1. Efficient Multiscale Waveform Tomography and Flooding Method* * *
C. Boonyasiriwat, P. Valasek, P. Routh, B. Macy,
W. Cao, and G. T. Schuster
* ConocoPhillips

2. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Multiscale Waveform Tomography: Efficiency
Results
Conclusions 1

3. Introduction: Waveform Tomography
No high frequency approximation.
Frequency domain: Pratt et al. (1998), etc.
Time domain: Mora (1987, 1989); Zhou et al. (1995), Sheng et al. (2006), etc.
Pratt and Brenders (2004) and Sheng et al. (2006) used early-arrival wavefields.
Bunks et al. (1995) and Pratt et al. (1998) used multiscale approaches. 9

4. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Multiscale Waveform Tomography: Efficiency
Results
Conclusions

5. Why Acoustic? Elastic wave equation is expensive.
Waveform inversion is also expensive.
Previous research shows acoustics is adequate.
Use acoustics and mute unpredicted wavefields. 11

6. Theory of Waveform Tomography
An acoustic wave equation:
The waveform misfit function is 12

7. Theory of Waveform Tomography
The waveform residual is defined by
The steepest descent method can be used to minimize the misfit function: 13

8. Theory of Waveform Tomography
The gradient is calculated by
where 14

9. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Multiscale Waveform Tomography: Efficient Smoothing
Results
Conclusions

10. Why Use Multiscale? Low Frequency Coarse Scale High Frequency
Model parameter (m)
Misfit function ( f )
Low Frequency
Coarse Scale
High Frequency
Fine Scale
Image from Bunks et al. (1995) 16

11. Our Multiscale Approach
Combine Early-arrival Waveform Tomography (Sheng et al., 2006) and a time-domain multiscale approach (Bunks et al., 1995).
Use a Wiener filter for low-pass filtering the data.
Use a window function to mute all energy except early arrivals.
Use multiscale V-cycles. 17

12. Why a Wiener Low-Pass Filter?
Original Wavelet
Target Wavelet
Hamming
Blackman
Wiener
Wavelet: Hamming Window
Wavelet: Wiener Filter
Lower frequencies, therefore dx, dt coarser 18

13. Multiscale V-Cycle High Frequency Fine Grid Low Frequency Coarse Grid19

14. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Multiscale Waveform Tomography: Efficient Smoothing & Frequency Band Selection
Multiscale Waveform Tomography: Efficient Smoothing
Results
Conclusions

15. Reflection Wavenumber Resolution Notation
G(x|s)G(x|g)= exp(i )=exp(i )exp(i ) s g xo
Parallel to t D
and length = w/c gx kg ks
Parallel to t D
and magnitude = w/c sx
txg= tx0g + dx txog D
Expand txs around xo
txs= tx0s + dx txos D
kz = kg + ks
txs+ txg = dx txos D)
+txos+ txog
+ txog
( D ks
+ kg w
kx= 0
Multiply by w
w(txs+ txg)
w(txs+ txg ) = dx (ks+kg)
+w(txos+ txog )
dx (ks+kg)
w(txos+ txog )

16. Reflection Wavenumber Resolution
d(g|s) = ∫G(x|s)G(x|g)m(x)dx
= ∫exp(iw[txs+txg])m(x)dx
= exp(iw[txos +txog ]) ∫exp(i[ks +kg ] dx) m(x)dx
(ignore geometrical spreading) s s g g
This is in form of a Fourier transform of m(x).
Therefore, d(g|s) ~ model spectrum M(k), i.e., after dividing d(g|s) by phase factor we get
kg=k(sina,cosa)
g=(xg,0); xo=(xo,zo)
sina=(xo-xg)/√(xo-xg)2+zo2
M((ks+kg)w/c)=F(m(x))=d(g|s) ^ xo kg ks
The data are Fourier
transform of model, so
inverse Fourier transform
of data is model.
ks=k(sina,cosa)
s=(xs,0); xo=(xo,zo)
sina=(xo-xs)/√(xo-xs)2+zo2
kz = kg + ks
kg + ks
Inverse transform of model spectrum/data=model
m(x)= ∫G(x|s)*G(x|g)*d(x|g)dg
A specified w and g -s pair will
reconstruct part of
wavenumber spectrum in model
= ∫exp(-i[ks +kg ] dx) d(g|s)dg

17. Wavenumber vector k determines spectrum M(kx,kz)
m(x)= ∫exp(-i[ks +kg ] dx) d(g|s)dg rs rg r
Source
Geophone k

18. Spatial Resolution Limit Formulag s
Dx = minimum separation of two
points so that they are distinguishable in image

19. Spatial Traveltime Resolution Limit Formula
source-receiver pairs where
the wavepath visits rx

20. Reflection Wavenumber Resolution
M((ks+kg)w/c)=F(m(x))=d(g|s) ^ s g kz c
= 2w kg
( |kg|=w/c) ks
( |ks|=w/c) w kz
kz = kg + ks

21. Reflection Wavenumber Resolution
M((ks+kg)w/c)=F(m(x))=d(g|s) ^ s g kz c
= 2w
( |kg|=w/c) kg ks
( |ks|=w/c) kz
kz = kg + ks w

22. Reflection Wavenumber Resolution
M((ks+kg)w/c)=F(m(x))=d(g|s) ^ s g kz c
= 2w
( |kg|=w/c) kg ks
( |ks|=w/c) kz
kz = kg + ks w

23. Reflection Wavenumber Resolution
M((ks+kg)w/c)=F(m(x))=d(g|s) ^ a
cos(a)=L/( L +z ) 2 L s g z kg ks
2|k|cos(a) kz c
= 2w
kz = 2|w/c|cos(a)
kz = 2|w/c|cos(a) kz w

24. Reflection Wavenumber Resolution
Efficient Strategy for Choosing w
(Sirgue&Pratt, 2004)
(Single Frequency Strategy: Sirgue&Pratt, 2004)
Just 2 frequencies & many offsets fill
the kz line; and therefore reconstruct
M(kx,kz) for wide range kz kz
We get this range of
kz’s for model, no need to
be redundant
In comparison, sloppy fine grid
of ws will require lots of CPU time
to cover same range of kz w
Choose next w w
Multiscale W(w)
(Multiscale x-t Strategy: Boonyasiriwat et al., 2004)
Wavelet Spectrum
½ power

25. Reflection Wavenumber Resolution
Efficient Strategy for Choosing w
(Sirgue&Pratt, 2004)
(Single Frequency Strategy: Sirgue&Pratt, 2004)
Just 2 frequencies & many offsets fill
the kz line; and therefore reconstruct
M(kx,kz) for wide range kz kz
We get this range of
kz’s for model, no need to
be redundant
In comparison, sloppy fine grid
of ws will require lots of CPU time
to cover same range of kz w
Choose next w w
Multiscale W(w)
(Multiscale x-t Strategy: Boonyasiriwat et al., 2004)
Wavelet Spectrum
½ power

26. Efficient Strategy for Choosing w
(Boonyasiriwat et al., 2007)

27. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Efficient Multiscale Waveform Tomography
Results: EWT vs MWT
Conclusions 1

28. Layered Model with Scatterers22

29. Initial Velocity Model23

30. TRT Tomogram
Gradient 24

31. EWT Tomogram using 15-Hz Data
Gradient 25

32. MWT Tomogram using 2.5-Hz Data
Gradient 26

33. MWT Tomogram using 5-Hz Data27

34. MWT Tomogram using 10-Hz Data28

35. MWT Tomogram using 15-Hz Data29

36. Layered Model with Scatterers30

37. Comparison of Misfit Function
15 Hz
15 Hz
5 Hz
10 Hz
2.5 Hz 31

38. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Efficient Multiscale Waveform Tomography
Results with Flooding
Conclusions 1

39. SEG Salt Velocity Model32

40. Initial Velocity Models
Depth (km)
4500
Traveltime Tomogram 4
Velocity (m/s)
Depth (km)
1500
v(z) Model 4
X (km) 16 9

41. TRT Tomogram
Gradient 33

42. MWT Tomogram (2.5,5 Hz)
TRT 34

43. Introduction: Waveform Tomography 1
Introduction: Waveform Tomography Not really good enough below salt Flooding Method: Flood salt below top, invert for bottom, flood sediment below salt bottom. FWI. 7

44. Waveform Inversion Results
Using Traveltime Tomogram
Depth (km)
4500 4
Velocity (m/s)
Using v(z) Model + Flooding
Depth (km)
1500 4
X (km) 16 10

45. Waveform Inversion Results
4500
Velocity (m/s)
True Model
Depth (km)
1500 4
X (km) 16 10

46. Flooding Technique Using v(z) Model w/o Flooding Depth (km) 4500 4
Depth (km)
4500 4
Velocity (m/s)
Waveform Tomogram after Salt Flood
Depth (km)
1500 4
X (km) 16 11

47. Flooding Technique Waveform Tomogram after Sediment Flood Depth (km)
Depth (km)
4500 4
Velocity (m/s)
Waveform Tomogram using v(z) and Flooding Technique
Depth (km)
1500 4
X (km) 16 12

48. SEG Salt Velocity Model35

49. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Efficient Multiscale Waveform Tomography
Results: Mapleton
Conclusions 1

50. Mapleton Model 36

51. TRT Tomogram 37

52. MWT Tomogram (30, 50, 70 HZ) 38

53. Mapleton Model 39

54. Mapleton Land Seismic Data
This is the zero-offset Mapleton land seismic data collected on an irregular surface in Mapleton, Utah.

55. Acquisition Geometry ~20 m ~78 m
With the topographic phase-shift method, the data are firstly converted to the data at z=0 (the highest geophone elevation).
In doing so, the exact geophone positions are used, and wavefield interpolation to a uniform grid is avoided.
Geophone intervals are uniformly 0.5 m in distance, however,
because of the relief, both dx, dz are nonuniform.

56. Tomography (Buddensiek and Sheng, 2004)

57. Wave-equation Migration
Without correction

58. Wave-equation Migration
With correction

59. Marine Data Results 40

60. Marine Data
480 Hydrophones
515 Shots
12.5 m
dt = 2 ms
Tmax = 10 s 41

61. Low-pass Filtering 42

62. Reconstructed Velocity43

63. Observed Data vs Predicted Data44

64. Waveform Residual vs Iteration Number
5 Hz
5 Hz
5 Hz
10 Hz
10 Hz
5 Hz
10 Hz 45

65. Common Image Gather
5 Hz
10 Hz 46

66. Outline Goal Introduction Theory of Acoustic Waveform Tomography
Multiscale Waveform Tomography
Results
Conclusions 47

67. Conclusions MWT partly overcomes the local minima problem.
MWT provides more accurate and highly resolved than TRT and EWT.
MWT is much more expensive than TRT.
Accuracy is more important than the cost.
MWT provides very accurate tomograms for synthetic data and shows encouraging results for the marine data. 48

68. Future Work
Use wider-window data and finally use all the data to obtain more accurate velocity distributions.
Take into account the source radiation pattern.
Apply MWT to land data. 49

69. Acknowledgment
We are grateful for the support from the sponsors of UTAM consortium.
Chaiwoot personally thanks ConocoPhillips for an internship and also appreciates the help from Seismic Technology Group at ConocoPhillips. 50