Explicit extrapolators and common azimuth migration

Transcription

1 Explicit extrapolators and common azimuth migration Eric Dussaud October 9th, 004

2 Agenda Downward continuation via explicit extrapolation. Design of explicit extrapolators. Design of 1-D convolution operators. Transformation filters. Extension to unequal sampling - Li correction. Application to common-azimuth migration. Conclusions and future work. 1

3 Explicit extrapolation (1/3) Downward continuation of zero-offset data: P z (k x,k y, z,ω) = ik z P(k x,k y, z,ω), k z = The analytical solution is given by: 4ω v k x k y. P(k x, k y,z + z,ω) = e ik z z P(k x,k y, z,ω) W(k x, k y, z, ω)p(ω,k x,k y,z). Inverse Fourier transform over k x and k y yields: p(x,y, z + z,ω) = w(x,y, ω) p(x,y, z, ω) = dx dy w(x, y, ω)p(x x,y y, z + z, ω)

4 Explicit extrapolation (/3) Lateral changes in v are accommodated by allowing w to vary laterally, i.e. write w(x, y, ω) w(x,y, x,y,ω) in the previous expression. In practice, substitute w with convolution operators f that are designed to fit the exact operator for a range of ω/v ratios. The operators f are optimized in such a way that their Fourier transform F over k x and k y approximate the exact phase-shift operator. 3

5 Explicit extrapolation (3/3) Major drawback of the method: computationally very expensive. Hale (1991): break the design procedure into two parts, computation of the coefficients of 1-D extrapolation filters that would be used for -D migration. computation of a small transformation filter that can be used to transform 1-D extrapolation operators into -D ones. Upshot: computational cost is greatly reduced, and the design is simplified. 4

6 Design of 1-D convolution operators Compute 1-D short convolution operators f(x, z,ω) with complex coefficients f n f n ( z, ω,v), and with a wavenumber spectrum: [ ] ω F(k x, z, ω) W(k x, z,ω) exp i z v k x, k x [0,k c ] The (discrete) Fourier transform of f(x, z,ω) is: F(k x, z,ω) (N 1)/ n=( N+1)/ f n e in xk x = f 0 + (N 1)/ n=1 f n cos (n xk x ). Optimization problem: find the coefficients f n so as to minimize F W. Do it for a range of ω/v ratios and store the coefficients in a file. 5

7 Principle of recursive extrapolation Courtesy: Jan Thorbecke. 6

8 Transformation filters (1/3) To transform 1-D filters into -D filters, use the fact that: W(k x,k y, z,ω) = W( k, z,ω), k = kx + ky Therefore: F(k x, k y, z, ω) f 0 + Use the Chebyshev recursion formula to write: (N 1)/ n=1 f n cos (n x k ) (1) cos(n x k ) = cos( x k ) cos [(n 1) x k ] cos [(n ) x k ] F(k x,k y, z,ω) (N 1)/ n=0 ˆf n cos n ( x k ) 7

9 Transformation filters (/3) To approximate cos( x k ), Hale (1991) suggested the following transform: ( ) cos x kx + ky G(k x,k y ) [1 + cos( xk x)] [1 + cos( xk y )] with c = c [1 cos( xk x)] [1 cos( xk y )] Denote by g(x,y) the corresponding 5x5 spatial stencil (obtained by IFT): c/8 0 c/4 0 c/8 0 1/8 1/4 1/8 0 g(x,y) c/4 1/4 (1 + c)/ 1/4 c/4 0 1/8 1/4 1/8 0 c/8 0 c/4 0 c/8 8

10 IDEAL HR9 HALE17 NORMALIZED CROSSLINE WAVENUMBER (CYCLES) NORMALIZED INLINE WAVENUMBER (CYCLES) Contours of constant amplitude and phase (constant k ) for the (scaled and shifted) Hazra & Reddy transformation compared to the improved McClellan transformation. 9

11 Transformation filters (3/3) The -D inverse Fourier transform of (1) is given by: N f(x, y, z, ω) = f 0 δ(x, y) + f n g n (x,y), where g 0 (x,y) δ(x, y), g 1 (x,y) g(x,y) and g n = g n 1 g g n, n > 1. n=1 The extrapolation step consists of convolving the wavefield with f(x,y, z, ω): p(x,y, z + z, ω) = f 0 p(x,y, z,ω) + N f n [g n p(x,y, z,ω)] n=1 10

12 0 CDP number Depth (km) IFP model: in-line section at constant cross-line coordinate y = 0m. 11

13 0 CDP number Depth (km) IFP model: migrated section using an explicit scheme. 1

14 0 CDP number Depth (km) IFP model: migrated section using GSP. 13

15 Extensions Accomodate unequal in-line and cross-line sampling intervals, following the idea set forth by Levin (1999, 004). Application of a correction filter (Li correction) to remove some of the phase error introduced by the Hale-McClellan tranformation filter (Etgen and Nichols, 1999). 14

16 0 CDP number Depth (km) Curve model: in-line section at constant cross-line coordinate y = 0m. 15

17 0 CDP number Two-way time (sec) 3 Synthetic data created with 101 cross-lines spaced 40m apart, and 401 in-lines spaced 5m apart. 16

18 0 CDP number Depth (km) Migrated section obtained using the explicit scheme. 17

19 0 CDP number Depth (km) Migrated section obtained using GSP. 18

20 0 CDP number Depth (km) Vertical slice through the migrated cube of an impulse response in a two-velocity medium. The operator used is a 5-point WLSQ operator, along with the Hale-McClellan transform. 19

21 0 CDP number Depth (km) Same result with the Li correction applied every 10 steps. 0

22 Application to common-azimuth migration Convenient restriction of the DSR operator for downward-continuing data that share a single azimuth (Biondi & Palacharla, 1996). After approximation with splitting, the dispersion relation associated with commonazimuth downward continuation is given by: ω ω 4ω k z = v s 1 4 (k m x k hx ) + v g 1 4 (k m x + k hx ) + } {{ } Convolution in x k vm m y }{{} Convolution in y ω v m 1

23 Downward-continuation step: Common-azimuth migration P(k mx,k my,k hx, z + z,ω) = D 1 (...)D (...)D 3 (...)P(k mx, k my, k hx,z, ω), with: ω D 1 (k mx,k hx, z, ω) = exp i z D (k mx,k hx, z, ω) = exp [ i z v s ω v g 4ω D 3 (k my, z, ω) = exp i z v m 1 4 (k m x k hx ) 1 4 (k m x + k hx ) ] km y ω v m

24 Implementation (1/) The operator D 3 is designed and implemented exactly as in the zero-offset case. Consider D 1. Setting k s = k mx k hx, we have: ω D 1 (k mx, k hx, z,ω) = exp i z 1 4 (k m x k hx ) = exp It can be approximated by the finite-length summation: v s D 1 (k mx, k hx, z,ω) d 0 + N h 1 n=1 where x represents the CMP in-line sampling interval. i z 4ω v s d n cos(n xk s ), () ks. 3

25 Implementation (/) The Chebyshev recursion can be used to write cos(n xk s ) in terms of an n-th order polynomial of cos( xk s ). Need to find spatial stencil corresponding to cos( xk s ). We have: G(k mx,k hx ) cos [ x (k mx k hx )] = cos ( xk mx ) cos ( xk hx ) + sin ( xk mx ) sin ( xk hx ) By inverse Fourier transform, we obtain: g(m x,h x ) = Note: the main anti-diagonal of the filter represents the impulse response of the cosk s filter, that is, the convolution is done in effect along the shot direction. 4

26 0 CMP In-line (km) Depth (km) 3 4 SEG-EAGE salt model: in-line section at constant cross-line coordinate y = 9, 80m. 5

27 0 CMP In-line (km) Depth (km) 3 4 Same section obtained using split COMAZ via the explicit scheme. 6

28 0 CMP In-line (km) Depth (km) 3 4 Same section obtained using COMAZ via GSP. 7

29 0 CMP Cross-line (km) Depth (km) SEG-EAGE salt model: cross-line section at constant in-line coordinate x = 7, 440m. 8

30 0 CMP Cross-line (km) Depth (km) 3 4 Same section obtained using split COMAZ via the explicit scheme. 9

31 0 CMP Cross-line (km) Depth (km) 3 4 Same section obtained using COMAZ via GSP. 30

32 Summary Broad overview of 3-D depth-extrapolation methods. Migration of common-offset common-azimuth data with an explicit scheme. Future work: extension to the full DSR operator? 31

33 Acknowledgments I would like to express my gratitude to: Henri Calandra and Biaolong Hua (Total) for their guidance and support thoughout the internship, and for allowing to show these results. William Symes (Rice University) for many useful suggestions and discussions and Jan Thorbecke (Delft University of Technology) for his help thoughout the summer. 3