Frequency domain seismic forward modelling: A tool for waveform ...

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$Fayek & Kyser 2000 uraninite fractionations.pdf - Queen's University$

The advantages of frequency domain schemes are utilised to the full extent in waveeld inversion approaches, in which only a few frequencies may be required (Pratt et al., 1995; Song et al., 1995); in this thesis I will deal exclusively with frequency-space domain methods and their application to inverse problems | hence the sub-title of this thesis, \A tool for waveeld inversion". In order to eciently invert eld data one needs an accurate and fast forward modelling technique. Computation time and accuracy normally tradeo against each other: Accuracy can usually be achieved by using very ne discretization grids, but this is costly in terms of computational resources. Sophisticated numerical methods achieve accuracy by optimized design of the numerical method, allowing the number of grid points to be reduced (for a given level of accuracy). In order to improve the performance of any forward modelling approach, the limitations of the particular scheme must be well understood. These limitations include those introduced by the original choice of the underlying wave equation (i.e., are we modelling acoustic or elastic waves, are we using one, two or three dimensions, are we accounting for viscous eects, etc), and those limitations caused by the numerical approximations (i.e., are the numerical methods suciently accurate). This understanding of the modelling method is important in order not to misinterpret the results, in order to build better models, and in order to be able to choose appropriate modelling techniques and modelling parameters. Simple frequency domain modelling codes, such as the one developed by Pratt (1989b) (in use when I started the project), are not suciently accurate to be able to handle large surveys. The software engineering problems associated with frequency domain forward modelling are rather dierent from those associated with time domain methods. Once the frequency domain equations are discretized, the solution (at a given frequency) is implicit in the solution of an extremely large matrix equation. The essential problem is to control the sparsity pattern of the matrix (itself controlled by the spatial extent of the dierencing operators), and to take full advantage of 23
this sparsity. As in time domain methods, however, the overiding concern is to limit the number of grid points per wavelength that are required. Thus, for frequency domain methods, it is critical to optimize the accuracy of the numerical operators, while also minimizing the spatial extent of these operators. In time domain forward modelling, accuracy can be achieved through the use of high order spatial operators; for large problems this is crucial. In frequency domain modelling, high order spatial nite dierence operators lead to large increases in computational costs that are not compensated for by the gain in accuracy (see chapters 2 and 3 for a full explanation). Instead we must seek other methods by which to increase the accuracy (and thereby reduce the costs). Two major improvements were implemented: The rst of these involved modications to speed up the numerical aspects of the matrix solver; the second involved modications to the numerical operators themselves. Together these two modications lead to a dramatic improvement in the computation times for acoustic wave equation modelling and inversion. These improvements were made use of in the inversion of a large, transmission seismic survey in which the use of extended parameter tests, involving a large number of modelling runs, was made possible by the improvements. During the inversion of the eld data, it became clear that the acoustic method needed to be replaced by an elastic method; in the nal stage of this project I developed extensions of the modelling methods to the elastic wave equation, paving the way for future elastic waveeld inversion methods. 1.3 Forward modelling in the frequency-space domain In this section the fundamental equations for seismic modelling in the frequencyspace domain are presented, and some of the basic considerations for frequencydomain methods are reviewed. I begin with the assumption that a particular wave equation has been selected for modelling purposes, and we have already discretized 24
Page 1 and 2: Frequency domain seismic forward mo
Page 3 and 4: Acknowledgements I would like to gr
Page 5 and 6: 2.5 Nested dissection ordering . .
Page 7 and 8: 6.1.4 Waveeld inversion . . . . . .
Page 9 and 10: 2.3 Two waydissected matrix S ~ = L
Page 11 and 12: 4.3 Two representative source gathe
Page 13 and 14: 4.15 Frequency domain modelled (pre
Page 15 and 16: 5.3 Numerical dispersion of the new
Page 17 and 18: Chapter 1 Introduction Modelling th
Page 19 and 20: Pratt (1995) and Song (1995) showed
Page 21 and 22: parameters is a daunting task. Extr
Page 23: element methods. 1.2 The signicance
Page 27 and 28: where the complex \impedance" matri
Page 29 and 30: 1.4 Fourier transforms and frequenc
Page 31 and 32: in order to unambiguously reconstru
Page 33 and 34: If the forward Fourier transform is
Page 35 and 36: ased on data setfrom an underground
Page 37 and 38: therefore one of the main applicati
Page 39 and 40: 2.3 Solving linear equation systems
Page 41 and 42: are always calculated by the time t
Page 43 and 44: Case 2 2 S = ~ 6 4 4 0 0 0 .5 0 .79
Page 45 and 46: Figure 2.2: Two-way dissected nite
Page 47 and 48: (n/2*n/2)/2 L 55 (n/2*n/2)/2 n*n/2
Page 49 and 50: (to within anorder of magnitude) ne
Page 51 and 52: D 4 (n; 0) (2n) 2 =2 + 2(2n=2) 2 =
Page 53 and 54: The memory required to perform LU d
Page 55 and 56: are of theorder of hundereds of kil
Page 57 and 58: 10000 1000 Time (s) Sequential orde
Page 59 and 60: Chapter 3 Visco-acoustic frequency
Page 61 and 62: Figure 3.1: Finite dierence operato
Page 63 and 64: However, analytical solutions do no
Page 65 and 66: 3.2.4 Lumped and consistent matrix
Page 67 and 68: +4c=1 (3.9) which makes the minimiz
Page 69 and 70: that the required memory is afuncti
Page 71 and 72: 1996). However quantitativeevaluati
Page 73 and 74: (1991). The low velocity regions (1
Page 75 and 76:
a) Time slice at 5s b) Time slice a
Page 77 and 78:
domain approach would have to gener
Page 79 and 80:
1988). Reviews have been provided b
Page 81 and 82:
Figure 4.1: Grimsel Pass areal phot
Page 83 and 84:
5 10 15 20 25 30 35 40 45 50 55 60
Page 85 and 86:
160m BOUS 85.003 BOBK 85.004 BOBK 8
Page 87 and 88:
will assume that rst n r n x n z (
Page 89 and 90:
where jxj represents represents the
Page 91 and 92:
or c J~ = S ~ ,1 G ~ (4.13) where F
Page 93 and 94:
Figure 4.6: Comparison of the trave
Page 95 and 96:
In order to initialize the waveeld
Page 97 and 98:
the trace-to-trace amplitude variat
Page 99 and 100:
manner these groups were identied.
Page 101 and 102:
km/s 4.80 4.85 4.90 4.95 5.00 5.05
Page 103 and 104:
4.6.2 Regularization tests From thi
Page 105 and 106:
RMS Roughness 0.8 0.7 0.6 0.5 5 10
Page 107 and 108:
Figure 4.15: Frequency domain model
Page 109 and 110:
y the best isotropic results. In al
Page 111 and 112:
a) b) c) Figure 4.19: Data residual
Page 113 and 114:
Figure 4.20: Anisotropic full wavee
Page 115 and 116:
km/s 4.80 4.85 4.90 4.95 5.00 5.05
Page 117 and 118:
Figure 4.24: Final waveeld inversio
Page 119 and 120:
can eect theimages (the wrong theor
Page 121 and 122:
In essence, our new scheme is the e
Page 123 and 124:
where ! = 2f is the angular frequen
Page 125 and 126:
dicult to formulate direct solvers
Page 127 and 128:
! 2 u 0 + v 0 + ( +2) + ( + ) @ u
Page 129 and 130:
\stars": ! @ 2 1 0 0 1 0 -2 0 @x 0
Page 131 and 132:
free to vary from one node point to
Page 133 and 134:
(see equations 5.24 and 5.25), exce
Page 135 and 136:
1 1 0.8 0.8 a 0.6 0.4 b 0.6 0.4 0.2
Page 137 and 138:
Numerical dispersion curves for σ=
Page 139 and 140:
Ifollow closely the dispersion anal
Page 141 and 142:
the opposite results. The fraction
Page 143 and 144:
identied; it is also possible to id
Page 145 and 146:
Real data Acoustic modelling result
Page 147 and 148:
observed data. However, even with t
Page 149 and 150:
Chapter 6 Conclusions and further w
Page 151 and 152:
compensated for by the accuracy gai
Page 153 and 154:
(1992)). I have also shown how sens
Page 155 and 156:
Extensions to more complex cases Th
Page 157 and 158:
Although waveeld inversions have be
Page 159 and 160:
to nd more dicult targets. An incre
Page 161 and 162:
Bathe, K. J., and Wilson, E. L., 19
Page 163 and 164:
Cole, J. B., 1994, A nearly exact s
Page 165 and 166:
Lailly, P., 1984, Migration methods
Page 167 and 168:
Peng, C., Cheng, C. H., and Toksoz,
Page 169 and 170:
Smith, W. D., 1974, The application
Page 171 and 172:
Appendix A Dispersion analysis for
Page 173:
and 2 = ( 1 R 2 + 2 )( 1 + 2 R 2
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