1. Signal Analysis and Imaging Group Department of Physics University of Alberta
Regularized Migration/Inversion Henning Kuehl (Shell Canada) Mauricio Sacchi (Physics, UofA) Juefu Wang (Physics, UofA) Carrie Youzwishen (Exxon/Mobil) This doc ->
Good afternoon everyone, I’m giving a talk on seismic wavefield reconstruction for AVA imaging .

2. Outline Motivation and Goals
Migration/inversion - Evolution of ideas and concepts
Quadratic versus non-quadratic regularization
two examples
The migration problem
RLS migration with quadratic regularzation
Examples
RLS migration with non-quadratic regularization
Summary

3. Motivation
To go beyond the resolution provided by the data (aperture and band-width) by incorporating quadratic and non-quadratic regularization terms into migration/inversion algorithms
This is not a new idea…
The motivation we regularize seismic data before wave equation, without the regulariztion. AVA imaging may lack of amplidued fidelity due to uneven distrubution of offset in CDP bins.
is AVA imaging, no matter ray tracing based Kirchhoff GRT, or wave equation based, they can have problem of lack of amplitude fidelity dues to uneven distribution of offset in CDP bins. And It’s often true for seismic data, especially in 3D. The problem oftenOne of the way to solve this problem is to use interpolation. In this talk, I will show you how interpolation of seismic data can be used to improve that we want to find a way to regularize seismic data before pre-stack 3D wave equation imaging. As we know that in 3D, AVA imaging is often lack of amplitude fidelity due to uneven distribution of offset in CDP bins

4. Evolution of ideas and concepts
Migration with Adjoint Operators [Current technology]
RLS Migration (Quadratic Regularization) [Not in production yet]
RLS Migration (Non-Quadratic Regularization) [??]
The motivation we regularize seismic data before wave equation, without the regulariztion. AVA imaging may lack of amplidued fidelity due to uneven distrubution of offset in CDP bins.
is AVA imaging, no matter ray tracing based Kirchhoff GRT, or wave equation based, they can have problem of lack of amplitude fidelity dues to uneven distribution of offset in CDP bins. And It’s often true for seismic data, especially in 3D. The problem oftenOne of the way to solve this problem is to use interpolation. In this talk, I will show you how interpolation of seismic data can be used to improve that we want to find a way to regularize seismic data before pre-stack 3D wave equation imaging. As we know that in 3D, AVA imaging is often lack of amplitude fidelity due to uneven distribution of offset in CDP bins
Resolution

5. Evolution of ideas and concepts
Migration with Adjoint Operators
1 RLS Migration (Quadratic Regularization)
2 RLS Migration (Non-Quadratic Regularization)
The motivation we regularize seismic data before wave equation, without the regulariztion. AVA imaging may lack of amplidued fidelity due to uneven distrubution of offset in CDP bins.
is AVA imaging, no matter ray tracing based Kirchhoff GRT, or wave equation based, they can have problem of lack of amplitude fidelity dues to uneven distribution of offset in CDP bins. And It’s often true for seismic data, especially in 3D. The problem oftenOne of the way to solve this problem is to use interpolation. In this talk, I will show you how interpolation of seismic data can be used to improve that we want to find a way to regularize seismic data before pre-stack 3D wave equation imaging. As we know that in 3D, AVA imaging is often lack of amplitude fidelity due to uneven distribution of offset in CDP bins
Resolution

6. Evolution of ideas and concepts
Two examples
LS Deconvolution Sparse Spike Deconvolution
LS Radon Transforms HR Radon Transforms
The motivation we regularize seismic data before wave equation, without the regulariztion. AVA imaging may lack of amplidued fidelity due to uneven distrubution of offset in CDP bins.
is AVA imaging, no matter ray tracing based Kirchhoff GRT, or wave equation based, they can have problem of lack of amplitude fidelity dues to uneven distribution of offset in CDP bins. And It’s often true for seismic data, especially in 3D. The problem oftenOne of the way to solve this problem is to use interpolation. In this talk, I will show you how interpolation of seismic data can be used to improve that we want to find a way to regularize seismic data before pre-stack 3D wave equation imaging. As we know that in 3D, AVA imaging is often lack of amplitude fidelity due to uneven distribution of offset in CDP bins
Quadratic Regularization
Stable and Fast Algorithms
Low Resolution
Non-quadratic Regularization
Requires Sophisticated Optimization
Enhanced Resolution

7. Deconvolution
Quadratic versus non-quadratic regularization

8. Convolution / Cross-correlation / Deconv
Deconvolution
Here is a toy example of the linear equation, Assume complete data consists of M=5 consecutive samples x1 x2 …, a sampling set N indicates the position of
Known sample are at 2 3 5, so the incomplete data y consists of x(2) x(3) x(5).
The sampling opertor is a 0/1 matrix.
It is obvious that The linear systerm is underdetermined. we have more unknowns than observation. The solutions to such a system is non-unique. In general, we solve this type of problem by restricting the class of solution by providing suitable prior information.

9. Quadratic and Non-quadratic Regularization
Cost
Quadratic
Here is a toy example of the linear equation, Assume complete data consists of M=5 consecutive samples x1 x2 …, a sampling set N indicates the position of
Known sample are at 2 3 5, so the incomplete data y consists of x(2) x(3) x(5).
The sampling opertor is a 0/1 matrix.
It is obvious that The linear systerm is underdetermined. we have more unknowns than observation. The solutions to such a system is non-unique. In general, we solve this type of problem by restricting the class of solution by providing suitable prior information.
Non-quadratic

11. Adj LS HR

12. Radon Transform
Quadratic versus non-quadratic regularization

13. LS Radon q h

14. HR Radon q h

15. Modeling / Migration / Inversion
De-blurring problem
Here is a toy example of the linear equation, Assume complete data consists of M=5 consecutive samples x1 x2 …, a sampling set N indicates the position of
Known sample are at 2 3 5, so the incomplete data y consists of x(2) x(3) x(5).
The sampling opertor is a 0/1 matrix.
It is obvious that The linear systerm is underdetermined. we have more unknowns than observation. The solutions to such a system is non-unique. In general, we solve this type of problem by restricting the class of solution by providing suitable prior information.

16. Back to migration
Inducing a sparse solution via non-quadratic regularization appears to be a good idea (at least for the two previous examples)
Q. Is the same valid for Migration/Inversion?

17. Back to migration
Inducing a sparse solution via non-quadratic regularization appears to be a good idea (at least for the two previous examples)
Q. In the same valid for Migration/Inversion?
A. Not so fast… First, we need to define what we are inverting for…

18. Migration as an inverse problem
Forward
Adjoint
Data
“Image” or
Angle Dependent Reflectivity

19. Migration as an inverse problem
There is No general agreement about what type of regularization should be used when inverting for m=m(x,z)… Geology is too complicated…
For angle dependent images, m(x,z,p), we attempt to impose horizontal smoothness along the “redundant” variable in the CIGs:
Kuehl, 2002, PhD Thesis UofA
url: cm-gw.phys.ualberta.ca~/sacchi/saig/index.html

20. Quadratic Regularization Migration Algorithm Features:
DSR/AVP Forward/Adjoint Operator (Prucha et. al, 1999)
PSPI/Split Step
Optimization via PCG
2D/3D (Common Azimuth)
MPI/Open MP
Kuehl and Sacchi, Geophysics, January 03 [Amplitudes in 2D]

21. 2D - Synthetic data example

22. Marmousi model (From H. Kuehl Thesis, UofA, 02)
AVA target
Velocity
Density

23. Migrated AVA CMP position [m] Ray parameter [mu s/m] 7000 7500 8000
200
400
600
800
0.5
0.5
1.0
1.0 ] ] m m k k [ [ h h t t p
1.5 p
1.5 e e D D
2.0
2.0
2.5
2.5

24. Regularized Migrated AVAp t h [ k m ]
0.5
1.0
1.5
2.0
2.5
7000
7500
8000
CMP position [m]
200
400
600
800
Ray parameter [mu s/m]

25. Migration RLSM 0.5 0.7 0.9 1.1 D e p t h [ k m ] 200 400 600 80010 20 30 40 50
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Angle of incidence (deg)
Reflection coefficient
0.5
0.7
0.9
1.1 D e p t h [ k m ]
200
400
600
800
Ray parameter [mu s/m]
Migration
RLSM

26. Incomplete CMP (30 %) Complete CMP 0.5 1.0 1.5 2.0 2.5 T i m e [ s c ]
0.5
1.0
1.5
2.0
2.5 T i m e [ s c ]
200
400
600
800
1000
1200
Offset [m]
Complete CMP
Incomplete CMP (30 %)

27. Migration RLSM 0.5 0.7 0.9 1.1 D e p t h [ k m ] 200 400 600 800
200
400
600
800
Ray parameter [mu s/m] 10 20 30 40 50
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Angle of incidence (deg)
Reflection coefficient
Migration
RLSM

28. Constant ray parameter image (incomplete)
0.5
1.0
1.5
2.0
2.5 D e p t h [ k m ]
3000
4000
5000
6000
7000
8000
9000
CMP position [m]
Constant ray parameter image (incomplete)

29. Constant ray parameter image (RLSM)
0.5
1.0
1.5
2.0
2.5 D e p t h [ k m ]
3000
4000
5000
6000
7000
8000
9000
CMP position [m]
Constant ray parameter image (RLSM)

30. 3D - Synthetic data example
(J Wang)
Parameters for the 3D synthetic data
x-CDPs : 40
Y-CDPs : 301
Nominal Offset: 50
dx=5 m
dy=10 m
dh=10 m
90% traces are randomly removed to simulate a sparse
3D data.

31. Incomplete Data Reconstructed data (after 12 Iterations)
4 adjacent CDPs
Reconstructed
data (after 12
Iterations)

32. A B C
A: Iteration 1 B: Iteration 3 C: Iteration 12

33. At y=950 m, x=130 m

34. Misfit Curve
Convergence – Misfit vs. CG iterations

35. Stacked images comparison (x-line 30)
A complete data
B. Remove 90% traces
C. least-squares

36. Field data example ERSKINE (WBC) orthogonal 3-D sparse land data set
dx=50.29m
dy=33.5m
Comment on Importance of W

37. CIG at x-line #10, in-line #71
Iteration 1
Iteration 3
Iteration 7
Cost Optimization

38. Stacked image, in-line #71
Migration
RLS Migration

39. Stacked image, in-line #71
Migration
RLS Migration

40. Image and Common Image Gather (detail)

41. Image and Common Image Gather (detail)

42. Stacked Image (x-line #10) comparison
Migration
LS Migration

43. Non-quadratic regularization applied to imaging

44. Non-Quadratic Smoothing
Non-quadratic regularization
But first, a little about smoothing:
Quadratic Smoothing
Non-Quadratic Smoothing

45. Quadratic regularization ->Linear filters
Non-quadratic -> Non-linear filters

46. Segmentation/Non-linear Smoothing

47. Segmentation/Non-linear Smoothing
1/Variance at edge position

48. 2D Segmentation/Non-linear Smoothing

49. GRT Inversion - VSP (Carrie Youzwishen, MSc 2001)*
To test the algorithm, we use a 2D VSP model. An area 1 km square with a maximum acoustic potential of 1 is evaluated. The source and receiver geometry is shown on the first figure. The middle box is the synthetic (diffracted) data with a SNR of 5. The remaining figure shows the damped least squares solution.

50. Dx Dz m

51. HR Migration (Current Direction)
Quadratic constraint (Dp) to smooth along p (or h)
Non-quadratic constraint to force vertical sparseness

52. Example: we compare migrated images m(x,z,h) for the following 3 imaging methods:
Adj LS HR

53. Acoustic, Linearized, Constant V, Variable Density
Synthetic Model
X(m)
True
Z(m)
m(x,z)
When the observations contain additive nosie rather than trying to fit exactly al observations, we attempt to fit the obsearvation in least squares sense. In this case we minimize a cost function that combins a data misfit function in conjuction with the model norm.
Acoustic, Linearized, Constant V, Variable Density

54. Pre-stack data d(s,g,t) Data
When the observations contain additive nosie rather than trying to fit exactly al observations, we attempt to fit the obsearvation in least squares sense. In this case we minimize a cost function that combins a data misfit function in conjuction with the model norm.
Shots
d(s,g,t)

55. Common offset images m(x,z,h) Adj Z(m)
CMP gather at 750m generated by a ray-tracing code. The code models
(cylindrical) geometrical spreading but no transmission effects. The offset for the
particular CMP ranges from 0m to 1280m. Middle: The same CMP
after randomly removing of 90\% the data. Bottom: The reconstructed CMP using WMNI,
the reconstructed offset ranges from 0m to 1200m.
We first test our alogrithm on a simple, horizontally layered model. The horizontally
layered model consists of four reflecting interfaces. The acoustic model
parameters in terms of compressional velocities and densities range from 1900 m/s to
2500 m/s and from 1.6 g/$\textrm{cm}^3$ to 2.25 g/$\textrm{cm}^3$ , respectively. All interfaces are well separated.
A ray tracing technique is used to generate the synthetic dataset. The
ray-tracer takes advantage of the fact that, in a stratified medium, the ray parameter is
constant for a particular ray. The geometrical spreading has been calculated
assuming a
cylindrical wavefront resulting in a $1 / \sqrt{r}$ amplitude scaling, where $r$ is the distance
travelled by the ray. Notes that transmission losses is neglected in the synthetics.
In Figure 1, top panel shows the CMP data at 750m. They exhibit a clear amplitude
variation versus offset (AVO). the offsets range from 0 to 1295m incrementing by 20m.
Figure 1 middle shows the same CMP after randomly removing $90$\% of the live traces.
The incomplete CMPs (total 100) are used as the input for the reconstruction. We first
perform Fourier transform along the time axis. Reconstructions are then carried at
temporal frequencies along two spatial (CMP and offset) coordinates simultaneously.
The output offset ranges 0m to 1200m for each CMP incrementing by 10m. Figure 1 bottom shows the reconstructed CMP at 750m using WMNI.
m(x,z,h)

56. Common offset images LS
Z(m)
m(x,z,h)

57. Common offset images HR
Z(m)
m(x,z,h)

58. Stacked CIGs
X(m)
Adj
Z(m)
m(x,z)

59. Stacked CIGs
X(m) LS
Z(m)
m(x,z)

60. Stacked CIGs m(x,z) HR X(m) Z(m)
CMP gather at 750m generated by a ray-tracing code. The code models
(cylindrical) geometrical spreading but no transmission effects. The offset for the
particular CMP ranges from 0m to 1280m. Middle: The same CMP
after randomly removing of 90\% the data. Bottom: The reconstructed CMP using WMNI,
the reconstructed offset ranges from 0m to 1200m.
We first test our alogrithm on a simple, horizontally layered model. The horizontally
layered model consists of four reflecting interfaces. The acoustic model
parameters in terms of compressional velocities and densities range from 1900 m/s to
2500 m/s and from 1.6 g/$\textrm{cm}^3$ to 2.25 g/$\textrm{cm}^3$ , respectively. All interfaces are well separated.
A ray tracing technique is used to generate the synthetic dataset. The
ray-tracer takes advantage of the fact that, in a stratified medium, the ray parameter is
constant for a particular ray. The geometrical spreading has been calculated
assuming a
cylindrical wavefront resulting in a $1 / \sqrt{r}$ amplitude scaling, where $r$ is the distance
travelled by the ray. Notes that transmission losses is neglected in the synthetics.
In Figure 1, top panel shows the CMP data at 750m. They exhibit a clear amplitude
variation versus offset (AVO). the offsets range from 0 to 1295m incrementing by 20m.
Figure 1 middle shows the same CMP after randomly removing $90$\% of the live traces.
The incomplete CMPs (total 100) are used as the input for the reconstruction. We first
perform Fourier transform along the time axis. Reconstructions are then carried at
temporal frequencies along two spatial (CMP and offset) coordinates simultaneously.
The output offset ranges 0m to 1200m for each CMP incrementing by 10m. Figure 1 bottom shows the reconstructed CMP at 750m using WMNI.

61. CIGs
CIG # CIG # CIG # 12
Adj LS
Z(m) HR
0m Offset m

62. Data
LS Prediction
T(s)
HR Prediction
d(s,g,t)
Shots

63. Conclusions
Imaging/Inversion with the addition of quadratic and non-quadratic constraints could lead to a new class of imaging algorithms where the resolution of the inverted image can be enhanced beyond the limits imposed by the data (aperture and band-width).
This is not a completely new idea. Exploration geophysicists have been using similar concepts to invert post-stack data (sparse spike inversion) and to design Radon operators.
Finally, it is important to stress that any regularization strategy capable of enhancing the resolution of seismic images must be applied in the CIG domain. Continuity along the CIG horizontal variable (offset, angle, ray parameter) in conjunction with sparseness in depth, appears to be reasonable choice.

64. Acknowledgments EnCana Geo-X Veritas Schlumberger foundation NSERC
AERI
3D Real data set was provided by Dr Cheadle