fixed-point computation and seismic waveform

FIXED-POINT COMPUTATION AND SEISMIC WAVEFORM TOMOGRAPHY

by Chaiwoot Boonyasiriwat

A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Computing

School of Computing The University of Utah August 2009

c Chaiwoot Boonyasiriwat 2009 Copyright All Rights Reserved

THE UNIVERSITY OF UTAH GRADUATE SCHOOL

SUPERVISORY COMMITTEE APPROVAL of a dissertation submitted by

Chaiwoot Boonyasiriwat

This dissertation has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory.

Chair:

Krzysztof Sikorski

Gerard T. Schuster

Mike Kirby

Martin Berzins

Elena Cherkaev

THE UNIVERSITY OF UTAH GRADUATE SCHOOL

FINAL READING APPROVAL

To the Graduate Council of the University of Utah:

I have read the dissertation of Chaiwoot Boonyasiriwat in its final form and have found that (1) its format, citations, and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the Supervisory Committee and is ready for submission to The Graduate School.

Date

Krzysztof Sikorski Chair: Supervisory Committee

Approved for the Major Department

Martin Berzins Chair/Director

Approved for the Graduate Council

David S. Chapman Dean of The Graduate School

ABSTRACT This dissertation is comprised of two parts: fixed-point computation and seismic waveform tomography. In the first part, I consider the problem of approximating fixed points of multivariate contractive and nonexpanding functions whose Lipschitz constant is close or equal to 1.

Both absolute and residual error criteria are

utilized as termination conditions. The circumscribed ellipsoid (CE) algorithm is shown to be able to approximate fixed points in a larger class of functions than possible in previous research. A function in this class is globally expanding but contractive or nonexpanding in the direction of fixed points. Line-search (LS) and circle-search (CS) methods are also developed for solving the fixed-point problem. The computational costs of both methods are lower than those of the CE method when the number of constraints is small. The LS method is well suited for almost linear functions while the CS method is well suited for rotational functions. To combine the strength of both methods, I developed an LS-CS hybrid method that in general is more efficient than either one of these methods. Various multivariate contractive and nonexpanding functions were implemented to test the performance of the proposed methods. The simple iteration (SI) and Newton-Raphson (NR) methods were compared with the CE, LS, CS, and LS-CS hybrid methods. The CE algorithm is an excellent method for low-dimensional functions with discontinuities and/or low regularity. However, the LS method is faster than the CE method in many cases because there is no need to construct an ellipsoid which requires solving an eigenvalue problem. However, the number of constraints for the LS, CS, LS-CS hybrid methods grows with the number of iterations. In the second part, I develop an efficient multiscale method for time-domain waveform tomography. I propose filters that are more efficient than the previously used Hamming-windowed filter. A strategy for choosing optimal frequency bands is proposed to achieve high computational efficiency in the time domain. A staggered-

grid, explicit finite-difference method with 4th -order accuracy in space and 2nd -order accuracy in time is used for forward modeling and the adjoint calculation. The adjoint method is utilized in the inversion for an efficient computation of the gradient directions. In the multiscale approach, multifrequency data and multiple grid sizes are used to partially overcome the severe local minima problem of waveform tomography. My method is successfully applied to both 1D and 2D heterogeneous models, and can accurately recover both the low and high wavenumber components of the velocity models. The inversion result for the 2D model also shows that the multiscale method is computationally efficient and converges faster compared to a conventional, single-scale method.

v

To my parents and my wife.

CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi CHAPTERS 1.

2.

3.

GENERAL INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1 Fixed-point Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Seismic Waveform Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

FIXED-POINT COMPUTATION . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1 Fixed-point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Two Classical Fixed-point Theorems . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Classes of Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fixed-point Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 A Constructive Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Simple Iteration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Ball Iteration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Newton-Raphson Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Proposed Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Circumscribed Ellipsoid Algorithm . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Larger Class of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Line-search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Circle-search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 LS-CS Hybrid Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 6 7 8 8 10 10 11 13 14 16 16 18 19 21 25 27 58 58

AN EFFICIENT MULTISCALE METHOD FOR TIME-DOMAIN WAVEFORM TOMOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2 Methods for Time-domain Waveform Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 Acoustic Waveform Tomography . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Efficient Low-pass Filtering for Multiscale Waveform Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Strategy for Choosing Optimal Frequency Bands . . . . . . . . . . . 3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 1D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Canadian Foothills Synthetic Data . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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63 65 68 74 75 79 83 86 88 90

LIST OF TABLES 2.1 Numerical results from applying the SI and NR methods to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . 29 2.2 Numerical results from applying the CE method to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . 29 2.3 Numerical results from applying the LS method to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . 30 2.4 Numerical results from applying the CS method to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . 30 2.5 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T1 : n = 5, x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T and ε = 10−6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Numerical results from applying the SI and NR methods to the test function T1 when the problem size n is varied: ρ = 1 − 10−6, ε = 10−6 , x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.7 Numerical results from applying the CE method to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6, x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Numerical results from applying the LS method to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6, x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.9 Numerical results from applying the CS method to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6, x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.10 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6 , x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . . . . . . . . . . . . . . . . 32 2.11 Numerical results from applying the SI and NR methods to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. . 33

2.12 Numerical results from applying the CE method to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. . . . . . . 34

2.13 Numerical results from applying the LS method to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. . . . . . . 34

2.14 Numerical results from applying the CS method to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. . . . . . . 35 2.15 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), c1 = 1.025, ρ1 = 0.9989885, B2 = B 2 ([2.2, −2.2]T , 1), c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.16 Numerical results from applying the SI and NR methods to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1−10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.17 Numerical results from applying the CE method to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.18 Numerical results from applying the LS method to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.19 Numerical results from applying the CS method to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.20 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . . . . . . . . . . . . . . . . . 38

2.21 Numerical results from applying the SI and NR methods to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). . . . . . . . . . . . . . . . . . . 40

2.22 Numerical results from applying the CE method to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). . . . . . . . . . . . . . . . . . . . . . . . 40 2.23 Numerical results from applying the LS method to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). . . . . . . . . . . . . . . . . . . . . . . . 41 2.24 Numerical results from applying the CS method to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). . . . . . . . . . . . . . . . . . . . . . . . 41

x

2.25 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). . . . . . . . 41 2.26 Numerical results from applying the SI and NR methods to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . . . . . 43 2.27 Numerical results from applying the CE method to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . . . . . . . . . . . . 43

2.28 Numerical results from applying the LS method to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . . . . . . . . . . . . 44

2.29 Numerical results from applying the CS method to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . . . . . . . . . . . . 44 2.30 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.31 Numerical results from applying the SI and NR methods to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). . . . . . . . . . . . . . . . . 47 2.32 Numerical results from applying the CE method to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). . . . . . . . . . . . . . . . . . . . . . . . 47 2.33 Numerical results from applying the LS method to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). . . . . . . . . . . . . . . . . . . . . . . . 48 2.34 Numerical results from applying the CS method to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). . . . . . . . . . . . . . . . . . . . . . . . 48 2.35 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). . . . 49 2.36 Numerical results from applying the SI and NR methods to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . 51 2.37 Numerical results from applying the CE method to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . . . . . . . . 52 2.38 Numerical results from applying the LS method to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . . . . . . . . 52 2.39 Numerical results from applying the CS method to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . . . . . . . . 53 2.40 Numerical results from applying the LS-CS Hybrid method with δ = 10−6 to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . 53 2.41 Numerical results from applying the SI and NR methods to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . 55 2.42 Numerical results from applying the CE method to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . . . . . . . . 55 xi

2.43 Numerical results from applying the LS method to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . . . . . . . . 56 2.44 Numerical results from applying the CS method to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . . . . . . . . . . . . . . . . . . . . . 56 2.45 Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). . . . 57 2.46 Fastest and slowest methods for all test functions. . . . . . . . . . . . . . . . . 58

xii

LIST OF FIGURES 2.1 The simple iteration algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The minimal ball enclosing the intersection of two balls. . . . . . . . . . . . 15 2.3 The ball iteration algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 The circumscribed ellipsoid algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 The line-search algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Illustration of the line-search algorithm. . . . . . . . . . . . . . . . . . . . . . . . 22 2.7 The circle-search algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.8 Illustration of the circle-search algorithm. . . . . . . . . . . . . . . . . . . . . . . 24 2.9 The LS-CS hybrid algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.10 Illustration of the linear function T1 in two dimensions: a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.11 Illustration of the complex function T2 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ). 33 2.12 Illustration of the linear function T3 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ). . . 36 2.13 Illustration of the saw-like function T4 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ).

39

2.14 Zoom view of the saw-like function T4 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ).

40

2.15 Illustration of function T6 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ). . . . . . . . . . . 46 2.16 Illustration of function T7 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ). . . . . . . . . . . 51 2.17 Illustration of function T8 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ). . . . . . . . . . . 54 3.1 Low-pass filtering test. a) Low-pass filters in the time domain. The solid line is a Hamming window, the dashed line is a Blackman-Harris window, and the dotted line is a Wiener filter in the time domain. This line style is also used in panels b), c), and d). b) The amplitude spectra of the low-pass filters in a). c) The original and filtered wavelets. The dash-dotted line is the original wavelet, which is a 20-Hz Ricker wavelet and was low-pass filtered such that the peak frequency is about 2 Hz. d) The amplitude spectra of the original and filtered wavelets in c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 The 1D basic scattering experiment with wavenumber illumination. The incident plane wave is reflected from a thin layer at the midpoint between source and receiver. A single frequency component of a single source-receiver pair generates a single wavenumber vector of the gradient image. The illuminated wave vector is given by the sum of the source-to-scatterer wave vector, k0 rˆs and the receiver-to-scatterer wave vector, k0 rˆg . (modified from Sirgue and Pratt [70]) . . . . . . . . . . . 70 3.3 Illustration of the strategy for choosing frequencies for frequencydomain waveform tomography. A range of vertical wavenumbers can be recovered by a single frequency component of the data by using a range of source-receiver pairs. A continuous coverage in vertical wavenumbers is the key for choosing the next frequency. (after Sirgue and Pratt [70]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4 A 15-Hz Ricker wavelet is shown in a) and its corresponding amplitude spectrum is shown in b). The frequency at half the maximum spectral amplitude at the lower end is set as the minimum frequency of the band fmin and the frequency at half the maximum spectral amplitude at the higher end is set as the maximum frequency fmax . . . . . . . . . . . 73 3.5 Illustration of the strategy for choosing optimal frequency bands for time-domain waveform tomography. Data within a limited range of frequencies are inverted, allowing the recovery of a range of vertical wavenumbers. A continuous coverage within the vertical wavenumber spectrum is the key for choosing the next frequency band. . . . . . . . . . 74 3.6 The model and data used in 1D inversion. a) One-dimensional velocity model. The target reflector is located at 1.15 km depth. b) A shot gather of the 1D model generated from a 10-Hz Ricker source wavelet. c) The shot gather low-pass filtered by the Wiener filter to the frequency band of a 2.5-Hz Ricker wavelet. . . . . . . . . . . . . . . . . . . 75 3.7 Source wavelets used in the 1D model are shown in a) the time domain and b) the frequency domain. The sequence of frequency bands generated by the strategy for choosing frequency bands for the 1D model is shown in c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.8 Time-domain multiscale waveform inversion results for the 1D model. In a) and b) the true velocity perturbations in the space and wavenumber domains, respectively, are shown as a dashed line while the final estimated velocity perturbations from the background are shown as a solid line. The contribution in the space and wavenumber domains from the two frequency bands are shown in c) and d) where the solid lines are the contribution from low-frequency-band data (peak frequency = 2.5 Hz) and the dashed lines are the contribution from high-frequency-band data (peak frequency = 10 Hz). . . . . . . . . . . . . . 78 3.9 The 2D velocity model used to test the applicability of the strategy for choosing optimal frequency bands to a 2D, heterogeneous case. . . . 79 xiv

3.10 An original shot gather of the 2D model generated from a 20-Hz Ricker source wavelet is shown in a) and the shot gather low-pass filtered by the Wiener filter to the frequency band of a 5-Hz Ricker wavelet is shown in b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.11 The initial velocity model used in the 2D model experiment is shown in a). This model was obtained using traveltime tomography. Its corresponding b) ray density diagram shows the number of rays passing through a model block. It also shows the areas in the model where waveform tomography will improve the inversion result using refraction energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.12 Time-domain multiscale waveform inversion results for the 2D model: a) the MWT velocity tomogram obtained after the inversion using 5-Hz peak-frequency data, b) the MWT velocity tomogram obtained after the inversions using 5-Hz and 20-Hz peak-frequency data, c) the SWT velocity tomogram obtained after the inversion using 20-Hz peak-frequency data, d) the true velocity model. . . . . . . . . . . . . . . . . . 82 3.13 Relative data-residual plot. The solid line shows the relative data residuals of MWT versus the iteration number. The residuals at the first 50 iterations are the residuals from the low-frequency inversion using data with a peak frequency of 5 Hz, and the last 20 residuals are from the high-frequency inversion using data with a peak frequency of 20 Hz. The dashed line shows the relative data residuals of SWT. . 83 3.14 An outcrop in the Canadian Foothills shows the rugged surface topography and complex near-surface structures. . . . . . . . . . . . . . . . . . . 84 3.15 A Canadian Foothills velocity model used to generate acoustic synthetic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.16 A synthetic shot gather generated by using the Canadian Foothills model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.17 Inversion results for the Canadian Foothills model. a) Initial velocity model. Waveform tomogram after the inversion using a) 5-Hz data, b) 10-Hz data, and c) 20-Hz data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xv

ACKNOWLEDGMENTS I truly thank my advisors Dr. Krzysztof Sikorski and Dr. Gerard T. Schuster. Dr. Sikorski introduced me to the interesting field of fixed-point problems. His advice was very useful and encouraging and helped me to learn the concepts of those problems. Dr. Schuster introduced me to exploration geophysics and taught me basic and advanced concepts in processing seismic data. I would also like to thank my committee members Dr. Mike Kirby, Dr. Martin Berzins, and Dr. Elena Cherkaev for their help and comments on my dissertation. Dr. Elena Cherkaev also gave me advice on solving inverse problems in geophysics. I greatly benefited from many discussions with my colleagues and would like to thank them for their assistance and encouragement. This work would not be accomplished without financial support from the Royal Thai Government and the sponsors of the University of Utah Tomography and Modeling/Migration Consortium. I would like to thank ConocoPhillips for giving me permission to use the Canadian Foothills synthetic data set in my dissertation. Finally, I am grateful to the following companies for giving me invaluable internships: Total, Vale Inco, and ConocoPhillips.

CHAPTER 1 GENERAL INTRODUCTION This dissertation focuses on two nonlinear problems. The first one is fixed-point computation for multivariate nonlinear problems. The second one is seismic waveform tomography for obtaining highly resolved velocity structures of subsurface from seismic data. In the first part, new nonlinear solvers are developed for specific classes of functions. In the second part, a nonlinear preconditioned conjugate gradient method is used to solve a nonlinear inverse problem.

1.1

Fixed-point Computation

In a fixed-point problem, a fixed point x∗ that satisfies the equation x∗ = f (x∗ ) for a given function f is approximated. Many problems can be formulated as fixed-point problems. For example, a root-finding problem for nonlinear equations f (x∗ ) = 0 can be rearranged as g(x∗ ) = f (x∗ ) + x∗ = x∗ , which is a fixed-point problem. The applications of fixed-point problems include economic equilibria [12, 56], game theory [12, 26, 55, 56, 86], boundary value problems [1, 3, 32, 85], and chaos and dynamical systems [20, 71]. A number of fixed-point theorems have been derived in the last century. Each theorem is focused on a specific class of functions. Banach’s fixed-point theorem [4] is the first theorem stating that any continuous and contractive function of a compact subset of ℜn into itself has a unique fixed point. Banach also demonstrated that the simple iteration (SI) algorithm, xi+1 = f (xi ), generates a Cauchy sequence that converges to the fixed point of any such function. Due to this restriction of the class of functions, there was a need for more general fixed-point theorems that focused on continuous functions only. Brouwer demonstrated that any continuous function from a nonempty, convex and compact subset of ℜn into itself has at

2 least one fixed point. However, Brouwer’s proof was nonconstructive. In 1967 H. Scarf [54] developed a simplicial algorithm for approximating fixed points for Brouwer’s maps from a simplex into itself. This was the first constructive proof of Brouwer’s fixed-point theorem. Since then, fixed-point computation has become an intensive research area and many new algorithms have been proposed including restart methods [39, 40], homotopy methods [23, 24] and ellipsoid algorithms [31, 65, 78]. As I have mentioned earlier, a fixed-point problem x = f (x) can be transformed into a root-finding problem g(x) = f (x) − x = 0. Therefore, effective root-finding algorithms such as Newton methods can also be used to solve fixed-point problems (see [2]). Algorithms dealing with fixed-point computation are iterative methods. I consider algorithms that are based on function evaluations, and assume that at least one function evaluation is required in each iteration. Most CPU time is consumed by function evaluations. Therefore, I can define the worst case cost of an algorithm, for a class of functions F , as the maximum number of function evaluations required to achieve a prescribed precision for all functions in F . The problem complexity is defined as the minimal cost among all possible algorithms for the class F . An algorithm is almost optimal if and only if its cost is almost minimal. The complexity depends on the class of functions and the selected error criterion, as shown in [9, 29, 63, 64, 66]. I consider contractive functions with contraction factor ρ close to 1, nonexpanding functions (ρ = 1), and then a larger class of functions that are contractive/nonexpanding only in the direction of fixed points but may be globally expanding (see [9, 80]). The contraction property is defined with respect to the Euclidean norm. The ε-approximations of fixed points are obtained by using the absolute or residual error criteria. Boonyasiriwat [8] developed a new implementation of the circumscribed ellipsoid (CE) algorithm that did not require the dimensional deflation used in the original CE method of Sikorski et al. [65]. This implementation was for the class of contractive or globally nonexpanding functions. In this dissertation I extend the applicablitiy of the CE method to a much larger class of functions that are nonexpanding only in the direction of fixed points but may be globally expanding.

3 The complexity bounds of the CE method stay the same in the larger class [9]. The CE algorithm enjoys almost optimal cost O(n2 log 1ε ) for obtaining residual solutions x : ||f (x) − x||2 ≤ ε for nonexpanding functions, and exhibits the cost

1 )) for obtaining absolute solutions x : ||x − x∗ ||2 ≤ ε for O(n2 (log 1ε + log 1−ρ

contractive functions with factor ρ < 1 [8]. In addition, I also develop three new algorithms: line-search (LS), circle-search (CS), and LS-CS hybrid methods for solving the fixed-point problem. Each iteration of these methods is considerably less expensive (for a small number of constraints) than that of the CE method since there is no need to solve an eigenvalue problem to obtain a circumscribed ellipsoid. In the infinity-norm case, Shellman and Sikorski [58, 59] developed a Bisection Envelope Fixed point (BEFix) and a Bisection Envelope Deep-cut Fixed point (BEDFix) algorithms for computing fixed points of two-dimensional nonexpanding functions. They also developed a recursive fixed-point algorithm (PFix) for computing fixed points of n-dimensional nonexpanding functions with respect to the infinity norm (See [60, 61]). In Chapter 2 two fixed-point theorems are presented and the fixed-point problem is formulated in the framework of information-based complexity theory. Various fixed-point algorithms are described including the algorithms that I develop, circumscribed ellipsoid (CE), line-search (LS) circle-search (CS), and LS-CS hybrid methods. The numerical results are presented. Finally, I draw conclusions and outline future research directions.

1.2

Seismic Waveform Tomography

Seismic waveform tomography is a method for inverting seismic data to obtain properties of the Earth’s subsurface structures. The properties to be recovered can be P-wave (compressional-, primary-wave) velocity Vp , S-wave (shear-, secondarywave) velocity Vs , density ρ, etc. In this dissertation I mainly focus on recovering the P-wave velocity using an inversion method based on the acoustic wave equation. Seismic waveform tomography is an inverse problem and that is why it is sometimes called seismic waveform inversion. Seismic data generated by a forward modeling algorithm or acquired by seismic equipment are inverted by an inversion

4 algorithm to obtain model parameters which in this case are the P-wave velocity values. Naturally, seismic wave propagation is described by the elastic wave equation. To reduce the nonlinearity of elastic waveform inversion, the acoustic approximation is usually used in the inversion of elastic data and S-waves are considered as coherent noise. In 1984, Tarantola [72] demonstrated that seismic inversion could be efficiently performed using the so-called adjoint method in which the computational cost for computing the gradient of the misfit function is about two times of that of forward modeling. This method is adopted here in this work. Though the adjoint method is computationally efficient for seismic waveform inversion, the inverse problem is still highly nonlinear and a waveform inversion algorithm is likely to converge to a local minimum if the starting model is not in the vicinity of the global minimum [27]. To partially overcome this local minima problem, Bunks et al. [16] proposed a multiscale method that sequentially inverted data from low to high frequencies. However, the low-pass filter used in their work is not the most efficient filter and there is no rule or strategy to choose frequency bands of data to be used. These can lead to computationally inefficient waveform inversion. The second part of my research presents an improvement to their method. In Chapter 3 the problem of waveform tomography is formulated. A multiscale, nonlinear, preconditioned conjugate gradient method is proposed for solving the problem. I improve the efficiency of the multiscale method of Bunks et al. [16] by using more efficient low-pass filters and a time-domain strategy for choosing optimal frequency bands. I show that these filters are more efficient than the FIR Hammingwindowed low-pass filter used by Bunks et al. [16] for time-domain multiscale waveform inversion. The strategy for selecting frequencies proposed by Sirgue and Pratt [70] is extended to time-domain waveform tomography. The modified strategy is verified by inversion results from both 1D and 2D heterogeneous models. True source wavelets were used in both experiments. In the 2D model experiment, I show that multiscale waveform tomography (MWT) can mitigate the local minima problem of waveform tomography and yield an accurate estimate of the velocity model, while conventional, single-scale waveform tomography (SWT) converges to

5 a local minimum, resulting in an inaccurate estimate of the true velocity model. Multiscale waveform tomography is also applied to Canadian Foothills synthetic data to investigate the possibility of velocity estimation in complex environments.

CHAPTER 2 FIXED-POINT COMPUTATION In Section 2.1 fundamental fixed-point theorems are reviewed. The fixed-point problem formulation is presented in Section 2.2. In Sections 2.3 and 2.4 I present previously developed algorithms and proposed algorithms for approximating fixed points, respectively. Numerical results are shown in Section 2.5. The conclusions are drawn in Section 2.6 and the future work is presented in Section 2.7.

2.1

Fixed-point Theorems

Fixed-point theorems deal with the existence and uniqueness of fixed points for specific classes of functions. I list two fixed-point theorems that are related to my work. 2.1.1

Two Classical Fixed-point Theorems

The Banach’s fixed-point theorem, established by Banach in 1922, for the specific case of the space ℜn is given by: Theorem 2.1.1 (Banach) [4] Suppose that Ω is a compact subset of ℜn and f : Ω → Ω satisfies ||f (x) − f (y)|| ≤ ρ||x − y||

∀ x, y ∈ Ω,

where || · || is an arbitrary norm defined on ℜn , and 0 < ρ < 1. Then, f has a unique fixed point x∗ = f (x∗ ).

Banach showed that the sequence xi generated by the simple iteration (SI) algorithm: xi+1 = f (xi ),

i = 0, 1, . . .

(2.1)

7 with the initial x0 ∈ Ω, uniformly converges to x∗ . A more general Brouwer’s fixed-point theorem is given by: Theorem 2.1.2 (Brouwer) [64, 78] Suppose that Ω is a compact and convex subset of ℜn . Let f : Ω → Ω be a continuous function. Then, there exists at least one fixed point x∗ = f (x∗ ) in Ω.

In this case the function f may have more than one fixed point. Although the Brouwer’s fixed-point theorem is more general, it was not immediately utilized in computation. This is because Brouwer used the degree theory in his proof and that made his result nonconstructive. The first constructive proof of the Brouwer’s fixed-point theorem was established by Scarf [54]. His proof is for any function g : S n → S n , where S n = {x ∈ Pn ℜn | i=1 xi ≤ 1 and xi ≥ 0} is a unit n-simplex. This proof can be easily extended to the domain Ω by the following arguments: A homeomorphism h : S n → Ω can always be constructed as shown, e.g., in [74]. Since the function

g = h−1 f h : S n → S n has already been proved to have a fixed point g∗ ∈ S n and

g(g∗) = h−1 f (h(g∗ )) = g∗ yields f (h(g∗ )) = h(g∗ ), I obtain that f has a fixed point x∗ = h(g∗ ) ∈ Ω. The proof of the Brouwer’s theorem is based on triangulation of S n and the Sperner’s Lemma (see [74]).

All algorithms based on Brouwer’s theorem that use function evaluations, such as Scarf’s, homotopy methods, etc., exhibit exponential cost (the number of function evaluations) in the worse case. This result holds for the residual error criterion in the class of functions satisfying the Lipschitz condition with constant ρ > 1, as shown by Hirsch et al. [29] (see also [22, 64]).

2.2

Problem Formulation

I consider fixed-point computation for the specific class of Lipschitz functions defined below. Both absolute and residual error criteria are utilized as termination conditions. The fixed-point problem is formulated in the framework of informationbased complexity theory [64, 76, 77].

8 2.2.1

The Classes of Lipschitz Functions

Suppose that Ω is a compact and convex subset of ℜn . A function f : Ω → Ω is called ρ-Lipschitz if ρ ∈ ℜ is the smallest constant such that f satisfies the Lipschitz condition ||f (x) − f (y)|| ≤ ρ||x − y||,

∀ x, y ∈ Ω,

(2.2)

where || · || is the Euclidean norm given by the formula: ||z|| =

q

z12 + z22 + · · · + zn2 ,

∀ z ∈ ℜn .

I will call ρ the Lipschitz constant (or factor) of f . The function f is contractive if ρ < 1, nonexpanding if ρ = 1, or expanding if ρ > 1. Let L(Ω, ρ) = {f : Ω → Ω|f is ρ−Lipschitz} denote the class of ρ-Lipschitz functions that map Ω into itself. The existence of fixed points of any function in L(Ω, ρ) is assured by the Brouwer’s theorem. Moreover, if ρ < 1, the fixed point of any function in L(Ω, ρ) is unique due to the Banach’s theorem. The domain Ω for the class L(Ω, ρ) in this work will be the unit ball B n = {x ∈ ℜn | ||x|| ≤ 1}. A ball with radius γ and center at c will be denoted

n as B n (c, γ) = {x ∈ ℜn | ||x − c|| ≤ γ}. I denote Bρ≤1 = L(B n , ρ) for n ≥ 2 and

0 < ρ ≤ 1. In the remaining part of this chapter, the symbol F denotes the class of functions and the symbol Ω denotes the domain of functions associated with F . 2.2.2

Formulation of the Problem

For a given class F and its associated domain Ω, our problem is to compute

ˆ to the fixed point x∗ for every f ∈ F , such that an absolute ε-approximation x

||ˆ x − x∗ || ≤ ε. To solve this problem I use arbitrary adaptive information operators

of the form: N(f ) = [f (x1 ), f (x2 ), . . . , f (xk )] where xi = αi (f (x1 ), f (x2 ), . . . , f (xi−1 ),

1 ≤ i ≤ k,

(2.3)

9 with each αi : Ωi−1 → Ω being an arbitrary function. The number k = card(N) is the cardinality of N, which indicates the number of function evaluations in N. An algorithm φ using information N is an arbitrary transformation φ : N(F ) → Ω,

(2.4)

such that φ(N(f )) approximates the fixed point x∗ = x∗ (f ). The error of φ is defined by e(φ) = sup ||φ(N(f )) − x∗ ||. f ∈F

I remark that the combination of (2.3) and (2.4) formulates the concept of fixedpoint algorithms based on function evaluations as used in [29, 43]. The cost of a fixed-point algorithm for F is measured by the maximal number of function evaluations needed by the algorithm to compute ε-approximation for any f in F . It is of interest to construct an algorithm φ that uses information N of the minimal cardinality card(N) and such that e(φ) ≤ ε. Such algorithms will be called optimal afterward since they use minimal number of function evaluations to compute ε-approximations. The complexity m(F , ε) is therefore defined as the minimal cost of all algorithms that compute ε-approximations for the class F . Apparently, m(F , ε) is equal to the cost of an optimal algorithm for F . In other words, m(F , ε) can be estimated if one can construct and analyze an optimal algorithm for F . The following notation is used in the information-based complexity theory [64, 76, 77, 78]. Let V (f ) = {g ∈ F |N(g) = N(f )} and U(f ) = {g∗ |g ∈ V (f )}. Then the adversary principle states that by knowing only N(f ), it is impossible to distinguish which fixed point x∗ or g∗ is being actually approximated for all g ∈ V (f ). This yields that a local error e(φ, f ) ≥ rad(U(f )),

∀ f ∈ F,

10 where e(φ, f ) = sup ||g∗ − φ(N(f ))|| g∈V (f )

for any algorithm φ using information N, where rad(U(f )) is the radius of the smallest ball enclosing U(f ). The radius of information N is defined by r(N) = sup rad(U(f ))

(2.5)

f ∈F

Then, r(N) = inf φ∈Φ(N ) e(φ)

(2.6)

where Φ(N) is the set of all algorithms that use information N. This means that r(N) is a lower bound on error of all algorithms in Φ(N). In particular, if r(N) > ε for every information N of the form (2.3), then m(F , ε) = ∞. This means that the problem cannot be solved with finite cost in the worst case.

2.3 2.3.1

Fixed-point Algorithms

A Constructive Lemma

The following lemma [65, 78] is the basis of the ball iteration (BI) algorithm [33, 45] and the circumscribed ellipsoid (CE) algorithm [63, 64, 65, 78]. n . Suppose that A ⊆ B n contains the fixed point x∗ . Lemma 2.3.1 Let f ∈ Bρ 0; imax (maximum number of iterations); function f ∈ Bρ≤1 ; i := 0; x0 := 0; while i ≤ imax do begin u := f (xi ); a := xi − u; if ||a|| ≤ (1 − ρ2 )ε/ρ then (2) return xi − a/(1 − ρ2 ) as an ε−approximation; if ||a|| ≤ ε then (3) return xi as an ε−approximation; b := (u + ρxi )/(1 + ρ); determine the intersecting point xC I xM := (b + xC )/2; I C xi+1 := x ∈ {b, xM , xC I } such that ||f (xi+1 )|| = min{||f (b)||, ||f (xM )||, ||f (xI )||} i := i + 1; end if i = imax then return failed to compute ε−approximation in imax iterations

Figure 2.5. The line-search algorithm.

2.4.4

Circle-search Algorithm

The circle-search (CS) method is an interpolatory method locally based on two iterations of the SI method. It is well suited for rotational functions. Given an initial ball B(c0 , γ0 ) and a starting point x0 , the SI method is used to find two new points x10 and x20 . If no termination criteria are satisfied at x0 and x10 , and points x0 , x10 , x20 are not collinear, then a fitting circle is constructed. If they are collinear then the next iterate x1 = x20 . The 2D Fitting-Circle (FC2D) and Householder Fitting-Circle (HFC) algorithms for computing a fitting circle in two dimensions and n dimensions, n > 2, respectively, are described below. If the center of the circle xCS at iteration i is inside the feasible region, then it is accepted as the next i iterate x1 . Otherwise, x1 = x20 . In order to be in the feasible domain, the center xCS must satisfy the condition i ||xCS − cj ||2 ≤ γj2 , i

(2.21)

22

Figure 2.6. Illustration of the line-search algorithm.

for j = 0, 1, 2, ..., k, and contractive functions (ρ < 1) where k + 1 is the number of constraining balls B(cj , γj ). For nonexpanding functions (ρ = 1), xCS must satisfy i the condition (xCS − bj ) · (f (xj ) − bj ) ≥ 0, i

(2.22)

for j = 0, 1, ..., k, and bj = (f (xj ) + xj )/2. The SI method is then reapplied to generate two new points and determine a new fitting circle to find the new iterate. This process is repeated until a new iterate satisfies one of the stopping criteria. The CS algorithm is illustrated in Figs. 2.7 and 2.8. The circle-search algorithm can be generalized by using m-dimensional spheres, 1 ≤ m ≤ n − 1, instead of circles with more SI-generated points used to determine the fitting spheres. FC2D Algorithm: Given two-dimensional, noncollinear points: P1 (x1 , y1 ), P2 (x2 , y2 ), P3 (x3 , y3), the center of the fitting circle is the intersection of two lines perpendicular to and passing through the midpoints of the lines P1 P2 and P2 P3 . The equations of lines P1 P2 and P2 P3 are of the form

23

n input ρ ≤ 1; ε > 0; imax (maximum number of iterations); function f ∈ Bρ≤1 ; i := 0; x0 := 0; while i ≤ imax do begin u := f (xi ); a := xi − u; if ||a|| ≤ (1 − ρ2 )ε/ρ then (2) return xi − a/(1 − ρ2 ) as an ε−approximation; if ||a|| ≤ ε then (3) return xi as an ε−approximation; w := f (u); b := (u + i h ρxi )/(1 + ρ); (xi −u)·(w−u) ; cos α := ||x i −u||||w−u|| if| cos α − cos(180)| = 0 then xi+1 := w; else if xCS lies within the feasible region then (see 2.20 and 2.21) i xi+1 := xCS i ; else xi+1 := w; i := i + 1;

end if i = imax then return failed to compute ε−approximation in imax iterations

Figure 2.7. The circle-search algorithm.

ya = ma (x − x1 ) + y1 ,

(2.23)

yb = mb (x − x2 ) + y2 ,

(2.24)

and

where the slopes ma =

y2 −y1 x2 −x1

and mb =

y3 −y2 . x3 −x2

The perpendicular of a line with

slope m has slope −1/m. Thus equations of the lines perpendicular to lines a and b and passing through the midpoints of P1 P2 and P2 P3 are ya′ = −

1 x1 + x2 y1 + y2 (x − )+ , ma 2 2

(2.25)

yb′ = −

x2 + x3 y2 + y3 1 (x − )+ , mb 2 2

(2.26)

and

24

Figure 2.8. Illustration of the circle-search algorithm.

These two lines intersect at the center. Solving for x gives x=

ma mb (y1 − y3 ) + mb (x1 + x2 ) − ma (x2 + x3 ) . 2(mb − ma )

(2.27)

If ma = 0 or mb = 0, the x value of the center of the fitting circle becomes x = (x1 + x2 )/2 or x = (x2 + x3 )/2, respectively. Substituting the x value into one of the equations of the perpendiculars gives the y value of the center of the fitting circle. If ma = mb , then both lines are coincident and thus no fitting circle exists. HFC Algorithm: Given n-dimensional noncollinear points: P1 , P2, P3 , these points are transformed onto a 2D plane spanned by the first and second components (x1 , x2 ) using the Housholder transformation. Then the FC2D algorithm is used to compute the center of the fitting circle in 2D. Since the Householder transformation matrix is symmetric and orthogonal, the 2D center is transformed back to n dimensions by using the same Householder transformation. A Householder matrix is of the form H = H(v) = I −

2 vT v

vvT ,

where the vector v is called a Householder vector.

(2.28) Householder matrices are

symmetric and orthogonal, and are rank-1 modifications of the identity. They

25 can be used to zero out selected components of a vector. Suppose we are given x ∈ ℜn and want Hx to be a multiple of e1 = (1, 0, ..., 0)T ∈ ℜn . By choosing v = x ± ||x||2e1 , we get Hx = ∓||x||2e1 . At the first step of the HFC algorithm, all points Pi for i = 1, 2, 3 are shifted such that P1 becomes the origin: Pi = Pi − P1 . Then the first Householder vector v1 is computed as v1 = P2 ± ||P2 ||2e1 . After applying this Householder transformation to P2 and P3 , the components of vector P2 become zero except the first component. Now we want to zero out the components of vector P3 except the first and second components. This is done by using the second Householder vector v2 = P′3 ± ||P′3 ||2e′1 where P′3 , e′1 ∈ ℜn−1 and P′3 = (P3 · e2 , ..., P3 · en )T where P3 · ei is the i-th component of P3 . After applying the second Householder transformation, all transformed points P1 , P2 , P3 are on a plane in two dimensions, and the FC2D algorithm is applied. Then the center of the circle is transformed back to ℜn by using inverse transformations to H(v2 ), H(v1 ) and the shift. 2.4.5

LS-CS Hybrid Algorithm

The LS-CS hybrid algorithm is a method developed to improve on the individual performance of the LS and CS methods. The LS method is very efficient for almost linear functions while the CS method is well suited for rotational functions. Numerical results illustrating these properties for test functions T1 , T7 and T8 are listed in Section 2.5. Given an initial ball B(c0 , γ0 ) and a starting point x0 , the SI method is used to find two new points x10 and x20 . If x0 , x10 , and x20 are not collinear and the angle α between vectors x0 −x10 and x20 −x10 satisfies the condition cos α−cos(180) > δ, then using these three points a fitting circle is constructed. Otherwise, the LS method is used to provide the next iterate. The FC2D and HFC algorithms described in the previous section are also utilized in this algorithm. If the center of the circle xCS i at iteration i is inside the feasible region, then it is accepted as the next iterate x1 . Otherwise, the LS method is also utilized to provide the next iterate xi+1 = xLS i . This process is repeated until a new iterate satisfies one of the stopping criteria. The LS-CS hybrid algorithm is shown in Fig. 2.9.

26

n input ρ ≤ 1; ε > 0; imax (maximum number of iterations); function f ∈ Bρ≤1 ; i := 0; x0 := 0; while i ≤ imax do begin u := f (xi ); a := xi − u; if ||a|| ≤ (1 − ρ2 )ε/ρ then (2) 2 return xi − a/(1 − ρ ) as an ε−approximation; if ||a|| ≤ ε then (3) return xi as an ε−approximation; w := f (u); b := (u + i h ρxi )/(1 + ρ); (xi −u)·(w−u) ; cos α := ||x i −u||||w−u|| if| cos α − cos(180)| < δ then xi+1 := xLS i ; else if xCS lies within the feasible region then (see 2.20 and 2.21) i xi+1 := xCS i ; else xi+1 := xLS i ; i := i + 1;

end if i = imax then return failed to compute ε−approximation in imax iterations

Figure 2.9. The LS-CS hybrid algorithm.

27

2.5

Numerical Results

In this section I compare the numerical results for several test functions using the SI, NR, CE, LS, CS, and LS-CS hybrid methods. The NR method is used to solve corresponding root-finding problems of the form f (x)−x = 0. I stress that the CE algorithm is designed to converge globally for noncontinuous, nondifferentiable functions, whenever Newton’s method requires C 1 smoothness and starting points that are sufficiently close to the solution. I include the tests of Newton’s method for illustrative comparison. The ACM TOMS implementation of the Newton’s method [13] is utilized in this work. For all algorithms I exhibit the total CPU time (in seconds), number of iterations and the number of the stopping criterion that resulted in termination. The numbers 1 and 2 indicate absolute termination and number 3 - residual termination (see section 6). I also exhibit (in the parentheses) the upper bounds (see Lemma 3.3, where δ = ε(1 − ρ) for the absolute termination case) on the number of iterations of the CE and SI algorithms. The speedup factors that represent the ratio in CPU time when using the CE, LS, CS, or LS-CS hybrid algorithm instead of the SI or NR algorithm are also included. All tests are carried out on a Linux operating system with Intel Pentium Dual 2 GHz processor. In the tests of functions 2 and 3, some initial balls are not unit balls. In these cases, the problems are defined on a general ball B n (c, γ). They can be transformed to the unit ball B n (0, 1) as follows. Let f : B n (c, γ) → B n (c, γ) denote the original function defined on a ball

B n (c, γ). The modified function f : B n (0, 1) → B n (0, 1) defined on a unit ball

B n (0, 1) is

f (x) =

1 (f (x · γ + c) − c) . γ

It turns out that if f ∈ L(B n (c, γ), ρ < 1) then f ∈ L(B n (0, 1), ρ < 1). The proof of this property is shown as follows. Let x1 , x2 ∈ B n (0, 1). Then 1 ||f (x1 · γ + c) − f (x2 · γ + c)|| γ 1 · ρ||γ · (x1 − x2 )|| ≤ γ

||f(x1 ) − f (x2 )|| =

28 = ρ||x1 − x2 ||, i.e., they satisfy Lipschitz condition with the same ρ and ||f(x)|| =

1 1 ||f (x · γ + c) − c|| ≤ · γ = 1, γ γ

i.e., f : B n (0, 1) → B n (0, 1).

Test 1. This test function (Figure 2.10) is a simple affine mapping f : ℜn → ℜn

given by T1 :

f (x) = ρx + (1 − ρ)s,

(2.29)

where the constant vector s is randomly chosen from B n (0, 1). Obviously, s is the unique fixed point of f . There are two experiments for this test function. In the first experiment, the problem size n = 5 and ε = 10−6 are fixed but the Lipschitz constant ρ is varied from 0.9 - 0.999999. Numerical results for this experiment are shown in Tables 2.1-2.5. It is worth noting that the upper bound of SI increases by an order of magnitude when the value of ρ is increased to be closer to 1. Both run times and numbers of iterations used also have the same characteristic. This is because this linear function with ρ close to one is almost the worst-case function for the SI method as the numbers of iterations used are very close to the upper bounds in the parentheses. In Table 2.2, there is an anomaly when ρ = 1 − 10−5 . This can sometimes happen since we cannot predict the behavior of the CE method for different values of parameters. In the second experiment, ρ = 0.999999 and ε = 10−6 are fixed and the problem size n is varied from two to five dimensions. Numerical results for this experiment are shown in Tables 2.6-2.10. For the NR, LS and LS-CS hybrid methods, this linear function only requires 1 iteration. The CE method also used small numbers of iterations compared to its upper bounds. This linear function is not well suited for the CS method since it cannot use a fitting circle and has to employ the SI method at every iteration. The cost per iteration is higher than that of the SI method so that the CS method is even slower than the SI method. It can be seen that the LS method is the fastest method for this function, Though the LS-CS hybrid method is more expensive than the LS method, it is also faster than the NR method.

29

Figure 2.10. Illustration of the linear function T1 in two dimensions: a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ). Table 2.1. Numerical results from applying the SI and NR methods to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . ρ SI NR −1 −6 1 − 10 6.58 × 10 , 119 (132), 2 2.90 × 10−5 , 2, 2 1 − 10−2 6.48 × 10−5, 1247 (1375), 2 3.24 × 10−5 , 3, 2 −3 −4 1 − 10 6.46 × 10 , 12531 (13809), 2 2.93 × 10−5 , 2, 2 1 − 10−4 6.44 × 10−3, 125361 (138149), 2 1.71 × 10−5 , 2, 2 1 − 10−5 6.51 × 10−2, 1253671 (1381545), 2 1.70 × 10−5 , 2, 2 1 − 10−6 6.57 × 10−1, 12536780 (13815504), 2 1.70 × 10−5 , 2, 2

Table 2.2. Numerical results from applying the CE method to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . ρ CE Speedup wrt SI Speedup wrt NR −1 −6 1 − 10 7.74 × 10 , 17 (1009), 2 0.8 3.7 −2 −6 1 − 10 8.26 × 10 , 18 (1147), 2 7.8 3.9 1 − 10−3 8.70 × 10−6 , 19 (1285), 2 74.3 3.4 −4 −5 1 − 10 3.40 × 10 , 30 (1424), 2 189.2 0.5 −5 −4 1 − 10 5.78 × 10 , 123 (1562), 2 112.5 0.03 1 − 10−6 9.79 × 10−5 , 41 (1700), 2 6705.8 0.2

30

Table 2.3. Numerical results from applying the LS method to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . ρ LS Speedup wrt SI Speedup wrt NR −1 −7 1 − 10 5.61 × 10 , 1, 2 11.7 51.7 1 − 10−2 5.48 × 10−7 , 1, 2 118.4 59.2 −3 −7 1 − 10 5.49 × 10 , 1, 2 1177.2 53.3 −4 −7 4 1 − 10 5.49 × 10 , 1, 2 1.17 × 10 31.2 1 − 10−5 5.65 × 10−7 , 1, 2 1.15 × 105 30.1 −6 −7 6 1 − 10 5.61 × 10 , 1, 2 1.17 × 10 30.2

Table 2.4. Numerical results from applying the CS method to the test function T1 : n = 5, ε = 10−6 and x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T . ρ CS Speedup wrt SI Speedup wrt NR −1 −5 1 − 10 1.86 × 10 , 61, 2 0.4 1.6 −2 −4 1 − 10 1.92 × 10 , 625, 2 0.3 0.2 1 − 10−3 1.90 × 10−3 , 6267, 2 0.3 0.02 −4 −2 1 − 10 1.97 × 10 , 62682, 2 0.3 8.7 × 10−4 1 − 10−5 0.19, 626837, 2 0.3 8.8 × 10−5 −6 1 − 10 1.93, 6268391, 2 0.3 8.8 × 10−6

Table 2.5. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T1 : n = 5, x∗ = [0.1, 0.3, 0.4, 0.1, 0.2]T and ε = 10−6 . ρ LS-CS Hybrid Speedup wrt SI Speedup wrt NR −1 −7 1 − 10 7.33 × 10 , 1, 2 9.0 39.5 −2 −7 1 − 10 7.25 × 10 , 1, 2 89.4 44.7 −3 −7 1 − 10 7.27 × 10 , 1, 2 889.4 40.3 1 − 10−4 7.27 × 10−7 , 1, 2 8854.2 23.6 −5 −7 4 1 − 10 7.40 × 10 , 1, 2 8.80 × 10 23.0 −6 −7 5 1 − 10 7.41 × 10 , 1, 2 8.87 × 10 22.9

31

Table 2.6. Numerical results from applying the the test function T1 when the problem size n is ε = 10−6 , x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . n x∗ SI 2 x∗1 4.24 × 10−1, 11971078 (13815504), 2 3 x∗2 5.15 × 10−1, 12448820 (13815504), 2 4 x∗3 5.40 × 10−1, 12467678 (13815504), 2 5 x∗4 6.57 × 10−1, 12536780 (13815504), 2

SI and NR methods to varied: ρ = 1 − 10−6 , = [0.1, 0.3, 0.4, 0.1]T , and NR 1.06 × 10−5 , 1.25 × 10−5 , 2.51 × 10−5 , 1.70 × 10−5 ,

2, 2, 2, 2,

2 2 2 2

Table 2.7. Numerical results from applying the CE method to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6 , x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . n x∗ CE Speedup wrt SI Speedup wrt NR ∗ −5 2 x1 4.91 × 10 , 54 (340), 2 8628.1 0.2 ∗ −4 3 x2 2.03 × 10 , 72 (680), 2 2535.1 0.06 4 x∗3 2.88 × 10−4 , 93 (1133), 2 1875.9 0.09 ∗ −5 5 x4 9.79 × 10 , 41 (1700), 2 6705.8 0.2

Table 2.8. Numerical results from applying the LS method to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6 , x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . n x∗ LS Speedup wrt SI Speedup wrt NR ∗ −7 2 x1 4.01 × 10 , 1, 2 1.06 × 106 26.4 ∗ −7 6 3 x2 4.45 × 10 , 1, 2 1.16 × 10 28.1 4 x∗3 5.06 × 10−7 , 1, 2 1.07 × 106 49.6 ∗ −7 6 5 x4 5.61 × 10 , 1, 2 1.17 × 10 30.2

Table 2.9. Numerical results from applying the CS method to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6 , x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . n x∗ CS Speedup wrt SI Speedup wrt NR ∗ 2 x1 1.32, 5985540, 2 0.3 8.0 × 10−6 ∗ 3 x2 1.56, 6224411, 2 0.3 8.8 × 10−6 4 x∗3 1.74, 6233840, 2 0.3 1.4 × 10−5 5 x∗4 1.92, 6268391, 2 0.3 8.8 × 10−6

32

Table 2.10. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T1 when the problem size n is varied: ρ = 1 − 10−6 , ε = 10−6 , x∗1 = [0.1, 0.3]T , x∗2 = [0.1, 0.3, 0.4]T , x∗3 = [0.1, 0.3, 0.4, 0.1]T , and x∗4 = [0.1, 0.3, 0.4, 0.1, 0.2]T . n x∗ LS-CS Hybrid Speedup wrt SI Speedup wrt NR ∗ −7 2 x1 5.51 × 10 , 1, 2 7.69 × 105 19.2 ∗ −7 5 3 x2 6.02 × 10 , 1, 2 8.57 × 10 20.8 4 x∗3 6.69 × 10−7 , 1, 2 8.08 × 105 37.5 ∗ −7 5 5 x4 7.41 × 10 , 1, 2 8.87 × 10 22.9

Test 2. This test function (Figure 2.11) is a function from [30], given by T2 : where

f (z) = g(g(z)),

z 2 + c cos2 z g(z) = , z + sin z cos z

(2.30)

(2.31)

z is a complex variable and c is a complex constant. I consider this test function as a 2D real function f : ℜ2 → ℜ2 , i.e., n = 2, but the function is evaluated as a 1D complex function. The problem is tested with two values of c : c = 1.025 and c = π/4 + 1.2 + i(π − 1.17). The fixed points of this problem are [0, 0.69032769]T

and [2.14062, −2.50683]T , respectively. Numerical results for this test function are

exhibited in Tables 2.11-2.15. When c = 1.025, the SI method could terminate with the absolute termination 2, but it terminated with the absolute termination 1 (the used number of iterations reaches the upper bound) when c = π/4+1.2+i(π−1.17). The NR method used at most 10 iterations. However, it failed to terminate with the absolute error criterion when ε = 10−6 for both values of c, although it terminated with the residual error criterion. The CE method used small number of iterations and was faster than the NR method in most cases. The LS and LS-CS hybrid methods used small numbers of iterations when c = 1.025 and were faster than the NR method. When c = π/4 + 1.2 + i(π − 1.17), they used much larger numbers of iterations and, consequently, they were slower than the NR method. The CS method employed the SI method and was the slowest method for this test function. According to the timing results, the CE method is the fastest method.

33

Figure 2.11. Illustration of the complex function T2 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ).

Table 2.11. Numerical results from applying the SI and NR methods to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. ε ρ c Ball SI NR −2 −4 10 ρ1 c1 B1 2.14 × 10 , 583 (4551), 2 7.98 × 10−5 , 8, 2 10−3 ρ1 c1 B1 8.48 × 10−4 , 2313 (6826), 2 8.62 × 10−5 , 9, 2 10−4 ρ1 c1 B1 1.68 × 10−3 , 4572 (9102), 2 8.64 × 10−5 , 9, 2 10−5 ρ1 c1 B1 2.51 × 10−3 , 6846 (11377), 2 9.30 × 10−5 , 10, 2 10−6 ρ1 c1 B1 3.34 × 10−3 , 9122 (13652), 2 3.72 × 10−5 , 9, 3 10−2 ρ2 c2 B2 1.44 × 10−3 , 2876 (2876), 1 3.82 × 10−5 , 4, 2 10−3 ρ2 c2 B2 2.16 × 10−3 , 4314 (4314), 1 3.83 × 10−5 , 4, 2 −4 10 ρ2 c2 B2 2.87 × 10−3 , 5752 (5752), 1 4.50 × 10−5 , 5, 2 10−5 ρ2 c2 B2 3.59 × 10−3 , 7190 (7190), 1 4.50 × 10−5 , 5, 2 10−6 ρ2 c2 B2 4.31 × 10−3 , 8628 (8628), 1 3.83 × 10−5 , 4, 3

34

Table 2.12. Numerical results from applying the CE method to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. ε 10−2 10−3 10−4 10−5 10−6 10−2 10−3 10−4 10−5 10−6

ρ ρ1 ρ1 ρ1 ρ1 ρ1 ρ2 ρ2 ρ2 ρ2 ρ2

c c1 c1 c1 c1 c1 c2 c2 c2 c2 c2

Ball B1 B1 B1 B1 B1 B2 B2 B2 B2 B2

CE Speedup wrt SI 4.67 × 10 , 5 (147), 2 45.9 5.29 × 10−6 , 6 (174), 2 160.4 7.78 × 10−6, 10 (202), 2 215.6 −6 9.69 × 10 , 13 (230), 2 258.8 1.30 × 10−5, 17 (257), 2 257.9 −5 2.00 × 10 , 14 (141), 1 72.1 −5 2.89 × 10 , 20 (169), 1 74.7 −5 3.63 × 10 , 25 (197), 1 79.1 4.52 × 10−5, 31 (224), 1 79.5 −5 5.41 × 10 , 37 (252), 1 79.7 −6

Speedup wrt NR 17.4 16.7 11.3 9.8 6.8 1.9 1.3 1.3 1.0 0.7

Table 2.13. Numerical results from applying the LS method to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. ε ρ c Ball LS Speedup wrt SI Speedup wrt NR −2 −6 10 ρ1 c1 B1 4.98 × 10 , 3, 2 43.0 16.3 10−3 ρ1 c1 B1 4.98 × 10−6, 3, 2 170.4 17.7 −4 −6 10 ρ1 c1 B1 5.00 × 10 , 3, 2 335.3 17.6 −5 −6 10 ρ1 c1 B1 6.92 × 10 , 4, 2 362.6 13.7 10−6 ρ1 c1 B1 6.93 × 10−6, 4, 2 482.1 12.7 −2 −4 10 ρ2 c2 B2 4.54 × 10 , 82, 2 3.2 0.09 −3 −4 10 ρ2 c2 B2 6.50 × 10 , 102, 2 3.3 0.06 10−4 ρ2 c2 B2 8.80 × 10−4 , 122, 2 3.3 0.05 −5 −3 10 ρ2 c2 B2 1.14 × 10 , 142, 2 3.1 0.04 −6 −3 10 ρ2 c2 B2 1.44 × 10 , 162, 2 3.0 0.03

35

Table 2.14. Numerical results from applying the CS method to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), B2 = B 2 ([2.2, −2.2]T , 1), c1 = 1.025, ρ1 = 0.9989885, c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. ε ρ c Ball CS Speedup wrt SI Speedup wrt NR −2 −4 10 ρ1 c1 B1 2.52 × 10 , 293, 2 0.8 0.3 10−3 ρ1 c1 B1 9.97 × 10−4 , 1158, 2 0.9 0.1 −4 −3 10 ρ1 c1 B1 1.97 × 10 , 2287, 2 0.9 0.04 −5 −3 10 ρ1 c1 B1 2.95 × 10 , 3424, 2 0.9 0.03 −6 −3 10 ρ1 c1 B1 3.93 × 10 , 4562, 2 0.9 0.02 10−2 ρ2 c2 B2 3.66 × 10−3 , 2936, 2 0.4 0.01 −3 −3 10 ρ2 c2 B2 4.58 × 10 , 3668, 2 0.5 8.5 × 10−3 10−4 ρ2 c2 B2 5.49 × 10−3 , 4400, 2 0.5 8.4 × 10−3 −5 −3 10 ρ2 c2 B2 6.40 × 10 , 5132, 2 0.6 7.2 × 10−3 10−6 ρ2 c2 B2 7.31 × 10−3 , 5864, 2 0.6 5.3 × 10−3

Table 2.15. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T2 : B1 = B 2 ([0.0, 0.1]T , 1), c1 = 1.025, ρ1 = 0.9989885, B2 = B 2 ([2.2, −2.2]T , 1), c2 = π/4 + 1.0 + i(π − 1.17), and ρ2 = 0.9984. ε ρ c Ball LS-CS Hybrid Speedup wrt SI Speedup wrt NR −2 −6 10 ρ1 c1 B1 6.42 × 10 , 3, 2 33.4 12.7 10−3 ρ1 c1 B1 6.40 × 10−6, 3, 2 132.5 13.8 −4 −6 10 ρ1 c1 B1 6.42 × 10 , 3, 2 261.5 13.7 −5 −6 10 ρ1 c1 B1 8.79 × 10 , 4, 2 285.3 10.8 10−6 ρ1 c1 B1 8.83 × 10−6, 4, 2 378.7 10.0 −2 −4 10 ρ2 c2 B2 5.15 × 10 , 82, 2 2.8 0.08 −3 −4 10 ρ2 c2 B2 7.25 × 10 , 102, 2 3.0 0.05 −4 −4 10 ρ2 c2 B2 9.72 × 10 , 122, 2 3.0 0.05 10−5 ρ2 c2 B2 1.25 × 10−3 , 142, 2 2.9 0.04 −6 −3 10 ρ2 c2 B2 1.56 × 10 , 162, 2 2.8 0.02

36 Test 3. This test function (Figure 2.12) is a parabolic, periodical function from [65], given by T3 :

f (x1 , x2 ) = [f1 (x1 , x2 ), f2 (x1 , x2 )],

(2.32)

where ρ ρ fi (x1 , x2 ) = (xi − 2m)2 + 1 − 2 2

(2.33)

for i = 1, 2 and 2m − 1 ≤ xi ≤ 2m + 1 and m is an arbitrary integer. This

two-dimensional contractive function has a unique fixed point at [1, 1]T . Numerical

results for this test function are exhibited in Tables 2.16-2.20. The SI and CS methods are the slowest while the LS and LS-CS hybrid methods are the fastest for this function. The CE method has a speed comparable with the NR method. It is worth noting that the CE algorithm can terminate with absolute error criterion even when ρ = 1, i.e., when the function becomes nonexpanding.

Figure 2.12. Illustration of the linear function T3 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ).

37

Table 2.16. Numerical results from applying the SI and NR methods to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5, and ρ3 = 1 − 10−15 . ε ρ Ball SI NR −3 −5 10 ρ1 B1 4.73 × 10 , 1120 (7598), 2 2.75 × 10−5 , 3, 2 −3 −4 10 ρ2 B1 4.98 × 10 , 11874 (760087), 2 7.15 × 10−5 , 7, 2 10−4 ρ1 B2 1.16 × 10−4 , 2768 (9900), 2 4.49 × 10−5 , 5, 2 10−4 ρ2 B2 1.57 × 10−3 , 37376 (990345), 2 8.41 × 10−5 , 7, 2 10−6 ρ2 B2 1.17 × 10−2 , 277744 (1450860), 2 8.48 × 10−5 , 7, 2 10−6 ∗ ρ3 B2 4.60 , 1.1 × 108 (1.3 × 1016 ), 2 3.60 × 10−5 , 3, 2 10−6 ∗ 1 B2 7.26 × 10−5 , 1673, 3 3.58 × 10−5 , 2, 3

Table 2.17. Numerical results from applying the CE method to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . ε ρ Ball CE Speedup wrt SI Speedup wrt NR −3 −5 10 ρ1 B1 3.39 × 10 , 34 (183), 2 1.4 0.8 10−3 ρ2 B1 4.43 × 10−5 , 45 (238), 1 12.2 1.6 −4 −5 10 ρ1 B2 4.74 × 10 , 47 (211), 2 2.5 0.9 −4 −5 10 ρ2 B2 5.36 × 10 , 54 (266), 2 29.4 1.6 −6 −5 10 ρ2 B2 7.93 × 10 , 79 (321), 2 147.1 1.1 10−6 ρ3 B2 8.38 × 10−5 , 87 (597), 1 5.5 × 105 0.4 −6 −5 10 1 B2 7.85 × 10 , 87 (∞), 1 0.9 0.5

Table 2.18. Numerical results from applying the LS method to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . ε ρ Ball LS Speedup wrt SI Speedup wrt NR −3 −6 10 ρ1 B1 5.67 × 10 , 9, 2 8.3 4.8 10−3 ρ2 B1 9.22 × 10−6 , 12, 2 54.1 7.8 −4 −6 10 ρ1 B2 9.23 × 10 , 12, 2 12.6 4.9 −4 −5 10 ρ2 B2 1.19 × 10 , 14, 2 131.3 7.0 10−6 ρ2 B2 1.49 × 10−5 , 16, 2 783.7 5.7 −6 −6 5 10 ρ3 B2 5.56 × 10 , 8, 2 8.3 × 10 6.5 −6 −6 10 1 B2 3.55 × 10 , 10, 3 20.5 10.1

38

Table 2.19. Numerical results from applying the CS method to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3 , ρ2 = 1 − 10−5 , and ρ3 = 1 − 10−15 . ε ρ Ball CS Speedup wrt SI Speedup wrt NR −3 −4 10 ρ1 B1 1.34 × 10 , 1121, 2 0.4 0.2 −3 −3 10 ρ2 B1 1.42 × 10 , 11875, 2 0.4 0.1 10−4 ρ1 B2 3.32 × 10−4 , 2769, 2 0.4 0.1 −4 −3 10 ρ2 B2 4.48 × 10 , 37377, 2 0.3 0.02 −6 −2 10 ρ2 B2 3.35 × 10 , 277745, 2 0.3 0.003 10−6 ρ3 B2 2.02 × 10−4 , 838, 3 2.3 × 104 0.2 −6 −4 10 1 B2 1.70 × 10 , 838, 3 0.4 0.2

Table 2.20. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T3 : B1 = B 2 ([0, 0]T , 2), B2 = B 2 ([0.1, 0.2]T , 2), ρ1 = 1 − 10−3, ρ2 = 1 − 10−5, and ρ3 = 1 − 10−15 . ε ρ Ball LS-CS Hybrid Speedup wrt SI Speedup wrt NR 10−3 ρ1 B1 5.67 × 10−6 , 9, 2 8.3 4.8 −3 −6 10 ρ2 B1 9.20 × 10 , 12, 2 54.2 7.8 −4 −6 10 ρ1 B2 9.23 × 10 , 12, 2 12.6 4.9 10−4 ρ2 B2 1.19 × 10−5 , 14, 2 131.4 7.0 −6 −5 10 ρ2 B2 1.48 × 10 , 16, 2 786.4 5.7 −6 −6 5 10 ρ3 B2 5.57 × 10 , 8, 2 8.2 × 10 6.5 −6 −6 10 1 B2 3.57 × 10 , 10, 3 20.6 10.1

39 Test 4. This fixed-point test function (Figure 2.13) is a saw-like, periodical function from [65], given by T4 :

f (x1 , x2 ) =

√

3/2 √ −1/2 1/2 3/2

f1 (x1 , x2 ) f2 (x1 , x2 )

(2.34)

where fi (x1 , x2 ) = min (ρ|xi − m − 10−2 j| + i/3) j=1,99

(2.35)

for i = 1, 2 and m ≤ xi ≤ m + 1 and m is an arbitrary integer. To clearly see the rapidly varying feature of this function, the zoomed view of this function is shown in Figure 2.14. This function has a unique fixed point at [−0.01946, 0.75933]T . Numerical results for this test function are exhibited in Tables 2.21-2.25. The SI also terminated when the numbers of iterations reached the upper bounds (termination 1) when ρ = 0.999999. This is the first time that the NR method used more than ten iterations. The speedup factors of the CE method with respect to the SI method are increased by four orders of magnitude when ρ is closer to 1. The CE method is superior to the NR method and is the fastest method for this test function. The LS and LS-CS hybrid methods are also faster than the NR method. Though the CS method is not the most efficient method in this case, it is interesting that the CS method is faster than the SI method when ρ is closer to 1.

Figure 2.13. Illustration of the saw-like function T4 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ).

40

Figure 2.14. Zoom view of the saw-like function T4 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ).

Table 2.21. Numerical results from applying the SI and NR methods to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). ρ Ball SI NR −2 −3 1 − 10 B1 1.12 × 10 , 1109 (1375), 2 4.48 × 10−4 , 20, 2 −2 1 − 10 B2 1.13 × 10−3 , 1109 (1445), 2 4.49 × 10−4 , 20, 2 1 − 10−2 B3 1.15 × 10−3 , 1141 (1445), 2 2.59 × 10−4 , 14, 2 1 − 10−6 B1 14.0, 13815504 (13815504), 1 2.12 × 10−4 , 13, 3 1 − 10−6 B2 15.1, 14508651 (14508651), 1 2.13 × 10−4 , 13, 3 1 − 10−6 B3 15.1, 14508651 (14508651), 1 2.85 × 10−4 , 16, 3

Table 2.22. Numerical results from applying the CE method to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). ρ Ball CE Speedup wrt SI Speedup wrt NR −2 −5 1 − 10 B1 6.159 × 10 , 36 (230), 1 18.3 7.3 −2 −5 1 − 10 B2 6.949 × 10 , 40 (238), 1 16.2 6.5 1 − 10−2 B3 7.039 × 10−5 , 41 (238), 1 16.4 3.7 −6 −5 5 1 − 10 B1 6.069 × 10 , 36 (340), 1 2.1 × 10 3.5 −6 −5 5 1 − 10 B2 7.049 × 10 , 41 (349), 1 1.9 × 10 3.0 1 − 10−6 B3 7.069 × 10−5 , 41 (349), 1 1.9 × 105 4.0

41

Table 2.23. Numerical results from applying the LS method to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). ρ Ball LS Speedup wrt SI Speedup wrt NR 1 − 10−2 B1 8.23 × 10−5 , 21, 2 13.7 6.5 −2 −4 1 − 10 B2 1.10 × 10 , 26, 2 10.4 4.8 −2 −4 1 − 10 B3 1.10 × 10 , 26, 2 10.4 2.7 1 − 10−6 B1 1.54 × 10−4 , 34, 2 9.4 × 104 1.6 −6 −4 5 1 − 10 B2 1.32 × 10 , 30, 2 1.1 × 10 1.9 −6 −4 5 1 − 10 B3 1.54 × 10 , 34, 2 9.8 × 10 2.2

Table 2.24. Numerical results from applying the CS method to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). ρ Ball CS Speedup wrt SI Speedup wrt NR −2 −3 1 − 10 B1 1.29 × 10 , 556, 2 0.9 0.4 1 − 10−2 B2 1.29 × 10−3 , 556, 2 0.9 0.4 −2 −3 1 − 10 B3 1.36 × 10 , 572, 2 0.8 0.2 −6 1 − 10 B1 8.50, 3593067, 2 1.7 2.9 × 10−5 −6 1 − 10 B2 8.48, 3593067, 2 1.8 2.9 × 10−5 1 − 10−6 B3 8.37, 3540986, 2 1.8 2.9 × 10−5

Table 2.25. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T4 : ε = 10−6 , x∗ = [−0.04, 0.74]T , B1 = B 2 ([0, 0]T , 1), B2 = B 2 ([0, 0]T , 2), and B3 = B 2 ([0.1, 0.2]T , 2). ρ Ball LS-CS Hybrid Speedup wrt SI Speedup wrt NR −2 −4 1 − 10 B1 1.09 × 10 , 21, 2 10.3 4.9 1 − 10−2 B2 1.42 × 10−4 , 26, 2 8.0 3.7 −2 −4 1 − 10 B3 1.42 × 10 , 26, 2 8.0 2.1 −6 −4 4 1 − 10 B1 1.96 × 10 , 34, 2 7.4 × 10 1.3 −6 −4 4 1 − 10 B2 1.70 × 10 , 30, 2 8.9 × 10 1.5 1 − 10−6 B3 1.98 × 10−4 , 34, 2 7.6 × 104 1.7

42 Test 5. This fixed-point test function is a nine-dimensional function from [84], given by f (x) = x − tF (x)

T5 :

(2.36)

where

and b=

with

F (x) = Ax + G(x) − b,   B −I 0 A =  −I B −I  , 0 −I B   4 −1 0 B =  −1 4 −1  , 0 −1 4 T 1 1 2 2 3 3 1 g , , x1 , g , , x2 , . . . , g , , x9 , G(x) = 8 4 4 4 4 4 4

,φ

2 ,0 4

,φ

3 ,0 4

(2.38)

(2.39)

(2.40)

+ φ 1, 14 , φ 0, 42 , 0, φ 1, 24 , T , φ 0, 43 + φ 14 , 1 , φ 42 , 1 , φ 34 , 1 + φ 1, 34

φ 0, 41 + φ

1 ,0 4

(2.37)

t g(s, t, u) = + |u − 1|3 , φ(s, t) = sin(0.5πst). s p I choose the value of t = L2 (1 − ρ2 )/L2 with L = 0.8234 and select the Lipschitz

constant ρ close to 1. This function has a unique fixed point at [8.89 × 10−4, 6.61 ×

10−2 , 0.184, 6.61 × 10−2 , 0.209, 0.415, 0.184, 0.415, 0.690]T . This choice implies that

f is contractive [78, 84]. Numerical results for this test function are exhibited in Tables 2.26-2.30. It turns out that the SI algorithm is faster than the CE algorithm when ρ is not too close to 1. This is because the cost of each iteration of the CE algorithm increases with the dimension as O(n3 ). However, when ρ is close to 1, the CE algorithm is much faster than the SI algorithm. The NR method is the fastest method for this test function and is much faster than the CE method. The LS method is faster than the NR method when ρ is not too close to 1 but it becomes slower than the NR method when ρ is closer to 1. The CS method is the slowest method in this case. Both the CS and CS-LS hybrid methods encountered a memory problems and could not use more than 10000 iterations because there will be 10000 constraints.

43

Table 2.26. Numerical results from applying the SI and NR methods to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . ρ SI NR 1 − 10−2 2.476 × 10−5 , 70 (1375), 2 1.333 × 10−4 , 5, 2 1 − 10−3 8.549 × 10−5 , 256 (13809), 2 1.134 × 10−4 , 4, 3 1 − 10−4 2.780 × 10−4 , 883 (138149), 2 1.613 × 10−4 , 6, 2 −5 −3 1 − 10 1.046 × 10 , 2990 (1381545), 2 1.585 × 10−4 , 6, 2 1 − 10−6 3.290 × 10−3 , 10041 (13815504), 2 1.566 × 10−4 , 6, 2 1 − 10−7 1.140 × 10−2 , 33571 (138155099), 2 1.298 × 10−4 , 6, 2 −8 −2 9 1 − 10 3.852 × 10 , 111858 (1.4 × 10 ), 2 1.460 × 10−4 , 6, 2 1 − 10−9 1.156 × 10−1 , 371639 (1.4 × 1010 ), 2 1.242 × 10−4 , 5, 2 1 − 10−10 3.806 × 10−1 , 1229543 (1.4 × 1011 ), 2 1.205 × 10−4 , 5, 2 1 − 10−11 1.321 × 100 , 4024132 (1.4 × 1012 ), 2 1.192 × 10−4 , 5, 2 −12 0 13 1 − 10 3.815 × 10 , 12391327 (1.4 × 10 ), 2 1.397 × 10−4 , 6, 2 1 − 10−13 1.193 × 101 , 38756581 (1.4 × 1014 ), 2 1.446 × 10−4 , 6, 2 1 − 10−14 3.654 × 101 , 118682979 (1.4 × 1015 ), 2 1.795 × 10−4 , 7, 2 1 − 10−15 1.101 × 102 , 357331990 (1.4 × 1016 ), 2 1.954 × 10−4 , 8, 2

Table 2.27. T5 : ε = 10−6 ρ 1 − 10−2 1 − 10−3 1 − 10−4 1 − 10−5 1 − 10−6 1 − 10−7 1 − 10−8 1 − 10−9 1 − 10−10 1 − 10−11 1 − 10−12 1 − 10−13 1 − 10−14 1 − 10−15

Numerical results from applying the CE method to the test function and ρ is varied from 1 − 10−2 to 1 − 10−15 . CE Speedup wrt SI Speedup wrt NR −2 2.151 × 10 , 1534 (3441), 2 1.151 × 10−3 6.197 × 10−3 −2 −3 2.922 × 10 , 2102 (3855), 2 2.926 × 10 3.881 × 10−3 3.115 × 10−2 , 2258 (4270), 1 8.925 × 10−3 5.178 × 10−3 3.224 × 10−2 , 2311 (4684), 1 0.03 4.916 × 10−3 −2 3.234 × 10 , 2325 (5099), 1 0.1 4.842 × 10−3 3.245 × 10−2 , 2330 (5513), 1 0.4 4.000 × 10−3 3.245 × 10−2 , 2327 (5928), 1 1.2 4.499 × 10−3 3.226 × 10−2 , 2326 (6342), 1 3.6 3.850 × 10−3 −2 3.247 × 10 , 2342 (6757), 1 11.7 3.711 × 10−3 3.167 × 10−2 , 2337 (7171), 1 41.7 3.764 × 10−3 3.229 × 10−2 , 2320 (7586), 1 118.1 4.326 × 10−3 −2 3.137 × 10 , 2338 (8000), 1 380.3 4.609 × 10−3 3.024 × 10−2 , 2331 (8415), 1 1208.0 5.936 × 10−3 3.089 × 10−2 , 2337 (8829), 1 3564.0 6.326 × 10−3

44

Table 2.28. Numerical results from applying the LS method to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . ρ LS Speedup wrt SI Speedup wrt NR 1 − 10−2 7.409 × 10−5 , 46, 2 0.3 1.8 −3 −5 1 − 10 8.589 × 10 , 52, 2 1.0 1.3 −4 −5 1 − 10 6.929 × 10 , 39, 2 4.0 2.3 1 − 10−5 8.579 × 10−5 , 47, 2 12.2 1.8 −6 −5 1 − 10 9.958 × 10 , 52, 2 33.0 1.6 −7 −4 1 − 10 2.414 × 10 , 97, 2 47.2 0.5 −8 −4 1 − 10 1.592 × 10 , 68, 2 242.0 0.9 1 − 10−9 1.335 × 10−4 , 57, 2 865.9 0.9 −10 −4 1 − 10 1.862 × 10 , 72, 2 2044.0 0.6 −11 −4 1 − 10 3.702 × 10 , 111, 2 3568.0 0.3 1 − 10−12 1.296 × 10−3 , 208 2944.0 0.1 −13 −4 4 1 − 10 1.795 × 10 , 67, 2 6.646 × 10 0.8 −14 −4 4 1 − 10 3.711 × 10 , 111, 2 9.846 × 10 0.5 1 − 10−15 1.535 × 10−2 , 672, 2 7173.0 0.01

Table 2.29. Numerical results from applying the CS method to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . ρ CS Speedup wrt SI Speedup wrt NR −2 −5 1 − 10 5.94 × 10 , 37, 2 0.4 2.3 1 − 10−3 2.24 × 10−4 , 142, 2 0.4 0.5 −4 −4 1 − 10 7.24 × 10 , 321, 2 0.4 0.2 −5 −3 1 − 10 2.72 × 10 , 1535, 2 0.4 0.1 1 − 10−6 5.89 × 10−3 , 4999, 2 0.5 0.03 1 − 10−7 1.03 × 10−2 , > 10000 N/A N/A −8 −2 1 − 10 1.01 × 10 , > 10000 N/A N/A −9 −2 1 − 10 1.01 × 10 , > 10000 N/A N/A 1 − 10−10 1.01 × 10−2 , > 10000 N/A N/A −11 −2 1 − 10 1.01 × 10 , > 10000 N/A N/A −12 −2 1 − 10 1.01 × 10 , > 10000 N/A N/A 1 − 10−13 1.01 × 10−2 , > 10000 N/A N/A −14 −2 1 − 10 1.01 × 10 , > 10000 N/A N/A −15 −2 1 − 10 1.01 × 10 , > 10000 N/A N/A

45

Table 2.30. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T5 : ε = 10−6 and ρ is varied from 1 − 10−2 to 1 − 10−15 . ρ LS-CS Hybrid Speedup wrt SI Speedup wrt NR 1 − 10−2 2.51 × 10−4 , 47, 2 0.1 0.5 −3 −4 1 − 10 4.07 × 10 , 65, 2 0.2 0.3 −4 −4 1 − 10 5.38 × 10 , 88, 2 0.5 0.3 −5 −3 1 − 10 1.31 × 10 , 179, 2 0.7 0.1 1 − 10−6 3.37 × 10−3 , 318, 2 1.0 0.05 1 − 10−7 3.50 × 10−4 , 64, 2 30.6 0.4 −8 −4 1 − 10 3.71 × 10 , 68, 2 96.1 0.4 1 − 10−9 3.36 × 10−4 , 64, 2 354.0 0.4 −10 −4 1 − 10 6.24 × 10 , 92, 2 629.1 0.2 −11 −3 1 − 10 2.34 × 10 , 190, 2 548.4 0.05 −12 1 − 10 5.6, > 10000 N/A N/A 1 − 10−13 5.6, > 10000 N/A N/A −14 1 − 10 3.3, > 10000 N/A N/A −15 1 − 10 5.5, > 10000 N/A N/A

46 Test 6. This fixed-point test function (Figure 2.15) is given by T6 : where gi (x) =

fi (x) = gi (x)2 +

1 4

xi − 41 1 . + 4 4||x − [ 41 , 41 ]||∞

(2.41)

(2.42)

for i = 1, 2. This function has a unique fixed point at [0.5, 0.5]T . Numerical results for this test function are exhibited in Tables 2.31-2.35. This test function has the Lipschitz constant ρ = 1. Therefore, only the residual error criterion is used in this experiment. The NR method is the fastest method for this test function in most cases. However, the SI method is faster than the NR method for large ε. The CE method is much faster than the SI method and is up to 7 times faster than the NR method for large values of ε. However, the CE method becomes slower than the NR method when ε is decreasing. The LS, CS and LS-CS hybrid methods are always slower than the SI methods so that they are the slowest method for this test function.

Figure 2.15. Illustration of function T6 : a) f1 (x1 , x2 ) and b) f2 (x1 , x2 ).

47

Table 2.31. Numerical results from applying the SI and NR function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). ε SI NR 10−2 3.68 × 10−7 , 8, 3 2.97 × 10−5 , 10−3 1.25 × 10−6, 33, 3 2.99 × 10−5 , 10−4 4.35 × 10−6 , 113, 3 2.97 × 10−5 , −5 −5 10 1.39 × 10 , 369, 3 2.97 × 10−5 , 10−6 4.42 × 10−5 , 1181, 3 2.97 × 10−5 , 10−7 1.41 × 10−4 , 3751, 3 2.97 × 10−5 , 10−8 4.45 × 10−4 , 11881, 3 3.51 × 10−5 , −9 10 1.40 × 10−3 , 37594, 3 4.01 × 10−5 , 10−10 4.52 × 10−3 , 118908, 3 4.02 × 10−5 , 10−11 1.41 × 10−2 , 376045, 3 4.01 × 10−5 , 10−12 4.49 × 10−2, 1189162, 3 4.01 × 10−5 , 10−13 0.14, 3761413, 3 4.01 × 10−5 , 10−14 0.45, 11898141, 3 4.01 × 10−5 , −15 10 1.41, 37578817, 3 4.00 × 10−5 ,

methods to the test

2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4,

3 3 3 3 3 3 3 3 3 3 3 3 3 3

Table 2.32. Numerical results from applying the CE method to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). ε CE Speedup wrt SI Speedup wrt NR −2 −6 10 5.0 × 10 , 5 (38), 3 0.667 6.8 10−3 1.4 × 10−5 , 13 (57), 3 0.750 2.6 −4 −5 10 2.4 × 10 , 23 (75), 3 2.000 1.4 −5 −5 10 3.2 × 10 , 32 (94), 3 4.250 1.0 −6 −5 10 4.0 × 10 , 40 (112), 3 12.25 0.8 10−7 4.8 × 10−5 , 48 (130), 3 33.20 0.7 −8 −5 10 5.9 × 10 , 57 (149), 3 123.5 0.7 −9 −5 10 6.5 × 10 , 65 (167), 3 583.3 0.7 10−10 7.5 × 10−5 , 75 (186), 3 1242 0.6 −11 −5 10 8.2 × 10 , 83 (204), 3 5470 0.5 −12 −5 10 9.1 × 10 , 92 (223), 3 12538 0.5 −13 −5 10 9.9 × 10 , 100 (241), 3 40625 0.4 10−14 1.07 × 10−4 , 108 (259), 3 163570 0.4 −15 −4 10 1.14 × 10 , 117 (278), 3 325000 0.4

48

Table 2.33. Numerical results from applying the LS method to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). ε LS Speedup wrt SI Speedup wrt NR 10−2 1.57 × 10−6 , 6, 3 0.2 19.0 −3 −6 10 3.20 × 10 , 9, 3 0.4 9.3 −4 −6 10 8.02 × 10 , 15, 3 0.5 3.7 10−5 1.60 × 10−5 , 22, 3 0.9 1.9 −6 −4 10 1.53 × 10 , 75, 3 0.3 0.2 −7 −4 10 7.42 × 10 , 171, 3 0.2 0.04 −8 −2 10 4.57 × 10 , 1344, 3 0.01 7.6 × 10−4 −9 −2 10 9.41 × 10 , 1936, 3 0.01 4.3 × 10−4 10−10 0.15, 2482, 3 0.03 2.6 × 10−4 10−11 0.30, 3466, 3 0.05 1.3 × 10−4 −12 10 0.30, 3466, 3 0.1 1.3 × 10−4 10−13 0.30, 3466, 3 0.5 1.3 × 10−4 10−14 0.53, 4551, 3 0.9 7.6 × 10−5 −15 10 0.53, 4553, 3 2.6 7.6 × 10−5

Table 2.34. Numerical results from applying the CS method to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). ε CS Speedup wrt SI Speedup wrt NR −2 −6 10 3.19 × 10 , 10, 3 0.1 9.3 10−3 1.22 × 10−5 , 30, 3 0.1 2.4 −4 −5 10 6.58 × 10 , 98, 3 0.1 0.5 −5 −4 10 5.02 × 10 , 325, 3 0.03 0.06 −6 −3 10 4.84 × 10 , 1083, 3 0.01 0.006 10−7 4.81 × 10−2 , 3497, 3 0.003 0.0006 10−8 0.39, > 10000 N/A N/A −9 10 0.39, > 10000 N/A N/A 10−10 0.39, > 10000 N/A N/A −11 10 0.39, > 10000 N/A N/A −12 10 0.39, > 10000 N/A N/A −13 10 0.39, > 10000 N/A N/A 10−14 0.39, > 10000 N/A N/A −15 10 4.16, > 10000 N/A N/A

49

Table 2.35. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T6 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0.1]T , 1). ε LS-CS Hybrid Speedup wrt SI Speedup wrt NR −2 −6 10 3.24 × 10 , 10, 3 0.1 9.2 −3 −5 10 1.24 × 10 , 30, 3 0.1 2.4 10−4 6.78 × 10−5 , 98, 3 0.1 0.4 −5 −4 10 5.24 × 10 , 325, 3 0.03 0.06 −6 −3 10 5.01 × 10 , 1083, 3 0.01 0.006 −7 −2 10 4.99 × 10 , 3497, 3 0.003 0.0006 10−8 0.4, > 10000 N/A N/A −9 10 0.4, > 10000 N/A N/A −10 10 0.4, > 10000 N/A N/A 10−11 0.4, > 10000 N/A N/A −12 10 0.4, > 10000 N/A N/A −13 10 0.4, > 10000 N/A N/A −14 10 0.4, > 10000 N/A N/A 10−15 0.4, > 10000 N/A N/A

50 α Test 7. This test function (Figure 2.16) is in the larger class Bρ≤1 , so it is

non-expanding (ρ = 1) in the direction of the unique fixed point [0.5, 0.5]T . This

is why for this test I utilize the residual stopping criterion. I observe that this function is discontinuous, and it is a rotation by angle θ around [0.5, 0.5]T of all points in the disk D = {(x1 , x2 ) : r ≤ R}, is a projection of any boundary point √ √ √ √ x, x 6= [1/ 2, 1/ 2]T of the unit circle C onto the point [1/ 2, 1/ 2]T , and is

a clockwise rotation around [0.5, 0.5]T of any interior point of C − D onto the boundary of C. It is given by  cosθ sinθ x 1   ;r ≤ R   −sinθ cosθ x 2   √   √ √ 1/√2   ; ||x||2 = 1, x 6= [1/ 2, 1/ 2]T  1/ 2 T7 : f (x1 , x2 ) = √ y    ; R < r ≤ 0.5 10 2 1/2    (1 − y )  p  √ √  y   ; 0.5 10 < r < 1.5 + 2 2 1/2 −(1 − y )

where

√ q √ C − 2 − C2 2 . r = ||x − [0.5, 0.5] ||2 , R = 1.5 − 2, C = 1.5 − r , y = 2 T

Tables 2.36-2.40 show the numerical results for this test function. The solutions computed by the CE method are within absolute distance = 10−2 , ..., 10−15 , to the fixed point [1/2, 1/2]T whenever ε = 10−2 , ..., 10−15 . For this test function the SI algorithm fails to obtain the fixed point. Though the NR method is mostly faster than the CE algorithm, the computed approximation is far away from the fixed-point (absolute distance from the fixed point is approximately equal to 0.707). The NR method converges to the boundary of the unit circle, in which neighborhood the function has arbitrarily small residual value. In addition, NR method fails to converge when ε ≤ 10−9 . The LS method is inefficient for this test function. In contrast, the CS and LS-CS hybrid methods used only two iterations to converge to the fixed point.

51


Table 2.36. Numerical results from applying the SI and NR methods to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε SI NR −2 10 Failed 2.6 × 10−5 , 2, 3 −3 10 Failed 2.6 × 10−5 , 2, 3 10−4 Failed 3.2 × 10−5 , 3, 3 10−5 Failed 3.3 × 10−5 , 3, 3 10−6 Failed 3.3 × 10−5 , 3, 3 10−7 Failed 4.1 × 10−5 , 4, 3 10−8 Failed 4.1 × 10−5 , 4, 3 10−9 Failed Failed −10 10 Failed Failed 10−11 Failed Failed −12 10 Failed Failed −13 10 Failed Failed 10−14 Failed Failed −15 10 Failed Failed

52

Table 2.37. Numerical results from applying the CE method to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε CE Speedup wrt SI Speedup wrt NR 10−2 2.5 × 10−5 , 9 (38), 3 N/A 1.04 −3 −5 10 4.1 × 10 , 27 (57), 3 N/A 0.63 −4 −5 10 5.5 × 10 , 40 (75), 3 N/A 0.58 10−5 6.7 × 10−5 , 53 (94), 3 N/A 0.49 −6 −5 10 7.8 × 10 , 66 (112), 3 N/A 0.42 −7 −5 10 9.4 × 10 , 81 (130), 3 N/A 0.44 −8 −4 10 1.2 × 10 , 94 (149), 3 N/A 0.34 10−9 1.1 × 10−4, 107 (167), 3 N/A N/A 10−10 1.3 × 10−4, 120 (186), 3 N/A N/A −11 −4 10 1.5 × 10 , 134 (204), 3 N/A N/A 10−12 1.7 × 10−4, 147 (223), 3 N/A N/A −13 −4 10 1.8 × 10 , 161 (241), 3 N/A N/A −14 −4 10 1.9 × 10 , 174 (259), 3 N/A N/A −15 −4 10 2.1 × 10 , 187 (278), 3 N/A N/A

Table 2.38. Numerical results from applying the LS method to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε LS Speedup wrt SI Speedup wrt NR −2 −5 10 1.79 × 10 , 23, 3 N/A 1.3 10−3 1.14 × 10−2 , 667, 3 N/A 0.002 10−4 4.13 × 10−2 , 1272, 3 N/A 6.7 × 10−4 10−5 8.96 × 10−2 , 1877, 3 N/A 6.7 × 10−4 −6 10 0.16, 2482, 3 N/A 6.7 × 10−4 10−7 0.24, 3086, 3 N/A 6.7 × 10−4 10−8 0.35, 3690, 3 N/A N/A −9 10 0.47, 4294, 3 N/A N/A 10−10 0.61, 4898, 3 N/A N/A −11 10 0.77, 5502, 3 N/A N/A −12 10 0.95, 6106, 3 N/A N/A −13 10 1.15, 6710, 3 N/A N/A 10−14 1.37, 7314, 3 N/A N/A −15 10 1.65, 8083, 3 N/A N/A

53

Table 2.39. Numerical results from applying the CS method to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε CS Speedup wrt SI Speedup wrt NR 10−2 9.4 × 10−7 , 2, 3 N/A 34.21 −3 −7 10 9.4 × 10 , 2, 3 N/A 34.21 −4 −7 10 9.4 × 10 , 2, 3 N/A 42.11 10−5 9.4 × 10−7 , 2, 3 N/A 43.42 −6 −7 10 9.4 × 10 , 2, 3 N/A 43.42 −7 −7 10 9.4 × 10 , 2, 3 N/A 53.94 −8 −7 10 9.4 × 10 , 2, 3 N/A 53.94 10−9 9.4 × 10−7 , 2, 3 N/A N/A −10 −7 10 9.4 × 10 , 2, 3 N/A N/A −11 −7 10 9.4 × 10 , 2, 3 N/A N/A 10−12 9.4 × 10−7 , 2, 3 N/A N/A −13 −7 10 9.4 × 10 , 2, 3 N/A N/A −14 −7 10 9.4 × 10 , 2, 3 N/A N/A −15 −7 10 9.4 × 10 , 2, 3 N/A N/A

Table 2.40. Numerical results from applying the LS-CS Hybrid method with δ = 10−6 to the test function T7 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε LS-CS Hybrid Speedup wrt SI Speedup wrt NR −2 10 1.0 × 10−6 , 2, 3 N/A 22.7 −3 10 1.0 × 10−6 , 2, 3 N/A 27.8 −4 −6 10 1.0 × 10 , 2, 3 N/A 27.7 −5 −6 10 1.0 × 10 , 2, 3 N/A 27.6 −6 −6 10 1.0 × 10 , 2, 3 N/A 33.1 10−7 1.0 × 10−6 , 2, 3 N/A 33.2 −8 −6 10 1.0 × 10 , 2, 3 N/A N/A −9 −6 10 1.0 × 10 , 2, 3 N/A N/A 10−10 1.0 × 10−6 , 2, 3 N/A N/A −11 −6 10 1.0 × 10 , 2, 3 N/A N/A −12 −6 10 1.0 × 10 , 2, 3 N/A N/A −13 −6 10 1.0 × 10 , 2, 3 N/A N/A 10−14 1.0 × 10−6 , 2, 3 N/A N/A −15 −6 10 1.0 × 10 , 2, 3 N/A N/A

54 Test 8. This test function (Figure 2.17) is similar to T7 , directionally nonexpanding function, and is given by  y1 ;r ≤ R      y2 ; r > R and ||y2 ||2 ≤ 1 T8 : f (x) =      y2 /||y2 ||2 ; ||y2 ||2 > 1 where

yi = Ri x, Ri =

cosθi sinθi −sinθi cosθi

T

, i = 1, 2; r = ||x − [0.5, 0.5] ||2 , R =

q

1.5 −

√

2.

The fixed point of this problem is [0.5, 0.5]T . In this test I use θ1 = 0.1 degree and θ2 = 1 degree. Tables 2.41-2.45 show the numerical results for this test function. In this case, the NR method converged to the correct fixed point and used only a few iterations while the SI method failed for this function. The CE method is slower than the NR method while the LS method failed in most cases. The CS and LS-CS hybrid methods used only one iteration to reach the fixed point and are much faster than the NR method.


55

Table 2.41. Numerical results from applying the SI and NR methods to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε SI NR 10−2 Failed 2.3 × 10−5 , 1, 3 10−3 Failed 2.3 × 10−5 , 1, 3 10−4 Failed 3.2 × 10−5 , 2, 3 10−5 Failed 3.1 × 10−5 , 2, 3 10−6 Failed 3.8 × 10−5 , 3, 3 10−7 Failed 3.9 × 10−5 , 3, 3 10−8 Failed 4.7 × 10−5 , 4, 3 10−9 Failed 4.6 × 10−5 , 4, 3 10−10 Failed 4.8 × 10−5 , 4, 3 10−11 Failed 4.6 × 10−5 , 4, 3 10−12 Failed 4.7 × 10−5 , 4, 3 10−13 Failed 4.7 × 10−5 , 4, 3 10−14 Failed 4.8 × 10−5 , 4, 3 10−15 Failed 4.8 × 10−5 , 4, 3

Table 2.42. Numerical results from applying the CE method to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε CE Speedup wrt SI Speedup wrt NR −2 −5 10 2.5 × 10 , 4 (38), 3 N/A 0.92 10−3 4.1 × 10−5 , 6 (57), 3 N/A 0.56 −4 −5 10 5.5 × 10 , 25 (75), 3 N/A 0.58 −5 −5 10 6.7 × 10 , 40 (94), 3 N/A 0.46 −6 −5 10 7.8 × 10 , 60 (112), 3 N/A 0.49 10−7 9.4 × 10−5 , 74 (130), 3 N/A 0.41 −8 −4 10 1.2 × 10 , 94 (149), 3 N/A 0.39 −9 −4 10 1.1 × 10 , 109 (167), 3 N/A 0.42 10−10 1.3 × 10−4, 129 (186), 3 N/A 0.37 −11 −4 10 1.5 × 10 , 144 (204), 3 N/A 0.31 −12 −4 10 1.7 × 10 , 164 (223), 3 N/A 0.28 −13 −4 10 1.8 × 10 , 184 (241), 3 N/A 0.26 10−14 1.9 × 10−4, 198 (259), 3 N/A 0.25 −15 −4 10 2.1 × 10 , 218 (278), 3 N/A 0.23

56

Table 2.43. Numerical results from applying the LS method to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε LS Speedup wrt SI Speedup wrt NR 10−2 3.58 × 10−6 , 9, 3 N/A 4.1 −3 −6 10 6.52 × 10 , 13, 3 N/A 2.3 −4 10 Failed N/A N/A 10−5 Failed N/A N/A −6 10 Failed N/A N/A −7 10 Failed N/A N/A −8 10 Failed N/A N/A 10−9 Failed N/A N/A −10 10 Failed N/A N/A −11 10 Failed N/A N/A 10−12 Failed N/A N/A −13 10 Failed N/A N/A −14 10 Failed N/A N/A −15 10 Failed N/A N/A

Table 2.44. Numerical results from applying the CS method to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε CS Speedup wrt SI Speedup wrt NR −2 −7 10 5.30 × 10 , 1, 3 N/A 27.8 10−3 5.31 × 10−7 , 1, 3 N/A 28.0 −4 −7 10 5.31 × 10 , 1, 3 N/A 27.8 −5 −7 10 5.31 × 10 , 1, 3 N/A 39.8 −6 −7 10 5.33 × 10 , 1, 3 N/A 39.8 10−7 5.32 × 10−7 , 1, 3 N/A 49.9 −8 −7 10 5.31 × 10 , 1, 3 N/A 50.3 −9 −7 10 5.31 × 10 , 1, 3 N/A 50.1 10−10 5.31 × 10−7 , 1, 3 N/A 50.1 −11 −7 10 5.31 × 10 , 1, 3 N/A 50.1 −12 −7 10 5.32 × 10 , 1, 3 N/A 50.1 −13 −7 10 5.31 × 10 , 1, 3 N/A 50.3 10−14 5.31 × 10−7 , 1, 3 N/A 50.2 −15 −7 10 5.31 × 10 , 1, 3 N/A 50.2

57

Table 2.45. Numerical results from applying the LS-CS hybrid method with δ = 10−6 to the test function T8 : x∗ = [0.5, 0.5]T , B1 = B 2 ([0, 0]T , 1.5). ε LS-CS Hybrid Speedup wrt SI Speedup wrt NR −2 −7 10 5.85 × 10 , 1, 3 N/A 25.2 −3 −7 10 5.87 × 10 , 1, 3 N/A 25.4 10−4 5.87 × 10−7 , 1, 3 N/A 25.2 −5 −7 10 5.88 × 10 , 1, 3 N/A 35.9 −6 −7 10 5.88 × 10 , 1, 3 N/A 36.1 −7 −7 10 5.87 × 10 , 1, 3 N/A 36.0 10−8 5.87 × 10−7 , 1, 3 N/A 45.1 −9 −7 10 5.88 × 10 , 1, 3 N/A 45.4 −10 −7 10 5.87 × 10 , 1, 3 N/A 45.3 10−11 5.89 × 10−7 , 1, 3 N/A 45.1 −12 −7 10 5.88 × 10 , 1, 3 N/A 45.2 −13 −7 10 5.87 × 10 , 1, 3 N/A 45.5 −14 −7 10 5.87 × 10 , 1, 3 N/A 45.4 10−15 5.87 × 10−7 , 1, 3 N/A 45.4

58 It can be noted that in all the tests the CE algorithm terminated with the absolute error criterion (1) or (2) much faster than indicated by the upper bounds. The summary of numerical results is shown in Table 2.46 to exhibit the fastest and slowest methods for all test functions.

2.6

Conclusions

Fixed points of functions that are contractive/nonexpanding only in the direction of fixed points but may be globally expanding can also be approximated by the circumscribed ellipsoid algorithm. The proposed methods were implemented and compared with the simple iteration and Newton-Raphson methods. The line- and circle-search methods are more efficient than the circumscribed ellipsoid method in many test cases. The Newton-Raphson method is very efficient for locally linear functions and higher-dimensional functions.

2.7

Future Work

I plan to develop a hybrid CE-LS-CS algorithm that combines the circumscribed ellipsoid, line-search, and circle-search algorithms in order to lower the total number of iterations and bound the number of linear and/or quadratic constraints bounding the feasible set of fixed points. Complexity bounds need to be developed for the line- and circle-search methods, and a new CE-LS-CS hybrid method. Finally, a

Table 2.46. Fastest and slowest methods for all test functions. Test Function F1 F2 F3 F4 F5 F6 F7 F8

Fastest Method LS CE/LS LS/LS-CS Hybrid CE NR NR CS CS

Second Fastest Method LS-CS Hybrid LS-CS Hybrid CE LS/LS-CS Hybrid LS CE LS-CS Hybrid LS-CS Hybrid

Slowest Method SI/CS SI/CS SI/CS SI/CS SI/CS LS/CS/LS-CS Hybrid SI SI

59 generalization of the circle-search method by using spherical methods should be developed and analyzed.

CHAPTER 3 AN EFFICIENT MULTISCALE METHOD FOR TIME-DOMAIN WAVEFORM TOMOGRAPHY In Section 3.1 an introduction to seismic waveform tomography is presented. Methods for time-domain waveform tomography are shown in Section 3.2 and numerical results are presented in Section 3.3. Finally, the conclusions are drawn in Section 3.4 and future research is proposed in Section 3.5.

3.1

Introduction

Traveltime tomography [38, 41, 44, 50, 57, 90] has proven to be a robust tool for estimating the subsurface velocity structure. By minimizing the difference between observed and calculated traveltimes, a velocity perturbation is calculated, and the velocity model is iteratively updated. Although this method is fast and cost effective, the high-frequency approximation utilized by traveltime tomography violates the finite-frequency bandwidth of the seismic source. Moreover, traveltime tomography only uses the traveltime of the first-arrival and neglects all of the other important information contained within the amplitude and phase of the recorded data. To overcome the high-frequency assumption, Fresnel-volume or finite-frequency traveltime tomography [18, 21, 57, 79, 83] and waveform tomography [34, 42, 72, 73] have been developed. Waveform tomography can be implemented in either the frequency domain [37, 49, 51, 52] or the time domain [16, 42, 72, 73, 88, 89]. A velocity tomogram obtained using waveform tomography that inverts all of the arrivals has higher accuracy and resolution than one obtained by traveltime tomography [62]. However, the waveform misfit function is strongly nonlinear, and

61 waveform tomography typically converges to a local minimum if the starting model is not in the vicinity of the global minimum [27]. Some efforts have been undertaken to overcome this local minima problem in both the time domain [16, 62] and the frequency domain [14, 70]. In the time domain, Sheng et al. [62] proposed a method called early-arrival waveform tomography (EWT) in which a time window is applied to seismic data to allow for the inversion of only early-arrival wavefields. The misfit function for EWT is more linear with respect to slowness because fewer events need to be fitted compared to conventional full waveform tomography. However, high-frequency data used in the inversion cause the misfit function to be highly nonlinear, and EWT can still suffer from the local minima problem. Alternatively, multiscale waveform tomography (MWT) proposed by Bunks et al. [16] for time-domain waveform tomography uses a finite-impulse response (FIR) Hamming-windowed filter for low-pass filtering of the seismic source wavelet and the data before inversion. This allows the inversion to proceed sequentially from lowfrequency data to high-frequency data. Since the misfit function at low frequencies is more linear with respect to slowness than at high frequencies, MWT is more likely to reach the global minimum [70]. The Hamming window function used for low-pass filtering in [16] is not the most efficient filter for time-domain MWT. Furthermore, data with arbitrarily chosen frequency bands are used in the inversion method of Bunks et al. [16]. For each frequency band, data were inverted for some number of iterations, but the drawback is that too many frequency bands can result in a huge amount of computation. In the frequency domain, Sirgue and Pratt [70] proposed a strategy for choosing optimal frequencies. This can greatly reduce the computational cost of waveform tomography, but a modification of their method is required before it can be used in the time domain. The multiscale approach can be naturally applied to frequency-domain waveform tomography because data have already been decomposed into separate frequency components using the Fourier transform. Combining the frequency selection strategy of Sirgue and Pratt [70] with the multiscale approach, a frequency-domain waveform tomography method becomes much more efficient than a time-domain

62 method, especially for 2D applications where forward modeling can be efficiently performed using the LU factorization, since there are many right-hand-side vectors due to a large number of sources in a seismic survey. However, for 3D experiments, the LU factorization used for solving the large, sparse linear system arising from frequency-domain forward modeling requires huge amounts of computer memory and computation time, and thus limits the problem size and the maximum frequencies to be used for 3D applications [48]. Instead, an iterative solver can be utilized to avoid the memory issue [82] but the efficiency of an iterative solver relies strongly on a preconditioner. Without a good preconditioner, several iterations will be required by an iterative solver causing inefficient forward modeling. Sirgue et al. [67, 68, 69] proposed a solution for 3D frequency-domain waveform inversion by using time-domain forward modeling and frequency-domain gradient computation. This hybrid method requires a discrete Fourier transform of the entire wavefield at each time step [68], which can be computationally expensive.

In contrast,

pure time-domain waveform tomography does not require such a large memory, preconditioner, and the use of a discrete Fourier transform. Therefore, it is desirable to develop a more efficient multiscale waveform inversion method in the time domain, which is the goal of this work. In this work, I improve the computational efficiency of the multiscale method of Bunks et al. [16] by using more efficient nonleaky low-pass filters and a time-domain strategy for choosing optimal frequency bands. The filters I used are shown to be more efficient than the FIR Hamming-windowed low-pass filter used by Bunks et al. [16] for time-domain multiscale waveform inversion. The strategy for selecting frequencies for frequency-domain waveform tomography proposed by Sirgue and Pratt [70] is extended to time-domain waveform tomography. This revised strategy is validated as an efficient method by inversion results with 1D and 2D, heterogeneous models. True source wavelets were used in both experiments. In the 2D model experiment, I show that MWT can mitigate the local minima problem of waveform tomography and yield an accurate estimate of the velocity model, while conventional, single-scale waveform tomography converges to a local minimum, resulting in an inaccurate estimate of the true velocity model. I also apply MWT to

63 Canadian Foothills synthetic data to show that MWT to provide a high-resolution velocity model for a complex geological setting [11].

3.2 3.2.1

Methods for Time-domain Waveform Tomography

Acoustic Waveform Tomography

In this section, I briefly review the theory of time-domain waveform tomography. The acoustic wave equation is used to describe wave propagation, given by 1 ∂ 2 p(r, t|rs ) 1 −∇· ∇p(r, t|rs ) = s(r, t|rs ), (3.1) κ(r) ∂t2 ρ(r) where p(r, t|rs ) is a pressure field at position r at time t excited by a source at rs ; κ(r) and ρ(r) are the bulk modulus and density distributions, respectively; and s(r, t|rs ) is the source function. The forward solution, p(r, t; rs ), of the twoway wave equation 3.1 is computed by a staggered-grid, explicit finite-difference method with 4th -order accuracy in space and 2nd -order accuracy in time [36]. The free-surface boundary condition is applied to the top boundary of the model, and the perfectly matched layer (PML) boundary conditions [6, 19, 25, 87] are utilized at the other boundaries. The solution to equation 3.1 can be written as Z p(r, t|rs) = G(r, t|r′, 0) ∗ s(r′ , t|rs )dr′,

(3.2)

where G(r, t|r′, 0) is the Green’s function and the symbol ∗ represents temporal convolution. Our inversion scheme is based on the adjoint method proposed by Tarantola [72]. The data residuals are defined as δp(rg , t|rs ) = pobs (rg , t|rs ) − pcalc (rg , t|rs ),

(3.3)

where pobs (rg , t|rs ) and pcalc (rg , t|rs ) are, respectively, the observed and calculated data recorded by a geophone at rg . The velocity model is updated by minimizing the misfit function, defined as the L2 norm of the data residuals, Z 1 XX (δp(rg , t|rs ))2 dt. E= 2 s g

(3.4)

A nonlinear preconditioned conjugate-gradient method [62] is used to minimize the p κ(r)/ρ(r). The gradient misfit function and obtain a velocity model c(r) =

64 of the misfit function with respect to the velocity model is computed by the zero-lag correlation between the forward-propagated wavefields and back-projected wavefield residuals [38, 62, 72, 88, 89], Z 2 X p(r, ˙ t|rs )p˙′ (r, t|rs )dt, g(r) = c(r) s

(3.5)

where p˙ denotes the time derivative of p, p(r, t|rs) represents the forward-propagated wavefields, and p′ (r, t|rs ) represents the back-projected wavefield residuals given by Z ′ p (r, t|rs ) = G(r, −t|r′ , 0) ∗ δs(r′ , t|rs )dr′, (3.6) and δs(r′ , t|rs ) =

X g

δ(r′ − rg )δp(rg , t|rs ),

(3.7)

where δ(r) is the Dirac delta function.

The velocity model is iteratively updated along the conjugate directions defined by dk = −Pk gk + βk dk−1 ,

(3.8)

where iterations k = 1, 2, ..., kmax , g = {g(r) for all image points r in the model}, and P is the conventional geometrical-spreading preconditioner [17]. At the first iteration d0 = −g0 . The parameter βk is obtained using the Polak-Ribiére formula [46] βk =

gkT · (Pk gk − Pk−1gk−1 ) . T · Pk−1 gk−1 gk−1

(3.9)

The velocity model is updated by

ck+1 (r) = ck (r) + λk dk (r),

(3.10)

where λk is the step length, which is determined by a quadratic line-search method [46, ], and dk (r) is the component of the conjugate-direction vector dk at position r. At each iteration, one forward-propagation and one back-projection are needed to compute the gradient direction. Additional forward modeling is required for the line search. In this work, the starting model c0 (r) is obtained using traveltime tomography with dynamic smoothing filters [44], and forward modeling and inversion

65 are simply parallelized by shot numbers and computed on a Linux cluster of AMD Opteron 1.8 GHz processor. Equations 3.8, 3.9, and 3.10 are applied iteratively until a stopping criterion is satisfied, which is either when the maximum number of iterations is reached or when there is no further decrease in data residuals. In practice, the RMS error or L2 norm of the data residuals may be used as a stopping criterion, e.g., the inversion stops when the error is less than a previously chosen tolerance value. 3.2.2

Efficient Low-pass Filtering for Multiscale Waveform Tomography

Conventional time-domain waveform tomography [72] using a single frequency band of the data and one finite-difference grid can lead to the local minima problem. By using several frequency bands of the data and varying grid sizes, the multiscale method of Bunks et al. [16] was very successful in inverting for the complex Marmousi model. Low-pass filtering is one of the crucial parts that make multiscale waveform inversion possible in the time domain. For a given frequency band, a finite-difference grid size and a time step can be determined by the numerical dispersion and stability conditions associated with the chosen explicit finite-difference method. The numerical dispersion condition for the finite-difference scheme used in this work requires at least 5 grid points per minimum wavelength [36], and a square grid (dx = dz) is utilized in my finite-difference scheme. Therefore, the grid size used in forward modeling is determined by dx ≤

cmin λmin , ≤ 5 5fmax

(3.11)

where dx is the grid size, λmin is the minimum wavelength, cmin is the minimum velocity, and fmax is the maximum frequency of the band. Once the grid size is determined, the time step dt is determined by the 2D numerical stability condition [16] dt < √ where cmax is the maximum velocity.

dx , 2cmax

(3.12)

66 In the original work of Bunks et al. [16], a Hamming window function [53] was utilized for low-pass filtering the source wavelet and the data. The source wavelet is presumably known or estimated prior to the inversion. Several methods for estimating a seismic source wavelet have been proposed [5, 35, 47, 81]. The Hamming window function is a leaky low-pass filter as shown in Fig. 3.1d by the solid line. Fig. 3.1d shows that the leaked high-frequency components are weak compared to large-amplitude frequency components, e.g., a dominant frequency component. However, these leaked high frequency components can cause the maximum frequency fmax to be a large number; therefore, a small grid size is required to satisfy equation 3.11 resulting in computational inefficiency. If a large grid size is used, numerical dispersion can occur in forward modeling, and may create artifacts in inversion results or even cause convergence problems. It is therefore desirable to implement an alternative low-pass filter for MWT that minimizes the leaking of the high frequency components. A number of suitable filters are considered for MWT including Wiener filter, Blackman-Harris window function [28], Nuttall window function [45], and Ricker function. The filtering results of the first two filters in Fig. 3.1 show that they are more efficient than the Hamming window function. The amplitudes of leaked high-frequency components from these filters are several orders of magnitude smaller than those from the Hamming filter (Fig. 3.1d). Nuttall window and Ricker functions also provide similar results but are not shown here. The Wiener filter is usually applied in the frequency domain, while the other filters are applied in the time domain. One of the merits of the Wiener filter is that it can filter one signal to closely match another target signal. A low-pass Wiener filter [10] can be computed by

† Wtarget (ω)Woriginal (ω) , fW iener (ω) = 2 |Woriginal (ω)| + ǫ2

(3.13)

where fW iener is the Wiener filter, Woriginal is the original wavelet, Wtarget is the low-frequency target wavelet, ω is the angular frequency, ǫ is a small parameter that prevents numerical over-flow, and † denotes the complex conjugate. In Fig. 3.1, the original and target wavelets are 20-Hz and 2-Hz Ricker wavelets, respectively. The

67 a) Low−pass Filters in the Time Domain 1 Spectral Amplitude

Amplitude

0.5 0 −0.5 −1 0

1 Time (s)

1.5

10

2

c) Original and Filtered Wavelets

0

20 40 Frequency (Hz)

60

d) Amplitude Spectra of Original and Filtered Wavelets 0 10 Spectral Amplitude

0.5 Amplitude

−5

10

−10

0.5

1

0 −0.5 −1 0

b) Amplitude Spectra of Low−pass Filters

0

10

−5

10

−10

0.5

1 Time (s)

1.5

2

10

0

20 40 Frequency (Hz)

60

Figure 3.1. Low-pass filtering test. a) Low-pass filters in the time domain. The solid line is a Hamming window, the dashed line is a Blackman-Harris window, and the dotted line is a Wiener filter in the time domain. This line style is also used in panels b), c), and d). b) The amplitude spectra of the low-pass filters in a). c) The original and filtered wavelets. The dash-dotted line is the original wavelet, which is a 20-Hz Ricker wavelet and was low-pass filtered such that the peak frequency is about 2 Hz. d) The amplitude spectra of the original and filtered wavelets in c). filtered wavelet obtained by Wiener filtering is not exactly the same as the target wavelet. However, the main purpose of the filtering is to obtain a wavelet with a low-frequency band and minimal spectral leakage. Fig. 3.1 shows the results of lowpass filtering a source wavelet. In time-domain multiscale waveform tomography, low-pass filtering is also applied such that both the filtered source wavelet and the data have roughly the same frequency range. In general, one can use either one of the proposed filters but in this work I use only the Wiener filter.

68 3.2.3

Strategy for Choosing Optimal Frequency Bands

The strategy proposed by Sirgue and Pratt [70] for choosing optimal frequencies for frequency-domain waveform tomography is now extended to the time domain in order to reduce the redundancy in the recovered wavenumber spectrum of a subsurface velocity structure. At a single frequency, the contribution of a single source-receiver pair has only a single wavenumber component, and a range of vertical wavenumber components of the velocity model can be updated by using a range of source-receiver pairs [70]. Time-domain waveform tomography uses multiple frequencies simultaneously during the inversion. Therefore, it can update a much wider range of wavenumbers than using only a single frequency at a time – as in frequency-domain waveform tomography. However, the band-limited nature of seismic data results in a band-limited range of recovered wavenumber components of the velocity model without sharp boundaries, i.e., each frequency component of the data has a different amplitude or strength. Strong frequency components of the data dominantly contribute to the wavenumber update while weak frequency components, especially at the low- and high-ends of a frequency band, have weak contributions and are ignored in our strategy. I now review the strategy for choosing optimal frequencies for frequency-domain waveform tomography proposed by Sirgue and Pratt [70]. In the frequency domain, the gradient of the misfit function can be written as n o XX g(r) = −ω 2 Re G†0 (r|rs )G†0 (r|rg )δp(rg |rs ) , s

(3.14)

g

where G†0 (r|rs ) and G†0 (r|rg ) are the Green’s functions for an excitation at the source and receiver positions in the reference medium. To proceed to a frequency selection strategy, Sirgue and Pratt [70] introduce the following assumptions with respect to the Green’s function: 1. Amplitude effects may be ignored. 2. The reference medium is homogeneous, with a velocity c0 . 3. We are in the far field, so that we may replace the Green’s functions with plane-wave approximations.

69 Under these assumptions, both Green’s function, G∗0 (r|rs ) and G∗0 (r|rg ), may be approximated by incident and scattered plane waves: G0 (r|rs ) ≈ exp(ik0 rˆs · r), G0 (r|rg ) ≈ exp(ik0 rˆg · r),

(3.15)

where k0 = ω/c0 is the wavenumber of the incident and scattered waves in the homogeneous reference medium, and rˆs and rˆg are, respectively, unit vectors in the incident propagation direction (source to scatterer) and the inverse scattering direction (receiver to scatterer). Inserting equation 3.15 into equation 3.14 yields XX g(r) = −ω 2 Re {exp(−ik0 (rˆs + rˆg ) · r)δp(r|rs )} . (3.16) s

g

According to equation 3.16, the contribution to the gradient image of a single source-receiver pair has only a single wavenumber component, given by k0 (rˆs + rˆg ). If the model is 1D, then all reflection points in the subsurface are midpoint reflection points as shown in Fig. 3.2. A common-shot gather is equivalent to a common-midpoint (CMP) gather. Each source-receiver pair records data from a given midpoint with a combination of symmetric plane waves. For a 1D earth, the incident and scattering angles are symmetric, so that k0 rˆs = (k0 sin θ, k0 cos θ), k0 rˆg = (−k0 sin θ, k0 cos θ).

(3.17)

From Fig. 3.2, we observe that cos θ = sin θ =

√ z , h2 +z 2 √ h , h2 +z 2

(3.18)

where h is the half offset and z is the depth of the scattering layer. Substituting equation 3.18 into equation 3.16, the components of the vector k0 (rˆs + rˆg ) are kx = 0, kz = 2k0 α,

(3.19)

with α = cos θ, z = √ , 2 h + z2 1 , = √ 1 + R2 where R = h/z is the half offset-to-depth ratio.

(3.20)

70

Figure 3.2. The 1D basic scattering experiment with wavenumber illumination. The incident plane wave is reflected from a thin layer at the midpoint between source and receiver. A single frequency component of a single source-receiver pair generates a single wavenumber vector of the gradient image. The illuminated wave vector is given by the sum of the source-to-scatterer wave vector, k0 rˆs and the receiver-to-scatterer wave vector, k0 rˆg . (modified from Sirgue and Pratt [70]) Using equations 3.18 and 3.19 for a range of offsets, we find that for a given surface seismic acquisition characterized by an offset range [0, xmax ], the vertical wavenumber coverage kz of a 1D thin layer for a given frequency is limited to the range [kzmin , kzmax ], where kzmin = 2k0 αmin , kzmax = 2k0 ,

(3.21)

with

where Rmax = hmax /z.

1 αmin = p , 2 1 + Rmax

(3.22)

The minimum and maximum wavenumber kzmin and

kzmax are produced by the furthest and nearest offsets, respectively. Expressing equation 3.21 in terms of frequency, we have kzmin = 4πf αmin /c0 , kzmax = 4πf /c0 ,

(3.23)

71 where f is the frequency and c0 is the velocity in the background medium. From relation 3.23, we may define the wavenumber coverage of a multioffset acquisition as ∆kz ≡ |kzmax − kzmin |, = 4π(1 − αmin )f /c0.

(3.24)

The strategy for choosing frequencies of Sirgue and Pratt [70] is determined as follows: each frequency has a limited, finite-band contribution to the image spectrum. In order to recover the target accurately over a broad range of wavenumbers, the continuity of the coverage of the object in the wavenumber domain must be preserved as the imaging frequencies are selected. They choose kzmin (fn+1 ) = kzmax (fn ),

(3.25)

where fn+1 is the next frequency to be chosen following the frequency fn . The principle (illustrated in Fig. 3.3) is that the maximum wavenumber of the smaller frequency must equal the minimum wavenumber of the larger frequency. This strategy thus relies on using the full range of offsets available. Using the condition defined in equation 3.25 and substituting equation 3.23, we arrive at the formula for choosing frequencies proposed by Sirgue and Pratt [70], fn+1 =

fn . αmin

(3.26)

In time-domain inversion, multiple frequencies for a given bandwidth are used simultaneously. Therefore, a criterion is needed for determining the minimum and maximum frequencies of the band that will be used to calculate the recovered wavenumber range. As mentioned earlier, frequency components of the data with strong amplitudes are more important than weak components. If weak frequency components are used to determine the minimum and maximum frequencies of a frequency band, the wavenumber components that are recovered from the current frequency band may be inaccurate. When higher-frequency data are used in the inversion to recover higher-wavenumber components of the velocity model, the inversion may fail to converge to the global minimum since lower-wavenumber components are not accurately recovered yet. Thus, my criterion is to use the frequencies at half the maximum spectral amplitude of the frequency band to

72

Figure 3.3. Illustration of the strategy for choosing frequencies for frequency-domain waveform tomography. A range of vertical wavenumbers can be recovered by a single frequency component of the data by using a range of source-receiver pairs. A continuous coverage in vertical wavenumbers is the key for choosing the next frequency. (after Sirgue and Pratt [70]) determine the minimum and maximum frequencies of the frequency band. This is illustrated in Fig. 3.4b where the amplitude spectrum of a 15-Hz Ricker wavelet (Fig. 3.4a) is shown. At half the maximum spectral amplitude, the lower frequency is set as the minimum frequency of the band fmin and the higher frequency is set as the maximum frequency fmax . Though frequency components with small amplitudes have insignificant contributions to the wavenumber recovery, their contributions can be considered as overlapped regions between two recovered wavenumber ranges from two consecutive frequency bands. The criterion for choosing minimum and maximum frequencies of a frequency band can be varied and depends on how much two recovered wavenumber ranges will be overlapped. Sirgue and Pratt [70] showed that more frequency components were required for inverting noisy data in frequency-domain waveform tomography. Similarly, in the time-domain case, larger

73 a) 15−Hz peak−frequency Ricker wavelet

b) Amplitude Spectrum of Ricker wavelet

1 25

20 Amplitude

Amplitude

0.5 15

10

0

5

−0.5 0

0.05

0.1 Time (s)

0.15

0.2

0 0

fmin 10

fmax 20 30 Frequency (Hz)

40

Figure 3.4. A 15-Hz Ricker wavelet is shown in a) and its corresponding amplitude spectrum is shown in b). The frequency at half the maximum spectral amplitude at the lower end is set as the minimum frequency of the band fmin and the frequency at half the maximum spectral amplitude at the higher end is set as the maximum frequency fmax . overlapped regions of recovered wavenumber components will be needed and may result in a larger number of frequency bands to be used in the inversion. In the n-th frequency band, the minimum and maximum frequencies of the band fmin (n) and fmax (n), respectively, determine the wavenumber range, kzmin (n) = 4πfmin (n)αmin /co , kzmax (n) = 4πfmax (n)/co .

(3.27)

In a similar manner to equation 3.25, at the next frequency band, n + 1, the lowest wavenumber to be updated is equal to the highest wavenumber at the current frequency band, n, kzmin (n + 1) = kzmax (n).

(3.28)

This principle is illustrated in Fig. 3.5. In summary, the computational efficiency of time-domain acoustic waveform tomography can be increased by using nonleaky low-pass filters and a strategy for choosing optimal frequency bands. Using a nonleaky filter, spatial and temporal intervals, dx and dt, can be larger than using a leaky filter, and the costs of forward modeling and inversion will be lower. Without a strategy for choosing

74

Figure 3.5. Illustration of the strategy for choosing optimal frequency bands for time-domain waveform tomography. Data within a limited range of frequencies are inverted, allowing the recovery of a range of vertical wavenumbers. A continuous coverage within the vertical wavenumber spectrum is the key for choosing the next frequency band. frequency bands, too many frequency bands of data may be used resulting to a computationally expensive inversion process which can provide about the same result as an inversion process using optimally chosen frequency bands.

3.3

Numerical Results

I now apply multiscale waveform tomography (MWT) to two synthetic data sets generated from 1D and 2D models. The strategy for choosing frequency bands is based on the 1D assumption in a homogeneous medium, originally developed for application in the frequency domain [70]. Thus, I need to validate my strategy in the time domain by applying it to a 1D model. The strategy is then applied to a 2D heterogeneous model, to show that it is also applicable to more realistic, geological settings. The 2D model experiment is also designed to show the advantages of MWT compared to the conventional, single-scale waveform tomography (SWT).

75 3.3.1

1D Model

The 1D velocity model (Fig. 3.6a) used in this experiment is similar to the one used by Sirgue and Pratt [70], whose strategy has been successfully applied to both 1D and 2D heterogeneous models. In this experiment, there are 81 sources and 401 receivers per shot, and the maximum offset is 4 km. The sources and receivers are evenly distributed along the surface with spatial intervals of 50 m and 10 m, respectively. The thin layer located at 1.15 km (the bottom of the layer) is the target to be imaged. Fig. 3.6b shows a shot gather from a source a) 1D Velocity Model 2500

Velocity (m/s)

2400

2300

2200

2100 0

0.2

0.4

0.6

0.8

1 1.2 Depth (km)

0

0.5

0.5

1

1

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2

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2.5 1

2 Offset (km)

3

1.8

2

1.5

2

0

1.6

c) Filtered Shot Gather

0

Time (s)

Time (s)

b) Original Shot Gather

1.4

4

0

1

2 Offset (km)

3

4

Figure 3.6. The model and data used in 1D inversion. a) One-dimensional velocity model. The target reflector is located at 1.15 km depth. b) A shot gather of the 1D model generated from a 10-Hz Ricker source wavelet. c) The shot gather low-pass filtered by the Wiener filter to the frequency band of a 2.5-Hz Ricker wavelet.

76 at x = 0 m. The original data were generated by using a 10-Hz Ricker wavelet, shown as the dashed line in Fig. 3.7a, with a total record length of 3 s and a grid size of 10 m. To satisfy the numerical stability condition, the time sampling interval is set to 2 ms. The source and recorded data are low-pass filtered using the Wiener filter to the bandwidth of a 2.5-Hz Ricker wavelet, shown as the solid line in Fig. 3.7a. Fig. 3.6b shows a shot gather after low-pass filtering. At this low-frequency band, I can use a larger grid size, dx = 50 m, and a larger time sampling interval, dt = 10 ms. Therefore, the inversion at this low-frequency band will be more computationally efficient, allowing us to compute a large number of iterations to recover the low-wavenumber components of the velocity model as accurately as possible. The total number of model parameters, i.e., velocity values, to be estimated at this band is 81 × 41 = 3321. Since the filtered data have a total record length of 3 s, 300 time steps will be required to generate the calculated data, Pcalc , for comparison with the observed data, Pobs . I now apply my strategy for choosing the frequency bands for the 1D experiment. The low-frequency band, the solid line in Fig. 3.7b, is the first band used in the inversion, n = 1, and the minimum and maximum frequencies at half the peak of this band are denoted as fmin (1) and fmax (1). The wavenumber range which can be recovered by this low-frequency band is shown in the solid area in Fig. 3.7c where the lines kzmin (f ) and kzmax (f ) are computed using equation 3.23 with c0 = 2200 m/s, z = 1.15 km, and h = 2 km. The next frequency band that satisfies equation 3.27 turns out to be that of the 10-Hz Ricker wavelet, the original source wavelet. Therefore, the grid size and time step used for this high-frequency band are the same as the parameters used to originally generate the observed data. The total number of model parameters to be estimated with this band is 401 × 201 = 80601, and the original data, with 1501 time samples per trace, are used in the inversion. The minimum and maximum frequencies of this second frequency band, fmin (2) and fmax (2), respectively, are shown in Fig. 3.7b, and its corresponding wavenumber range is shown in Fig. 3.7c (the solid area in the high-frequency range).

77 a) Ricker wavelets 1

Amplitude

0.5

0

−0.5 0

0.5

1

1.5

Time (s) b)

Amplitude spectra

−3

x 10 5

Amplitude

4 3 2

c)

8 10 12 Frequency (Hz)

fmax(2)

fmin(2)

(1)

6

f

f

4

max

2

min

0 0

(1)

1

14

16

18

20

Sequence of frequency bands 20

k

(f)

15

10

k

(f)

zmin

8 10 12 Frequency (Hz)

fmax(2)

6

fmin(2)

4

max

f

f

2

min

0 0

(1)

5

(1)

Wavenumber (1/km)

zmax

14

16

18

20

Figure 3.7. Source wavelets used in the 1D model are shown in a) the time domain and b) the frequency domain. The sequence of frequency bands generated by the strategy for choosing frequency bands for the 1D model is shown in c).

78 The inversion results of this experiment are shown in Fig. 3.8. With the lowfrequency-band data, the inversion is iterated for 50 iterations starting from an initial homogeneous velocity model (co = 2200 m/s). Twenty iterations are carried out with the high-frequency-band data. The vertical profile of the final results at x = 2 km is shown in Fig. 3.8a, and its corresponding wavenumber components are shown in Fig. 3.8b. Fig. 3.8c and 3.8d show the contributions for each frequency band to the final image. The low-wavenumber components of the velocity model can be recovered quite accurately using the low-frequency band while the high-frequency

a) Velocity Perturbation in Space Domain

b) Velocity Perturbation in Wavenumber Domain 4000

200

3000

Amplitude

300

100 0 −100 0

2000 1000 0 0

0.5

1 1.5 2 Depth (km) c) Velocity Contribution in Space Domain

10 15 20 25 30 Wavenumber (1/km) d) Velocity Contribution in Wavenumber Domain

5

200 Amplitude

Velocity Contribution (m/s)

Velocity Perturbation (m/s)

band recovers the moderate- and high-wavenumber components.

100 0 −100 0

0.5

1 Depth (km)

1.5

2

3000 2000 1000 0 0

5

10 15 20 Wavenumber (1/km)

25

30

Figure 3.8. Time-domain multiscale waveform inversion results for the 1D model. In a) and b) the true velocity perturbations in the space and wavenumber domains, respectively, are shown as a dashed line while the final estimated velocity perturbations from the background are shown as a solid line. The contribution in the space and wavenumber domains from the two frequency bands are shown in c) and d) where the solid lines are the contribution from low-frequency-band data (peak frequency = 2.5 Hz) and the dashed lines are the contribution from high-frequency-band data (peak frequency = 10 Hz).

79 3.3.2

2D model

The 2D model used in this experiment consists of several small-scale velocity anomalies embedded within a background of slowly varying macro layers (Fig. 3.9). These small-scale anomalies are very difficult to recover using conventional traveltime tomography. The seismic data for this experiment consist of 201 sources and 201 receivers per shot, and the maximum offset is 4 km. The sources and receivers are evenly distributed along the surface with spatial intervals of 20 m. Fig. 3.10a shows a shot gather from a source at x = 0 m. The source signature used to generate the data is a 20-Hz Ricker wavelet. The grid size of 5 m and the time sampling interval of 0.5 ms were used to generate the observed data with a total record length of 2.5 s. Both SWT and MWT were applied to this data set. In SWT, the original source and data were used in the inversion, while multiple frequency bands of the source and the data were used in MWT. In this experiment, the starting frequency band has a peak frequency of 5 Hz. By applying my strategy for choosing frequency bands to this experiment, only two frequency bands are needed: the frequency bands of the 5-Hz Ricker wavelet and the 20-Hz Ricker wavelet, i.e., the original source wavelet. Fig. 3.10b shows a shot gather after low-pass filtering. With the low-frequency band, I can use a grid size of 20 m and a time sampling interval of 2 ms without risk of numerical dispersion and instability. The total number of Velocity (m/s) 2D Velocity Model

Depth (km)

0

4000 3500

0.5

3000 2500

1

0

0.5

1

1.5

2 2.5 Location (km)

3

3.5

4

2000

Figure 3.9. The 2D velocity model used to test the applicability of the strategy for choosing optimal frequency bands to a 2D, heterogeneous case.

80 b) Filtered Shot Gather 0

0.5

0.5

1

1

Time (s)

Time (s)

a) Original Shot Gather 0

1.5

1.5

2

2

0

2 Offset (km)

4

0

2 Offset (km)

4

Figure 3.10. An original shot gather of the 2D model generated from a 20-Hz Ricker source wavelet is shown in a) and the shot gather low-pass filtered by the Wiener filter to the frequency band of a 5-Hz Ricker wavelet is shown in b). model parameters to be estimated at this band is 201 × 51 = 10251, and the filtered data with the same total record length have 1249 time samples per trace. With the high-frequency band, the total number of model parameters to be estimated has increased to 801 × 201 = 161001, and the original data with 5000 time samples per trace were used. The initial velocity model used by both SWT and MWT is obtained by the traveltime tomography method of Nemeth et al. [44] with a square smoothing filter of size 50 m. One hundred iterations were carried out to obtain the traveltime velocity tomogram which is incapable of resolving the small-scale anomalies (Fig. 3.11a). The ray density diagram from traveltime tomography shown in Fig. 3.11b can be used to determine regions in the traveltime tomogram that are reliable. Regions with high ray densities are more reliable than those with low densities. The inversion results of this experiment are shown in Fig. 3.12. The data were inverted for 50 iterations with the low-frequency band, followed by 20 iterations with the high-frequency band. Inversion of low-frequency data is completed in

81 Velocity (m/s) 4000

a) TRT Velocity Tomogram

Depth (km)

0

0.5

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2000 0

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b) Ray Density

Depth (km)

0 10000 0.5

6000 2000

1 0

0.5

1

1.5

2 2.5 Location (km)

3

3.5

4

Figure 3.11. The initial velocity model used in the 2D model experiment is shown in a). This model was obtained using traveltime tomography. Its corresponding b) ray density diagram shows the number of rays passing through a model block. It also shows the areas in the model where waveform tomography will improve the inversion result using refraction energies. 2 hours on a Linux cluster with 100 AMD Opteron 1.8 GHz processors, while the high-frequency inversion is completed in 12 hours. The velocity tomogram obtained after the low-frequency inversion is shown in Fig. 3.12a. The tomogram obtained after both the low- and high-frequency inversions, the final inversion result from MWT, is shown in Fig. 3.12b. Fig. 3.12c illustrates the final inversion result of SWT after 45 iterations. Although the maximum number of iterations is set to 70, the inversion is stopped at iteration 45 due to no further decrease in misfit. This is likely indicative of local minima problems encountered by SWT. The final model from SWT in Fig. 3.12c shows an effect of the local minima problem, and the recovered model is much inferior to that obtained using MWT. The total runtime of SWT is 23 hours, longer than MWT by 64 %, without any additional benefit to the inversion result, as evident by comparing the SWT result to the MWT tomogram. I also compared relative data residuals from SWT and MWT where the relative data residual is defined by

82 Velocity (m/s)

Depth (km)

a) MWT Velocity Tomogram after Low−Frequency Inversion 0

4000

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Figure 3.12. Time-domain multiscale waveform inversion results for the 2D model: a) the MWT velocity tomogram obtained after the inversion using 5-Hz peak-frequency data, b) the MWT velocity tomogram obtained after the inversions using 5-Hz and 20-Hz peak-frequency data, c) the SWT velocity tomogram obtained after the inversion using 20-Hz peak-frequency data, d) the true velocity model.

Erelative =

||δp(rg , t|rs )|| × 100 , ||pobs (rg , t|rs )||

where δp(rg , t|rs ) = pobs (rg , t|rs ) − pcalc (rg , t|rs ), and ||p(rg , t|rs )|| =

XXZ s

g

p2 (rg , t|rs )dt.

(3.29)

83 The relative data-residual plot shown in Fig. 3.13 shows that MWT residual is lower than SWT residual by 16 %. Comparing the tomograms to the true model suggests that MWT converged to the global minimum while SWT did not. By successfully applying multiscale waveform tomography to a 2D model, and comparing the results to conventional, single-scale waveform tomography, it is clear that multiscale waveform tomography has more computational efficiency and a faster convergence rate than single-scale waveform tomography. 3.3.3

Canadian Foothills Synthetic Data

Seismic imaging in compressional thrust belts such as the Canadian Foothills is very challenging due to complex geological structures, rugged surface topog40 MWT SWT

Relative Data Residuals (%)

35 30 25 20 15 10 5 0 0

10

20

30 40 Iteration Number

50

60

70

Figure 3.13. Relative data-residual plot. The solid line shows the relative data residuals of MWT versus the iteration number. The residuals at the first 50 iterations are the residuals from the low-frequency inversion using data with a peak frequency of 5 Hz, and the last 20 residuals are from the high-frequency inversion using data with a peak frequency of 20 Hz. The dashed line shows the relative data residuals of SWT.

84 raphy, and highly-variable near-surface conditions. A thrust belt is a series of mountainous foothills that form due to compression. Seismic sections across the Canadian Foothills are usually progressively more distorted when approaching the Canadian Foothills region. Fig. 3.14 shows the degree of structural complexity and topographic variations which are in part responsible for the deteriorated imaging in the thrust belt. Accurate velocity models of subsurface structures are critical for improving seismic images of thrust belts in both the time domain (e.g., tomostatics) and the depth domain (e.g., prestack depth migration). Velocity estimation methods using refracted waves, e.g., refraction traveltime tomography, are usually ineffective because this energy is trapped by the complex near-surface structures and consequently has limited depth penetration. In contrast, waveform tomography which uses both refraction and reflection energies can accurately estimate both near-surface and deep velocity structures. Time-domain multiscale waveform tomography is used here in this work to investigate the possibility of velocity estimation in complex environments. Synthetic acoustic data were generated from a geologically realistic velocity model to test the potential of waveform tomography to reconstruct velocity models in complex areas. Results show that multiscale waveform tomography can provide an accurate and highly resolved velocity tomogram especially at depth. This test

Figure 3.14. An outcrop in the Canadian Foothills shows the rugged surface topography and complex near-surface structures.

85 also validates that waveform tomography can effectively handle rugged surface topography. The velocity model used to generate synthetic data for this study was based on actual surface geology, topography, seismic and well data from the Canadian Foothills (Fig. 3.15). Synthetic, long-offset, split-spread, acoustic data were generated from this model. A representative shot gather is shown in Fig. 3.16. A 20-Hz Ricker wavelet was used to generate 204 shots with 1601 receivers per shot. Sources and receivers are located 10 m below the topographic surface with source and receiver intervals of 100 m and 10 m, respectively. Each shot gather has a maximum offset of 8 km, and a total record length of 5 s. An initial model was obtained by combining a traveltime tomogram, which provides the near-surface velocity structure, and a 1D average of the true model (Fig. 3.17a). The refraction energy is trapped in the near-surface area and does not provide a good starting model for the deep region. Full-waveform data with 3 frequency bands (peak frequencies of 5 Hz, 10 Hz, and 20 Hz) were used in multiscale waveform tomography without any time window and the inversion results are shown in Fig. 3.17b-d. Compared with the true model (Fig. 3.15), the final tomogram (Fig. 3.17d) is accurate and highly resolved. Thin-layer structures in the near surface and the thrust-fold structures are accurately recovered.

m/s

True Velocity Model Depth (km)

0 6000 2 5000 4 6 0

4000 5

10 15 Horizontal Position (km)

20

3000

Figure 3.15. A Canadian Foothills velocity model used to generate acoustic synthetic data.

86

A Synthetic Shot Gather 0 0.5 1 1.5 Time (s)

2 2.5 3 3.5 4 4.5 5 −8

−6

−4

−2

0 Offset (km)

2

4

6

8

Figure 3.16. A synthetic shot gather generated by using the Canadian Foothills model.

3.4

Conclusions

I have improved a computational efficiency of a previously proposed method for time-domain waveform tomography by using a Wiener filter and the strategy for choosing optimal frequency bands. The Wiener filter applied in the experiments is one of nonleaky filters for efficient low-pass filtering for multiscale waveform tomography in the time domain. By applying the strategy for choosing optimal frequency bands, a small number of frequency bands is needed and unnecessary computational costs from using too many frequency bands are avoided. At low frequencies, forward modeling and inversion are very efficient because larger grid sizes and time steps can be used than at higher frequencies. By progressively recovering higher wavenumber components of the velocity structure, in a low to high fashion, the multiscale approach can improve the convergence property of

87

m/s

a) Initial Velocity Model Depth (km)

0 6000 2 5000 4 6 0

4000 5

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b) Reconstructed Velocity using 5−Hz Data Depth (km)

0 6000 2 5000 4 6 0

4000 5

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m/s

c) Reconstructed Velocity using 5−Hz and 10−Hz Data Depth (km)

0 6000 2 5000 4 6 0

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d) Reconstructed Velocity using 5−Hz, 10−Hz, and 20−Hz Data 0

3000

m/s 6000

2 5000 4 6 0

4000 5

10 15 Horizontal Position (km)

20

3000

Figure 3.17. Inversion results for the Canadian Foothills model. a) Initial velocity model. Waveform tomogram after the inversion using a) 5-Hz data, b) 10-Hz data, and c) 20-Hz data.

88 waveform tomography and partially overcome the problem of encountering local minima in the misfit function. The frequency-band selection strategy is validated in an experiment with a 1D model. By using the two frequency bands as calculated by my strategy, both low- and high-wavenumber components of the model are accurately recovered. This result agrees with previously published work for waveform inversion in the frequency domain. The strategy can be effectively used in a 2D heterogeneous model. Results from an experiment with a 2D model also suggest that the multiscale method is more computational efficiency than a conventional waveform tomography method and can converge to an accurate solution. Time-domain multiscale waveform tomography is successfully applied for velocity estimation in a synthetic, complex environment. Both near-surface and deep, complex structures are accurately recovered. These promising results show that waveform tomography can be a method of choice for velocity estimation in complex environments. In field data cases, there are additional challenges including random noise, surface waves, nonuniform source radiation, elastic effect, attenuation, and anisotropy, which make waveform tomography extremely difficult to work.

In

such circumstances, waveform tomography with early-arrival muting windows and intensive data preprocessing will be strongly required.

3.5

Future Work

To speed up the convergence of waveform tomography, a good initial model is crucial. I plan to apply quasi Monte Carlo methods to improve the accuracy of initial models from traveltime tomography.

Optimization methods are also

another factor that affects the convergence rate. Quasi-Newton methods will be implemented and tested to improve the current waveform tomography method. Waveform tomography is computationally expensive and requires a large number of CPUs to obtain results in reasonable time. I also plan to implement waveform tomography on fast GPUs (Graphics Processing Units).

89 Currently, waveform tomography is applied to only 2D data which may introduce some bias in the inversion results due to the out-of-plane effect of 3D wave propagation. I will extend waveform tomography for 3D applications.

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fixed-point computation and seismic waveform

fixed-point computation and seismic waveform

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