The FOCI method of depth migration. Gary Margrave Saleh Al-Saleh Hugh Geiger Michael Lamoureux POTSI

Transcription

1 a The FOCI method of depth migration Gary Margrave Saleh Al-Saleh Hugh Geiger Michael Lamoureux POTSI

2 Outline Wavefield extrapolators and stability A stabilizing Wiener filter FOCI Dual operator tables Spatial resampling Post stack testing Pre stack testing

3 Wavefield Extrapolators The phase-shift extrapolation expression 1 ikzz ikt xt ψ x, z, ω ϕ k, z 0, ω e i e dk ( ) = ( = ) n 1 ( ) n 1 2π T T T ψ( x z ω) output wavefield T,, ϕ kt, z= 0, ω ( ) Fourier transform of input wavefield

4 Wavefield Extrapolators The phase-shift operator ik z e ˆ z = W z n ( ) k z ω > v v = i k > v ω 2 k, 2 T k 2 T ω 2 ω T, k 2 T 2 v wavelike evanescent

5 Wavefield Extrapolators The PSPI extension k z ik z e ˆ z = W z ( ) ω > v( x ) ( ) = i k > n ω 2 k, 2 T k 2 T T v xt ω 2 ω T, k 2 T 2 v( x ) ( ) T v xt

6 Wavefield Extrapolators Wˆ2 ( z) ω v wavenumber + ω v

7 Wavefield Extrapolators phase Wˆ2 ( z) ω v wavenumber + ω v

8 Wavefield Extrapolators In the space-frequency domain ψ( x, z, ω) = ψ( xˆ, z= 0, ω) W ( x xˆ, z, v, ω) dxˆ n 1 T T n T T T ( ω) ( 2π) where 1 ˆ ( ˆ ) W x x z W k z e dk ( ω) ikti xt xt n T ˆ T,, =,, n 1 n 1 n T T

9 Wavefield Extrapolators In the space-frequency domain imaginary real meters

10 Wavefield Extrapolators Back to the wavenumber domain F ΩW ( ) 2 Wˆ 2 wavenumber

11 Wavefield Extrapolators Back to the wavenumber domain wavenumber

12 Stabilization by Wiener Filter Two useful properties ˆ ( ) ˆ z,,,, ˆ z Wn kt z ω = Wn kt ω W n kt,, ω 2 2 Product of two half-steps make a whole step. Wˆ k z Wˆ k z k k 1 ( ) * ( ) 2 2 n T,, ω = n T,, ω, > x The inverse is equal to the conjugate in the wavelike region.

13 Stabilization by Wiener Filter A windowed forward operator for a half-step W z W z n ( 2) = Ω ( 2) n Solve by least squares W ( z 2) WI F 1 ˆ = W ( z 2) n n n 0 η 2 η

14 Stabilization by Wiener Filter WI n is a band-limited inverse for Both have compact support Wn ( z/2) Form the FOCI approximate operator by W z = WI * W z W z nf n n n ( ) ( 2) ( ) FOCI is an acronym for Forward Operator with Conjugate Inverse.

15 Linear System to Solve W W W 0 D 2,0 2,1 2, p 2,0 W2,1 W2,0 W2,1 0 WI 2,0 D 2,1 W 2,1 W 2,0 W 2, p WI 2,1 D 2,2 W 2, p W 2,1 = 0 W 2, p W 2,1 W WI 2, m 2,0 D 0 2, m + p

16 Properties of FOCI operator n inv = length( WI ) n length W ( z /2) n Let for = ( ) n Then ( ( )) = = + 1 length W z n n n nf op for inv

17 Properties of FOCI operator n for determines phase accuracy. ninv determines stability. Empirical observation: n inv 1.5n for

18 Properties of FOCI operator Amount of evanescent filtering is inversely related to stability η 0 no evanescent filtering ( 1000 steps) = 1 half evanescent filtering ( 100 steps) 2 full evanescent filtering ( 50 steps)

19 Operator tables Since the operator is purely numerical, migration proceeds by construction of operator tables. kmin k k min + k + k min 2 WnF nf k ( ) min W k + k ( ) min W k + k nf ( ) min 2 k max WnF k ( ) max k min = ω v min max k = ω mean v () k max = ω v max min

20 Operator tables We construct two operator tables for small and large η. The small η table is used most of the time, with the large η being invoked only every n th step. η 0 no evanescent filtering ( 1000 steps) = 1 half evanescent filtering ( 100 steps) 2 full evanescent filtering ( 50 steps)

21 Operator Design Example

22 Improved Operator Design

23 Spatial Resampling Frequency ω max Wavelike ω ( 2 x) 1 kx = v min Wavenumber ω k = x v min ( 2 x) 1

24 Spatial Resampling Frequency ( 2 x) 1 Wavenumber ( 2 x) 1 In red are the wavenumbers of a 7 point filter

25 Spatial Resampling Frequency ( 2 x) 1 ( ) 1 2 x ( ) 1 2 x ( 2 x) 1 Downsampling for the lower frequencies uses the filter more effectively

26 Spatial Resampling Frequency ( 2 x) 1 ( 2 x) 1 Spatial resampling is done in frequency chunks.

27 Operator in Space

28 Improved Operator in Space

29 Run Time Experiment

30 Run Times log-log Scale PS slope 1.07 FOCI slope 1.03 FOCI slope 1.05 Slopes for last three points

31 Phase Shift Impulse Response

32 FOCI Impulse Response nfor=7, ninv=15, (21 pt), no spatial resampling

33 FOCI Impulse Response nfor=7, ninv=15, (21 pt), with spatial resampling

34 Marmousi Velocity Model

35 Exploding Reflector Seismogram

36 FOCI Post-Stack Migration nfor=21, ninv=31, nwin=0

37 FOCI Post-Stack Migration nfor=21, ninv=31, nwin=51

38 FOCI Post-Stack Migration nfor=21, ninv=31, nwin=21

39 FOCI Post-Stack Migration nfor=7, ninv=15, nwin=0

40 FOCI Post-Stack Migration nfor=7, ninv=15, nwin=7

41 Marmousi Velocity Model

42 FOCI Pre-Stack Migration nfor=21, ninv=31, nwin=0, deconvolution imaging condition

43 FOCI Pre-Stack Migration nfor=21, ninv=31, nwin=0, deconvolution imaging condition

44 FOCI Pre-Stack Migration Shot 30

45 FOCI Pre-Stack Migration Shot 30

46 FOCI Pre-Stack Migration Stack +50*Shot 30

47 Detail of Pre-Stack Migration

48 Marmousi Reflectivity Detail

49 Marmousi Velocity Model

50 FOCI Pre-Stack Migration Velocity model convolved with 200m smoother

51 FOCI Pre-Stack Migration Velocity model convolved with 400m smoother

52 FOCI Pre-Stack Migration Exact Velocities

53 FOCI Pre-Stack Migration Exact Velocities

54 FOCI Pre-Stack Migration Velocities smoothed over 200m

55 FOCI Pre-Stack Migration Velocities smoothed over 400m

56 Detail of Pre-Stack Migration Exact Velocities

57 Detail of Pre-Stack Migration Velocities smoothed over 200m

58 Detail of Pre-Stack Migration Velocities smoothed over 400m

59 Run times Full prestack depth migrations of Marmousi on a single 2.5GHz PC using Matlab code. 20 hours for the best result 1 hour for a usable result

60 Conclusions Explicit wavefield extrapolators can be made local and stable using Wiener filter theory. The FOCI method designs an unstable forward operator that captures the phase accuracy and stabilizes this with a band-limited inverse operator. Reducing evanescent filtering increases stability. Spatial resampling increases stability, improves operator accuracy, and reduces runtime. The final method appears to be ~O(NlogN). Very good images of Marmousi have been obtained.

61 Research Directions Better phase accuracy. Extension to 3D. Extension to more accurate wavefield extrapolation schemes. Migration velocity analysis. Development of C/Fortran code on imaging engine.

62 Acknowledgements We thank: Sponsors of CREWES Sponsors of POTSI NSERC MITACS PIMS A US patent application has been made for the FOCI process.

63 Detail of Pre-Stack Migration

64 FOCI Pre-Stack Migration nfor=21, ninv=31, nwin=0, deconvolution imaging condition

65 FOCI Pre-Stack Migration nfor=21, ninv=31, nwin=0, deconvolution imaging condition