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

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

that may fail to adequately modelthewaveforms in complex media. Asymptotic ray theory assumes a high-frequency wave behaviour; this puts certain constraints on the model complexity asa function of the lowest wavelength. If velocity discontinuities are reached, explicit boundary conditions must be applied in order to divide the ray into reected and transmitted (i.e., refracted) rays, each of which are further traced through the model. The high frequency restriction limits the use of the technique to simple models with relatively few data phases, usually specied in advance. The second group of modelling methods comprise the numerical methods based on partial dierential or integro-dierential wave equations, without the use of a high frequency approximation. These methods are usually formulated as nite dierence or nite element problems. Such wave equation methods equation guarantee the simulation of all possible phases (within the assumptions built into the initial wave equation). The generation of mode conversions, reections and refractions is not determined by the choice of input parameters (as in asymptotic ray theory), but is instead an integral feature of the modelling itself. [An exception to this are the numerical methods of (Madariaga, 1984), based on matrix propagator methods. However these methods are usually only available for 1-D models]. As a result, relating the phases in the seismic record to individual features in the model may not be straightforward in complex models. Wave equation methods can be further sub-divided intoanumber of classes, depending on the domain in which the initial wave equation is solved. Possible choices of domain include any combination of time/frequency, space/wavenumber or other domains, such as the , p transform domain. Each domain has its own advantages and disadvantages. For 2-D earth models, time domain methods have dominated the literature. In contrast, this thesis will be largely concerned with numerical modelling in the frequency-space domain. The primary reason for this is that the modelling algorithm is tightly coupled to a formal method for the automatic, frequency-space domain inversion of seismic waveform data. The results obtained by 17
Pratt (1995) and Song (1995) showed the great potential of frequency-space domain waveeld inversion. Unfortunately, the size of the geological experiments in which these techniques could potentially be applied were limited by the ineciency of the forward modelling technique. The overall objective of the project described in this thesis was to develop improved forward modelling methods and to incorporate these into frequency-space inverse methods, thereby increasing the maximum model size that can be handled in these methods. 1.1 Historical overview Seismologists began using wave equation based numerical methods in the late 1960's. Most of this inital work was based on nite dierence techniques. Numerous discrete solutions for the second order wave equation in homogeneous regions by use of explicit time integration were published (Alterman and Kornfeld, 1968; Alterman and Karal Jr, 1968; Alterman and Rotenberg, 1969; Alterman and Loewenthal, 1970). In Alterman's work, a homogeneous wave equation formulation was used and interfaces were treated using explicit boundary conditions. This early work was only of limited value due to the limited computational resources available at the time, and due to the limitations on model complexity due to the necessity of treating each interlayer boundary explicitly. Nevertheless, these experiments produced a deep theoretical understanding of wave propagation in homogeneous and two layer media, and proved that a numerical methods were useful in the innitely many earth modesl for which no analytical solution is available. Today exploration geophysicists attempt to model much more complex, realworld media that include irregular boundaries and laterally varying model parameters in all directions. In order to predict the response in such cases, the interlayer boundary conditions had to be built implicitly into the modelling scheme. It become common practice in the mid 1970's to use a heterogeneous wave equation formulation 18
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: Chapter 1 Introduction Modelling th
Page 21 and 22: parameters is a daunting task. Extr
Page 23 and 24: element methods. 1.2 The signicance
Page 25 and 26: this sparsity. As in time domain me
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
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1988). Reviews have been provided b
Page 81 and 82:
Figure 4.1: Grimsel Pass areal phot
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5 10 15 20 25 30 35 40 45 50 55 60
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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
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km/s 4.80 4.85 4.90 4.95 5.00 5.05
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Figure 4.24: Final waveeld inversio
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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
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\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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