Theoretical and computational acoustics 2005: Hangzhou, China, 19-22 September 2005

Theoretical and Computational Acoustics 2005 Editors

Alexandra Tolstoy Er-Chang Shang Yu-Chiung Teng

World Scientific

Theoretical and Computational Acoustics 2005

Theoretical and Computational Acoustics 2005 Hangzhou, China

19 - 22 September 2005

Editors

Alexandra Tolstoy ATolstoy Sciences, USA

Er-Chang Shang University of Colorado, Boulder, USA

Yu-Chiung Teng Femarco, Inc., USA

\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

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THEORETICAL AND COMPUTATIONAL ACOUSTICS 2005 Proceedings of the 7th International Conference on ICTCA 2005 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

The seventh International Conference on Theoretical and Computational Acoustics (ICTCA) was held September 19-23, 2005 in Hangzhou, China. This meeting was sponsored by the China Hangzhou Association for International Exchange of Personnel (China), the U.S. Office of Naval Research (ONR), the U.S. Naval Undersea Warfare Center (NUWC), Columbia University (U.S.A.), Zhejiang University (China), Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (Italy), Hangzhou Applied Acoustics Research Institute (China), the Key Laboratory of Geophysical Exploration (CNPC, China), and the Hangzhou Municipal Government (China). The objective of this conference was, as usual, to provide a forum for active researchers to discuss state-of-the-art developments and results in theoretical and computational acoustics and related topics. It brought together researchers from numerous areas of science to exchange ideas and stimulate future research. The website is located at: www.ictca2005.com. Approximately one hundred scholars, scientists, and engineers from numerous countries participated in this event. The presented lectures examined topics in Underwater Acoustics, Mathematics, Scattering and Diffraction, Seismic Explorations, Genetic Algorithms, Reverberation, IFEM, Radon Transforms, Wavelet Statistics, Applications to the Oil Industry, Visualization, Shallow Water Acoustics, Gaussian Beams, Ocean Acoustic Inversion, and the Parabolic Equation (PE) with special emphasis on the work of the late Frederick D. Tappert. The conference committee wishes to particularly thank Dr. Ding Lee (NUWC and Yale University) as the founder and Honorary Chair of the conference. Dr. Lee continues to be the primary person in charge of where, when, and how these meetings will take place. Additionally, we would like to send special thanks to Anna Mastan for her tireless help in administrative tasks. Special thanks also go to the local organizing chairs: Xianyi Gong, Yonggusang Mu, Yu-Chiung Teng, and Sean Wu. We also would like to thank our three keynote speakers: Profs. Dan Givoli ("High-Order Absorbing Boundary Conditions for Exterior Time-Dependent Wave Problems"), Oleg Godin ("Sound Propagation in Moving Media"), and Michael Buckingham ("Inversions for the Geoacoustic Properties of Marine Sediments Using a High-Doppler, Airborne Sound Source"). The special sessions were of particular interest, and we gratefully acknowledge Drs. Sean Wu, Ding Lee, and E.C. Shang for the PE sessions. Additionally, there were sessions by Drs. Marburg and Nolte (FEM), Borovikov (Noise), Hui-Lian Ge (Scattering), J.M. Chiu (Seismic Acoustics), Godin (Shallow Water), Chapman (Applications), Taroudakis (3-D), Wu (Inverse Problems), Bjorno & Bradley (Wave Interactions), Gong (Modeling), Hanyga (Inversion), Chen (Underwater), Mo (Computational Acoustics), and Wang (Environmental Acoustics). We look forward to the eighth meeting scheduled in 2007 for Iraklion, Greece to be organized by Prof. Michael Taroudakis. v

ORGANIZING COMMITTEES AND SPONSORS Honorary Chair: Ding Lee U.S. Naval Undersea Warfare Center Newport Conference Chairs: Alexandra Tolstoy ATolstoy Sciences, USA Er-Chang Shang ORES, University of Colorado Boulder, CO, USA Yu-Chiung Teng Femarco, Inc., USA

Local Chairs:

Xianyi Gong Hangzhou Applied Acoustics Research Institute Zhejiang University Hangzhou, China Yongguang Mu Key Laboratory of Geophysical Exploration (CNPC) Beijing, China Yu-Chiung Teng Femarco, Inc., USA Sean Wu Wayne State University, USA Guohai Zhu Hangzhou Association for Science and Technology, China

Point-Of-Contact: Anna Mastan U.S. Naval Undersea Warfare Center Newport

Local Organizing Committee: China Hangzhou Center for International Exchange of Personnel Zhu Xue Feng Hou Wei Jie vii

Vlll

Sponsors: China Hangzhou Association for International Exchange of Personnel U.S. Naval Undersea Warfare Center (NUWC) U.S. Office of Naval Research (ONR) Columbia University Zhejiang University Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (SGS), Italy Hangzhou Applied Acoustics Research Institute, China Key Laboratory of Geophysical Exploration (CNPC), China Hangzhou Municipal Government

CONTENTS

Preface

v

Reconstruction of Sound Pressure Field by IFEM R. Anderssohn, St. Marburg, H.-J. Hardtke and Chr. Grossmann

1

Seabed Parameter Estimation by Inversion of Long Range Sound Propagation Fields W. Chen, L. Ma and N. R. Chapman High Resolution Radon Transform and Wavefield Separation /. Chen, Q. Li, P. Wu and B. Zhang

5 15

Three-Dimensional Acoustic Simulation on Acoustic Scattering by Nonlinear Internal Wave in Coastal Ocean L. Y. S. Chiu, C.-F. Chen and J. F. Lynch

23

Estimation of Shear Wave Velocity in Seafloor Sediment by Seismo-Acoustic Interface Waves: A Case Study for Geotechnical Application H. Dong, J. M. Hovem and S. A. Frivik

33

The Optimum Source Depth Distribution for Reverberation Inversion in a Shallow-Water Waveguide T. F. Gao and E. C. Shang

45

Semi-Automatic Adjoint PE Modeling for Geoacoustic Inversion J.-P. Hermand, M. Meyer, M. Asch, M. Berrada, C. Sorror, S. Thiria, F. Badran and Y. Stephan Modeling 3D Wave Propagation in the Ocean Coupled with Elastic Bottom and Irregular Interface L.-W. Hsieh, D. Lee and C.-F. Chen

53

65

Reflections from Steel Plates with Doubly Periodic Anechoic Coatings S. Ivansson

89

Seismic Characterization and Monitoring of Thin-Layer Reservoir L. Jin, X. Chen and J. Li

99

The Energy-Conserving Property of the Standard PE D. Lee andE.-C. Shang Estimation of Anisotropic Properties from a Surface Seismic Survey and Log Data R. Li and M. Urosevic Using Gaussian Beam Model in Oceans with Penetrating Slope Bottoms Y.-T. Lin, C.-F. Chen, Y.-Y. Chang and W.-S. Hwang IX

119

127 135

X

Application Niche Genetic Algorithms to AVOA Inversion in Orthorhombic Media M. -H. Lu and H. -Z. Yang Reconstruction of Seismic Impedance from Marine Seismic Data B. R. Mabuza, M. Braun, S. A. Sofianos and J. Idler

145 153

Characterization of an Underwater Acoustic Signal using the Statistics of the Wavelet Subband Coefficients M. I. Taroudakis, G. Tzagkarakis and P. Tsakalides

167

Some Theoretical Aspects for Elastic Wave Modeling in a Recently Developed Spectral Element Method X. M. Wang , G. Seriani and W. J. Lin

175

Inversion of Bottom Back-Scattering Matrix J. R. Wu, T. F. Gao and E. C. Shang New Methods of Scattering Coefficients Computation for the Prediction of Room Acoustic Parameters X. Zeng, C. L Christensen and J. H. Rindel

199

209

RECONSTRUCTION OF SOUND PRESSURE FIELD B Y IFEM R. ANDERSSOHN, ST. MARBURG and H.-J. HARDTKE Institut fr Festkrpermechanik, Dresden University of Technology, D-01062 Dresden, Germany anderssohn@ifkm. mw. tu-dresden. de CHR. GROSSMANN Institute of Numerical Mathematics, Dresden University of Technology, D-01062 Dresden, Germany This talk discusses an inverse problem of acoustic. T h e aim is t o reconstruct the sound pressure field of a cavity based on a small number of measurements. In the calculation, arbitrary admittance boundary conditions are considered. Therefore, the inverse formulation requires t o include the boundary admittance as a coefficient of the Robin boundary condition for the Helmholtz differential equation. In order to support a minimization of the necessary number of measurements, the new approach is based on an inverse formulation of the finite element method for the acoustical boundary value problem, of which its facility to extract a modal solution can be advantageous.

1

sure itself my be caculated from pressure measurements in the interior domain by solving a Dirichlet problem and computing an ill-conditioned inversion in a second step, but without quantifying the admittance boundary condition [11]. Hence, an algorithm is investigated to firstly calculate the surface sound pressure based on sound pressure measurements in the interior domain using a FEM formulation. The governing equations of the damped FEM acoustics as well as its well known forward solution are refereed to now, before we face the actual inverse problem. The boundary value problem

Introduction

This contribution reports about the progress on a FEM-based approach to solve the inverse acoustic problem of an internal space considering admittance boundary condition, called IFEM. In spaces bounded by structures with complex geometry it is difficult to measure directly the boundary admittance, that is an essential parameter for acoustic simulations. The authors have not found any methods in literature to globally estimate the boundary admittance of arbitrarily shaped cavities by using inverse methods. Apparently, in inverse acoustics two major types of algorithms are developed usually to detect sources. There is the near-field acoustic holography (NAH) [1,2] and the inverse frequency response function (IFRF), also called inverse boundary element method (IBEM) [3-8]. A so-called hybrid NAH [9] was developed to combine the advantages of NAH and IBEM. Although this method is applicable to arbitrarily shaped surfaces, it does still not consider or calculate admittance boundary conditions. One major motivation for our rather exceptional FEM-based investigations in inverse acoustics is based upon the capability of FEM to extract modal information. This feature shall be used to decrease the experimental expenses. Owing to its properties, an orthogonal modal basis might be better suited for sound field reconstruction than other basis functions. Further, it is shown in principal that boundary admittance can be explicitly evaluated based on the surface sound pressure [10]. The surface sound pres-

Ap(x) + k2p(x) = 0, p,n{x) = sk[vs(x) + Y(x)p(x)],

i£flcHd x 6 r , s — ipoc, (1)

i.e. the Helmholtz differential equation together with the Robin boundary condition, describes the sound pressure field p{x) at wavenumber k of a one-way structure-fluid interaction model for a cavity. The fluid properties are given by the density p0 and the speed of sound c. The boundary condition incorporates the surface velocity vs as well as damping, elasticity and mass influence of the boundary T through the complex coefficient Y, the boundary admittance. The discretisation of the acoustic boundary value problem by means of FEM results in {K -k2M

-ikD)p

= b,

(2)

with the stiffness, mass and damping matrices K, M and D, respectively. The matrices are of size

1

2 NxN where N is the number of nodes. T h e excitation appears in 6 = skFve, where F denotes the b o u n d a r y mass matrix. T h e damping m a t r i x can be formally written as D = pocYF, whereas D is actually a superposition of element boundary mass matrices with Y being constant on each b o u n d a r y face. Herein, we assume the admittance to be constant over frequency to enable a modal solution. T h e modal forward solution of Eq. (2) 2N~N7„d

p=- E

wTTkwr'-

Inverse P r o b l e m

T h e objective of this section is t h e inverse formulation of Eq. (2) which is to be used to reconstruct t h e whole sound pressure field from pressure measurements pm located at internal grid points. T h e pressure at the remaining internal nodes pj and at the boundary pb are to be estimated without knowledge a b o u t the b o u n d a r y admittance. Eq. (2) is rearranged in terms of the type of t h e nodes [bb + ifcD b b p b ~ = 0 Pf \j>m\ L 0

\Pb~\

(4)

with submatrices G y = K,j — k2Mij, {i,j} = {b, / , m} . Motivated by t h e observations t h e Dirichlet problem in [11] we extract only t h e lower row of submatrices and have to solve t h e incomplete but linear system of equations G ffff Gf„ Gmf Grr

Gfh Gmb

Pb-

(5)

Gn An eigenvalue analysis of the symmetric system matrix GD of the Dirichlet problem provides us with a set of global and orthogonal basis functions t h a t are used t o express G^, 1 . After some rearrangement we end up at the linear equation (A-h?B)pu

= q

Solution Techniques

A and B are static matrices so as to allow for a modal solution. But they are, depending on the overor under-determination of t h e problem, rectangular and as most inverse problems heavily ill-conditioned, impeding sensible results of a modal superposition. Instead, a Tikhonov regularization

\\QPv-q\\2

(ft ^ 0).

' G j i Gbf Gbml Gfb Gff Gfm _Gmb Gmf Gmm\

3

pj].

(3)

is obtained by a superposition of eigenvectors Vi t h a t are computationally produced via t h e solution of t h e general linear eigenvalue problem of the state space transform of Eq. (2). Nrea defines the depth of modal reduction; « ; and ft are the eigenpairs t h a t are connected to t h e eigenvalues A; through Ai = Minimize

(4)

Derivate m to obtain the least-squares solution: m = (LTL)~1

LTd = (llLYm0

(5)

m 0 is low resolution velocity stack computed by transposed or adjoint operator. m is relatively high resolution result by least-squares inversion.

2.2

High resolution hyperbolic RT

Hyperbolic RT calculated by least-squares inversion method still possesses low resolution. In order to improve the transform resolution further, hyperbolic RT domain is

17 converted to sparse pulse equation, namely, solving sparse solution. For hyperbolic RT, value and constraint of sparse solution are solved in X — q domain, generally there are many standards to determine sparse property. This paper takes Cauchy class standard, make object function minimum:

7=||rf-L»i| 2 +//^ln(l + m t 7*)

(6)

k

Here mk is the element of m obtained by RT, JU and b distributed parameter accordingly, object function derivates m, we obtain: LTLm-LTd+Qm=0 Here Q is a diagonal matrix, the diagonal element in Q is:

& ~V7^

(7)

(8)

b+mt m may be get from (7):

m=(llL + QY LTd = (LTL + Q)~lm0

(9)

Here mo = L d is the result of low resolution RT gained by adjoint or conjugate transposed operator, namely without removing space convolution effect. Multiplying operator IL L + QI

, we get the result of high resolution RT.

mk =(LTL + Qk_ym0 Expression (8) substitutes in expression (10), we get specific solution: mk=

2u * + i»i?-iJ

IJL + -—?—— m0

(10)

(11)

k is iteration times. For practical CSP trace, matrix size and amount of calculation are very great if solving directly, when introducing conjugate gradient method, matrix L and iJ need not to be stored, even the matrix may be omitted and operates vector directly. One point of operator L in T —q domain corresponds to one hyperbolic event in t — x domain, data along hyperbolic trajectory in t — x domain is scanned and stacked into a series of points by operator L . Given primary problem: y — Lx, x' = L y,taking conjugate gradient method to solve min\ |Lx— v|| \, to overdetermined problem we give least squares method solution, to under-determined problem we give least norm solution. 2.3

Conjugate gradient (CG) algorithm

CG algorithm is also called conjugate inclined survey method, it is an orthogonal projection method, its convergence is assured and processing procedure needs very limited work space only, iterative operand is also very limited each time and matrix multiplying or dividing can be avoided, which make it compute rapidly and efficiently.

18 Let b replace y, A replace L ,so y = Lx becomes b = Ax , the following three steps describe the procedure of CG algorithm: Take arbitrarily *' 0 ' e Rn Let r®=b-Ax®,P®=r® While k = 0,1,..., do iterations:

(r(k)yk)) x'k+l)=x(k)+akp{k) r(*+D = r (*>

A=

-akAp(k)

(r{k+l),r(k+i)) (r(*\r(t))

/,=r ( k + 1 ) +&/, If r W = 0 (/>W,ApW) = 0 ,

or (p(*),Ajp'*)j = 0 , the algorithm stopped, If residual vector let

JCW=JC*.

If ( p W , A p W ) = 0 , we have p W = 0 , namely

(/•(*),rWj = (rW,pWj = 0, and rW = 0 , since A is forward defined. Obviously, the problem with n equations can be solved with CG method by n steps theoretically and exact solution will be reached, It is a direct algorithm actually. 2.4

Workflow of high resolution hyperbolic RTfor wavefield separation

The following eight steps describe the Work flow of high resolution hyperbolic RT for wavefield separation (1) Enter t-x seismic record (2) Set initial value and compute operator matrix (3) Compute initial velocity stack matrix m (4) Solve sparse linear set of equations using CG iteration method (5) Output (T-q) (6) Discriminate and separate wavefield in {z — q) (7) Radon inverse transformation (8) Output seismic record having separated in t-x domain 3 3.1

Testifying of the Theoretical Models and Practical Data Theoretical models Figure 1 is a three horizontal layers geologic model. The depth of each layer is H x =

450 m, H 2 = 1050 m, H 3 =1550 m, P-P-wave velocity of the first layer is Vpi = 2500 m/s, that of p-s-wave is Vsl = 1000 m/s.

19

450 1050 1550

Vp = 2500m/s Vs = 1000mIs

-+x(m)

Vp = 3000mIs Vs = 1500mIs Vp = 3500m Is Vs= 2000m /s

z(m) Fig.l. Theoretical model with three horizontal layers.

Second

layer

is

VP2 = 3000 m/s,

Vs2 = 2000 m/s

and

third

layer

IS

Vp3 = 3500 m/s , VS3 = 2000 m/s . Figure 2 is synthetic theoretical CSP data from figure 1. Each layer generates both P-P reflected wave and P-SV converted wave, and we artificially add a very strong linear interference with low velocity (900 m/s), other parameters of the data include: 120 traces, 40 m minimum offset, 25 m trace interval, 2 ms sampling rate, and total 2000 ms recording length. Figure 3 is the result of hyperbolic Radon forward transform from the CSP gather. Because q value of the linear interference is great in hyperbolic RT, it can be easily separated, P-P reflected waves and P-SV converted waves are indicated by different "points" in hyperbolic RT domain. All of which can be clearly distinguished. Figure 4 and figure 5 are P-P reflected wave and P-SV converted reflected wave separated respectively by hyperbolic RT, it can be seen that the separation effect is quite satisfactory, the waveshape being distorted very slightly with high fidelity, and the linear interference being eliminated completely. The OBC multi-component seismic data were acquired along all offset line in China. Trace /No 12 24 36 48 60 72 84 96 108

Fig. 2. Synthetic seismic record of CSP gather from Fig. 1.

750

Fig. 3. Hyperbolic RT from the CSP data of Fig. 2.

20 Trace /No 12

24

36

48

60

Trace /No 72

84

96

108

10 0

20

30

40

50

60

70

-

200400600-

Fig.4. P-P reflected wave separated by hyperbolic RT. Fig.5. P-SV reflected wave separated by hyperbolic RT.

3.2

Practical data

Seismic exploration data picked up by multi-component in sea domain. Figure 6 is time stacked section of the vertical component. Figure 7 is the section after adding hyperbolic RT in processing. Compared Fig. 7 with Fig. 6, we can see that the resolution get higher obviously at 600ms, 1500ms, 2000ms, and the resolution is higher obviously, especially within the three ellipses in Fig. 7.

I

CDP/No Fig.6. Time stacked section before RT (OBC data, z component).

21

CDP/N Fig.7. Time stacked section after RT, P-SV converted wave of z component has been suppressed the resolution has been improved. Such as the three ellipses I, II, III.

4

Conclusions

With theoretical models and practical data, we get the concluded remarks, hyperbolic RT is a wavefield separation method which can keep true energy and owns high resolution, owing to t0 and q are different in different wavefields, they can be easily separated in the transformed domain, and this method has the characters as follows: (1) Noises can be eliminated simultaneously when separating wavefields, nonreflected interferential signals can be suppressed. (2) Not only reflected p-wave can be separated, so can be done to converted waves and multiples. (3) Keep energy fidelity in processing. Because each trace ^transformed domain corresponds to determined stack velocity, velocity can be analyzed when we take wavefield separation in transformation. Reflected waves are completely focused in transformed domain, so imaging can be done directly in transformed domain and needs not to be transformed into t-x domain. Reference 1. Q. Li, High Resolution Hyperbolic RT Multiple Removal, The University of Alberta, 2001. 2. Y. X. Liu and Mauricio D.Sacchi, De-multiple via a Fast Least-squares methods Hyperbolic RT, SEG IntT Exposition and 72nd Annual Meeting, 2002.

22 3. Q. S. Cheng, Mathematical Principle of Digital Signal Processing, Petroleum Industry Press (1982). 4. S. Y. Xu, Wavefield separation with T — q transform method, China Offshore Oil and Gas (Geology) 13(5) (1999) 334-337. 5. X. Y. Sun, The separation of P- and S-wave fields using X — q transform method, Petroleum Geology & Oilfield Development in Daqing 21(4) (2002) 76-79. 6. X. W. Liu, High resolution Radon transform and its application in seismic signal processing, Progress in Geophysics 19(1) (2004) 8-15.

THREE-DIMENSIONAL ACOUSTIC SIMULATION ON ACOUSTIC SCATTERING BY NONLINEAR INTERNAL WAVE IN COASTAL OCEAN LINUS Y. S. CHIU, CHI-FANG CHEN Department of Engineering Science and Ocean Engineering, National Taiwan University E-mail: cvs(3),uwaclab. na. ntu. edu. tw JAMES F. LYNCH Woods Hole Oceanography

Institute

Nonlinear internal wave (NIW) packets cause ducting and whispering gallery effects in acoustic propagation. The acoustic energy restricted within the internal wave crests (crest-crest) on the shelf is the ducting effect, and the energy confined along the crest when the source is located upslope from the NIW crest is the whispering gallery effect. This paper presents the simulation results concerning the phenomena of whispering gallery by FOR3D wide-angle version. It appears that energy emerges right before and along the wave crest and then vanish right in the back of the wave crest and then converges again, especially with the lower frequency band (150Hz~ 600Hz).

1

Introduction

The influence on the amplitude and phase of an acoustic field propagating through the shallow water waveguides is significant over relatively short ranges while the sound speed fluctuations in the region are typically less than one percent of the mean speed [15]. Our interest in this behavior stems from two points related to underwater communication system or the sonar performance. It is that signal detection is a function of the signal-to-noise ratio and is affected by transmission loss (TL) variability caused by sound speed perturbations with internal wave. Recent papers and experiments address the acoustic field is fluctuated by the nonlinear internal waves (NIW). We have seen the energy distribution has specified modification due to the acoustic mode coupling while the sound propagates across the internal wave. They also cause ducting and whispering gallery effects as the sound propagates along the internal waves. The acoustic energy restricted within the internal wave crests (crest-crest) is the ducting effect [6], and the energy confined along the crest when the source is located at the upslope region relatives to the NIW crest is called the whispering gallery effect. Computer simulation offers a practical method for systematic assessment of TL and coherence degradation in complex ocean environments. This approach is applied here, where the TL and azimuthal spatial coherence are estimated for frequencies band of 50800 Hz as a function of range, depth, and azimuth in shallow water, continental shelf environment under summer condition. Sound speed fluctuations considered in this paper are induced by an internal gravity wave field that perturbs the thermocline. Some recent theoretical efforts have considered the effect of internal wave induced phase decorrelation on horizontal arrays in both deep and shallow water environments under a variety of modeling assumptions [6]. Our analysis differs from those of previous studies in that we employ a simplified, data-constrained internal wave model which is observed in the ASIAEX experiment, South China Sea (SCS) component that includes a azimuthal anisotropic component, and 23

24 apply 3D acoustic modeling techniques (FOR3D with wide angle version) to estimate TL and azimuthal coherence in this environment. This paper presents evidence that acoustic field can be significantly affected in an environment supporting oceanographic features that break azimuthal symmetry. Such affection might not be predicted by N x2D calculations since they ignore horizontal refraction and may thus produce misleading TL and azimuthal coherence in these environments. This paper also addresses and quantifies the whispering Gallery Effect induced by the internal solitary wave in the typical continental slope region. We have two basic results: 1. Enhanced energy emerges right before and along the wave crest and then vanishes right in the back of the wave crest and then converges again, especially with the lower frequency band (150Hz~ 600Hz). 2. Scattering of sound due to the internal solitary waves brings about much worse azimuthal coherence in the time scale of 25 minutes. The azimuthal coherence is better in lower frequencies and increasing depth. In Sec. II we briefly review the simulation scenario and analysis approach. In Sec. Ill, we give the results and implementation of numerical experiments, which estimate energy distribution in the 3D field, the adaptive depth-averaged acoustic energy and azimuthal coherence under several conditions. The summary and conclusions are presented in Sec. IV.

2

Simulation Approach

Three dimensional effects on underwater acoustic propagation have been frequently reported [7-9]. The acoustic propagation model is based on the 3D parabolic approximation to the Helmholtz equation implemented in the computer code FOR3D [9]. This code implements a finite difference solution scheme, using discretized differential operators to represent wide-angle propagation in elevation and narrow-angle azimuthal coupling. The major causes for the 3D effects are variations in azimuth of bottom topography and/or water column properties [10-13]. Experiment site in South China Sea is of a similar nature, in that both bathymetry and horizontally anisotropic water column properties contribute to horizontal refraction of energy. Details of the simulation scenario and parameters are given in the Fig.l (a). The acoustic point source which placed in the upper water column is assumed to be a tow acoustic source. The emitted sound propagates in the wedge bathymetry which slope is equal to the 1/20. Superimposed on the sound speed volume is an observed internal wave in South China Sea which causes very large thermocline depressions even to 85 meters from thermo. Internal wave propagated onshore from 2 kilometers far from the source until the 0.5 kilometers. The dynamic elevations of the internal wave due to the onshorepropagating are ignored in this time scale of only 25 minutes. Finally the bottom parameters are constant in range and selected from a somewhat very hard, sandy bottom. And the density is set to be twice that of the water density.

2.1

Transmission loss

Nonlinear internal wave fields introduce significant azimuthal transfer of energy. Acoustic field calculations performed through a set of 2D range/depth planes (a.k.a. Nx2D computations) for different azimuthal directions allow for variations in sound speed within range/depth planes but ignore horizontal refraction of energy between adjacent

25 planes. The 3D calculations presented here include such azimuthal coupling, if present, and can be used to assess the relative importance of horizontal refraction in complex oceanographic environments. A rather simple means of estimating the amount of azimuthal energy transfer is outlined here and used to interpret TL and coherence results. Define a adaptive depth-averaged acoustic energy density (ADAAE) E: ) \u\ I p(z)dz (1) C,H where H is the total depth(or arbitrary depth) of water column and sediment, and c 0 is a nominal reference sound speed. The depth averaged, or mean TL, 7Xz([l] and [4]), is E = E{r,$)

TLZ =10 1og 1 0 £

(2)

where E has unit of energy per area. Source Depth

/

^ ^ '

8 km

U

~~~~~\

2 km

\n/v :

1.2 km 400m

= 20m

Slope

-1/20

Frequency

- 50-800 Hz

dr

= A/10

dz

= A/10

e

- 180'

dG

= 1"

Amplitude of I W s

= 85 meters

(from thermalcline)

j

Ave. Phase speed of I W s = 0.8 m/s IW's propagating timing = 25 min.

Li

(a)

IWs

BHtt! (b)

Figure 1. The details of simulation scenario and parameters. Internal wave propagated onshore from 2 kilometers far from the source until the 0.5 kilometers. 2.2

Azimuthal Coherence

Azimuthal coherence is a three-dimensional function which includes the parameters r, z, P (range, depth and azimuth). The complex pressure field U (r, z, /?) (azimuthally

26 across the slope) is correlated with its value at j3 = 0 (along the slope) and temporally averaged and then normalized as: u(r,z,9(f)u(r,z,JJ))\

C

^^P)-

(3)

^|M(r,z,90f)|2J}(| u{r,z,P)

See also Fig.l (b). The angle brackets represent the time average over environment snapshot (-25 minutes). This dependence on integration time is due to the non-stationary nature of the sound speed filed induced by the internal waves. 3 3.1

Implementation and Results Results of Transmission loss

This section presents some results of acoustic calculations for TL, ADAAE, effect of whispering gallery and azimuthal coherence. TL examples presented here are the single environment snapshot while the internal wave is right at 1 km from source and the frequencies are of 50 Hz and 150 Hz. They are shown in Fig.2 (a), (b) as a function of range and azimuth at specified depth. ((a)30 meters, (b)50 meters). The first column in each figure is the case for imposed internal wave; the middle is the case for background sound speed profile (without the internal wave) and the last is the difference between the previous two columns. In (a) and (b), the middle ones are the typical bench mark of three-dimensional wedge problem. Clear see that energy distribution is no longer circle-like but curved and bend down to the deeper water region. But the internal wave comes in (propagates onshore), they may induce the oceanic waveguide so as to concentrate the energy near or along the boundary, as shown in the left column of Fig. 2. Such concentration of energy near the boundary is completely analogous to the whispering gallery modes. The right column in Fig. 2 shows the enhanced horizontal refraction induced by the internal wave since the phasing and the amplitude of the interference pattern has changed.

SOHz

S!S

l

150Hz

1

I I I

27

(b) Figure 2 Transmission losses at specified depths, ((a) 30 meters, (b) 50 meters). The left columns show the case of incoming internal wave; the middle ones are the cases of background sound speed profile and the right ones are the differences. 3.2

Results ofADAAE

In order to clearly see the redistribution of energy caused by the internal wave, the ADAAE is utilized where H = 50 meters and shown in Fig. 3. The averaging depth of 50 meters is chosen to see the acoustic scattering of upper water column induced by the incoming internal wave. Fig. 3(a) and (b) are the cases for increasing frequencies; the cases for the incoming internal wave are shown in the left column and the ones for the background sound speed profile are shown in the right column. Only the results of 150Hz, 200Hz, 700Hz and 800Hz are shown here. Fig.3 illustrates the enhanced energy occurring near and along the boundary which is regarded as the oceanic waveguide induced by the nonlinear internal wave, especially in Fig. 3(a). For lower frequencies (50-600Hz), the modal interference pattern of energy and the scattering effect are clearly seen since the source may excite only lower modes, but the pattern are getting disordered (see Fig.3. (b)) due to the higher modes excited at higher frequencies. The effect of enhanced energy along the boundary of the internal wave has been gradually smeared (not shown here) while the averaging depth is increasing. This tells that the whispering gallery effect mainly occurs in the upper water column so that the effect is smeared with the increasing averaging depth. 3.3

The Effect of Whispering Gallery

The quantity has been defined for describing the effect of whispering gallery since the enhanced energy is horizontally stratified induced by the internal wave. The parameter WG (y k, w) is defined as the function of the distance relative to the source, yk , and frequencies G7, see also Fig. 4. f^DAAE&^m) WG(yk,m) = ^ N

, * = 1,2,3,...

(4)

28 PFGL = 1 0 1 o g 1 0 ( ^ G / ^ ) - 1 0 1 o g 1 0 ( ^ G S G )

(5)

where the energy distribution of ADAAE is summarized and averaged. Since the WG (yk,m ) i s t 0 describe the energy horizontally distributed in the computed sound volume while the averaging depth H = 50 meters, the difference between logarithmic scale of WG (yt,m) of the case with imposed internal wave and of the background case can clearly represent the scattering effect induced by the internal wave. Fig.5 clearly displays the effect of whispering gallery along the boundary and the shadow region in the back of wave-crest. Enhanced energy emerges right before and along the wave crest and then vanishes right in the back of the wave crest and then converges again, especially with the lower frequency band (150Hz~ 600Hz).

I

IW's

BG

far-

I

(a) BG

IW's

i&» I

?°

-§

800Hz

i 750Hz 5pfcV

i

£

*$$£]

fc

I

?Jq£

(b) Figure 3. (a) and (b) are the results of ADAAE for increasing frequencies; the left columns are the results of incoming internal wave and right ones are the results of background sound speed profile. The averaging depths are 50 meters.

29

£M£,(V„(»)

y* %DAAE,(yk,co) -,

*=5

Figure 4. The parameter WG{yk, w) is defined as the function of the distance relative to the source, yk , and frequencies XU . Whispering Gallery Effect (Y, Freq)

Freqency (Hz)

Figure 5. The effect of whispering gallery, a function of yk and frequencies. Enhanced energy emerges right before and along the wave crest and then vanishes right in the back of the wave crest and then converges again, especially with the lower frequency band (150Hz~600Hz). 3.4

Results ofAzimuthal Coherence

As shown in Fig. 1 (b), the azimuthally coherence is obtained by correlating the complex sound field in each slice of different azimuth with the values of the slice of u (r, z, 90 ") and averaged over 25 minutes. The azimuthal coherence is to see the signal coherence between the acoustic channel along the same isobaths of the source and the others which are in the upward (or downward) slope bathymetries. The azimuthally coherence will provide important information for communication or detection about how they treating the targets at different azimuths and evaluating the results, especially in the threedimensional environment.

30 Fig. 6(a) and 6(b) show the cSw(r\>G>i>...>wm; O is a zero matrix with dimension (mx(n-m)); u, is the z'th column of U and w,- they'th column of W. There is an effect of ill-conditioning in the numerical solution of this inverse problem and regularization theory is used to reduce the effect with the modified normal equation v / ={TTT + yHTHy{TT\c. (5) where H, a square matrix with dimension (mxm), is a general operator embedding the a priori constraints imposed on the solution, and the regularization parameter y > 0. The regularized solution is given by v;=rfvc (6) with r f = W(Cl + i1-'y(HW)T (HW)Yl UT

(7)

In this practical case, we have used H = I, unit matrix, and hence 7* = W(Q. + jQ'[)"' UT. Figure 2 shows a flowchart of the inversion algorithm. The input data consist of the measured dispersion curves from the previous processing described above, and the structure of an earth model. This earth model has a given number of sediment layers; the parameters are the compressional and shear wave velocities and the densities. In the

37 current application the layer thickness, here is 2 meters, the compressional velocities and the densities are fixed; only the shear wave velocities of the layers are assumed unknown at the initialization. The parameters of the earth model are inputted to a forward acoustic model (based on the scheme by Takeuchi H. et al [12]) which calculates a synthetic dispersion curve. The measured and synthetic dispersion curves are compared by an objective function, $ys) = ||7Vs-vc||2 and the objective function is minimized by varying the free earth model parameters, in this case the shear wave velocities of the layers. This iteration is continued until an acceptable fit between the measured and the synthetic dispersion curve is obtained. The results are estimates of the shear wave velocities of the sediment layers together with an uncertainty analysis. M e a s u r e d rlisnersion curve from

Initialization

analysis of the interface waves

'' Adjusted parameters

''

Synthetic data by forward modeling

Objective function

i L U

4

Feedback

No

Yes

Inversion results

Figure 2. Flowchart of the inversion algorithm with five parts: initial environmental parameters' setting, acoustic forward modeling, comparison of the synthetic data computed by forward acoustic model and observed data, here the dispersion curve of the group/phase velocity extracted from the interface waves, by an objective function, minimizing the objective function by varying the environmental parameters and uncertainty analysis for inversion results, here shear wave velocity.

5

Data analysis

Figure 3 shows in the left panel the time signal traces recorded on the 24 hydrophone array from one particular shot PI-86 at the Steinbaen site. The left panel shows the raw data with the full frequency bandwidth. The middle panel shows the high pass filtered and zoomed version of the same traces, the high frequency filtering emphasizes the refracted arrivals and we can observe an refracted arrival which determines the compressional wave velocity of the upper sediment layer to approximately 1515 m/s. In the right panel the raw data have been low pass filtered, which brings out the interface waves. The amplitudes of the interface waves are weak compared to the water borne modes (compare left and right panels in Fig 3), but one can clearly observe that the velocity of the interface waves is in the range of 40 to 100 m/s The low pass filtered versions of each of the recorded traces are used in die subsequent dispersion analysis using the time-frequency analysis of the Wavelet Transform (WT), and time is converted to velocity since the distance between source and the receiver is known. An example of the analysis of trace no 10 is shown in figure 4 as group velocity as function of frequency. Both panels show the same results but presented in two different ways. The left panel shows a colormap and in the right panel the data are plotted in contour plot. From the latter representation it is easy to detect the maximum values along the each contour as indicated by the stars in the left panel of figure 4. The detected maximum values are the samples of the dispersion curve that will be used as the measured data in the inversion algorithm of figure 2.

38

B fli^lf J [1 '

"J iffi

° 0.05 0.1

1

i

I

0.15

! 1

,~t-

"5 0.5

4 6 Frequency [Hz]

100 150 Shear wave velocity [m/s]

200

2 4 6 Principal components of T

Figure 9. Inversion results of multi-trace analysis of shot PI-86 of Steinbaen site. Top left: Extracted group velocity vs. frequency from the dispersion contour plot (stars) and the model fit as continuous line; Top right: Singular values from SVD and the singular values to the left of the blue thick line are larger than the regularization parameter y. The corresponding singular vectors, that constitute the resulting shear wave velocity vector, are marked with blue in the bottom right panel; Bottom left: Shear wave velocity vs. depth with blue thick line, and error estimates with red thin line.

42 In figure 10 the results of the inversion are compared with the measured values from the core samples. The inverted shear wave velocity by multi-trace method is represented by a blue line with squares, the inverted results of trace-10 and trace-15 by single-trace method are plotted in red line with triangles and pink line with circles. The results from the core testing are denoted by black star and diamond. As mention before we had only two values of the shear wave velocity from the core samples, at depths 2.5 and 2.7 meter, but these compared quite well with the inversion results. The difference is in order around 12% compared with the results of the single-sensor method and about 24% compared with the multi-sensor method. In both cases the measured core values are higher than the values of obtained by inversion of the interface waves.

20

Array Trace 10 Trace 15 Core GCS7 samp.1 Core GCS7 samp.2

•&—Hi -

24

•26

30

60

80

100 120 Shear wave velocity [m/s]

140

160

Figure 10. Comparison of the shear wave profiles from analyzing data from Pl-86 at Steinbaen and the geotechnical results at the same location. Blue line with squares for multi-trace method; Red line with triangles for trace-10 by single-trace method; Pink line with circles for trace-15 by single-trace method; Black star for geotechnical site testing of core GCS7 sample 1 and black diamond for geotechnical site testing of core GCS7 sample 2.

6

Conclusion

This paper has used different methods to extract the dispersion curve of the interface wave and estimated shear wave velocity profile as function of depth by inverting the dispersion curves. The single-trace method gives the dispersion of the group velocity by using the wavelet transform which has a continuously varying filter bandwidth and provides better velocity-frequency resolution imaging compared with the Gabor analysis. The single-sensor method can also be utilized to study velocity variations with range. The multi-trace method estimates dispersion of phase velocity by principal components method and assumes seabed parameters are range independent beneath the receiving array. The estimated shear wave velocities from the different dispersion curves by using single-sensor and multi-sensor method, respectively, are in good agreement and the

43 comparison with the geotechnical site testing shows a good agreement with an uncertainty of less than 12% in the single-sensor method and less than 24% in the multi-sensor technique. Most important is that the analysis and inversion of recorded interface waves give estimates of shear wave velocity as function of depth in the bottom, in this case down to 10 meters in the sediment. 7

Acknowledgements

We would like to acknowledge valuable help from Robert Hawkins and his team in FUGRO LTD and Ole Chr. Pedersen and Arild Olsen in Geomap AS for acquiring the data. Furthermore, we appreciate valuable discussions with Rune Allnor and 0ystein Korsmo. References 1. Allnor, R., Seismo-acoustic remote sensing of shear wave velocities in shallow marine sediments, PhD thesis, Rapport no.: 420006, Norwegian University of Science and Technology, (2000). 2. Caiti, A., Akal, T. and Stoll, R.D., "Estimation of shear wave velocity in shallow marine sediments", IEEE Journal of Oceanic Engineering (1994), 19, pp. 58-72. 3. Dziewonski, A.S., Bloch, S. and Landisman, M. A., "A technique for the analysis of transient seismic signals", Bull. Seismol. Soc. Am. 59, pp. 427-444 (1969). 4. Frivik, S.A., Determination of shear properties in the upper seafloor using seismoacoustic interface waves, PhD thesis Norwegian University of Science and Technology, (1998). 5. Fugro LTD, Field and in-situ testing report: Geotechnical site investigation, Asgardstrand, Jeloya and Steinbaen, Oslofjord, Fugro report 55083-2 (Final report), London (1999). 6. Fugro LTD, Advanced Laboratory Report: Geotechnical site investigation, Asgardstrand, JelBya and Steinbaen, Oslofjord, Fugro report 55083-3, London (2000). 7. Jensen, F. B., and Schmidt, H., "Shear properties of ocean sediments determined from numerical modeling of Scholte wave data" In Ocean Seismo-acoustics, Low frequency underwater acoustics, pp. 683-692, ed. by Akal, T. and Berkson, J. M. (Plenum Press, 1986) 8. Kritski, A., Yuen, D.A. and Vincent, A. P., "Properties of near surface sediments from wavelet correlation analysis", Geophysical Research letters 29, (2002). 9. Land, S. W., Kurkjian, A. L., McClellan, J. H., Morris, C. F. and Parks, T. W., "Estimating slowness dispersion from arrays of sonic logging waveforms", Geophysics, 52(4) (1987) pp. 530-544. 10. Mallat, S., A Wavelet tour of Signal Processing, Academic Press, USA, (1998). 11. Raugh D., "Seismic interface waves in coastal waters: A review". Technical Report SR-42 (SACLANT ASW Research Centre, La Spezia, Italy, 1980). 12. Takeuchi H. and M. Saito, "Seismic surface waves". In Methods in Computational Physics ed. by B. A. Bolt, (Academic Press, New York, 1972) 11 pp. 217-295. 13. Korsmo, 0., Wavelet and complex trace analysis applied to the seismic surface waves, Master thesis Norwegian University of Science and Technology, (2004).

THE OPTIMUM SOURCE DEPTH DISTRIBUTION FOR REVERBERATION INVERSION IN A S H A L L O W - W A T E R WAVEGUIDE T. F. GAO Institute ofAcoustics, Chinese Academy of Science, Beijing, China E-mail: [email protected] E. C. SHANG CIRES, University of Colorado, Boulder, USA E-mail: [email protected] Abstract An approach of extracting the modal back-scattering matrix from the reverberation data in shallow-water is proposed recently (Shang, Gao and Tang, 2002). The kernel matrix of the inversion is constructed by the square of the modal function. The singularity of this matrix (or the stability of the inversion ) is the crucial issue to be considered. In this paper, we discuss this issue analytically for a Pekeris waveguide with limited mode number M. The method that we used for singularity analysis is to calculate the maximum value of the determinant of this kernel matrix. We found that there is an optimum source depth distribution corresponding to the maximum value of the determinant of the kernel matrix. That means that by choosing the optimum source depth distribution we can get the most stable inversion. The conclusion is that under a quite tolerant condition the matrix is not singular. 1. Introduction The inversion of reverberation data is very attractive because reverberation data is easy to obtain and a lot of environmental information can be retrieved. In [1], an approach of extracting the modal back-scattering matrix from the reverberation data is proposed, some numerical simulations are conducted in [2] and the inversion based on reverberation data is presented in [3]. In this paper, the stability of the inversion and the optimum source-depth distribution with an ideal waveguide is discussed. Up to the Born approximation the reverberation field can be expressed as [1] : MM

p\zs,z\rc)

= {2nlk,rS£Zm{zs)Sz)Smn m

jdv-Tj(r)cxp{i(km

+*>}

(1)

n

where <j>m(z) is the normalized mode function, km is the modal wave number, zs is the source depth, z is the receiver depth, rc is the center range of the scattering area, Smn is the matrix which describes the mode coupling feature at the scattering element and rj(r) describes the random fluctuation of the scattering element which could be the interface roughness or the volume inhomogeneities. By using the mode filter at the receiving array, we can get they'-th mode component: -

P1(zs;rc)=\ps(zs,z;rc)>2) sin2 (.My,) 2 2 2 tf(zl2) X{k),

y

Y:0^ M J - Y(j),

(8)

W:P^ i

y

H+

M W{i),

'

'

where X{k) represents the index of the module for which k is the index of one of its input variables, Y(j) the index of the module for which j is the index of one of its output variables, and W(i) returns the corresponding index of the module for which i is the index of one of the module parameters. With these definitions the module Mn can be formally defined as Vj e y _ 1 ( n ) , yj = fj ({xk)kex-i(n),

K)iew-i(n)) •

(10)

This is the formal statement of the constraint that each output variable yj of a given an module Mn be defined as a function fj of the input variables {x^kex-1^) d the 1 parameters (iWt)»6W- (n) of that module. At the same time each input variable xk of the module Mn is required to emanate from one and only one output variable of a preceding module MijKL + 1 - j)Ar, zb]

(21)

with the convolution coefficients gij and 6

Pb

l1 + 1 " )

V

is simplified here by dropping the range coordinate and using ipj{zb) = ip[(L + 1 — j)Ar, zi,]. Furthermore v2 = 4i/koAr, and the subscripts w and b indicate the water column and bottom, respectively. The finite difference implementation of the direct problem given in Eq. (20) is an implicit Crank-Nicolson scheme and the NLBC in Eq. (21) is treated as a first order ODE in depth. Integration with respect to the depth z yields the calculation13 of the field on the boundary (H = zb) tP(H) = e'^ 1 / 2 **) ip(H -

+ ie

i/3(l/4Az)

l/2Az)

sm(l/4pAz)Y,9i,3

WJ{H) + *l>i(H ~ 1/2Az)] .

(23)

Following the discretization of the direct WAPE system, the forward model can then be decomposed according to the modular graph concept described in Sec. 2. 3.1. Modular

decomposition

The resulting modular graph (Fig. 3) is divided into four blocks (a)-(d), each of which can be further subdivided vertically and/or horizontally. Given a finite difference discretization with NZ and NR gridpoints in depth and range respectively • Space (a) is of the dimension NZxl and is used to initialize the tridiagonal finite difference matrices (Crank-Nicolson scheme), which are represented by the modules diaGt and diaG respectively. The sound speed profile, the depth-dependent density and sound attenuation in the water column are represented accordingly by the by modules pw, c(z) and aw. Furthermore, also the LU decomposition 10 of the finite difference system which is represented by modules bet and gag is initialized in this space. b

A similar treatment of Papadakis' original spectral integral formulation of the NLBC (Neumann to Dirichlet map) is proposed in Eqs. (2.20)-(2.23) in Ref9.

60

z—Az

Z+Az

r-Ar

Figure 3. Modular graph representation of the WAPE NLBC model. The nomenclature is consistent with the notation in Sec. 3. Modules with the superscript "LU" or "UN" implement the LU-decomposition 10 and the Crank-Nicolson scheme, respectively. Module "J^" refers to the summation of the boundary-field values in Eq. (23).

Space (b) is of the dimension NZxNR, and in this space the acoustic field represented by module ip is calculated by solving the numerical system for each range step r via LU-decomposition (modules res and ixu). Space (c) is of the dimension l x l and it mainly serves for the initialization of the sediment geoacoustic parameters {pb,Cb,ab} and the calculation of related variables, such as e.g., refractive index Nb and parameters /?, ei,e2 of the NLBC (Eqs. 21-23). Space (d) is of the dimension lxNR, and is used to calculate the NLBC at the water-sediment interface in order to determine the acoustic field at the bottom (Eq. 23).

Horizontal layering within a block indicates adjacent finite difference depth cells (z, z±Az) and vertical subdivision represents successive range steps (r —Ar, r). The dashed arrow further indicates that the module J^ which represents the summation of the boundary-field values in Eq. (23) depends on all the known values (history) of the source modules at previous range steps, not just on the actual value of the current instance.

61 4. O p t i m i z a t i o n With YAO the cost function is calculated automatically from the module that is declared as cost module and from observations that are loaded from an external file. An example of a multiple frequency cost function0 with two regularization terms is given by m -.

J(x) = E 2 [ (G ' (X) ~~ ^ o b s > l ) T i?"1 ( G l ( x ) " ^obs'i} + - a (x - x a p r ) T B'1 (x - x a p r ) + ^HVx||2,

(24)

where the index i denotes different source frequencies and Tpobsj, i = 1 , . . •, rn are the corresponding observations at each frequency. The parameter x a p r is included in the cost function as an a priori estimate of the desired solution x, R and B represent the covariance matrices for the field and the control parameter, respectively and (a, b) are the two regularization parameters. With a cost function specified in Eq. (24) the numerical implementation of the direct model (Sec. 3) can be differentiated using YAO in reverse mode to generate the adjoint code. Equation (19) then allows the computation of the gradient of the cost function with respect to the control variable. A Taylor test ensures that the derivatives generated with the adjoint code agree with the corresponding finite difference approximations for different directions of perturbation of the control variable. Minimization is generally accomplished through the use of standard iterative gradient methods like e.g. conjugate gradient or Newton-type methods 12 . The routine M2QN1, which is used for the optimization process in the following example, is a solver of bound constrained minimization problems and implements a quasi-Newton (BFGS) technique with line-search. As an illustrative test case the numerical adjoint approach is briefly demonstrated for the geoacoustic characterization of a shallow water environment (Fig. 4). The control variable x is determined in this case by the geoacoustic parameters {pb,Cb,ab} of the sediment. Acknowledgments The research reported in this paper is supported by the Royal Netherlands Navy and the Service Hydrographique et Oceanographique de la Marine Francaise (EPSHOM). The research work contributes to the Flux3 sub-component of AQUATERRA integrated project funded by the European 6th Framework Programme, research priority 1.1.6.3 Global change and ecosystems, European Commission. c

An extensive analytic treatment of multiple-frequency adjoint-based inversion of a locally reacting impedance boundary condition for the standard P E can be found in Ref. 11

62

1470

1480

1490

1500 1510 1520 Sound speed (m/s)

1530

1540

0

20 40 Iteration number

Figure 4. Adjoint-based geoacoustic characterization of a shallow water environment: Acoustic fields for the three source frequencies 200, 400 and 500 Hz (a)-(c); acoustic fields at 9 km range, environmental input data and experimental configuration (d); evolution of the estimated parameters vs. iteration number (e)—(g).

Appendix A. Tangent linear model As a counterpart to the reverse calculation of the Lagrange multipliers in the adjoint model (Fig. 2), the following illustration explains the tangent linear model, which operates forward in the sense that it determines a gradient with respect to output from a gradient with respect to input.

63

pi

y3

f)2 P3

y4

P7

«» = £* 0tSfa/dxt y6

P8

yi

a6

xlO ill xi;>

pio Pii

J

P12

a

7 = Et P f t / * l

P9

y2

y7

Ml(i+2)

0t = at predecessor

fj =

J2t0hdfj/dxk

Figure 5. Modular graph: Tangent linear model. Forward calculation of the Lagrange multipliers

References 1. R. M. Errico. What is an adjoint model? Bulletin of the American Meteorological Society, 78:2577-91, 1997. 2. C. Sorror and S. Thiria. YAO User's guide. Version 1.0. Technical report TR-123, LOCEAN, Paris, France, 2005. 3. G. Madec, P. Delecluse, M. Imbard, and C. Levy. OPA 8.1 Ocean General Model reference manual. Technical note 11, LODYC/IPSL, Paris, France, 1998. 4. S. Ouis, Y. Bennani, S. Thiria, F. Badran, and L. Memery. Assimilation de donnees de traceur oceanique: Une methodologie neuronale. Technical report, LODYC, Paris, France, 1999. 5. J. Noilhan and J.-F. Mahfouf. The ISBA land surface parameterization scheme. Global and Plan. Change, 13:145-59, 1996. 6. M. Meyer and J.-P. Hermand. Optimal nonlocal boundary control of the wide-angle parabolic equation for inversion of a waveguide acoustic field. J. Acoust. Soc. Am., 117(5):2937-48, May 2005. 7. J.F. Claerbout. Coarse grid calculations of waves in inhomogeneous media with application to delineation of complicated seismic structure. Geophysics, 35:407-18, 1970. 8. D. Yevick and D. J. Thomson. Nonlocal boundary conditions for finite-difference parabolic equation solvers. J. Acoust. Soc. Am., 106(l):143-50, July 1999. 9. D. J. Thomson and M. E. Mayfield. An exact radiation condition for use with the a posteriori PE method. J. Comp. Acoust, 2(2):113-32, 1994. 10. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes. Cambridge Univ. Press, Cambridge, U.K., 2nd edition, 1992. 11. M. Meyer, J.-P. Hermand, M. Asch, and J.-C. Le Gac. An analytic multiple frequency

64 adjoint-based inversion algorithm for parabolic-type approximations in ocean acoustics. Inverse Probl. Sci. Eng., 14, 2005. (accepted for publication). 12. J.C. Gilbert and C. Lemarechal. Some numerical experiments with variable-storage quasi-newton algorithms. Mathematical Programming, 45:407-35, 1989.

M O D E L I N G 3 D W A V E P R O P A G A T I O N IN T H E O C E A N C O U P L E D W I T H ELASTIC BOTTOM AND IRREGULAR INTERFACE

LI-WEN HSIEH 1 , DING LEE 2 , AND CHI-FANG CHEN 1 ' Department of Engineering Science and Ocean Engineering, National Taiwan University http.V/uwaclab. na. ntu. edu. tw 2 Naval Undersea Warfare Center DingLeel (a),aol. com In the past few decades the elastic properties of ocean bottom were usually ignored to simplify problems by assuming a fluid seabed. Nevertheless, while it is acceptable to make such assumptions in deep water, the effects of shear waves can never be omitted as long as sound waves penetrates into ocean bottom, especially in shallow water where interactions between sound waves and elastic bottom are very frequent. Hence, seabed has to be considered as elastic solids to correctly reveal the propagating behavior of sound waves. A novel mathematical model and an implicit finite difference method to obtain a numerical solution for predicting wave propagation in a 3D ocean coupled with irregular fluid/solid interface are presented and developed into a computer code. Theoretical and computational aspects of the proposed parabolic equation solution procedure are investigated. Several numerical examples are included to show satisfactory results after comparing to known reference solutions with shear effects.

1

Introduction

In 1989, Shang and Lee [2] introduced a model to treat the two-dimensional fluid/solid horizontal interface following Ref. [11]. This model is limited to solving narrow-angle, two-dimensional horizontal interface problems. Moreover, no solution of elastic PE was incorporated into the fluid model. Later In 1998, Lee et al. [3] extended the Shang-Lee model to handle the horizontal fluid/solid interface three-dimensionally. Their approach is to transform the fluid/solid interface for the Helmholtz equation into the conditions suitable for the PE. A mathematical model was formulated to predict wave propagation in a coupled three-dimensional fluid/solid media. In 1999, a numerical solution to this horizontal fluid/solid interface model was introduced by Sheu et al. [4] who used a finite difference technique to solve the above wave equation using a predictor-corrector procedure. In 2002, Nagem and Lee [6] extended the horizontal fluid/solid interface model to handle the irregular fluid/solid interface. However, after closely following their procedure, serious mistakes are found so that their results can not be adopted. Therefore this dissertation is based on the same fundamental relations and theories with Nagem's work but subsequent derivation is novel. An efficient numerical model for 3D wave propagation in the ocean coupled with elastic bottom and irregular interface by a PE method and a stable ODE solver is to be developed. First the mathematical model is formulated, and then a computational model which can generate a satisfactory solution using an accurate and stable numerical method is developed in this dissertation. This model is designed and capable not only for coupled 3D ocean acoustic wave propagation, but also for propagation in pure fluid or elastic solids, provided the initial and boundary conditions as well as other environmental variables are properly defined. Results of some examples with analytic solutions are also reported in this dissertation to validate the model and to show the shear wave effects. A 3D test case is also given to exhibit 3D effects. 65

66 The paper is organized as follow. Section 2 derives the representative fluid/solid coupled wave equations written in operator form. Following the theoretical formulation summary, Sec. 3 briefly presents the development of the computational model. The theoretical and computational aspects of the numerical algorithm and the resultant difference equations are given. Section 4 is devoted to validate the numerical model by several test cases. Summary of this paper is given in Sec. 5 remarking the major conclusions and directions for future works. This paper is partly extracted from the first author's Ph.D. dissertation [16].

2

Theoretical Derivation

In this section, the theoretical part for the proposed numerical model is briefly reviewed. A mathematical model has been developed by Lee et al. [3] which introduced a set of 3D fluid/solid coupled wave equations. However, an alternative mathematical model is derived in this paper instead of direct applying their result. A summary of this set of equations is given in operators form of a set of parabolic equations. This summary outlines the mathematical model involving the fluid wave equation, a set of interface vector equations, and the elastic wave equations. To be adapted for parabolic equation approximation, the displacement potentials written in cylindrical coordinates can be related to the elastic potential functions by [12]

= r-"Aeik'-r,

2.1

ik

¥r=r'"Bre

^,

yf^r^B^,

r

¥z=r~"Bz^

.

(2.1)

Parabolic Elastic and Fluid Wave Equations in Operator Form

If the potential functions expressions (2.1) are substituted into the wave equations, and considering the zero-divergence condition along with far-field approximation, rearrangement of the results gives d2A „ . , 8A d2A 1 82A n 82Bz , . . dBz 82BZ 1 d2Bz

dr2

T

2

dr

dz2 2

d Be 2

dr

dBe d Bg T

dr

dz

2

r2 dO2

dz

ld2Bg__2 2

r 86

2

8Br r

3

86

Equations in (2.2) are second-order partial differential in the variable r, but each can be separated into two uncoupled first order parabolic equations [12], one equation representing waves which propagate in the direction of increasing r , and the other equation representing waves which propagate in the direction of decreasing r . The separation gives the elastic outgoing wave equations in operator form 1 ( „dB,\ 2

-+ikL-ikLjurLjA=o, ^ - ^ V ^ = ^ ^ [ — + ikTT -ikTsjl + Lj. \BZ= 0, — + ikT-ikT^\ + LT \Be dr *^V—rj-.-". ydr™r «TS-"rp 2ikTJui;

where the operators are defined as

& / (23)

67 (

L

a2

dz2

'-Tf

+

1 d2 r2 dd2

f 52

\

,

LT —

dz2

KT

+

1 d2 r2 dd2

\

(2.4)

Rearrangement of Eq. (2.3) gives the parabolic elastic wave equations in a matrix form 0

(

A}

0

_3_ Bz 8r Br

0

\BeJ

0

0

0

0

B,

0

0

0

(

A\ B B

(2.5)

i a r 3 89

B,

where the operators in coefficients are defined as

AL=ikL(-l

+ Jl + LL),

AT=ikT(-l

+ Jl + LT),

BT=-^-j=.

(2.6)

A fluid can be regarded as an elastic material of no rigidity, therefore the fundamentals and derivations of the wave equations for both media are the same. Thus for fluid, Eq. (2.5) reduces to

£M=MK).

(2.7)

where the operator is defined as f »22

Af - ikf

d

v&

2

I

d2

"\

+ r2 d62 j

(2.8)

and kf is the fluid wave number. 2.2

Parabolic Interface Vector Equations in Operator Form

For the formulation of the irregular fluids-elastic interface conditions, a set of unit vectors must be defined which describes the geometry of an arbitrary orientation. This set of vectors is defined by: T

ly1r>'He>Tlz)

t[tr,te,t2)

ls t n e un

^ vector normal to interface;

is the unit vector in plane of the interface;

S (S r , Se, S2 ) is the second vector in the tangent plane of the interface perpendicular to t . Along with the orientation vectors, the irregular fluid/solid interface conditions are formulated by means of tensor vectors as u

In = u /n> pf = -T\GT%

t r oi] = o, sroti = o.

(2.9)

In practice, the geometry can be simplified by introducing the cylindrical sloping interface where the angle to the slope is &, and the specific orientations are defined in following, as shown in Fig. 2.1.

68 • For the horizontal interface, ^ = (0,0,1), t = (l,0,0), and s = (0,l,0). • For the irregular cylindrical sloping interface,

n

= (-sin 3,0,cos 3), t = (cos 3,0,sin 3),

and s = (0,1,0) • If 3 = 0, the irregular interface cases are all reduced to the case of horizontal interface. Note that r]) + rjz = rjztr - r/rtz = cos2 3 + sin2 3 = 1.

V r2

V

\ J

s = (0,1,0) cos 5,0, sin .9)

(r2»°»^5 Fig. 2.1Schematic of irregular cylindrical sloping interface

For the orientations introduced in the above, the general irregular interface conditions are simplified to give explicitly sin 9u, + cos 3w - sin &u„ + cos i9w. (2.10) v / -pf

= sin2 9afi

&[kfAz)

-

2{kfrA0)'

ikf

2[kfAz)

2{kfAz)

ikf

ik, "•/m '

ikf

,

«

/

•

=

3ikf

(3.9) ikf

-

(kfAz)

4(kfAz)

(kfrA0)

The subscripts i, j represent the i th grid point in depth, and j th grid point in azimuth. Note that if Ad is chosen so large as 1/A# approaches zero, the equations reduce to two-dimensional case. In equation for Br, the operator coefficient of Bz is approximated using Eq. (3.7) and resulting in 33 \ 1 53 3 ds 1 d_ 1 (3.10) 3- + - 4 5 2 2 2 dz 2k\ dz Sk T dz 2k Tr 80 dz ' dz lH J- \ j 1 ~t" J_ty In equation for Bg, the operator coefficient of Br is also approximated using Eq. (3.7) and resulting in 33 ^ 3 3 1 ( 1 d )_ i 1 1 5 (3.11) 2 4 A 2 3 ikTyJl + LT I r3dd)~r3kT 80 2k] 80dz - + 8A:r 808z 2k\r 80 As a summary, with the square root approximations and proper discretization, the numerical ODE formulations for elastic and fluid wave equations like Eq. (3.8) and the interface equations can be obtained. The complete results are referred to the first author's Ph.D. dissertation [16] and not shown here. Results for wave equations in Ref. [5] are similar except for its equation of Br where an error occurs in their derivation of wave equation for Br. 3.2

Computational Aspects

In order to develop the computer code, a number of computational aspects have to be taken into consideration. For simplicity, taking a horizontal interface problem as an example, Fig. 3.1 schematically shows computation grids and settings. As shown in Fig. 3.1, the upper boundary which usually refers to the ocean surface is assumed to be flat and pressure released A,\ =0. (3.12)

74 The lower boundary denoted as rigid bottom is assumed to be flat and force the first derivatives with respect to depth of all quantities to be zero

h°

(3.13)

= 0,

resulting to be a total reflecting boundary. The bottom boundary can also be set to zero for simplicity.

T (marching direction)

"SjiilklKety

r

Pressure Release Surface Z = 0

V

z 1

Initial]

Port

Starboard

0=0

(9 = Q

t

Inii-rt'iici' Z — lit

t Pressure Release Bottom Z = 0

e

Fig. 3.1 Schematic of computation grids and settings

In order not to use nonphysical points outside the boundaries in difference equations for the grid points near boundaries, forward/backward difference formulas are applied. For example, at z = Az the second order forward difference formula of the forth depth derivative,

&4

is given as

75 ^-9pl+\6p2~\4p3

+ 6p4~p5 (A,)

(3.14)

4

where p0 vanishes if pressure release condition is considered. This formula induces inconsistency error which will be shown in a test case in next section. Nevertheless, the mirror effect, /? . = -p., at pressure release boundaries, /?0 ~ 0, can be applied so that central difference formula can still be applicable without using nonphysical points as Fig. 3.2 shows. Therefore the second order central difference formula of the forth depth derivative, P - i ^ + 6Pi -4/>2 + P 3

d4p &4 =

is given as

~py + 6 A ~4p2 + p3

(Az)4

(Azf

P-2

Po=°»

=

5/7, -4p2

+p3

(3J5)

(Azf

>- Nonphysical points

P-i=~Pi > d4p n atz = Az, —~- = ? dz4

Fig* 3.2 Schematic of pressure release boundary in discrete space.

Comparing Eqs. (3.14) and (3.15), it can be noted that considering mirror effect at pressure release boundaries not only maintains the consistency of using central ditTerence formulas but also reduces the number of grid points from five to three at z = Az. For the port/starboard sidewall boundaries, they are not pressure released or rigid under most circumstances except for numerical tests. In realistic situations, they are not known and have to be computed and provided as boundary conditions of Dirichlet type QL,a=fo,a2D (r-z)

(3-16)

solutions may be the most straightforward answers provided for sidewall

boundaries. A computer code is developed to implement the marching implicit scheme (3.2). However, it must be mentioned that the mathematic and numerical development of the proposed model does not contain the density variation and other capability enhancement, the computer code is basically a research code. It still needs some efforts to turn this code into a practically working code like other well known models, say, FOR3D. The geometry of propagation has been presented in Fig. 3.1. The data structure should be particularly noticed since this model deals with a heterogeneous problem. That is, at a

76 single grid point, there are more than one unknown quantities. Each field quantities Af,Ae,Br,Be,Bz is stored in separate matrices at the beginning and in its final form. However, during the calculation stage, solving the unknowns requires these field quantities to be organized as a single vector at each range step. The coefficient matrix corresponding to this unknown vector is thus constructed as the following figure shows along with the structure of the unknown vector. The figure shows an example of 9 azimuth sections (side-wall boundaries excluded). Blue dots represent nonzero elements which are the coefficients of the difference equations. It can be seen that the matrix is formed as a band-matrix. Dark lines indicate different azimuth sections whereas the red dashed lines showing the five interface equations between the fluid and the elastic wave equations which are colored with light blue and olive boxes, respectively. The unknown vector is a column shown at right hand side of the coefficient matrix.

Fig. 3.3 Data structure of the coefficient matrix and unknown vector.

4

Model Appraisal

In this section, several test cases are investigated to validate the model and also to show the model's application and ability. Exact solutions to the coupled 3D wave propagation problems are practically unachievable due to complexities in environment and boundaries. To validate models claiming to solve these problems is therefore limited. In the following, several examples will be presented to start from the simplest two-dimensional range-independent problem for waves in fluid only, and finally to the coupled 3D wave propagation model with irregular interface.

77 4.1

2D Fluid Waves in Range-Independent Environment

Being the first step to validate the model, this example is focused on testing the numerical marching scheme Eq. (3.2). Also the effect of considering the mirror effect on pressure release boundaries as prescribed in previous section is presented. Considering a two-dimensional ( r z ) plane, the upper and lower boundaries are both flat and pressure-released, and the medium in between the two boundaries is pure fluid only without any energy absorption. The environment is totally range-independent. Numerical ODE formulation for the 2D fluid wave equation can be obtained by applying the square root approximation (3.4) and (3.5) resulting similar to Eq. (3.6) as dAf aO (4.1) -ik, 4 V Sk) dz 2 ~dr~ 2k) dz

_Li_.

To find the general solution to the above equation, one can use the method of separation of variables given the initial condition A (o z) = sin — I a n d obtain (TVIH)1

= exp < -ir

2kf

(7tlH)A

Sk}

(4.2)

•sin — .

Using the presented numerical model, a banded matrix is formed to solve the problem with computation parameters set as Az = 0.1/L,Ar = 3/L, where sound speed c is 1500 m/s, sound frequency / is 200 Hz, water depth H is 60 m, wavelength ; is 7.5 m. Figure 4.1 compares the absolute value of computed solutions using (l,l)/(2,2) Pade scheme and sided/central difference near boundaries for range 0 - 1 2 km. It can be clearly seen that errors accumulate from the upper and lower boundaries if mirror effect is not considered and sided difference is applied. Also higher order scheme, such as (2,2) Pade scheme, is more sensitive to such errors from boundaries. However, if mirror effect is taken into account, then the errors are removed and (2,2) Pade scheme produces better results as it is expected. error n o r m = 3 . 3 7 8 6 e - 3

error n o r m = 9 5 4 9 2 e - 1 - 0.4 -

: PA -. u.a

-r — „

n n

04

-

(1,1) P a d e + s i d e d d i f f e r e n c e near b o u n d a r i e s

(1,1) P a d e -*• central difference with mirror effect n e a r b o u n d a r i e s

error norm - 6.3677e+O20

error n o r m = 1 . 7 9 7 8 e - 3

0"4

no -

— : -*aL

\

-

-1--1-H _ ,

(2,2) P a d e + s i d e d difference near b o u n d a r i e s

(2.2) P a d e + central difference w i t h m i r r o r effect n e a r b o u n d a r i e s

Fig. 4.1 Comparison between the computed solutions using (l,l)/(2,2) Pade" scheme and sided/central difference near boundaries.

78

By this test case, the marching scheme using (l,l)/(2,2) Pade approximations are validated. The improvement of considering mirror effect near pressure release boundaries is also presented. This suggests the consideration of mirror effect in all computations when pressure release boundaries occur. 4.2

2D Elastic Waves in Range-Independent Environment

The elastic wave governing equations are coupled for 3D propagation as two individual systems, {Ae} and {Br,Be,Bz}. For 2D propagation, #-derivatives are dropped so that Br is uncoupled from the equation of B$, therefore the governing equations are further uncoupled into three independent sets, lAe}, {Be\, and iBr,Bz\.

Provided there is no

coupling mechanism on boundaries, 2D elastic waves problem can be resolved by separately finding the solutions of the three systems. Given the same flat and pressure-released upper and lower boundaries as in the previous case for fluid wave problem, since equations of iA\ and {Be} are of the same form as that of \Af], similar analytic solutions and results presented in the previous case can be obtained. Hence here we will focus on the two-variable coupled system, {Br,Bz\ • Note that under these pressure-released boundary conditions, if Bz is initially unexcited, i.e., B\

= 0, then B will be decoupled from the equation of B • Therefore the problem

is reduced to the same form of \Af] , and the whole Bz field will be completely silent, thus the equation of Bz turns out to be trivial and its calculation may be saved. After applying square root approximations (3.6) and (3.7), the two-variable coupled system, iBr,Bz\ writes ( 1 82 ^ = ikTT dr 2k2 dz2 (

dr

1 d4^ Sk4 dz*

JLJ?L__L_3l 1

=)2

2

dz2

y2k

1

a4

"\

%k* dz4

(4.3)

^i

rs i• ff Pi

dz

-X dPi3 3 5F)5 ^ - 4 dz5 B_. 2k* dzs - + Sk 1

1

a33

o

a5

5 A

As before, the analytic solution to Eq. (4.3) is desired. The strategy to solve Eq. (4.3) is summarized as two steps. The first step is to solve the parabolic equation for Bz, and this is followed by solving the parabolic equation for Br with the part containing derivatives of B known as the inhomogeneous source term. By the method of separation of variables, Bz can be derived in the form of Bz=e\p(-ikTg2r)lciexp(£lz)+C2exp(-llz)

+ Cisin(£2z) + C4cos(£2zjj,

(4.4)

where the functions with underline are the eigenfunctions, and £l,£2 are related to the eigenvalues as

79

••yfikrjj:l +

2?2+l.

(4.5)

Note that the given boundary conditions Bz (r,0)=Bz (r,H) = 0 are not sufficient to define the coefficients C ; . However, with proper initial conditions input, the coefficients can be determined. It is obvious that s j n | r^Lz ] for any integer n can satisfy the zero boundary

U J conditions. Further more, it is of the same form with one of the eigenfunctions, sin(^2z) • Therefore, if the initial filed is given as Q s m TLZ I, where C0 is a constant, then ° \H ) Bz(0,z) = C 1 exp(V) + C2exp(-Az) + C 3 sin(V) + C4cos(^2z) = C 0 sin(— Z\

^ ^

the coefficients in Eq. (4.4) can be determined as C, = C2 = C4 = 0, and C3 = C0 a known nn , so that the eigenvalues are given as H r nn \ nn Sn \kTH j ykTH j

constant. Also it leads to

(4.7)

and thus the analytic solutions to B is then written as Bz = C0expl-ikT

A4 1f nn nn + —l K,rp±l J ykTH j

I

. fnn sin — z

(4.8)

The analytic solution to A, in the previous case can be verified by the above equation as well where « = 1,C0=1Next step is to substitute Eq. (4.8) into Eq. (4.3) to solve the equation for Br. After the substitution, it reduces to 55, dr

2 2 K2k Tdz

+C0iexp(-ikTg2nr)

%kATdz" j

B,

nn 1 nn V kTH - + 2—\kTH j

nn

V

\kTH j

(4.9) f

rm_ ^ cos \H j

Equation (4.9) is an inhomogeneous PDE which can be solved by eigenfunction expansion method [13]. If n = 2,C 0 =l is chosen to have the initial condition for B being Bz (r, 0) = sin[ — z 1» and a static initial condition for Br, i.e., Br (r, 0) = 0, the solution to Eq. (4.9) is obtained as

Br=

4/i4o(e"v-e"v) ' nn —r^—rz ^-sin Z 2 H n=w,..fr(n -4)(AH-A2)

^

(4.10)

80 where f

71

4>=»

yKjH

j

+4

f

\

n

\ K-pi~t

\ K-T-JLI.

j

nn

nn \

r + 12

V 71

ykTH

f y /\r\

ZAlXj

j

\2

{

j

\ Kjii

ykTH

y 71

71

i

(4.11)

,n = l,3,5,---.

yKj-H j

I

18m

mmmmmmm

ItlllttHIHiMU • 0« U\\ ' H

-

s !,

I

*

M

i

tllllllllMlKltM IIIIIMIItlliltMl 0

5

10

15

20

Range (KM) Range (KM) Fig. 4.2 Absolute value, real part, and imaginary part of the numerical solutions to 2D elastic wave problem (4.3). Comparison of computed and analytic solutions of B -(1,1) Pade (2,2) Pade -Analytic Solution

Range (km) Fig. 4.3 Comparison of (1,1 )/(2,2) Pade schemes and analytic solutions of B

at depth of 18 m.

81 Given the depth H = 99 m, wave frequency / = 100 Hz, transverse wave speed cT = 900 rn/s, the numerical solutions of Bz and Br are plotted in Fig. 4.2 for range 0 20 km considering the absorption coefficient being 0.5 dB per wavelength. Setting the computational parameters Az = 0.2/1,. and Ar = 8 ^ , the numerical results of Br agree well with the analytic solution at selected depth of 18 m shown in Fig. 4.3. However, (1,1) Pade scheme does not provide as good prediction as (2,2) Pade scheme for this case. 4.3

Coupled 2D Waves with Irregular Interface

In this test case, at first a down-slope wedge is considered then an up-slope wedge. Both slope ratios are 1:20, or about +2.86°, and the water depth at source location is 200 m. A Greene's source is placed at depth of 30 m, the computation parameters are set as Az = 0.18m, Ar = 3.6ra , and the absorption coefficient being 0.5 dB per wavelength. Figure 4.4 shows the computed solutions in water column, \A,1, in the upper two plots. Up-slope

jy.lA^dz/H, :

' . "*" '•^^^i ~ - - E 3 ~ _ _ i ^ ^ ^ — Shear £ffect^*%vj'-^

...

!HP

- ™ - ~ Down-sfope. C T = 4 0 m/s (A.: p. ~ 1800 ; 1} Down-stope. C T = 900 m/s (A.: n ~ 3 : 2)

yU

Up-slope, C T = 900 m/s (X : p. ~ 3 : 2) ~" "0

05

1

1.5

2

2.5

3

3 5

Range (km) Fig. 4.4 Comparison between the calculated results of up/down-slope wedge with low/high shear wave speed.

The lower plot compares the depth averaged energy along the propagation range for four difference situations. The green solid line represents the result calculated in down-slope wedge with very low shear wave speed (CT = 40 m/s) comparing to the red dashed line where the shear wave speed is 900 m/s. These two line does not differ too much, i.e., shear effect is not obvious in down-slope wedge. In up-slope wedge, energy in water column is expected apparently decreasing due to more interaction between wave and bottom as shown in the figure. The blue dotted line represents the result calculated in

82 up-slope wedge with shear wave speed being 40 m/s whereas the black solid line is the case with cT = 900 m/s. It can be seen from the figure that there is noticeable difference due to shear effect for up-slope wedge. More energy of water column is transferred into bottom as shear wave. 4.4

Coupled 3D Waves with Irregular Interface

It has to be noticed as mentioned before that the analytic solutions to such problems are inaccessible. For simplicity, the upper and lower boundary conditions are pressure released boundaries, and the two side-wall boundary conditions are zeros as well. Given the environment setting being the same as before except for total depth of H = 70 m and a range-dependent bathymetry H,(r)- For sound frequency of 25 Hz, the initial field is placed at r 0 = 1082.3 ni to satisfy far-field approximation ft.r>100, i = f,L,T computational

parameters

are

• The

set as: Ar = OAAj = 14.4 m, A8 = 0.5° , and

Az = 0.015/^ =0.54 m. It must be noted that Az is chosen so small to have accurate solutions because of the interface effect. Starting from r0, the initial field is propagated 150 m which is about 10 range steps, and the computation span is 20 degrees which consists 21 sections including two side-wall boundaries. The bathymetry is defined as // ; (r 0 ) = 35,//,(r 0 +50)=36,// ; (r 0 +100) = 34, and H, (r0 +150) = 37, as shown in Fig. 4.5. It must be emphasized that the bathymetry is given of axial symmetry to be consistent with the irregular interface defined and shown in Fig. 2.1. Ar = 0.4A,. = 14.4m A6> = 0.5°

Az = 0.015^ =0.54m -34. — -35.

20"

1 . -36, -37

K

-300

__—-"~T --H 1220 1200

1240

Range (X-axis, m; Range (Y-axis, m)

Fig. 4.5 Schematic of 3D bathymetry and other computational settings.

To have a 3D initial field instead of an Nx2D field, all the field is initially static except for Af(r0,d = 0,z) is excited by a normalized sine function. The calculated results are plotted in Figs. 4.6 - 4.7 where the absolute values of Af,Ae,Br,Be,Bz

are shown. The

83 solutions are selected from three specified ranges: the next step to the initial field, halfway on the propagation path which is five steps from r 0 , and the final step of the computation. IA,l©r 0 +Ar

0

2

IA,l@r0+5A,

£

10

Q

30 -2

0

2

IA,l@r 0 + 1 0 A r

Azimuth (degree) Fig. 4.6 Absolute values of A r at selected ranges.

From Fig. 4.6 it can be observed that the energy of Af from the initial field is gradually propagated towards the two side-wall yet a great part of the energy remains at 9 = 0°. Note that since the environment is also symmetric with respect to the 0 = 0° plane, the solutions perfectly reveal this symmetry as well. In Fig. 4.7 the transmitted energy from fluid to solid layer is clear displayed. The energy is continuously input to Ae along the propagation range so that the absolute values are keeping increasing. 3D propagation is also obviously noticed. The results of Br shown in Fig. 4.7(b) present a major difference from what observed in previous plots of A, and Ae. The energy does not focus on the central plane but spread out from the plane. Also it is very interesting that at midway on the propagation bath, the energy is less than at the first range step. This has revealed that the energy can not only be transferred into but also output through coupling. Recalling 2D problems where the unknowns can be grouped as two sets, the mathematic or numerical formulations of the wave equations and interface equations have indicated that there is no coupling mechanism between lAf,Ae,Be) and [Br,B\, and this coupling only exists in 3D problems. Similar outcome can be expected in results of Bz presented in subsequent figure after the plot of Be • Similar to the results of Ae plotted in Fig. 4.7(a), the results of Be shown in Fig. 4.7(c) reveal the characteristics of energy spreading along the interface but not deep into the bottom. As mentioned in the above discussion of the results of Br, Fig. 4.7(d) shows expected feature similar to B • From

84

interface equations or numerical formulation, this feature shared by B and B is due to azimuth coupling at interface and thus is a kind of 3D effect.

mf

•

mm •5.—-r—r~

llliflll

m *»

t

~4

(a)

(b)

HI :" ',

AzfmuHiEdtgnM)

(c)

.

.1

B*

Ml

AzmttKtfegnw)

(d)

Fig. 4.7 Absolute values of (a) Ae (b) Br (c) 5 t f (d) # , at selected ranges.

The 3D example tested in this section is highly restrained and simplified to focus on the primary concerns, i.e., realization of the proposed model and how it works with coupled fluid/solid medium with irregular interface. It has to be emphasized that a major difference between considering fluid bottom and real elastic bottom is the demand in computation resources especially the CPU time. This is due to the increasing of the number of physical quantities in elastic bottom, from single one to four. The additional three, Br,Be,Bz, account for shear waves. Under the condition of same grid points, the coefficient matrix considering shear waves is 4 x 4 times larger than fluid bottom. In other words, the range of interested problem is therefore practically limited. Nevertheless, this kind of technical shortness can be expected to be resolved just like decades ago, and the emphasis must be placed on pursuing the completeness in describing the problem and its solution. 5

Summary

This paper has introduced a modified mathematical model to 3D coupled fluid/solid wave propagation problem and also developed its computational model and a research code. The numerical results produced by this computer code has presented good agreements with analytic solutions which reveals that this computer code produces satisfactory results. The

85 validation has also shown that the stable marching scheme which implements implicit finite difference method is accurate. The emphasis has been placed on the development of the numerical model which can solve 3D fluid, elastic, or fluid/solid coupled wave propagation problems. The underlying idea is applying parabolic displacement potential functions to rewrite the wave equations and interface equations, and then using implicit finite difference method to solve the ODE system. Pade series expansion has been used to improve the accuracy in range direction. Since the proposed numerical model and computer code are new, analytical validation of the scheme has been conducted for several problems. The proposed model has been successfully applied to simulate the fluid/solid coupled wave problem. From both mathematical and numerical modeling, two major differences can be found in 2D/3D comparison, with/without shear effect. A 2D problem is a simplified special case from 3D problems where all the five unknowns [Af,Ae,Br,Bg,Bz\ representing the parabolic functions of displacement potentials are coupled together and must be solved simultaneously. The simplest case is 2D problem without shear effect and its unknowns are \Af, Ae) only, that means the energy will be shared by the two quantities only. This is also true for 3D problem without shear effect. If shear effect is to be considered in a 2D problem with only Af excited by a waterborne source, then the unknowns become lAf,Ae,Bg) which means the energy will now be shared by one more quantity, Bg. Note that if {Br,B\

is not initially static, then this set of unknowns should also be solved yet as an

independent problem so that the solution of j Af,Ae, Be ] will not be affected. For the most general case, a 3D problem sustaining shear effect includes all the unknowns iAf,Ae,Br,Bg,B2] , and the energy will shared by these five quantities. In other words, the energy of compressional waves {A, and Ae) can be overestimated if shear effect is ignored whether in 2D or 3D problems. Also the energy can be overestimated even when shear effect is considered in 2D case, because the energy coupling mechanism between {A,, A ,Bg) and iBr,B\ is missed. Table 5.1 compares 2D/3D cases with/without shear effect. Shear Effect X

1 ahle 5 1 ('ninpiirjinn nl 21) .»l) CUM'.- with uilhuiil .••hear effect. ... _ Compressional Wave Unknown Quantities _,. r _ . 2D/3D Energy Overestimated 2D/3D

K'4}

2D

{Af,Ae,Bg} and {Br,Bz}

3D

{Af,Ae,Br,Bg,Bz)

@ O

X

3D effects are found to occur from four sources. First, the initial field can decisively affect the propagation pattern including how the waves spread in azimuth direction. Second, the #-coupling terms (derivatives with respect to azimuth) in the governing equations and interface conditions reflect the constitutional properties of 3D wave propagation. Third, the environment, including the geometry and the acoustic parameters of the medium, has direct influence on the wave propagation path by reflection, refraction, and scattering. Hence a

86 3D environment will definitely induce 3D effects. Finally, 3D distributed boundaries will induce 3D effects as well. Although the irregular fluid/solid interface investigated in this dissertation is range-dependent, £?-variation is not considered in the formulation. As Fig. 2.1 shows, the second unit tangent vector s(sr,sg,sz} is set to (0,1,0) so that a cylindrical irregular interface is obtained. In other words, this interface is of axial symmetry and this is why it is drawn as a frustum of right circular cone in Fig. 2.1. This assumption is a serious drawback of the model since such interface will only reflect waves in fixed 8 planes, i.e., geometric #-coupling at interface is ignored. To deal with a real 3D problem, the interface has to be generalized to include variation of bathymetry in 0 direction. For example, if the interface as shown in Fig. 2.1 is counterclockwise rotated an angle q> respective to unit direction vector t , then three unit direction vectors are given by r\] f-sin>9cos#>, -sin t -=| cos.9, 0, sin,9 I. ( 4 - 12 ) sj [ sini9sin, cos9sin#> Although a modified mathematical formulation and a novel numerical model for 3D fluid/solid coupled wave propagation problem considering irregular interface this dissertation has been developed and coded as a research prototype program C4PM, it is only a beginning for this challenging topic. There are several issues regarding mathematical and numerical enhancements to the modeling and theoretical completeness, namely, wide angle expansion in azimuth, proof of the energy-conserving property such as the proof for LSS wave equation given in Ref. [15]. Each of these issues can be a great improvement and validity proof of the proposed model. Acknowledgement This work is supported by National Science Council of Republic of China. The authors would like to thank Dr. Yu-Chiung Teng for her encouragement and discussion. References 1. Lee, D., Nagem, R. J., Teng, Y.-C, and Li, G. (1996) "A Numerical Solution of Parabolic Elastic Wave Equations," in Proc. 2nd Int'l Conf. Theo. And Comp. Acoust., eds. D. Lee, Y.-H. Pao, M. H. Schultz, and Y.-C. Teng, World Scientific Pub. Co., Singapore. 2. Shang, Er-Chang and Lee, Ding. (1989) "A Numerical Treatment of the Fluid/Elastic Interface Under range-dependent Environments," J. Acoust. Soc. Am., Vol. 85, No. 2, pp. 654 - 660. 3. Lee, D„ Nagem, R. J., Resasco, D. C, and Chen, C.-F. (1998) "A Coupled 3D Fluid/solid Wave Propagation Model: Mathematical Formulation and Analysis," Applicable Analysis, Vol. 68, pp. 147 - 178. 4. Sheu, T. W.-H., Chen, S.-C, Chen, C.-F., Chiang, T.-P., and Lee, D. (1999) "A Space Marching Scheme for Underwater Wave Propagation in Fluid/solid Media," J. Comput. Acoust., Vol. 7, No. 3, pp. 185 - 206.

87 5. Lee, D., Nagem, R. J., and Resasco, D. C. (1997) "Numerical Computation of Elastic Wave equations," J. Comput. Acoust., Vol. 5, No. 2, pp. 157 - 176. 6. Nagem, R. J. and Lee, D. (2002) "Coupled 3D Wave Equations with Fluid/solid Interface: Theoretical Development," J. Comput. Acoust., Vol. 10, No. 4, pp. 431 444. 7. Lee, D. and Schultz, M. H. (1995) NUMERICAL OCEAN ACOUSTIC PROPAGATION IN THREE DIMENSIONS, World Scientific, Singapore. 8. Jensen, Finn B., William A. Kuperman, Michael B. Porter, and Henrik Schmidt, (2000) Computational ocean acoustics, Springer-Verlag, New York. 9. Lee, Ding, Pierce, Allan D., and Shang, Er-Chang (2000) Parabolic equation development in the twentieth century, J. Comput. Acoust., Vol. 8, No. 4, pp. 527 637. 10. Lee, Ding, and McDaniel, S. T. (1987) Ocean acoustic propagation by finite difference methods, Comp. Maths Applic, Vol. 45, No. 5, special hardcover issue, published by Pergamon, New York (1988). 11. McDaniel, S. T. and Lee, Ding (1982) A finite-difference treatment of interface conditions for the parabolic wave equation: The horizontal interface, J. Acoust. Soc. Am., Vol. 71, No. 4, pp. 855 - 858. 12. Nagem, R. J., Lee, Ding, and Chen, T. (1995) Modeling elastic wave propagation in the ocean bottom, J. Math. Modeling and Scientific Computing, Vol. 2, No. 4, pp. 1-10. 13. Farlow, Stanley J. (1982) Partial Differential Equations for Scientists and Engineers, John Wiley & Sons, Inc., Singapore. 14. Lee, Ding (1974) "Nonlinear multistep methods for solving initial value problems in ordinary differential equations," Ph.D. paper, Polytechnic University of New York. 15. Chen, C.-F., Lee, D., Hsieh, L.-W., and Wang, C.-W. (2005) "A discussion on the energy-conserving property of a three-dimensional wave equation," J. Comput. Acoust., to appear in Vol. 13, No. 4. 16. Hsieh, Li-Wen (2005) "Modeling 3D Wave Propagation in the Ocean Coupled with Elastic Bottom and Irregular Interface," Ph.D. dissertation, National Taiwan University, Taiwan, R.O.C.

Journal of Computational Acoustics © IMACS

R E F L E C T I O N S F R O M STEEL P L A T E S W I T H DOUBLY PERIODIC ANECHOIC COATINGS SVEN IVANSSON Swedish Defence Research Agency, SE-164-90 Stockholm, Sweden [email protected]

A thin rubber coating with cavities in a doubly periodic lattice can redistribute sound energy, normally incident on a steel plate, in t h e lateral direction. At high frequencies, propagating reflected beams appear in a discrete set of nonnormal directions in t h e surrounding water. T h e phenomenon is illustrated by pulse measurements in a water t a n k . T h e results are modeled by adapting modern computation techniques for electron scattering and b a n d gaps in connection with photonic and phononic crystals. At lower frequencies, with only one propagating reflected b e a m in t h e water, differential evolution and winding-number integral algorithms are applied to design coatings with low reflectance. A stochastic resampling algorithm is a d a p t e d for accurate characterization of t h e p a r t s of parameter space with favorable properties. Keywords:

invariant embedding; multiple scattering; tank measurements; nonlinear optimization.

1. Introduction and Summary Already during the second world war, rubber coatings with air-filled cavities were used on submarines, for anechoic purposes. 1 Such coatings are said to be of Alberich type. When sound from an active sonar enters the coating, Fig. 1, energy that is scattered by the cavities can be absorbed by the rubber material, and the reflection amplitude can be reduced significantly. The mechanism of the echo reduction has been discussed by Gaunaurd et al., based on resonance theory, but multiple scattering among the cavities was not included in a rigorous way.2 More recent homogenization approaches, 3 ' 4 are also deficient in this respect. In the present paper, reflections of normally incident plane waves by steel plates with Alberich coatings are modeled numerically with a semi-analytical method briefly reviewed in Sec. 2. It has been borrowed from atomic physics 5 and applied in recent years to studies of band gaps for photonic and phononic crystals. 6,7 Sound propagation through a sequence of layers, with or without cavities, is handled recursively by the invariant embedding or Riccati method. 8 The wave field scattered by each cavity is expanded in spherical wave functions, and multiple scattering among the cavities is incorporated in a rigorous self-consistent way. Transformation formulas between spherical and plane waves provide the coupling to the plane waves needed for the recursive invariant embedding treatment of multi-layered cases. A basic computational example is given in Sec. 3. It is for the fundamental case with spherical cavities, but other cavity shapes can be handled as well. 89

90 normally incident plane-wave sound energy t

water O

t

coating

T Q

T Q

steel plate water

y

Fig. 1. Left: A steel plate in water is covered with an Alberich rubber coating with spherical cavities. Right: The cavity lattice with period d is viewed from another perspective. Lateral xy coordinates are introduced.

At high frequencies, the theory predicts propagating nonnormal beam or plane-wave arrivals, after the initial specularly reflected arrival. The phenomenon is related to the Praunhofer diffraction patterns for a multiple-slit aperture in optics. Pulse measurements together with modeling results are presented in Sec. 4, where these later arrivals show up. The computational technique in Sec. 2 is fast compared to purely numerical methods like the finite element method. 9 Numerical coating design with many objective function evaluations becomes feasible. Two different techniques are applied in Sec. 5: a differential evolution (DE) algorithm to achieve low reflectance within a wide frequency band, and a winding-number integral technique to obtain vanishing reflectance at a specified frequency. The identification of the parts of the parameter space resulting in favorable anechoic properties is aided by a stochastic resampling algorithm borrowed from inverse theory. 10 Reflection elimination is illustrated with a coating not much thicker than 5 % of the wavelength, and an apparent splitting of a reflected pulse is explained. 2. Computational M e t h o d 2.1.

Invariant

embedding

A right-hand Cartesian xyz coordinate system is introduced, with xy in the interface plane between two homogeneous solid or fluid half-spaces. Sound waves with time dependence exp(—\wi) are considered, where u is the angular frequency. In the solid case, because of three possible polarizations, P,SV,SH, an incident plane wave in the positive z direction gives rise to three reflected and three transmitted waves. 8 Including the three incident-wave polarizations, a 3 x 3 reflection-coefficient matrix TZB and a 3 x 3 transmission-coefficient matrix TB may be formed. These matrices depend on the lateral wavenumber, which is, however, constant among all seven waves according to Snell's law. For an incident plane wave in the negative z direction, the matrices TZA and TA are introduced analogously. An "interface" at z = 0 is considered next, with a two-dimensional periodic array of cavities. The centers of the scatterers, right panel in Fig. 1, have xy coordinates given by R = (a;, y, 0) = m • {d, 0, 0) + n • (0, d, 0)

,

m,neZ

,

(1)

where d is the lattice period. R / T (reflection/transmission) matrices TZB-TB and 1ZA,T~A can still be defined. However, reflected and transmitted waves appear with lateral wavenumber

91 vectors different from that of the incident wave, ley. By a Fourier expansion in x,y, the appearing lateral wavenumber vectors are k|| + g, where g belongs to the reciprocal lattice g = (kx,ky,0)

= m • (27r/d, 0,0) + n • (0,2n/d,0)

,

m,n£Z

.

(2)

Displacement coefficients are used for the R / T matrices. With r = (x, y, z), the displacement vector for an incident, reflected or transmitted plane wave is a multiple of u(r)=exp(i.K^-r)-e, .

(3)

The time dependence exp(—itot) is suppressed, and j = 1,2,3 for a wave of type P,SV,SH, respectively. Furthermore, s = +(—) for a wave in the positive (negative) z direction, and K±. = k , | + g ± [ ^ 2 / c ? - | k | | + g | 2 ] 1 / 2 - ( 0 , 0 , 1 ) = -(sin0cos0,sin0sin of K * are defined by (4), with a possibly complex cosfl. The vectors e , = e j ( K r \ ) , finally, are defined as usual by e i = (sin 0 cos 0, sin 0 sin 0, cos#), e 2 = (cos 0 cos ^>, cos#sin^>, — sin#), e3 = (— shi(/>, cos, 0). Including one scatterer interface within the rubber layer, four interfaces are involved in the left panel of Fig. 1. Individual R / T matrices can be combined recursively. With R / T matrices 1ZBI,1~BI,

1ZAI,T~AI and 1ZB-2,1~B2, T^-A2,TA2 for two interfaces given in the order

of increasing z, and phase shifts included to account for layer thicknesses, the following formulas are easily established for the total R / T matrices 1ZB,T~B:5'S TIB = 1IBI + TAi • 1ZB2 • [I - TlA\ • ^ s 2 ] " 1 • TB1 TB

= TB2 • [I - KAI • KBI}-1

• TB\

(5)

,

(6)

where each I denotes the appropriate identity matrix. Formulas for TZA,TA are analogous. 2.2. Interface

with periodically

distributed

scatterers

Explicit expressions for the R / T matrices in Sec. 2.1 are well known for an interface between two homogeneous half-spaces. 8 To handle an interface with periodically distributed scatterers, the following spherical vector solutions to the wave equations can be used: 7

ufm(r) = ^.V(/ i (W«W n (e,0))

,M,,_.,,,,, ,m • = i/,(^)

O)

7

W

< ( r ) = ^ •V x < ( r )

i

W

)

(7)

( i^ air(M) „_,_, wry, - ^ •e (0 - ^

•(

2

• e3(r))

(8) (9)

CO

where r = |r| and 6,

uhn 1S u s e c ^ f° r t n e t w o basic cases with ft as the spherical Bessel function jj and fi as the spherical Hankel function h ; + , respectively.

92 For an incident plane wave as in (3), the total scattered field u s c can be written u S c(r) = £ ( H? £ e l k " R • O Plm V R

- R) ) /

,

P = L,M,N.

(10)

The incoming field on the scatterer at the origin has two parts: the incident plane wave of type (3) and the scattered field from all the other scatterers. Both parts can be expanded in terms of u ^ , P = L,M,N, with expansion coefficients denoted a ^ and b^, respectively. It follows by a T-matrix (transition matrix) argument that hm

=

Z.^ Tlm;l'm' ' (al'm' + h'm') P'l'm'

(H)

with explicit expressions for a ^ and the T-matrix T^Vm, for a spherical scatterer. 6,7 A second equation system is derived by translating each wave bf • u t , (r — R ) to the origin: 7 ,'p _ sr^ c,pp' E^w^:, °lm

(12)

where the computable matrix fi^(/m/ depends on ku, the lattice, and on to/a and to/(3. Inserting Eq. (12) in Eq. (11), a linear equation system for bfm is obtained. In order to obtain the R / T matrices, the expansion (10) must be transformed to plane waves of the type (3). The following relation is crucial for this purpose: 5 ' 7 £

e i k " ' R h + M r - R | / C j ) Y, m (r - R) = £

R

g

2

~^=^

>T(K± ) e ^ V

.

(13)

%j*

Here, Kl"- should be used for z > 0 while K~- is needed for z < 0. A caret indicates the angular variables of the indicated quantity, and K^- is the z component of Kl_-. Using Eq. (13), a plane-wave representation of (10) is easily obtained 6 by expressing derivatives of Y";m in terms of itself and y ; m ± 1 . As anticipated from Sec. 2.1, it is the reciprocal lattice (2) that provides the changes of the lateral wavenumber vectors. 3. Basic Example A computer implementation has been made, with an existing program for photonic crystals 11 (the electromagnetic case) as a useful starting point. A basic example of the type in the left panel of Fig. 1 is now considered, with a 4 mm thick steel plate covered with a 3.5 mm rubber coating immersed in water with sound velocity c = 1480 m/s. In the middle of the rubber layer, spherical cavities with diameter 2 mm appear in a doubly periodic quadratic pattern with period d = 10 mm. The steel parameters are 5850 and 3230 m/s for the compressionaland shear-wave velocities, respectively, and 7.7 kg/dm 3 for the density. Only the rubber is anelastic, a viscoelastic solid with shear-wave velocity and absorption given by 100 m/s and 17.5 dB/wavelength, respectively. The corresponding compressional-wave parameters are 1500 m/s and 0.1 dB/wavelength, respectively, while the rubber density is 1.1 kg/dm 3 .

93 Curve (c) in Fig. 2 shows the frequency dependence of the corresponding reflectance. As compared to curves (a) and (b) for an uncoated and a homogeneously coated reference case, respectively, significantly reduced reflectance appears in the 10-60 kHz interval. At very low frequencies, the plate is thin compared to the wavelength and the reflectance drops. According to (2)-(4), with k|| = 0, only the normal beam is propagating in the water below c/d = 148 kHz. Nonnormal beam quartets become propagating at 148 kHz for (m,n) = (±1,0) and (0,±1) in (2), and at c^/2/d = 209.3 kHz for (m,n) = ( ± 1 , ± 1 ) . The corresponding small contributions to curve (c) are shown in curves (d) and (e), respectively.

Fig. 2. Variation with frequency of time- and space-averaged reflected energy flux, in dB relative to the timeaveraged normally incident plane-wave flux. The almost coinciding curves (a) and (b) show such reflectancies for two reference cases, an uncoated steel plate and a plate with a homogeneous rubber coating without cavities, respectively. Curves (c)-(e) concern the basic example as specified in the text. Curve (c) shows total reflectance, and curves (d) and (e) show the contributions from the two first nonnormal beam quartets.

4. Nonnormally Reflected B e a m s Pulse measurements in a water tank were designed to verify the existence of the nonnormal beam quartets from Fig. 2. A hydrophone at a distance of about 1 dm from a coated plate registered direct and reflected waves from a distant source. The hydrophone was moved laterally in x steps of 2.5 mm, covering 1.5 d = 15 mm for a constant y.

x=—d

x=0

x=+d/2

x=—d

x=0

x=+d/2

Fig. 3. Measurements (left panel) and modeling results (right panel) for pulse insonification centered at 177.5 kHz. The indicated tick-mark times are relative to a somewhat arbitrarily chosen reference time (zero, at the upper horizontal line). The seven traces in each panel correspond to the different lateral (x) hydrophone positions covering 1.5d = 15 mm. The direct arrival is denoted 'dir', the normally reflected beam 'rflO', and the first reflected beam quartet 'rfll'.

Figure 3 shows experimental data and modeling results for a source pulse centered at

94 177.5 kHz. The direct arrival 'dir' is followed by the normally reflected beam 'rflO', after about 0.135 ms. The pulse frequencies are higher than c/d = 148 kHz, allowing the existence of a propagating beam quartet corresponding to (m,n) = (±1,0) and (0, ±1) in (2)-(4). Indeed, a late arrival 'rfll' can be observed in both panels of Fig. 3. The geometrical and material parameters for the modeling, performed by Fourier synthesis, are exactly as in the basic example of Sec. 3, except that the cavities are adjacent to the steel to better match the actual coating. Since the water pressure is proportional to div(u), it follows from (2)-(4) that the lateral xy dependence of the pressure contribution by the beam quartet is given by ,.2nx. .2irx. ,.2ny. . 27ry. exp(i—) + exp(-i—) + exp(i—) + exp(-i—)

2-rrx 2iry\ •_ + « » - * ) .

, . (14)

For a constant y, varying constructive and destructive interference with x period d appears, as also observed for the 'rfll' arrival in the right panel of Fig. 3, computed for a particular y. The normal wavenumber of the beam quartet is kz = ^/w2/c2 — (2n/d)2, as obtained from (2)-(4), corresponding to a separated late arrival with normal group velocity du dkz

2it

C

(15)

d w

The expected lateral variations for the nonnormal beam quartet 'rfll' are not clearly seen in the measurements in the left panel. Contributing factors could be imperfections in the cavity lattice geometry, and that it was difficult keep the y value and achieve good accuracy during the desired 2.5 mm x translations of the hydrophone.

£=—d

x=0

x=+d/2

x=—d

x=0

x=+d/2

Fig. 4. As Fig. 3 but with a pulse centered at 250 kHz and two nonnormal beam quartets, 'rfll' and 'rfl2'.

Figure 4 is similar, but for a pulse center frequency of 250 kHz. Two nonnormal beam quartets are propagating in the water in this case, the previous 'rfll' quartet and an 'rf!2' quartet. By (15), the 'rfll' arrival gets an increased normal group velocity when the frequency is increased, and it now appears as a tail to the normally reflected beam 'rlfO'. The later beam quartet 'rfl2' consists of the four plane waves with representation (m,n) = ( ± 1 , ± 1 ) , according to (2)-(4). Results corresponding to (14)-(15) can easily be derived. The period in x for a fixed y is now halved to d/2. Both nonnormal beam quartets are weak in this case, as seen in both panels of Fig. 4. Noting that the spatial averaging in Fig. 2 causes cancellation of lateral energy flux, relative amplitudes in Figs. 3 and 4 are consistent with curves (d) and (e) in Fig. 2.

95 5. Design of Anechoic Coatings At lower frequencies, with only the normally reflected beam propagating in the water, Fig. 2 shows that an Alberich coating can provide significant echo reduction. Results of the same character have been given by Cederholm, 4 who computed reflection coefficients as functions of frequency based on parameter matching to certain experimental data. Unfortunately, direct measurements of the anechoic properties cannot be presented in an open publication. Anechoic coatings can be designed by numerical methods. The results obtained by two such methods, allowing certain variations to the basic example in Sec. 3, are shown as curves (d) and (e) in the left panel of Fig. 5. Curve (d) was obtained by differential evolution (DE) minimization. Simulating annealing and genetic algorithms have been popular global optimization methods during the last decade. DE is related to genetic algorithms, but the parameters are not encoded in bit strings, and genetic operators such as crossover and mutation are replaced by algebraic operations. For applications to underwater acoustics, DE has recently been claimed to be much more efficient than genetic algorithms 12 and comparable in efficiency to a modern adaptive simplex simulated annealing algorithm. 13 m s

/ > P3

HR

^ % r \^L^^^

VI

l.Or

LOU

-

10 20

«

\(e)/ 20

0.5 100

\i&f 10

'Jc^c'

. 30

.

Tf\

kHz

1450

m / s , j>2

1500

1550

O.Q[ 1.25 2

mn

3

?'.?6.' 4 5

Fig. 5. Left: Reflectancies as functions of frequency. Curves (a)-(c) are exactly as in Fig. 2, but for a restricted frequency interval. Curve (d) was obtained with DE to minimize the maximum reflectance in the band 15-30 kHz. Curve (e), obtained with the analytic design method of Sec. 5.1, exhibits a reflectance null at 22.5 kHz. Middle and right: ^-function characterizations of coating models with maximum 15-30 kHz band reflectance below -17 dB, jointly in terms of j>2,P3 (middle panel) and J>6,J>7 (right panel). The £^ level-curve values are 1,5,10,20,30,.., reaching 60 in the middle panel and 50 in the right panel. The two dashed arcs in the right panel represent cavity diameters of 2.7 mm (lower dashed arc) and 3.6 mm (upper dashed arc).

The objective function for the DE minimization was specified as the maximum reflectance in the frequency band 15-30 kHz. Starting from the basic example of Sec. 3, eight parameters, denoted Pi,P2,--,P8, were varied within a reasonable search space: rubber density [pi, 0.9-1.3 kg/dm 3 ] and compressional-wave velocity [p2, 1450-1550 m/s], rubber shear-wave velocity [j?3, 70-150 m/s] and absorption [p4, 7-27 dB/wavelength], and lattice period \p$ = d, 7-20 mm], coating thickness \pe, 1.25-5 mm], cavity diameter [0.5mm+p7 • (p^—1.25mm)], outer coating thickness between water and cavities [0.75mm+P8-(P6—1.25mm) ]. The parameters P7,p$ were defined as fractions, with nonnegative values such that pr + ps < 1. An echo reduction of at least 17.5 dB can be achieved throughout the band 15-30 kHz, as seen to the left in Fig. 5, curve (d). The corresponding rubber parameters are p i = 0.90 kg/dm 3 , p2 = 1455.3 m/s, ps = 149.8 m/s, p 4 = 26.9 dB/wavelength. The optimal geometrical parameters are p$ = d = 14.9 mm, pe = 4.98 mm, and p~t = 0.76, p% = 0.01. Improved echo

96 reduction could be obtained by also varying the rubber compressional-wave absorption. Some 40000 coating models were tested at this DE optimization. More information is contained in the search ensemble than just the optimal model. For example, let A be the set in the search space corresponding to coatings with maximum reflectance below -16.8 dB in the band 15-30 kHz, with characteristic function XA{PI,P2, --TPS)- Estimation is possible of certain dimensionless functions, generically denoted £4, of various parameters, such as . /

x

{nidP2dPz) IIIIIIIS

• IIJIIIxA(pi,P2,-,P8)dpidpidp5dp6dp7dp8 XA(PI,P2, -,PS) dpidp2dp3dpidp5dp6dp7 dp&

for the parameters P2 and P3, where each integral involves the whole corresponding searchspace cross-section. Estimates directly based on the DE search ensemble may be misleading, however, since the DE sampling is typically biased with an unknown sampling distribution. For Bayesian inverse problems, Sambridge 10 has proposed a resampling algorithm to estimate a posteriori probability density (PPD) function marginals. A neighborhood approximation 10 to the PPD, from a DE search ensemble, for example, can easily be evaluated along lines in parameter space. The new ensemble is constructed by random walks in directions parallel to the axes (Gibb's sampling), without further objective function calls. Here, a neighborhood approximation to \A is specified, and the Sambridge algorithm is adapted to produce an accordingly resampled ensemble with some 200000 models. The function £A{P2,P3) from (16), with average 1, is subsequently estimated in a straightforward way. The result is shown in the middle panel of Fig. 5. Most of the favorable coatings have rubber compressional-wave velocities below 1500 m / s , reasonably close to the velocity in the water, and rubber shear-wave velocities above 120 m / s . Without the resampling, the ^-function diagram would have appeared differently with higher values up to the left, indicating a DE-sampling bias as compared to the desired sampling here, controlled by XACharacterization in terms of a similarly produced £4 function of pe and p7 is made in the right panel of Fig. 5. It is natural that thick coatings (large p$) are preferred, but additional low-reflectance coatings appear within a large part of the region of p6iP7-space with cavity diameters, 0.5mm+p7-(p6 — 1.25mm), between 2.7 and 3.6 mm. Diameter-dependent singlecavity resonances, as modulated by multiple-scattering effects, appear to be essential for the loss mechanism and the frequency dependence of the reflectance. 5.1.

Designing

vanishing

reflectance

at an isolated

frequency

The second numerical design method is based on analytic function theory. It was used to produce curve (e) in the left panel of Fig. 5, with vanishing reflectance at 22.5 kHz. For a constant rubber density p and a varying complex rubber shear-wave velocity (3, the normal plane-wave reflection coefficient for waves from the water, now denoted 1Z, is an analytic function of the shear modulus fi = pf32 of the rubber material. The analyticity allows zeroes of lZ{n) to be identified by numerical winding-integral techniques, whereby the argument variation of 1Z is determined around search rectangles in the ji plane. Adaptive splitting of these search rectangles is applied until exactly one zero is enclosed. The secant method is finally used to refine the estimate of an isolated zero.

97 With carefully implemented error control, the existence of zeroes can actually be proved. The argument variation of H{n) around a closed path in the fj, plane is an integral multiple of 27r. The exact value is of course not obtained numerically, but a value close to 2w, for example, implies that one zero is enclosed. For the example in curve (e) in Fig. 5, exactly vanishing reflectance at 22.5 kHz is obtained at a rubber fi corresponding to a shear-wave velocity of about 98.3 m / s and a shear-wave absorption of about 26.7 dB/wavelength. All remaining parameters are kept at their values from the basic example in curve (c) of Fig. 2. Figure 6 shows corresponding time domain results obtained by Fourier synthesis, for a pulse with spectrum in the band 18-27 kHz. The reflected pulse as viewed at the water/rubber interface (left panel) is weak except in close connection to a spherical cavity at (x, y) = (0, 0) (the central trace). The corresponding energy is built up by evanescent waves with an exponential drop-off in the normal direction. At a distance of 1 m into the water, right panel, such waves are no longer discernible.

3

0.0ms-

— 0.0ms - 0.5ms

0.5ms -

- 1.0ms

-d/2

x=+d/2

=0

=+d/2

=0

x=-d/2

Fig. 6. Results of pulse computations by frequency synthesis, corresponding to curve (e) in Fig. 5. The reflected pulse is shown at the water/rubber interface (left panel) and 1 m into the water (right panel). There is a horizontal line for a common reference time (zero), where the center of the incident pulse has reached the water/rubber interface. Seven traces are drawn in each case, covering the overall period rf=10 mm along the x axis. The incident pulse is actually very similar in shape to the central trace in the left panel, but its amplitude is more than four times as large.

An apparent splitting of the pulse can be noted in the right panel of Fig. 6. To explain this effect, consider a general function g(t) of time t, with Fourier transform G(u>) = / g(t) exp(iwi) dt. It is the input to a linear filter with real-valued impulse function h(t) and transfer function H(tv) = J h(t) exp(iwi) dt. Thus, the output is given by the convolution (h * g){t) with Fourier transform H{LS)G(LO),

and H{—UJ) =

H*(LS).

For the notch filter from curve (e) of Fig. 5, there are real constants a,b such that H(w)

«

(17)

a (\ui\ — LOQ)+ib(u> — LUQsgn(uj))

in the vicinity of ±wo, where u>o = 2TT • 22500 Hz. For a function g(t) with spectrum concentrated to neighborhoods of ±WQ and Hilbert transform (TLg){t), it follows that

(h*g)(t)

-a ([Hg)'{t)+u0g(t))-b(g'{t)-uQ{Hg){t))

.

(18)

For a particular g(t) with G(u>) — $(o> — wo) + $(w + LOQ), where <E>(o;) is a real-valued, nonnegative, symmetrical function that vanishes for |w| > LJQ, g(t) = 2 cos(u>ot) tp(t)

and

(Tig)(t) = — 2 sin(uiot) ip(t)

(19)

98 w h e r e tp(t) is t h e inverse Fourier t r a n s f o r m of $ ( w ) . It follows by (18) t h a t (h*g)(t)

«

-2(p'(t)

(-a

sm(uJ0t) + b cos(uj0t))

.

(20)

T h e m o d u l a t i n g factor tp'(t) is small for small t, since tp(t) is a s y m m e t r i c a l function implying t h a t 4%vsJoc%

15

JO

25

30

incidence angle

(a)

35

change change change change change

—— 0 % velocity change — 1 % velocity change - • - 2 % velocity change -*>~ 3 % veloctty change -**_4%vetocaychange

--1.S5-

00

-1.65

«

15

30

25

30

35

40

45

50

incidence angle (b) Figure 6 Amplitude versus incidence angle for different velocity change (a) and phase versus incidence angle for different velocity change (b) for saturation =1 and thickness=0.125wavelength

4 Seismic attributes versus scale for thin-layer Wavelet transform is helpful in analyzing energy and frequency difference[12]. Seismic attributes versus scale is proposed and tested in thin-layer analysis. The theory of wavelet is not discussed here. We use continuous wavelet transform and morlet wavelet is chosen. Figure7 is amplitude versus scale for different bed thickness. Both amplitude maximum and corresponding scale are different for bed thickness change. So the new attributes can better delineate thin-layer bed thickness. Figure8 is amplitude versus scale for different incidence angle. When incidence angle increases, amplitude increases and scale decreases. Figure0- is amplitude versus scale for different velocity change. Velocity change mainly affects amplitude.

Scale (a)

(b)

Figure7 Amplitude versus scale for different bed thickness when incidence angle is 0 (a) and Amplitude versus scale for different bed thickness when incidence angle is 30(b).

Scale (b)

Figure8 Amplitude versus scale for different incidence angle when bed thickness is 1/4 wavelength (a) and Amplitude versus scale for different incidence angle when bed thickness is l/8wavelength (b).

Scale

(a)

0

10

20

30

40

ao

60

TO

Scale

(b)

Figure9 Amplitude versus scale for different velocity change when bed thickness is 1/4 wavelength (a) and Amplitude versus scale for different velocity change when bed thickness is l/8wavelength (b).

5 Reflection coefficient spectrum for thin-layer thickness and velocity change Spectral decomposition has been successfully used in bed thickness estimation and fluid discrimination[13][14]. The basis of spectral decomposition is reflection coefficient spectrum dependence on thickness and velocity change. Using reflection coefficient spectrum, thickness and velocity change can be separated in thin-layer. Figure 10-11 is reflection coefficient spectrum for different bed thickness and different velocity change. Bed thickness mainly affect frequency of reflection coefficient spectrum maximum. Velocity change mainly affect amplitude of reflection coefficient spectrum maximum. Using these two attributes, the bed thickness and velocity change can be discriminated.

(a)

Ill

(b)

Figure 10 Reflection coefficient spectrum for different bed thickness when incidence angle is 0(a) and Reflection coefficient spectrum for different bed thickness when incidence angle is 30(b).

Frequency{Hz)

(a)

112

0,3 -

20% velocity artf densfty cjiange }.

(b)

Figure 11 Reflection coefficient spectrum for different velocity change when bed thickness is l/4wavelength(a) and Reflection coefficient spectrum for velocity change when bed thickness is l/8wavelength (b). 6 Time-frequency analysis for thin-layer Time-frequency analysis can remove the tuning effect. The generalized S transform is used in the analysis. Figure 12 is the reflection coefficient and seismic trace. The seismic trace is affected by tuning. Figure 13 is the generalized S transform of the seismic trace. When the frequency increases, the spectrum has better correlation with reflection coefficient. Figure 14 is the comparison of one frequency spectrum and reflection coefficient. It can be shown that the position of maximum of spectrum can indicate the position of reflection coefficient. Figure 15 is the recovered reflection coefficient using time-frequency analysis. Time-frequency analysis is used to delineate the structure of seismic trace and combined with amplitude of the trace to form the recovered refection coefficient.

113 4)

!

"

o:

c

oi—fL

g-0.5 0) 0= -if9) 0

0.02

T~^7 0,04

O.06

0.08

0,1

0.12

0,14

0.16

0.18

0.2

time

Figure 12 Simple reflection coefficient and seismic trace

100

ISO

frequency

Figure 13 Spectrum for the modeled seismic trace

i;

1

S05i

T

•S-osi-

^

i

!f= I zx

1 /-I

ro, ZX

_ M

u ' =-i — smidAl -i—smidA2-i ax ax w(> =-i — cos idA{-i— ax ax aM = -P{0)2

— cosisB2 (3X

cosidA2-i—smisB2 f3x

cos(2r')4 1 - px0)2 cos{2is)A[ + A » 2 sin(2/])^

1

CO1 .

m

2ju{

T(£ =

(3)

,

CO2 .

., ,,

,

., ,,

CO

sin i\ cos i\A\ + — s i n i\ cos i\A\ + —— cos(2/s1 )B\ or, or, 2p,

• &>

.

.3 ,3

. #?

.3

n

3

- P ' - - ? — s i n / Ja 4 1 - z — cos/^5, 0

(3)

«"3

/"3

•

ou ^ U = °'•

(2-3)

121 3. Energy-Conserving Property Writing Eq. (2.1) in the form ur = a(n2(r, z) — \)u + buz

(3.1)

where ik0

a = —i

b = „ ,

2

(3.2)

.

2ik0

From Eq. (3.1), d2u ur = a(n2(r.z) — l)u + b

(3.3)

we have uru = a(n2(r, z) — l)uu + b

(3.4)

dz*

and —

-,

2/

\

T fd2U

-.s-

.

(3.5)

iru = a{n (r.z) — \)uu + b I —— J u Then

uru + uru =

d2u

d2i

u dz2

d2

£ W = 1(\U\>).

u

(3.6)

We want to examine whether or not \u\zdz = 0.

dr

(3.7)

Making use of expressions in Eq. (3.6), we have fZb

d2u\_

fz

r)

fd2u\

'

dr

dz.

(3.8)

Then, saving of the writing of ^ and the constant b, the first integral of the righthand-side of Eq. (3.8) can be evaluated by means of integration by parts; i.e. Zb / 2

(dus

d u\_1

du\

.

r-dudu,

dz

Tzn--LTzd-z -

/

,

x

(3 9)

'

Similarly, the second integral of the right-hand-side of Eq. (3.8) becomes czb /oa( ^ )

du\ UdZ

.

fdu\

dz~)uU+{dz')uL+

. /I

fZbdudu, dz -~^T~ dz dz

(3-10)

122 The term (§^)w|Zs in Eq. (3.9) and the term ( | j ) u | 2 s in Eq. (3.10) all go to zero due to the surface boundary condition, (2.2). Similarly, the term {%)u\Zb in Eq. (3.9) and the term ( f f H 2 „ in Eq. (3.10) all go to zero due to the bottom boundary condition, (2.3), therefore; d

fz\

,2,

d

A [Zb f

du9a\

,

fZbdudUl]

n

. „,

Then, the energy-conserving property of the standard PE with the prescribed boundary conditions, (2.2) and (2.3), is proved.

4. Remarks The Standard PE is a two-dimensional model with a narrow angle capability. These days the three-dimensional models have become more realistic in real applications. Not many users in the scientific community are using the two-dimensional model. Why bother to study the energy-conserving property for the Standard PE? There are a few answers for this question: 1. Because of the interest in three-dimensional problems, the Standard PE may not be used often in the acoustics community, to report this theoretical result to the public, we believe, may still interest the readers. 2. We selected the Tappert model to show it is energy-conserving, on the other hand, is to remember the late Tappert for what he did for the scientific community. 3. The technique, we used to examine the energy-conserving property, can be used to examine the energy-property of other PE models.

5. Conclusion The PE influence to the acoustic community is huge. All further-developed PE's are in use widely in the acoustifics community; they were all derived from the standard PE which benefited the scientific community a great deal. The Standard PE, even now-a-days is having limited use, it must not be forgotten; interestingly, the energy-conserving property of the Tappert model should not be unmentioned. This procedure may be applied to investigate the energy-conserving property for all other PE's, PE-like models, or other types of wave propagation models.

5.1.

Dedication

The impact of the PE to the scientific community is huge. In recognition of the PE contribution to the acoustic community, we cannot forget the Standard PE; and Frederick D. Tappert must be remembered. This paper is written in memory of our long time colleague Frederick D. Tappert.

123

Acknowledgments This research of the first author was supported by the U.S. Naval Undersea Warfare Center (Newport) Independent Research project.

References 1. Tappert, F. D., The Parabolic Equation Approximation Method, in Wave Propagation and Underwater Acoustics, ed. J. B. Keller and J. S. Papadakis, Lecture Notes in Physics 70, Springer-Verlag, Heidelberg, 1977, 224-287. 2. Lee, D. and S. T. McDaniel, BOOK. 3. Lee, D., A. D. Pierce, and E. C. Shang, Parabolic Equation Development in the Twentieth Century, J. Corny. Acoist. 8(4), 2000, 527-628.

A Dedication to Professor Tappert

Professor Frederick D. Tappert, who introduced the parabolic equation approximation to the acoustic community, passed away in 2001. Professor Michael I. Tarodakis and Dr. Finn B. Jensen organized a special memorial session for Prof. Tappert at the 6th International Conference on Theoretical and Computational Acoustics (ICTCA) at Hawaii, Honolulu, U.S.A. August 11-15, 2003. Professor Tarodakis and Dr. Jensen further encouraged the session speakers to contribute their articles to be included in the Proceedings of Theoretical and Computational Acoustics 2003. Their efforts in organizing this memorial session is appreciated by all of us. In January, 2000,1 visited Prof. Tappert in Miami, Florida, U.S.A. He expressed interest in contributing a paper to the 6th ICTCA. At that time, I started writing a paper on Revolutionary Influence of the Parabolic Equation Approximation to honor him. I continued to make progress on this article. At that stage, it was an article but, I left room for expansion. After the shocking news regarding Prof. Tappert, I immediately started writing another article entitled The Energy-Conserving Property of the Standard PE and dedicated it in memory of Prof. Tappert. Suddenly I was diagnosed with age-related Macular Degeneration. I had difficulty reading and writing. I was forced to stop writing this article, which I had planned to submit at the 2003 Hawaii ICTCA in the memorial session for Prof. Tappert, organized jointly by Prof. Tarodakis and Dr. Jensen. I was unable to submit the Energy Conserving paper on time. I felt very guilty for not being able to present this paper. After the conference, I was determined to complete writing this article, if possible. Prof. Er-Chang Shang came to help. With his help, this article has been completed. I thank Prof. Shang for his help and thank the committee chairs for giving me the opportunity to present this article at the 2005 Hangzhou ICTCA. Professor Frederick D. Tappert has gone; his PE contribution will be remembered. This article is dedicated in memory of my long-time colleague, Frederick D. Tappert. Ding Lee

124

Fredrick D. Tappert April 21, 1940 - January 9, 2001

Frederick D. Tappert was born April 21, 1940 in Philadelphia, Pennsylvania. His parents, the Reverend Dr. Theodore G. Tappert and Helen Louise Carson Tappert, raised their family of four children in the Lutheran Theological Seminary in Philadelphia, where the Reverend Dr. Tappert was a noted theologian. Fred showed an early penchant toward mathematics and science and attended Central High School in Philadelphia, which recognized outstanding young men in this area. From there he went on to study engineering at Penn State University, funded by the Ford Foundation, where he graduated with a B.S. in Engineering Science with honors in 1962. Fred went on to pursue his Ph.D. in theoretical physics from Princeton University with a full scholarship from the National Science Foundation. He earned his Ph.D. in 1967. Upon graduation Dr. Tappert was hired to the Technical Staff at Bell Laboratories in Whippany, New Jersey from 1967 - 1974, where he worked on plasma physics and high altitude nuclear effects, UHF radar propagation, solitons in optical fiber, and ocean acoustic surveillance systems. He left Bell Labs and became a Senior Research Scientist at the Courant Institute, at New York University from 1974 - 1978, where he performed research on controlled fusion and nonlinear waves, as well as ocean acoustics. It was at the Courant Institute that Fred first realized the impact that he could have upon students and thus his future took on even more meaning as the great professor and advisor began to emerge in Fred. Fred realized his potential as an educator and scientist when he left the Courant Institute and joined the faculty at the University of Miami's Rosenstiel School for Marine and Atmospheric Science in August 1978. At RSMAS he taught graduate courses in ocean acoustics, occasional undergraduate courses in physics, and supervised the research of more than 25 awardees of M.S. and Ph.D. degrees. In addition, Professor Tappert carried out a vigorous program of sponsored research in the areas of ocean acoustics, and wave propagation theory and numerical modeling. Dr. Tappert was a major participant in the ONR-sponsored initiative on "Chaos and Predictability in Long Range Ocean Acoustics Propagation." In this research he applied a recently developed 4-D (three space dimensional plus time) full-wave fully rangedependent parabolic equation (PE) ocean acoustic model to determine the limits of predictability of sound propagation and scattering. Since Dr. Tappert's most cited research was the original development of the PE numerical model, and he was also one of the originators of the concept of "ray chaos" in ocean acoustic propagation, this was a natural evolution for his research. Previously, Professor Tappert was a major participant in the ONR-sponsored "Acoustic Reverberation Special Research Project," the goal of which was to gain a scientific understanding of long-range low-frequency ocean surface and bottom reverberation by comparing numerical model predictions to measured 125

126 acoustic data, taking into account high resolution environmental data. In that research Professor Tappert developed a PE model of bistatic reverberation, the predictions of which compared favorably with measurements. In addition to his university research, Dr. Tappert was a consultant to many organizations involved in applied projects related to wave propagation theory and numerical modeling. This includes the DANTES project, sponsored by DARPA, in which he developed a novel technique call Broadband Matched Field Processing (BMFP) that localizes sources of acoustic transient signals using a back-propagation method. In October 2001, Fred was awarded the Superior Public Service Award from the Office of Naval Research. It was at this time that he was undergoing the rigors of chemotherapy in hopes that he would have more time in his fight against pancreatic cancer. This recognition brought tremendous joy to Fred. Unfortunately, he succumbed to the cancer only three months later on January 9, 2002. In November 2002, he was also posthumously awarded the Pioneers in Underwater Acoustics Award by the Acoustical Society of America. His wife, Sally, and two sons, Andrew and Peter, were present in Cancun, Mexico, to receive this award in his honor. Sally Tappert

ESTIMATION OF ANISOTROPIC PROPERTIES FROM A SURFACE SEISMIC SURVEY AND LOG DATA RUIPING LI, MILOVAN UROSEVIC Department of Exploration

Geophysics, Curtin University of Technology,

Australia.

[email protected]

Routine P-wave seismic data processing is tailored for isotropic rocks. Such assumption typically works well for small incidence angles and weak anisotropy. However, in the last decade it has become clear that seismic anisotropy is commonplace. Moreover, its magnitude often severely violates the presumptions taken for routine processing. Consequently reservoir characterization may be significantly distorted by anisotropic effects. In particular the intrinsic shale (often sealing rock) anisotropy often has first order effect on AVO gradient. Hence an assessment of the shale properties from surface seismic data may be of the primary importance for quantitative interpretation. There are several inversion approaches which require full set of geological information. In reality we expect to have at least the log and surface seismic data available for such a task. We present here a newly developed hybrid inversion method which is suitable for the recovery of anisotropic parameters of sealing rocks under such conditions. The effectiveness of this approach was successfully tested on seismic data recorded in the North West Shelf, Australia.

1

Introduction

Inversion of surface seismic data for the elastic properties of sealing rocks can impact on the accuracy of the reservoir characterisation. Since shales, which are intrinsically anisotropic, comprise often sealing rocks, an inversion has to at least incorporate recovery of the full set of anisotropic parameters for a transversely anisotropic medium. The shale anisotropy and its variation across an oil or gas field could have first order effect on Amplitude Versus Offset-and-azimuth analysis (AVOaz) [6; 1]. An example incorporating weak shale anisotropy is shown in Figure 1. Shale anisotropy in this case affects reflectivity curve on moderate to far angles. This "deviation" of the reflectivity curve could potentially impact onto our ability to accurately predict fluid type and its distribution across the field. Thus it is clear that before attempting detailed analysis for reservoir properties it is highly desirable to analyze and determine the magnitude of the seal anisotropy. Consequently an assessment of the shale properties from surface seismic data may be of the primary importance for quantitative interpretation of reservoir rocks. Thomsen [7] derived a convenient five-parameter model to describe seismic wave propagation in a transversely isotropic medium. There are many methods proposed to recover these elastic parameters, for example, the slowness surfaces method [2], the ray velocity field method from VSP surveys [4], the anisotropic moveout method from reflection events [8; 5]. Each of the above inversion method has been tested on field data sets separately provided enough information was available. However, we often have only surface seismic data and log data available for such inversion. In such case the existing methods fail to recover the elastic parameter accurately. For example the slowness method recovers the elastic parameters for an interval layer. The existence of a heterogeneous layer between successive receivers may produce errors in slowness surface determination. Deviation of the borehole, near surface inhomogeneities or topography of the surface also makes calculation of the slowness surfaces more

127

128 difficult. Because errors in slowness are in inverse proportion to the layer's thickness, errors for a thin interval layer will be larger due to the small time differences involved [3]. Using anisotropic NMO analysis, we may obtain information about overall anisotropy. We still need more constraints to determine the individual layer parameter values. For the ray velocity field method, the elastic parameters for an overall or interval layer may be estimated when the exact values for reflector depths are measured beforehand. Such method uses large number of observations, thereby statistically reducing the errors in the inverted parameters from measurement errors. However, any errors in the depth determination may produce inaccurate velocity field, which result in accumulated errors for the recovered parameters.

Incidence angles

Figure 1. Reflectivity curves for b-j\'A VTI and isotropic shale sealing an isotropic reservoir rock.

In the absence of suitable information, a new inversion approach which combines positive merits of different methods may be required. We present here a newly developed hybrid inversion which is suitable for the recovery of anisotropic parameters of sealing rocks (shales). The effectiveness of this approach was tested on seismic data recorded in the North West Shelf, Australia. 2

Recovery of elastic parameters using joint inversion method

We first discuss the inversion for the parameters for an overall layer, and then we will show how to recover the interval layer parameters. 2.1

Parameter for an overall layer to a reflector

For a reflection event, we use the anisotropic moveout velocity approximation [8] as below: 1 t2(x) = :

tl+^\

+4At^

(1)

Here, a represents the horizontal velocity. A is a newly defined parameter and its approximate value to the second order expressed in terms of Thomsen's anisotropy parameters eand £is [5]:

129 A*2-(e-8)-{ 2 /

Here,

/=

i-K

- i y + ( 3 + — )s2 -(4-y)«y. If'

(2)

with a0, /? S 0 •c 1

Parameter A for different CMPs

** *

0.2 0.15

.".

-

«

•

0.1 0.05

r 4185

4190

'•

~~ 4195

4200

4205

4210

0 -. 4180 4185

4190

4195 CMP

CMP

Figure 5. Anisotropy parameter ,4 and horizontal velocity a change along the seismic line (fixed to).

Surface seismic section

Figure 6. The surface seismic data, log data and the anisotropic semblance analysis.

Log data

4200

4205

4210

132 Even we have the analytical relation between anisotropic parameter A and s, 8 [5], it is still hard to obtain the anisotropic parameters e, 8directly from parameter^ because we lack enough information for the depth or vertical velocity. Making an assumption may cause big errors due to the sensitivity of the anisotropic parameters. The hybrid inversion which combines the ray velocity field method [4] and the anisotropic moveout method is then employed. We first pick the TWTs for different offsets for a reflection event. Then the hybrid inversion program is executed with the input of the TWTs and the recovered parameters a, A as a constraint. For the overall layer above the top of shale, we have £]=0.175, 8i=0.086. The reflector depth and the vertical velocity are also inverted. For the overall layer above the top of reservoir, we have s2=0.192, S2=0.081. Figure 7 shows the two-way-travel times from the measurements in circles (o) for the top layer. The asterisk (*) denotes the TWTs calculated using the inversion results. Both data sets match very well and the inversion results for the overall layer are quite satisfactory.

2.2

t 26 measurements calculated from the inversion results

2.7 28,

500

100D

ISOCI 2000 OUset (m)

2500

3000

3500

Figure 7. Comparison of the TWTs from the measurements and calculated from the inversion results. Very good agreement between these two sets of data indicates that the inversion is successful.

Subsequently, for the interval shale property, we apply the ray velocity field method based on a twolayer's model [4]. The anisotropic parameters obtained for the shale above the reservoir are: s=0.224, 8=0.108. Such results are also verified by the slowness surface plot in Figure 8. Notice that the thickness will affect the inversion so that for very thin shale layer at this CMP, the measured slowness surface in figure 8a is of low quality. Figure 8b shows another example with a thicker shale layer in another CMP position. The anisotropic parameters s and 8 for the overall layer to the top and the bottom of the shale are inverted first. Subsequently, the interval parameters e and 8 for the interval shale layer are then successfully recovered. The inverted anisotropic parameters can then be used in the AVOaz analysis aimed at the reservoir characterization.

133

(a) Slowness surface for a thin layer

(b) Slowness surface for a thick layer

Figure 8. The comparison of the slowness surfaces from the measurements and calculated from the inversion results.

4

Conclusions

From the log data and anisotropic semblance analysis, the reflection events at different two way travel times are analysed, as well as the horizontal velocities a and the anisotropic parameter A. From a seismic section, the two-way travel times for different offsets for a CMP location are manually picked. With the constraint of the parameter^ and horizontal velocity a values, a new hybrid inversion method is developed to recover anisotropic parameters e, S, reflector depth and the vertical velocity from the observations of two way travel times for different offsets. As the velocity field at different ray angles can be converted using the inverted reflector depth, verification procedure is carried out. The calculated values of TWT for different offsets using the recovered parameter values should coincide with the log measurements. Apparent differences between the measured and estimated values may suggest misfit of the seismic section with the log data. After obtaining the apparent average parameter for the top and the bottom sealing layer or reservoir, the interval anisotropy parameters are obtained from the velocity field data using two-layer model approach [4]. From the travel time picks, the slowness surface for the interval layer is also constructed which allows us again to recover the interval anisotropy parameters. These two estimates should match each other. The application of our new hybrid inversion methods to the field petroleum data suggests that the method is robust and should consequently result in reliable parameter estimates. 5

Acknowledgments

This is a project supported by the Curtin Reservoir Geophysics Consortium (CRGC). We thank CRGC for providing the field data. Thanks also go to Mr. Said Amiri Besheli for his help with the filed seismic data and log data.

134 6

References 1. Banik, N. C, An effective anisotropy parameter in transversely isotropic media: Geophysics, Soc. of Expl. Geophys., 52 (1987) pp. 1654-1664. 2. Hsu, K., Schoenberg, M. and Walsh, J. J., Anisotropy from polarization and moveout: 61st Ann. Internat. Mtg., Soc. of Expl. Geophys., (1991) pp. 1526-1529. 3. Kebaili, A., Le, L. H. and Schmitt, D. R., Slowness surface determination from slant stack curves, in Rathore, J. S., Ed., Seismic anisotropy: Soc. of Expl. Geophys., (1996) pp. 518-555. 4. Li R., Uren N. F., McDonald J. A. and Urosevic M., Recovery of elastic parameters for a multilayered transversely isotropic medium: J. Geophys. Eng., 1 (2004) pp. 327-335. 5. Li R. and Urosevic M., Analytical relationship between the non-elliptical parameter and anisotropic parameters from moveout analysis: (2005) being prepared for publication. 6. Ruger, A., Variation of P-wave reflectivity with offset and azimuth in anisotropic media, 66th Ann. Internat. Mtg: Soc. of Expl. Geophys., (1996) pp.1810-1813. 7. Thomsen, L., Weak elastic anisotropy. Geophysics 51 (1986) pp. 1954- 1966. 8. Zhang, F. and Uren, N, Approximate explicit ray velocity functions and travel times for p-waves in TI media: 71th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, (2001) pp. 106-109. 9. Zhang, F., Uren, N., and Urosevic, M., Anisotropic NMO corrections for long offset P-wave data from multi-layered isotropic and transversely isotropic media: 73rd Ann. Internat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, (2003) pp. 133-136.

USING GAUSSIAN BEAM MODEL IN OCEANS WITH PENETRATING SLOPE BOTTOMS Y I N G - T S O N G LIN*, CHI-FANG CHEN, Y U A N - Y I N G CHANG, WEI-SHIEN H W A N G Department

of Engineering

Science

and Ocean Engineering,

E-mail:

National

Taiwan

University

[email protected]

A numerical code using Gaussian Beam Model (NTUGBM) is developed for underwater acoustic propagation at high frequency (larger than 1 kHz) in oceans with penetrating slope bottom. Several test cases are used to benchmark NTUGBM. Cases include continental shelf and continental slope. The results of NTUGBM are compared with results using EFEPE and FOR3D (Nx2D version). Results of NTUGBM agree well with those of both codes.

1

Introduction

In order to accurately and efficiently simulate the acoustic field, some sorts of numerically methods have been developed. In this paper, a numerical model called NTURAY, which is developed using the Gaussian Beams Method, is illustrated [1]. The propagation models deduced from the Helmholtz Equation are classified in Fig. 1, which are divided into the range-dependent and the range-independent models. Our goal is to establish a high-frequency, range-dependent numerical model with the capability to accomplish the long-range ray tracing and transmission loss calculations in the laterally varying multi-layered ocean environment. According to the requirements, only the Ray method is efficient enough to handle the high-frequency and ray tracing computation. f=-

v$-


!

d> + kld

NM

= 0

RdllJtC llL'pilull'Il! Hflniholl/. liquation

0 - /•'(.-Jf/li )

Ray o~Flx.y,:)i' I": Amrtifji- .•''-r.--'i;;i

!?

i I

I . • M I - ' U 1 :.«.-..!• : - H I -

PE 4-nr,0a)O(r)\ t f parabolic Equation i G. Bess^/H^iitsl Function. \ "iT-'p"

FD/FE.

•.'.. : i y j »••:

Pirate Eianenf.

Figure 1. The propagation models deduced from the Helmholtz Equation.

* Current position: Post-Doc fellow of Woods Hole Oceanographic Institute 135

136 A serious drawback is using the Ray Method in the vicinity of caustics, and Gaussian Beams Method can overcome this problem and effectively calculate the transmission loss in caustics and shadow zone as well. Cerveny et al. [2-4] first applied the Gaussian Beams Method in geophysics, and then this method is used in underwater acoustics application by Porter and Bucker [5] and Weinberg and Keenan [6]. All the applications introduced above dealt with a flat bottom, so the contribution of this paper is to apply the Gaussian Beans Method in cases of slope bottom and laterally varying layered bottom. In section 2, we will introduce the theory of Gaussian Beams Method. The verification of the NTURAY model is discussed in section 3, and section 4 will talk about the calculation of the layered bottom. Finally, section 5 will give brief discussions and conclusions. 2

Gaussian Beam Method

The linear acoustic wave equation is written as V

1 d2P 1 = l- ^ T '~c dt

'2

_

P

(1)

For a harmonic wave, the solution to the linear acoustic wave equation is

P(x,t)

= A{x)eim[t~T(s)\

(2)

where W\t — T\x))is the constant phase surface, G7 is the frequency and A(x)is amplitude. Substitute Eq. (2) into Eq. (1), we can obtain the following equation, UV2A-(O2A\VT\2

+^A)

+ i(26)VA-VT + coAVh)ieia('-T) =0

the

(3)

The real part and imaginary part are equal to zero as following,

y2A

,

.

-A\VT\

,2

A + ~T = 0

(4)

C

CO

2V^-Vr + ^V2r

= 0

(5)

Thus, if the amplitude changes slightly with the space and if the frequency is high enough, Eq. (4) becomes Eq. (6) |Vr|2=^ c

(6)

Because the directional vector of the ray is

dx

Vr

combining this directional vector with Eq. (6), the Eikonal Equation can be deduced to be Eq. (8). r

^dx}

ds c ds

(8) •

>

Thus we can obtain the geometry of acoustic rays by solving the functions in cylindrical coordinate (Nx2D calculation, eliminate the 0 coupling)

d(\dr\

j_dc_

ds ye ds_

c2 dr

d (\_dz^

]_dc_ c2 dz

ds c ds j 1 dr

(9)

„ 1 dz q and = Q , Eq. (9) becomes

Giving that

c ds dr(s)

c ds

=c(S)-m

ds dz(s) ds

am

1

dc(s)

ds

c2(s)

dr

d£(s)

1

dc(s)

2

ds

(10)

c (s)

dz

Thus we can solve the equation system simultaneously with the initial conditions to obtain the ray traces. If we rewrite the solution of the standard linear wave equation as P(x,t)=

^ ( x > ' r a [ ' - r ( i ) ] = u{x)eim
is identical to x except for Xm = 0. Using Eqs. (14)-(17), the components xm of the reflectivity sequence can be re-sampled, one at a time.

157 2.

Resampling

of the seismic wavelet

In order t o re-sample the seismic wavelet h, we deduce from the Bayes rule t h a t P(h\a2,x,z) oc P(z\a2,x,h), given P(h) = 1, where P(z\a2,x, h) is given by Eq. (12). Moreover, it is easy t o check the following identity:

_\\z~h*xf

=_^{h_ii)TR_l{h_^

(19)

where » = {XTX)-1XTz,

(20)

R = (XTX)-1a2l,

(21)

and

where X is the Toeplitz matrix of size (N + M — 1, N) such that Xh = h*x.

(22)

This allows us to conclude t h a t the posterior probability of h is a multivariate Gaussian with mean vector n and with covariance matrix R. T h e latter probability is easy t o sample according t o h = fi + Qe, where e is a normalized Gaussian white noise and QT is a square root matrix of R (that is, such t h a t R = QQT), such as the one resulting from the Cholesky decomposition.

3.

Re-sampling of the hyperparameter a

Given P(<J 2 ) = 1, it is also true that P(a2\z,x, takes the form P(z\a2,x,\)

h, A) ex P{z\a2,x,

h). As a function of a 2 , Eq. (12)

= pjSTTexp(-/3/a2)

up to a multiplicative constant, with a = (N + M — l ) / 2 — 1 and f3 = \\z — h * x\\2/2, which means that the posterior probability of a2 follows an inverse gamma distribution of parameters (a, /3). T h e latter can be easily sampled by taking the inverse of a gamma random generator output with the same parameters.

4-

Re-sampling of the hyperparameter A

T h e reflectivity sequence x gathers all the information about A contained in (z,x,h, a2), t h a t is, P(X\z,x,h,a2) = P(\\x). Following [13], let us remark that the Bernoulli sequence q can b e retrieved from x with probability one according t o q^ = 1 if Xt ^ 0, q^ = 0 otherwise. Thus, P(\\x) = P(X\q), the latter being proportional t o P(q\X) since we assumed a flat prior P(A) = 1. Finally, according to Eq. (6), we get P(A|z,x,/i,a2)cxAn(l-A)M-n,

(23)

which belongs to the family of b e t a probability densities B(a,f3) with a = n+1 and (3 = M — n + 1.

158 III.

I N V E R S E REFLECTION P R O B L E M

The one-dimensional seismic wave equation for the elastic displacement u is given by [18] p

d2u

d (

2du

W-dz{pC8-z)=°'

W

where t is the time, z is t h e space coordinate along the direction of propagation, p = p{z) is the density of the medium, and c = c(z) is the speed of t h e seismic wave. We are considering here a longitudinal displacement in t h e ^-direction. T h e Marchenko integral equation is directly applicable to t h e case of inversion with a seismic wave normally incident on a planar stratified medium, provided t h a t t h e one-dimensional seismic wave equation is converted t o the Schrodinger equation. Thus, the coordinate variable z is changed t o the travel time £ defined by

dz= c^y •

(25)

When integrating Eq. (25) we obtain

*=La£)"'

(26)

which is the travel time for a pulse to move from the origin to position z. Upon using this relation we can rewrite t h e wave equation as , ,d2u

d ( ,^du\

, s (27)

"«> *' II |I>*' 11 ''*'

-50 -100

FIG. 2: 240th seismic trace.

~ -

162 0 T

3 4 -

5 E-i

6 7 0

50

100

150

Number of traces FIG. 3: The seismic data without moveout correction.

0

-1 + -2

w' ,

-20

i

0

0.5

i

1

i

1.5

i

2

l

2.5

l

3

i

3.5

FIG. 8: Scaled reflectivity sequence in Fig. 6.

[19] W. C. Chew, in Waves and fields in inhomogeneous media, Van Nostrand, Reinhold, New York, 1990, p.49-52, 532-547. [20] J. M. Reynolds, in An introduction to Applied and Environmental Geophysics, John Wiley and Sons, Singapore, 1997, p. 218, 226, 233, 360. [21] F. B. Jensen, W. A. Kuperman, M. B. Porter, H. Schmidt, in Computational ocean acoustics, American Institute of Physics, New York, 1994, p. 41-54. [22] E. L, Hamilton, Geoacoustic modeling of the sea flour, J. Acoust. Soc. Am. Vol. 68, 1980, p. 1313-1340. [23] E. L. Hamilton, Acoustic properties of sediments, in Acoustics and Ocean Bottom, edited by A. LaraSaenz, C. Ranz-Guerra and C. Carbio-Fite (C.S.I.C, Madrid, Spain, 1987), p. 3-58.

165

FIG. 9: Estimated seismic impedance corresponding to the scaled reflectivity sequence in Fig. 8.

FIG. 10: Comparison between the estimated seismic impedances in Figs. (9) and (7) with and without a scaling factor respectively.

Journal of Computational Acoustics © IMACS

C H A R A C T E R I Z A T I O N OF A N U N D E R W A T E R A C O U S T I C S I G N A L U S I N G T H E STATISTICS OF T H E WAVELET S U B B A N D COEFFICIENTS MICHAEL I. TAROTJDAKIS Department of Mathematics, University of Crete, Institute of Applied and Computational Mathematics, FORTH, P.O.Box 1385, 711 10 Heraklion, Crete, Greece taroud@iacm. forth, gr GEORGE TZAGKARAKIS and PANAGIOTIS TSAKALIDES Department of Computer Science, University of Crete, Institute of Computer Science, FORTH, P.O.Box 1385, 711 10 Heraklion, Crete, Greece {gtzag, tsakalid} @ics. forth.gr

A novel statistical scheme for the characterization of underwater acoustic signals is tested in a shallow water environment for the classification of the bottom properties. The scheme is using the statistics of the 1-D wavelet coefficients of the transformed signal. For geoacoustic inversions based on optimization procedures, an appropriate norm is defined, based on the Kullback-Leibler divergence (KLD), expressing the difference between two statistical distributions. Thus the similarity of two environments is determined by means of an appropriate norm expressing the difference between two acoustical signals. The performance of the proposed inversion method is studied using synthetic acoustic signals generated in a shallow water environment over a fluid bottom.

1. Introduction Recently, a new method for the classification of the underwater acoustic signals has been proposed by the authors, aiming at the definition of an alternative set of "observables" to be used for geoacoustic inversions l. The study was motivated by the fact that it is not always possible to obtain a set of identifiable and measurable properties of the acoustic signal to be used in the framework of an inversion process. As the efficiency of an inversion procedure is directly related to the character of the observables, a major task on a specific physical problem is to define observables which will be more sensitive to changes of the environmental parameters and easily identified in noisy conditions. In previous works *'2 it was shown that the modelling of the statistics of the wavelet subband coefficients of the measured signal, provides an alternative way for obtaining a set of observables which is easily calculated and has the necessary sensitivity in changes of the environmental parameters, so that its use for inversions to be well justified. Here, this method is tested in shallow water environments for the recovery of the bottom parameters. The inversion is based on an optimization scheme utilizing the Kullback-Leibler divergence to measure the similarity between the observed signal and a signal calculated using a candidate set of bottom parameters.

167

168 2. T h e classification scheme In the framework of the proposed approach, an acoustic signal is classified using the statistics of the subband coefficients of its 1-D wavelet transform. In particular, the measured signal is decomposed into several scales by employing a multilevel 1-D Discrete Wavelet Transform (DWT) 3 . This transform works as follows: starting from the given signal s(t), two sets of coefficients are computed at the first level of decomposition, (i) approximation coefficients Al and (ii) detail coefficients D l . These vectors are obtained by convolving s(t) with a low-pass filter for approximation and with a high-pass filter for detail, followed by dyadic decimation. At the second level of decomposition, the vector Al of the approximation coefficients, is decomposed in two sets of coefficients using the same approach replacing s(t) by A l and producing A2 and D2. This procedure continues in the same way, namely at the k-th level of decomposition we filter the vector of the approximation coefficients computed at the (k-l)-th level. 2.1. Derivation

of the statistics

of the wavelet

subband

coefficients

The Feature Extraction (FE) step is motivated by previous works on image processing 4 ' 5 , 6 . The signal is first decomposed into several scales by employing a 1-D DWT as described above. The next step is based on the accurate modelling of the tails of the marginal distribution of the wavelet coefficients at each subband by adaptively varying the parameters of a suitable density function. The extracted features of each subband are the estimated parameters of the corresponding model. For the acoustical signals studied, the wavelet subband coefficients are modelled as symmetric alpha-Stable (SaS) random variables. The SaS distribution is best defined by its characteristic function 7 ' 8 : (Kw)=exp(? 0) is the dispersion of the distribution. The characteristic exponent is a shape parameter, which controls the "thickness" of the tails of the density function. The smaller the value of a, the heavier the tails of the SaS density function. The dispersion parameter determines the spread of the distribution around its location parameter, similar to the variance of the Gaussian. In general, no closed-form expressions exist for the SaS density functions. Two important special cases of SaS densities with closed-form expressions are the Gaussian (a = 2) and the Cauchy (a = 1). Unlike the Gaussian density, which has exponential tails, stable densities have tails following an algebraic rate of decay (P(X > x) ~ Cx~a, as x —> 00, where C is a constant depending on the model parameters), hence random variables following SaS distributions with small a values are highly impulsive. 2.2. Feature

Extraction

After the implementation of the 1-D wavelet transform, the marginal statistics of the coefficients at each decomposition level are modelled via a SaS distribution. Then, to extract

169 the features, we simply estimate the (a, 7) pairs at each subband. Thus, a given acoustic signal S, decomposed in L levels, is associated with the set of the L + l pairs of the estimated parameters: S ^ { ( a i , 71), (a2, 72), • • •, («L+i, 7 i + i ) } .

(2)

where (o^, 7$) are the estimated model parameters of the i-th subband. Note that we follow the convention that i = 1 corresponds to the detail subband at the first decomposition level, while i = L + 1 corresponds to the approximation subband at the L-th level. The total size of the above set equals 2(_L + 1) which means that the content of an acoustic signal can be represented by only a few parameters, in contrast with the large number of the transform coefficients. As it has already been mentioned, the FE step becomes an estimator of the model parameters. The desired estimator in our case is the maximum likelihood (ML) estimator. The estimation of the SaS model parameters is performed using the consistent ML method described by Nolan 9 , which provides estimates with the most tight confidence intervals. 2.3. Similarity

Measurement

In the proposed classification scheme, the similarity measurement between two distinct acoustic signals was carried out by employing the Kullback-Leibler divergence (KLD) 10 . As there is no closed-form expression for the KLD between two general SaS densities which are not Cauchy or Gaussian, numerical methods should be employed for the computation of the KLD between two numerically approximated SaS densities. In order to avoid the increased computational complexity of a numerical scheme, we first transform the corresponding characteristic functions into valid probability density functions and then the KLD is applied on these normalized versions of the characteristic functions. Due to the one-to-one correspondence between a SaS density and its associated characteristic function, it is expected that the KLD between normalized characteristic functions will be a good similarity measure between the acoustic signals. If 4>{UJ) is a characteristic function corresponding to a SaS distribution, then the function

fa) = ^

(3)

is a valid density function when oo 4>(u>) duo.

/

-00

For the parameterization of the SaS characteristic function given by Eq. (1) and assuming that the densities are centered at zero, that is 5 = 0, which is true in the case of wavelet subband coefficients since the average value of a wavelet is zero, the normalization factor is given by

170 By employing the KLD between a pair of normalized SaS characteristic functions, the following closed form expression is obtained 4 :

^ll^) = l n ( ^ ) - ^ + ( j r . ^ l \CXJ

«i

V71/

(5)

r ( —)

where (on, %) are the estimated parameters of the characteristic function <j>i(-) and C; is its normalizing factor. It can be shown that D is the appropriate cost function for our application as DfyiWfa) > 0 with equality if and only if (on, 71) = («2> 72)Thus, the implementation of an L-level DWT on each underwater acoustic signal results in its representation by L + 1 subbands, (D\, D2, • • •, DL,AL), where Di, A{ denote the i-th level detail and approximation subband coefficients, respectively. Assuming that the wavelet coefficients belonging to different subbands are independent, Eq. (5) yields the following expression for the overall distance between two acoustic signals Si, S2: L+l

£(S1||S2) = ^£(R . For the physical variables denoted by u(x,y,z,t) , constrained to T , the boundary of Q."" , and (0,7) , we sayu(x,y,z,t):rx(0,T)—>R , where T i s either the part of £2"" or the whole boundary 1 of CI"" . In order to avoid intensive mathematical difficulties, we assume that the functions of interest belong to the / / ' space, i.e., a classical Sobolev space that belongs to a Hilbert space ]} , or a space of functions, with square Integrabel generalized first derivatives. This assumption requires that, each physical variable, is a kind of function such that it is piece-wise continuous, and the integration of the square of its first derivative on the defined space domain is finite, which is generally true for our linear elastodynamic problem. With this in mind, the boundary value problem for elastic wave equation can be expressed as the followings Given ft, g.,h t , uoi, uoj, find w. : Q,"" X (0,T) —» M such that p(x,y,z)uii„(x,y,z,t) ui(x,y,z,t)

= (TUj(.x,y,z,t) + fi(x,y,z,t)

= gi(x,y,z,t)

on £>"" x(0,T),

on Tgj x(0,T),

R, «„,,,(*, y,z,0): are given functions for each i,\ K. (1.3c) They denote the given body force, displacement and surface stress components, respectively. The density, p: Q."" —» M, assumed to be positive, needs also to be specified in the present case. Note that the boundary T for the domain 0."" may be decomposed into two basic sets in accordance with the given boundary conditions, i.e.,r = r f t [ J r g .The stresses are given on Th ,while the displacements are given on r However, r ^ l j r ^ = 0 , which is an empty set .The constitutive equation, cr. = cjjklu(kl) , links stress and displacement

through

with

+ M

" ( U ) =("*./

*,/)

/ 2

geometric

equations

or strain-displacement

equations,

•

2.2. Weak formulation of elastodynamic equation In spectral element methods, the strong formulation in (1.1) is converted into weak formulation, and then it is converted into Galerkin weak formulation in which the weighted function and trail solutions are expressed by finite terms of orthogonal basis functions. These functions can be constructed with orthogonal either Chebyshev or Legendre polynomials. The corresponding weak formulation is Given f, g, h, u oi , iioi, find u e LI,,f G (0,T), such that for all test function w e PL (w,/ni) + a(w,u) = (w,f) + (w,h) r , (w,/7u|,=0) = (w,pu 0 ),

(2.1)

(w,pu| (=0 ) = (w,pu 0 ). In the above equations, the inner products are defined as (w,u) - f w^u.dQ.,

(2.2a)

(w,h)r=J|^,Vr,

(2.2b)

a(w u) =

' L wu.J)cmuiu)dQ' K" J ,u| rs =g} , and n = {w6ff'(fl"'):fl"' -»R"";w| r =0} . Some times we denote (r, ,?)„„,, = \ rsdQ. where r and s are two scalar functions with the same properties as the uv.

180 Note that the choice of the test function in such a way, as we may see, will bring a great convenience in the following mathematical and numerical treatments. It results in natural satisfactions of traction free boundary conditions if the displacements are prescribed on the whole boundary. If we take the whole domain as one element, the semi-discrete Galerkin formulation of the elastodynamic problem is (Hughes, 1987) given f,g,h,u 0 , and ii 0 , find uh = \h + g \ u * ( 0 e UhJe [0,7], such that for all w* e I T ,

(w\/7vVa(w%v*)Kw\f)+(w\h)r-(w,pg*)-a(w\g*), (w\/7v*(O)) = (w\ / 9u o )-(w\ / 0g' i (O)), (w\/>V*(0)) = ( W \/?U o )-(w\/?g''(0)). (2.3) In the above equations, w*, v* and g* can be expanded by finite numbers of basis functions denoted by NA(x) w.'•=

I NA(x)c,A(t), ten-v.,

v,"= £

(2.4a)

NA(x)diA(t),

(2.4b)

S,*=5>,(x)gM(0,

(2.4c)

where A belongs to the node point in the element Q"* . Note that 77 E {1,2,..., N} is the set of node numbers and 77g is the set of nodes at whichui = g.. Also, wh = whiei, vh - v'i'ei, and gh = g'.e!. In order to solve the weak formulations as shown in Equation (2.1), the space domain is discretized into a finite number of non overlaying elements Nd , the eth element is denoted by£2 e , and Q."J ={J&e whereee [l,2,...,Nel],

and Ne[ is the total

element number. In each element, there are a number of nodes to be used in order to interpolate the related function values. Equation (2.3) for all of the elements is converted into a matrix equation, that is Given F : [0.7] -^ K, find the d : [ 0 , T ] ^ R such that Md + Kd = F, (2-6a)

181

(2.6b) and the associated stiffness matrixes K and k e K=A(kc),

(2.6c)

ke=[F


MV)> (3-3) It is easy to show that (Na,Nb)n,=(^t)]^^

(3-4)

where the inner product is defined as

In this case, the element mass matrix m =[mepq] is diagonal, which can be see from equation (2.6b), where <pa is the 1-D basis functions. 3.1. Basis functions constructed using Legendre polynomials When Legendre polynomials Pn (£) are chosen for constructing the basis functions with n = 0, 1,2, ... N, the element mass matrix is always diagonal, which reduces computation time in forming element mass matrix. For stiffness matrix, by checking(N a ^,N br ) Qr , we have Waj'Ni.Jv = MatfWMh-m&MfiXr):,; • Let P = ^, and y = t], the above equation will become

Waj'^a;

= ((hAZWMAifr&^K', •

(3-6a)

< 3 - 6b )

When we use $,(£) = c a P a (£) as a basis function, it is easy to show that (^./^A; = ^aP^Ah-ic.P^c^ .

(3.6c)

Since PB4Pb={a + mPaPb-PB+xP„)l{\-e), We may convert it into a sum of truncated series Pa A =(« + l)Z[