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

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

small real number. This computation is easily implemented in frequency domain modelling, and it has the advantage of yielding, after the inverse DFT, the time function 1X n=,1 h(t + nt max )e ,(t+ntmax) : (1.12) Thus, by using a complex value for the frequency, the time function has been eectively multiplied by a decaying exponential function. Each successive alias component of the time function is multiplied by a smaller value. To recover an approximation of the original, desired function we multiply this result by e t and produce the result 1X n=,1 h(t + nt max )e ,ntmax : (1.13) For the orginal, unaliased component (n = 0), the original signal is recovered. For all positive values of n, the signal is attenuated by an ever smaller factor { the aliased signal is still there, but it is attenuated. The method fails if the time series has non-zero values for negative times (n
If the forward Fourier transform is dened by H(!) = 1 ,1 h(t)e i!t dt, then Z 1 ,1 h(t , x=c)e,i!t dt = Z 1 ,1 h()e,i!(t+x=c) d = e ,i!x=c Z 1 ,1 h()e ,i! d = e ,i!x=c H(!): (1.15) The quantity x=c is a simple time shift, and thus we make use of the shifting property of the Fourier transform (in which the dummy integration variable is changed from t to ). Equation (1.15) shows that we can move the time window of interest by multiplying the frequency domain result by e ,i!x=c before performing the inverse DFT. This is an important result in frequency domain modelling, since we often need to simulate the seismic time domain response at far oset receivers in large experiments. Let us take for example a source-receiver oset of 300 km. In this case the full wave propagation time can be up to 30 to 40 seconds, while the required signal may be only a few seconds long. Due to the shifting property above, we have the option in frequency domain modelling to compute only a ve second time window, from, e.g., 35 to 40 seconds. We do this by setting the time shift equal to 35 seconds in equation (1.15). A ve second window can be completely represented using a frequency sample interval of f =1=5=0:2 Hz. This should be compared with the frequency sample interval required for the full 40 second record, f =1=40 = 0:025 Hz. The number of samples required for a given maximum frequency is reduced by 87.5%. In the time domain, it would be necessary to simulate the full 40 seconds in order to generate the same, nal, ve seconds of useful data. In this way we may directly calculate the time domain data in reduced time, in order to decrease the number of frequencies required for forward modelling. If reduced time output is used in conjunction with anti time-aliasing, it is important to ensure that the rst output time sample occurs before the rst data arrival. This is becaues non-zero signal arriving before the desired time window begins will be 32
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 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: in order to unambiguously reconstru
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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