube. It is clear that the term 6 P

Ui

i¼1

6 is the average value of the potential at the point p. Thus, the Laplacian can be written as 6 ð1:64Þ r2 U ¼ 2 ½U av ðpÞ  UðpÞ h that is, it is again a measure of the difference between the average value of the function and its value U at the same point. For example, if the average value exceeds the value of the function, the Laplacian is positive. In the case of the gravitational field the latter is always negative, that is, UðpÞ4U av ðpÞ, and this

Principles of Theory of Attraction

25

follows from the fact that the right hand side of Equation (1.41) is negative, if d6¼0. Now, making use of Equation (1.64) we obtain the simplest form of Laplace’s equation: U av ðpÞ  UðpÞ ¼ 0

ð1:65Þ

Therefore, if the function U satisfies the Laplace’s equation, then it possesses a remarkable interesting feature, namely, its average value calculated around some point p is exactly equal to the value of the function at this point. A certain class of functions has this feature only, and such functions are called harmonic. Correspondingly, we conclude that the potential of the attraction field is a harmonic function outside the masses. In accordance with Laplace’s equation the sum of the second derivatives along coordinate lines, x, y, and z, equals zero, provided that U(p) is a harmonic function. At the same time we know that in the one-dimensional case there is a class of functions for which the second derivative is equal to zero, that is, d 2U ¼0 dx2 From this comparison of the behavior of the second derivatives, it is natural to consider harmonic functions as an analogy of the linear functions and expect that they have similar features. Let us outline some of them. 1. It is clear that if values of a linear function are known at terminal points of some interval of x, then it can be calculated at every point inside. In the same manner, if a harmonic function is given at each point of the boundary surface surrounding the volume, it can be determined at any of its point. 2. If a linear function has equal values at terminal points of the interval, then it has the same value inside it, that is, the linear function is constant. By analogy, if a harmonic function has same value at all points of the boundary surface, then it has the same value at any point within the volume. Of course, both statements can be proved from the theorem of uniqueness for the attraction field. In addition, it is appropriate to comment: a linear function reaches its maximum at terminal points of the interval. The same behavior is observed in the case of harmonic functions, which cannot have their extreme inside the volume. Otherwise, the average value of the function at some point will not be equal to its value at this point, and, correspondingly, the Laplacian would differ from zero. At the same time, saddle points may exist.

1.9. THEOREM OF UNIQUENESS AND SOLUTION OF THE FORWARD PROBLEM Suppose, first, that a distribution of masses is known everywhere. Then, as was shown earlier, the potential of the attraction field is Z dðqÞ UðpÞ ¼ k dV ð1:66Þ V Lqp

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Methods in Geochemistry and Geophysics

(b)

(a) S

S

V

V •p

U = ϕ1

•

p

(c) S

•

p

U=C

U = ϕ2 n

M

Fig. 1.8. (a) Dirichlet’s problem, (b) Neumann’s problem, (c) the third boundary value problem.

and at regular points this function obeys Poisson’s equation DU ¼ 4pkd

ð1:67Þ

Besides, the potential and its first derivatives are continuous at boundaries where a volume density is discontinuous function. It is obvious that in this case the solution of the forward problem is unique. Now consider a completely different situation, when a density d(q) is given only inside some volume V surrounded by a surface S, Fig. 1.8a. Inasmuch as the distribution of masses outside V is unknown, it is natural to expect that Poisson’s equation does not uniquely define the potential U, and in order to illustrate this fact let us represent its solution as a sum: UðpÞ ¼ U i ðpÞ þ U e ðpÞ

ð1:68Þ

where Ui(p) and Ue(p) are potentials in the volume V caused by masses inside and outside the volume V, respectively, DU i ðpÞ ¼ 4pkd and

DU e ðpÞ ¼ 0

ð1:69Þ

Therefore, we can write DU ¼ DðU i þ U e Þ ¼ DU i þ DU e ¼ 4pkd

ð1:70Þ

This means that Poisson’s equation defines the potential with an uncertainty of a harmonic function Ue. Regardless of a distribution of masses outside the volume the potential Ue remains harmonic function inside V and, correspondingly, there are an infinite number of potentials U which satisfy Equation (1.70), and they can be represented as: Z ddV UðpÞ ¼ k þ U e ðpÞ ð1:71Þ V Lqp where the last term on the right hand side is unknown. It is clear that along with Poisson’s equation we need additional information, which allows us to find the

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Principles of Theory of Attraction

harmonic function Ue(p). It turns out that a behavior of the potential on the boundary S gives this information, and the latter are called boundary conditions. In essence, they contain the information about the distribution of masses outside the volume V. To find these conditions we will proceed from Gauss’s theorem, which is the natural ‘‘bridge’’ between values of the field inside the volume and those at the boundary surface: Z I I div XdV ¼ XdS ¼ X n dS ð1:72Þ V

S

S

Here n is the unit vector perpendicular to the surface S and directed outward, and Xn the normal component of an arbitrary vector X, which is a continuous function within volume V. As was pointed out, Equation (1.67) has an infinite number of solutions; let us choose any pair of them, U1(p) and U2(p), and form their difference: U 3 ðpÞ ¼ U 2 ðpÞ  U 1 ðpÞ

ð1:73Þ

To derive the boundary conditions, we introduce some vector function X(p): X ¼ U 3 grad U 3 ¼ U 3 rU 3 Substitution of Equation (1.74) into Equation (1.72) gives Z I rðU 3 rU 3 ÞdV ¼ U 3 rn U 3 dS V

ð1:74Þ

ð1:75Þ

S

where rnU3 is the component of the gradient along the normal n, and rn U 3 ¼

@U 3 @n

ð1:76Þ

Since r is a differential operator, we have rðU 3 rU 3 Þ ¼ U 3 r2 U 3 þ rU 3 rU 3 ¼ ðrU 3 Þ2

ð1:77Þ

because r2 U 3 ¼ DU 3 ¼ DðU 2  U 1 Þ ¼ 4pkd þ 4pkd ¼ 0 Taking into account Equations (1.76 and 1.77), we can rewrite Equation (1.75) as Z I @U 3 2 dS ð1:78Þ ðrU 3 Þ dV ¼ U 3 @n V S

This equality is a special form of Gauss’s theorem, and it will allow us to find several boundary conditions, which provide uniqueness of a solution of the forward problem. First, we make three comments: a. The volume V can be enclosed by several different surfaces. b. The integrand of the volume integral in Equation (1.78) is non-negative and this fact plays a vital role for deriving boundary conditions.

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Methods in Geochemistry and Geophysics

The equality (1.78) relates the values of the function inside the volume V to its values on the boundary surface S, and the harmonic function U3 is the difference of two arbitrary solutions of Poisson’s equation. Now we are prepared to formulate boundary conditions for the potential of the attraction field, which uniquely define this field inside the volume V. With this purpose in mind suppose that the surface integral on the right hand side of Equation (1.78) equals zero. Then Z ðrU 3 Þ2 dV ¼ 0 ð1:79Þ c.

V

and taking into account the fact that its integrand cannot be negative, we have to conclude that at every point of the volume grad U 3 ¼ 0

ð1:80Þ

This means that the derivative of function U3 in any direction l is zero @U 3 ¼0 @l that is, this function is constant, and therefore the derivatives of solutions of Poisson’s equation are equal to each other @U 1 ðpÞ @U 2 ðpÞ ¼ @l @l In other words, if the surface integral in Equation (1.78) vanishes, these solutions can differ by a constant only: U 2 ðpÞ ¼ U 1 ðpÞ þ C

ð1:81Þ

where C is constant, that is, the same for all points of the volume V, including the surface. In particular, this constant can be zero. Next, we will define conditions under which the surface integral vanishes: I @U 3 dS ¼ 0 ð1:82Þ U3 @n S

and correspondingly Equation (1.81) becomes valid. Several such conditions are described below. 1.9.1. The first boundary value problem Suppose that the potential U(p) is known on the boundary surface; that is, UðpÞ ¼ j1 ðpÞ on S

ð1:83Þ

and we consider a solution of Poisson’s equation that satisfies the condition (1.83). Let us assume that there are two different solutions to this equation inside the volume, U1(p) and U2(p), which coincide on the boundary surface: U 1 ðpÞ ¼ U 2 ðpÞ ¼ j1 ðpÞ on S

Principles of Theory of Attraction

29

Then their difference U3 on this surface becomes zero: U 3 ðpÞ ¼ 0 on S and, consequently, the surface integral in Equation (1.82) vanishes. Therefore, in accordance with Equation (1.81) solutions of Poisson’s equation satisfying the condition (1.83) can differ from each other by a constant only. Moreover, this constant is known and it is equal to zero since on the surface S all solutions should coincide. In other words, we have proved that equations: DU ¼ 4pkd inside V and UðpÞ ¼ j1 ð pÞ on S

ð1:84Þ

uniquely define the potential U as well as the attraction field g, since g ¼ grad U Equation (1.84) form Dirichlet’s boundary value problem, which can be either exterior or internal one, Fig. 1.8a, and it has several important applications in the theory of the gravitational field of the earth. It is worth to notice that in accordance with Equation (1.83) we can say that along any direction tangential to the boundary surface, the component of the field gt is also known, since gt ¼ @U=@t. Consequently, the boundary value problem can be written in terms of the field as curl g ¼ 0 div g ¼ 4pkd and gt ¼

@j1 on S @t

ð1:85Þ

This first case vividly illustrates the importance of the boundary condition. Indeed, Poisson’s equation or the system of field equations have an infinite number of solutions corresponding to different distributions of masses located outside the volume. Certainly, we can mentally picture unlimited variants of mass distribution and expect an infinite number of different fields within the volume V. In other words, Poisson’s equation, or more precisely, the given density inside the volume V, allows us to find the potential due to these masses, while the boundary condition (1.83) is equivalent to knowledge of masses situated outside this volume. It is clear that if masses are absent in the volume V, the potential U is a harmonic function and it is uniquely defined by Dirichlet’s condition. 1.9.2. The second boundary value problem Now let us assume that two arbitrary solutions of Poisson’s equation within the volume V; U 1 ðpÞ and U 2 ðpÞ, have the same normal derivatives on the surface S;

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Methods in Geochemistry and Geophysics

that is, @U 1 @U 2 ¼ ¼ j2 ðpÞ @n @n

on

S

ð1:86Þ

where j2 ðpÞ is a known function. From this equality it instantly follows that the normal derivative of a difference of these solutions vanishes on the boundary surface: @U 3 ¼0 @n

on

S

Therefore, the surface integral in Equation (1.78), as in the previous case, equals zero and correspondingly inside the volume we have rU 3 ¼ 0 This means that any solutions of Poisson’s equation, for instance U1(p) and U2(p), can differ from each other at every point of the volume V by a constant only, if their normal derivatives coincide on the boundary surface S. Thus, this boundary value problem defines also uniquely the field of attraction, and it can be written as ðinside V Þ

DU ¼ 4pkd and

@U ¼ j2 ðpÞ on S @n

ð1:87Þ

or curl g ¼ 0 div g ¼ 4pkd and gn ¼ j2 ðpÞ on S

ð1:88Þ

and it is called Neumann’s problem. Unlike the previous case, Equation (1.87) define the potential only to within a constant, but of course the field of attraction is determined uniquely. 1.9.3. The third boundary value problem We suppose that the boundary surface S is equipotential, that is, UðpÞ ¼ C

ð1:89Þ

on the boundary surface S, and C is some constant. In addition, it is assumed that the total mass M in the volume V, surrounded by the surface S, is known, Fig. 1.8c. As follows from the Gauss’s theorem: I I I @U dS ¼ 4pkM ð1:90Þ gdS ¼ gn dS ¼ @n S

S

S

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Principles of Theory of Attraction

This means that

I

@U dS ¼ j3 @n

ð1:91Þ

S

where j3 ¼ 4pkM is known. Now we will show that two solutions of Poisson’s equation, U1(p) and U2(p), satisfying Equations (1.89 and 1.90), can differ from each other by a constant only. Consider again the surface integral in Equation (1.78): I @U 3 dS U3 @n S

Inasmuch as the boundary surface is an equipotential surface for both potentials U1 and U2, their difference is also constant on this surface and correspondingly we can write 8 9 I I I