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

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

elds by explicitly dening each back scattered event, but reverberations and surface waves travelling perpendicular to the paraxial direction cannot be modelled at all. Even with this serious disadvantage, the approach has been very popular as a migration algorithm (Claerbout and Doherty, 1972; Loewenthal et al., 1976; Berkhout and Van Wulten Palthe, 1979; Berkhout, 1985), since in post-stack migration the propagation is required in only one direction (down), and the computational costs are much lower than full wave equation modelling. Full wave equation methods, however have been used extensively in migration from the late 70's (Hemon, 1978; Beysal et al., 1983; Loewenthal and Mufti, 1983). In many disciplines the nite element formulation is the primary choice of numerical method. However, seismic modelling the nite element method has never taken over from nite dierences as a main stream technique. Although the earliest papers on seismic modelling used the nite element method, (Smith, 1974), the essential diculty remains with us today: The lack of a mesh generator which will utilise the full advantage of nite elements, distorting the grid where possible and still providing a sucient number of node points for accurate wave equation modelling. There is another reason why nite element seismic modelling is not used more often: Since wave propagation problems demand that the model be sampled at a very ne scale, using exact, irregular boundaries will not signicantly aect the result. In most practical cases the knowledge of the model itself is only known at a relatively long scale length, much coarser than the model parametrization, so that exact boundaries cannot be dened. Finite elements may have certain advantages in the case of theoretical, simple models in which only a limited number of homogenous regions represent the model and an exact solution is required, but in applied cases where the model is highly heterogeneous and the shape of the \homogeneous" elements is not known the main advantages of nite elements appear to be of little use. The main nite element work is still on square (rectangular) grids. In this case there is no particular advantage of using either the nite dierence or the nite 21
element methods. 1.2 The signicance of the frequency-space domain It is clear from the review given above that frequency domain methods are less common than time domain methods in seismic wave propagation modelling. An early exception was (Lysmer and Drake, 1972), and the fundamental advantages of frequency domain modelling (especially for multi-source inverse problems) was clearly pointed out by Marfurt (1984a; 1984b), by Marfurt and Shin (1989) and by Pratt (1989a). Time domain methods are suitable if the full time domain seismic section for a single source, or for a small number of sources is required. On the other hand, frequency domain methods are ecient in cases in which a limited number of single frequency data are required, or in cases in which a time response for a large number of sources is required. As I will show in this thesis, in these circumstances a frequency domain approach can produce results at a fraction of the computational costs required by time domain schemes. Recently, much attention has been focussed on the modelling of seismic waves in models that include visco-elastic losses. Because the attenuation due to viscoelastic losses is thought to be related to time dependent creep and relaxation effects (see for example Kjartansson (1979)) that lead to integral terms in the wave equation, special techniques are required to include these eects into time domain numerical simulations (Emmerich and Korn, 1987; Carcione et al., 1988; Robertsson et al., 1994). A solution to the diculty of the representation of the integral terms in the time dependent wave equation is to transform the equations into the frequency domain, and model the resultant Helmholtz type equations, in which case frequency dependent attenuation can be easily represented by complex valued elastic parameters (Muller, 1983), without any additional computational eort. 22
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: parameters is a daunting task. Extr
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
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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