1. Sparse Least-Squares Reverse Time Migration using Seislets
Gaurav Dutta
King Abdullah University of Science and Technology
October 22, 2015

2. Outline Motivation Theory of LSRTM using Seislets Numerical Examples
Conclusions

3. Outline Motivation Theory of LSRTM using Seislets Numerical Examples
Conclusions

4. Motivation LSRTM: Problems: 1) Sparsely sampled data → low SNR.
𝜙(𝑚)= 𝐿𝑚−𝑑 2
Problems: 1) Sparsely sampled data → low SNR.
2) Crosstalk with blended sources. 2 10
X (km) 1
Z (km)
RTM 2 10
X (km)
LSRTM

5. Motivation LSRTM: Problems: 1) Sparsely sampled data → low SNR.
𝜙(𝑚)= 𝐿𝑚−𝑑 2
Problems: 1) Sparsely sampled data → low SNR.
2) Crosstalk with blended sources.
Solution:
𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
RTM 2 10
X (km)
LSRTM 1 2
Z (km) 2 10
X (km)

6. Motivation LSRTM: Problems: 1) Sparsely sampled data → low SNR.
𝜙(𝑚)= 𝐿𝑚−𝑑 2
Problems: 1) Sparsely sampled data → low SNR.
2) Crosstalk with blended sources.
Solution:
𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
RTM with Seislets 2 10
X (km)
LSRTM 1 2
Z (km) 2 10
X (km)

7. Motivation LSRTM: Problems: 1) Sparsely sampled data → low SNR.
𝜙(𝑚)= 𝐿𝑚−𝑑 2
Problems: 1) Sparsely sampled data → low SNR.
2) Crosstalk with blended sources.
Solution:
𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
RTM with Seislets 2 10
X (km)
LSRTM with Seislets 1 2
Z (km) 2 10
X (km)

8. Outline Motivation Theory of LSRTM using Seislets Numerical Examples
Conclusions

9. Background
Wavelet transforms are useful for formulating efficient signal processing and imaging algorithms.
They exploit the directional properties of an image through suitable basis functions.
DWT, curvelets, POCS, seislets are some examples.
Foster et al., 1994; Dessing, 1997; Wapenaar et al., 2005; Abma and Kabir, 2006; Candes et al., 2006; Douma and de Hoop, 2006; Hermann et al., 2007, 2009, 2012, etc., have shown several useful applications with sparse transforms.

10. Background
In the context of LSM and FWI, sparsity promoting imaging has been shown to produce high quality images at a reasonable cost.
Computational cost for LSM for 3D is an issue.
Two possible solutions:
1) phase-encoded migration (Romero et al., 2000; Dai et al., 2010, 11,12).
2) sparsity promoting imaging (Wang and Sacchi 2005, 2007; Herrmann et al., 2007, 2009; Herrmann and Li, 2012).
Sparsity promoting imaging using seislets is a possible alternative.

11. Seislet Framework
Seislet transform (Fomel and Liu, 2010) follows the lifting scheme used in wavelet transform (Sweldens, 1995) as:
1) Divide the data into even and odd traces, 𝑒 and 𝑜.
2) Find the residual, 𝑟.
𝑃= prediction operator
𝑟=𝑜−𝑃[𝑒]
𝑃 𝑒 𝑘 = 𝑒 𝑘−1 + 𝑒 𝑘 /2
3) Find a coarse approximation, 𝑐, of the data.
U = update operator
𝑐=𝑒+𝑈[𝑟]
𝑈 𝑟 𝑘 = 𝑟 𝑘−1 + 𝑟 𝑘 /4
4) The coarse approximation, 𝑐, becomes the new data and repeat steps 1-4 at the next scale.
𝑃 𝑒 𝑘 = 𝑆 𝑘 (+) [𝑒 𝑘−1 ]+ 𝑆 𝑘 (−) [ 𝑒 𝑘 ] /2
𝑆 𝑘 + , 𝑆 𝑘 (−) = predict a trace from local slopes
𝑈 𝑟 𝑘 = 𝑆 𝑘 (+) [ 𝑟 𝑘−1 ]+ 𝑆 𝑘 − [ 𝑟 𝑘 ] /4

12. Theory
Aim: Given observed data, 𝑑, we seek to find a reflectivity model, 𝑚.
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆𝑅(𝑚)
Objective function:
𝜙 𝑚 ≈𝜆𝑅(𝑚)
λ is very large:
The prior information is the solution
λ is very small:
𝜙 𝑚 ≈ 𝐿𝑚−𝑑 2
Unconstrained problem
Works well in a lot of cases.

13. Theory
Aim: Given observed data, 𝑑, we seek to find a reflectivity model, 𝑚.
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆𝑅(𝑚)
Objective function:
𝜙 𝑚 ≈𝜆𝑅(𝑚)
λ is very large:
The prior information is the solution
λ is very small:
𝜙 𝑚 ≈ 𝐿𝑚−𝑑 2
Unconstrained problem
Works well in a lot of cases.

14. Theory
Aim: Given observed data, 𝑑, we seek to find a reflectivity model, 𝑚.
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆𝑅(𝑚)
Objective function:
𝜙 𝑚 ≈𝜆𝑅(𝑚)
λ is very large:
The prior information is the solution
λ is very small:
𝜙 𝑚 ≈ 𝐿𝑚−𝑑 2
Unconstrained problem
Works well in a lot of cases.
Not sufficient when the data are very sparse, have noise.
Does not lead to geologically meaningful solutions.

15. Theory
Aim: Given observed data, 𝑑, we seek to find a reflectivity model, 𝑚.
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆𝑅(𝑚)
Objective function:
𝜙 𝑚 ≈𝜆𝑅(𝑚)
λ is very large:
The prior information is the solution
λ is very small:
𝜙 𝑚 ≈ 𝐿𝑚−𝑑 2
Unconstrained problem
Works well in a lot of cases.
Not sufficient when the data are very sparse, have noise.
Does not lead to geologically meaningful solutions.

16. LSRTM with Seislets
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
Express the reflectivity using seislet basis functions:
𝑚 =𝑆𝑚
𝑆= Seislet Transform (Fomel, 2010)
𝑆 ∗ = Pseudo-inverse of 𝑆
𝑚= 𝑆 ∗ 𝑚
Reformulate the problem:
𝑚 𝑚𝑖𝑛 𝑚 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑚 = Sparse transform of 𝑚
𝜙 = Tolerance level
Objective: Find 𝑚 that has the smallest 𝑙 1 norm when represented in seislet frames ( 𝑚 ).

17. LSRTM with Seislets 𝜙 𝑚 = 1 2 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝑚 𝑚𝑖𝑛 𝑚 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑚 𝑚𝑖𝑛 𝑅 𝑠 ( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅ 1 5
Z (km)
200
1000
Scale

18. LSRTM with Seislets 𝜙 𝑚 = 1 2 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝑚 𝑚𝑖𝑛 𝑚 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑚 𝑚𝑖𝑛 𝑅 𝑠 ( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅ 1 5
Z (km)
200
1000
Scale

19. LSRTM with Seislets 𝜙 𝑚 = 1 2 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝑚 𝑚𝑖𝑛 𝑚 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑚 𝑚𝑖𝑛 𝑅 𝑠 ( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅ 1 5
Z (km)
200
1000
Scale

20. LSRTM with Seislets 𝜙 𝑚 = 1 2 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝜙 𝑚 = 𝐿𝑚−𝑑 2 +𝜆 𝑅(𝑚)
𝑚 𝑚𝑖𝑛 𝑚 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑚 𝑚𝑖𝑛 𝑅 𝑠 ( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑅 𝑠 ( 𝑚 ) = Penalty term that promotes sparsity
𝑅 𝑐 ( 𝑚 ) = Penalty term that promotes image updates along dips
(Hale, 2009; Liu et al., 2010)
𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑅 𝑚 = 𝛼𝑅 𝑠 𝑚 +𝛽 𝑅 𝑐 ( 𝑚 )
𝛼+𝛽=1

21. Multisource LSRTM with Seislets
For multisource LSRTM,
𝜙 𝑚 = 𝑁𝐿𝑚−𝑁𝑑 2 +𝜆 𝑅(𝑚)
𝑁= Phase-encoding matrix
𝐿= Linear modeling operator
𝑚= Reflectivity model
𝑑= Data
New problem:
𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑁𝑑−𝑁𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑅 𝑚 = 𝛼𝑅 𝑠 𝑚 +𝛽 𝑅 𝑐 ( 𝑚 )
𝛼+𝛽=1

22. Workflow Algorithm: 𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑅 𝑚 = 𝛼𝑅 𝑠 𝑚 + 𝛽𝑅 𝑐 ( 𝑆 ∗ 𝑚 )
𝛼+𝛽=1
Algorithm:
𝑘←0; 𝑚 𝑘 ← 𝑚 0 ; 𝑚 𝑘 =𝑆 𝑚 𝑘
while condition do
𝑟 𝑘 ←𝐿 𝑆 ∗ 𝑚 𝑘 −𝑑
𝑟= Residual
𝛿 𝑚 𝑘 ← arg min 𝑚 𝑅( 𝑚 ) 𝑠.𝑡. 𝑟 𝑘 2 ≤ ∅
𝑚 𝑘+1 = 𝑚 𝑘 +𝜆𝛿 𝑚 𝑘
𝜆= Step-length
end while
𝑚= 𝑆 ∗ 𝑚

23. Outline Motivation Theory of LSRTM using Seislets Numerical Examples
Conclusions

24. True Reflectivity Model
Numerical Examples 3 7
0.5
1.5
Z (km)
X (km)
True Reflectivity Model
Acquisition
Background Velocity: 3500 m/s
10 shots evenly spaced at 1 km on the surface
50 receivers spaced 200 m apart.
Source: Ricker ( 𝑓 𝑝𝑒𝑎𝑘 =20 Hz)

25. Numerical Examples
RTM 3 7
X (km)
LSRTM
0.5
1.5
Z (km) 3
X (km) 7

26. Numerical Examples RTM with Seislets LSRTM 0.5 1.5 Z (km) 3 X (km) 7 3

27. Numerical Examples RTM with Seislets LSRTM with Seislets 0.5 1.5
Z (km) 3
X (km) 7 3
X (km) 7

28. Numerical Examples 256 shots evenly spaced at 50 m on the surface
Acquisition
256 shots evenly spaced at 50 m on the surface
512 receivers spaced 25 m apart.
Recording time: 10 s
Source: Ricker ( 𝑓 𝑝𝑒𝑎𝑘 =20 Hz)
True Velocity Model 2 3 4
km/s 1 5
Z (km) 2 14
X (km)

29. Migration Velocity Model
Numerical Examples
Acquisition
256 shots evenly spaced at 50 m on the surface
512 receivers spaced 25 m apart.
Recording time: 10 s
Source: Ricker ( 𝑓 𝑝𝑒𝑎𝑘 =20 Hz)
Migration Velocity Model 2 3 4
km/s 1 5
Z (km) 2 14
X (km)

30. Numerical Examples 256 shots evenly spaced at 50 m on the surface
Acquisition
256 shots evenly spaced at 50 m on the surface
512 receivers spaced 25 m apart.
Recording time: 10 s
Source: Ricker ( 𝑓 𝑝𝑒𝑎𝑘 =20 Hz)
RTM 2 3 4
km/s 1 5
Z (km) 2 14
X (km)

31. Estimated Dips from RTM Image
Numerical Examples
Acquisition
256 shots evenly spaced at 50 m on the surface
512 receivers spaced 25 m apart.
Recording time: 10 s
Source: Ricker ( 𝑓 𝑝𝑒𝑎𝑘 =20 Hz)
Estimated Dips from RTM Image 2 3 4
km/s 1 5
Z (km) 2 14
X (km)

32. Numerical Examples 20 iterations Multisource RTM Multisource LSRTM 214
X (km)
Multisource LSRTM 1
Z (km) 5 2
X (km) 14

33. Multisource RTM with Seislets
Numerical Examples
20 iterations
Multisource RTM with Seislets
Multisource LSRTM 1
Z (km) 5 2
X (km) 14 2
X (km) 14

34. Numerical Examples 20 iterations Multisource RTM with Seislets
Multisource LSRTM with Seislets 1
Z (km) 5 2
X (km) 14 2
X (km) 14

35. Numerical Examples Acquisition 515 shots at an interval of 37.5 m
GOM CSG
Time (s) 6 1
Offset (km)
Acquisition
515 shots at an interval of 37.5 m
Cable length = 6 km
480 receivers at an interval of 12.5 m
Recording time = 10 s

36. Estimate the wavelet by stacking the water bottom reflections
Data Processing
3D to 2D correction (scaling amplitudes by √𝑡 for geometrical spreading and phase correction with a filter 𝑖/𝜔 )
Estimate the wavelet by stacking the water bottom reflections
Filter the CSGs with a Wiener filter to transform the original wavelet to a Ricker of 10 Hz
Estimate the velocity model by multisource waveform inversion (Huang 2013)

37. Waveform Tomogram (Huang, 2013)
GOM Data
GOM CSG
Waveform Tomogram (Huang, 2013)
2.5
2.0
1.5
km/s
0.5
2.5
Z (km) 1
X (km) 18

38. Standard RTM (all shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
Standard RTM (all shots)

39. RTM with Seislets (all shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
RTM with Seislets (all shots)

40. Standard LSRTM (all shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
Standard LSRTM (all shots)

41. LSRTM with Seislets (all shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
LSRTM with Seislets (all shots)

42. LSRTM with Seislets (all shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
LSRTM with Seislets (all shots)

43. LSRTM with Seislets (all shots)
GOM Data
LSRTM with Seislets (all shots) 2 10
X (km) 1
Z (km)

44. Standard LSRTM (all shots)
GOM Data
Standard LSRTM (all shots) 2 10
X (km) 1
Z (km)

45. LSRTM with Seislets (all shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
LSRTM with Seislets (all shots)

46. LSRTM with Seislets (all shots)
GOM Data
LSRTM with Seislets (all shots) 11 17
X (km) 1 2
Z (km)

47. LSRTM with Seislets (all shots)
GOM Data
LSRTM with Seislets (all shots) 11 17
X (km) 1 2
Z (km)

48. Standard LSRTM (all shots)
GOM Data
Standard LSRTM (all shots) 11 17
X (km) 1 2
Z (km)

49. Standard LSRTM (sub-sampled shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
Standard LSRTM (sub-sampled shots)

50. LSRTM with Seislets (sub-sampled shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
LSRTM with Seislets (sub-sampled shots)

51. LSRTM with Seislets (sub-sampled shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
LSRTM with Seislets (sub-sampled shots)

52. LSRTM with Seislets (sub-sampled shots)
GOM Data
LSRTM with Seislets (sub-sampled shots) 2 10
X (km) 1
Z (km)

53. Standard LSRTM (sub-sampled shots)
GOM Data
Standard LSRTM (sub-sampled shots) 2 10
X (km) 1
Z (km)

54. LSRTM with Seislets (sub-sampled shots)
GOM Data
LSRTM with Seislets (sub-sampled shots) 2 10
X (km) 1
Z (km)

55. Standard RTM (all shots)
GOM Data
Standard RTM (all shots) 2 10
X (km) 1
Z (km)

56. LSRTM with Seislets (sub-sampled shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
LSRTM with Seislets (sub-sampled shots)

57. LSRTM with Seislets (subsampled shots)
GOM Data
LSRTM with Seislets (subsampled shots) 8 12
X (km)
0.5
2.5
Z (km)

58. Standard LSRTM (subsampled shots)
GOM Data
Standard LSRTM (subsampled shots) 8 12
X (km)
0.5
2.5
Z (km)

59. LSRTM with Seislets (sub-sampled shots)
GOM Data 1 18
X (km)
0.5
2.5
Z (km)
LSRTM with Seislets (sub-sampled shots)

60. LSRTM with Seislets (sub-sampled shots)
GOM Data
LSRTM with Seislets (sub-sampled shots) 11 17
X (km)
0.5
2.5
Z (km)

61. Standard LSRTM (sub-sampled shots)
GOM Data
Standard LSRTM (sub-sampled shots) 11 17
X (km)
0.5
2.5
Z (km)

62. LSRTM with Seislets (sub-sampled shots)
GOM Data
LSRTM with Seislets (sub-sampled shots) 11 17
X (km)
0.5
2.5
Z (km)

63. Standard RTM (all shots)
GOM Data
Standard RTM (all shots) 11 17
X (km)
0.5
2.5
Z (km)

64. Outline Motivation Theory of LSRTM using Seislets Numerical Examples
Conclusions

65. Conclusions
A sparse multisource LSM technique using seislets is presented.
𝑚 𝑚𝑖𝑛 𝑅 𝑠 ( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
A penalty term that promotes image updates along dips was used.
𝑚 𝑚𝑖𝑛 𝑅( 𝑚 ) 𝑠.𝑡. 𝑑−𝐿 𝑆 ∗ 𝑚 2 ≤ ∅
𝑅 𝑚 = 𝛼𝑅 𝑠 𝑚 +𝛽 𝑅 𝑐 ( 𝑚 )
𝛼+𝛽=1

66. Conclusions Good quality images with sub-sampled data. Standard LSM2 10
X (km) 1
Z (km)
Standard LSM
LSM with Seislets

67. Multisource LSM with Seislets
Conclusions
Less crosstalk with multisource LSM. 2 14
X (km) 1 5
Z (km)
Multisource LSM
Multisource LSM with Seislets
Good estimation of dips required.

68. Acknowledgements SEG for providing this platform.
Madagascar open-source software package.
Jianhua Yu and Matteo Giboli for discussions on this subject.
Sponsors of the CSIM consortium.
KAUST Supercomputing Laboratory (KSL) and IT research computing group.

69. Thank You !!!