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

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

3.2 Numerical dispersion curves for frequency domain acoustic forward modelling using ordinary second order nite dierence operators. a) Phase velocity dispersion. b) Group velocity dispersion. . . . . . . . 61 3.3 Numerical dispersion curves for frequency domain acoustic forward modelling using rotated nite dierence operators. a) Phase velocity dispersion. b) Group velocity dispersion. . . . . . . . . . . . . . . . 65 3.4 Dierence between the numerical velocity produced with and without the additonal coecient, d. a) Dierence in group velocity. b) Dierence in phase velocity. See text for detail explanation. . . . . . 67 3.5 a) Model used for wide angle forward modelling, from McCarthy et al. (1991). b) c) and d) The shaded regions depict the size of the models that one could simulate without nested dissection and/or rotated nite dierences if the same equipment were used. . . . . . . . . . . 71 3.6 a) Synthetic data section from the model on gure 3.5. b) Common shot gather from the eld data. c) One of the models suggested by McCarthy et al. (1991) showing the ray paths used in their modelling approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.7 Time slices generated by forward modelling true the model on Figure 3.5(a) at 5, 10, 15, 20, 25 and 30 seconds. . . . . . . . . . . . . . . . 74 4.1 Grimsel Pass areal photo. . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Inside of the Grimsel Rock labaratory tunnel. . . . . . . . . . . . . . 80 9
4.3 Two representative source gathers of VSP data from Field 2, astrue amplitude displays. a) A VSP source gather with large oset. The spurious variation of amplitude from trace to trace is evident, as is the consistency of alternate traces. The data were recorded in two passes, with intermediate traces recorded during a later, \in-ll" survey. b) A near oset VSP source gather, on which the dramatic change in amplitude with receiver depth is evident. These variations in amplitudes cannot be modelled using the 2D acoustic method. In order to invert these data I apply a normalization to each trace separately. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 A representative common receiver gather of the Field 2 data, following windowing and trace normalization. The receiver was in borehole 3. The rst portion of the gather was recorded with sources in borehole 2, and thus represents a portion of the cross borehole data. The second section was recorded with sources in the tunnel, and thus represents a portion of the VSP data. The data have been windowed and tracenormalized. The random static shifts in the cross borehole data, and the systematic static shifts in the VSP data are evident. The labels indicate the VSP source groups that were identied, in order to solve for the source consistent static shifts. . . . . . . . . . . . . . . . . . 83 4.5 Map of the Field 2 study area at the Grimsel Test Site. The seismic data were acquired using the tunnel and boreholes BOUS85.002 and BOUS85.003 (\boreholes 2 and 3"). The remaining boreholes are exploratory boreholes in which velocity information is available and is used to test the waveform images. The scale of this map is 1:1000, a representative square area 160m 160m is shown for reference. . . 84 10
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: 2.3 Two waydissected matrix S ~ = L
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 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
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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
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1996). However quantitativeevaluati
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(1991). The low velocity regions (1
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a) Time slice at 5s b) Time slice a
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domain approach would have to gener
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1988). Reviews have been provided b
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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
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will assume that rst n r n x n z (
Page 89 and 90:
where jxj represents represents the
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or c J~ = S ~ ,1 G ~ (4.13) where F
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Figure 4.6: Comparison of the trave
Page 95 and 96:
In order to initialize the waveeld
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the trace-to-trace amplitude variat
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manner these groups were identied.
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km/s 4.80 4.85 4.90 4.95 5.00 5.05
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4.6.2 Regularization tests From thi
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RMS Roughness 0.8 0.7 0.6 0.5 5 10
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Figure 4.15: Frequency domain model
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y the best isotropic results. In al
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a) b) c) Figure 4.19: Data residual
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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
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In essence, our new scheme is the e
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where ! = 2f is the angular frequen
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dicult to formulate direct solvers
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! 2 u 0 + v 0 + ( +2) + ( + ) @ u
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\stars": ! @ 2 1 0 0 1 0 -2 0 @x 0
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free to vary from one node point to
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(see equations 5.24 and 5.25), exce
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1 1 0.8 0.8 a 0.6 0.4 b 0.6 0.4 0.2
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Numerical dispersion curves for σ=
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Ifollow closely the dispersion anal
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the opposite results. The fraction
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identied; it is also possible to id
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Real data Acoustic modelling result
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observed data. However, even with t
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Chapter 6 Conclusions and further w
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compensated for by the accuracy gai
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(1992)). I have also shown how sens
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Extensions to more complex cases Th
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Although waveeld inversions have be
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to nd more dicult targets. An incre
Page 161 and 162:
Bathe, K. J., and Wilson, E. L., 19
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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,
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Smith, W. D., 1974, The application
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Appendix A Dispersion analysis for
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and 2 = ( 1 R 2 + 2 )( 1 + 2 R 2
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