Transcript Least Squares Migration Combined with a Deblurring Filter

Fast Least Squares Migration
with a Deblurring Filter
30 October 2008
Naoshi Aoki
1
Outlines
•
•
•
•
•
•
Motivation
Deblurring filter theory
A numerical result of the deblurring filter
Deblurred LSM theory
Numerical results of the deblurred LSM
Conclusions
2
Outlines
•
•
•
•
•
•
Motivation
Deblurring filter theory
A numerical result of the deblurring filter
Deblurred LSM theory
Numerical results of the deblurred LSM
Conclusions
3
Forward and Inverse Problems
for Acoustic Wavefield
• Forward problem:
d  Lm,
where d is data, L is forward modeling operator, and
m is reflectivity model.
• Inverse problem:
-1
T


m   L L  L d,
T
where LT is an adjoint of forward modeling operator,
and [LTL]-1 is the inverse of Hessian.
4
Alternatives to Direct Inversion
• Migration
mmig = L d  = L Lm 
T
T
• LSM (e.g., Nemeth, Wu and Schuster,1999)
mn1 = mn   gn ,
where gn  L (Lmn - d),  n 
T
 gn , gn 
 Lgn , Lgn 
5
The U Model Test
5
Model Description
● Source
● Receiver
TWT (s)
0
3D U Model
Data
0
1.8
X (m)
U model is designed for testing Prestack
3D LSM with arbitrary 3D survey
geometry.
• Model size:
– 1.8 x 1.8 x 1.8 km
• U shape reflectivity anomaly
Depth (m)
Reflectivity
250
1
500
-1
750
1
1000
-1
1250
1
• Cross-spread geometry
– Source : 16 shots, 100 m int.
– Receiver : 16 receivers , 100 m
6
int.
Depth Slices from
Migration and LSM
(a) Actual Reflectivity
(b) Test geometry
Kirchhoff Migration Images
(c) Z = 250 m
(e) Z = 750 m
(g) Z=1250m
LSM Images after 30 Iterations
(d) Z=250m
(f) Z=750m
(h) Z=1250m
● Source
● Receiver
7
Challenges in LSM Processing
• Estimation of modeling operators
– Velocity Model
– Source wavelet
• Computational Cost
– LSM typically requires 10 or more iterations.
– Each LSM iteration requires about 3 times
higher computational cost than that of the
migration.
8
Outlines
•
•
•
•
•
•
Motivation
Deblurring filter theory
A numerical result of the deblurring filter
Deblurred LSM theory
Numerical results of the deblurred LSM
Conclusions
9
An Alternative to LSM
• Deblur the migration image with a local
non-stationary filtering
– Migration deconvolution (Hu and Schuster,
2001),
– Deconvolution of migration operator by a local
non-stationary filter (Etgen, 2002, Guitton
2004),
– FFT based approach(e.g., Lecomte(2008);
Toxopeus et al, (2008)).
10
Deblurring Filter Theory
• Actual Migration Image:
T
T
L d = L Lm
The computational cost
is about one iteration of
LSM
• Compute a reference migration image from a reference model
m’:
T
T
L d' = L Lm '
• Find a deblurring operator with a matching filter (He, 2003) :
T

F  L d' = m '
• Apply the operator F  L L 


T
-1
F L d   m
to the actual migration image
T
11
Outlines
•
•
•
•
•
•
Motivation
Deblurring filter theory
A numerical result of the deblurring filter
Deblurred LSM theory
Numerical results of the deblurred LSM
Conclusions
12
Point Scatterer Model Test
Actual Reflectivity Model
▼▼▼▼▼▼▼▼▼▼▼▼▼
1.8
TWT (sec)
0
Z (km)
CSG Example
Scatterer:
50 m x 50 m
V=1000 m/s
2.5
0
-0.1
X (km)
0
2.5
0.1
2.8
0.5
X (km)
1.5
Fdominant = 5 Hz; λ=200 m13
Migration Image
Actual Reflectivity Image
Migration Image
Z (km)
0
Z (km)
0
2.5
2.5
0
X (km)
2.5
0
X (km)
2.5
The Rayleigh resolution limit = 200
14 m
-0.1
0
0.1
Deblurred Migration Image
Actual Reflectivity Image
Deblurred Migration Image
Z (km)
0
Z (km)
0
2.5
2.5
0
X (km)
2.5
0
X (km)
2.5
15
-0.1
0
0.1
LSM Image
Actual Reflectivity Image
LSM Image after 30 Iterations
Z (km)
0
Z (km)
0
2.5
2.5
0
X (km)
2.5
0
X (km)
2.5
16
-0.1
0
0.1
Horizontal Image of the Scatterer
Reflectivity
0.1
0
0.5
X(km)
1.5
17
Migration Deblurring Test Summary
• Deblurring filter improves spatial resolution
of migration image about double.
• The computational cost is about one
iteration of LSM.
• The deblurred migration image is slightly
noisier than that in the LSM image.
18
Outlines
•
•
•
•
•
•
Motivation
Deblurring filter theory
A numerical results of the deblurring filter
Deblurred LSM theory
Numerical results of the deblurred LSM
Conclusions
19
Deblurred LSM Theory
•
•
DLSM is a fast LSM with a deblurring filter.
2 types of DLSM algorithms are proposed:
1. Regularized DLSM (or RDLSM)
mn1 = mn   gn ,
g n  LT (Lm n - d)    m - mapri  ,  n 
gn
2
Lg   g n
2
2
where mapri is a skeletonized version of FLTd ,
and γ is a regularization parameter.
2. Preconditioned DLSM (or PDLSM)
Fg n , g n 

mn1 = mn   Fgn ,  n 
.
2
LF g n
20
,
Outlines
•
•
•
•
•
•
Motivation
Deblurring filter theory
A numerical results of the deblurring filter
Deblurred LSM theory
Numerical results of the deblurred LSM
Conclusions
21
Numerical Results
• A synthetic data set from the Marmousi2
model.
• A 2D marine data set from the Gulf of
Mexico.
22
Marmousi2 Model
Geological Cross Section
(Martin et. al., 2006)
23
Velocity and Density Models
0
0
Z (km)
Density Model
Z (km)
P wave Velocity Model
3
3
0
X (km)
1.5
Velocity (km/s)
15
4.5
0
X (km)
1
Density (g/cc)
15
2.6
24
Traveltime Field Computation
0
0
Z (km)
Traveltime Field Example
Z (km)
P wave Velocity Model
3
3
0
X (km)
1.5
Velocity (km/s)
15
4.5
0
1
X (km)
Velocity (km/s)
15
4
25
(UTAM ray- tracing code written by He, 2002)
Reflectivity Model and Data
Source Wavelet
Reflectivity Model
0
Z (km)
Amplitude
2000
0
-2000
3
0
6
-0.2
X (km)
0
12
300
Time (msec)
Fdom = 25 Hz
0.2
26
Reflectivity Model and Data
Reflectivity Model
Zero-offset Data
0
Z (km)
TWT (s)
0
3
3
6
-0.2
X (km)
0
12
0.2
6
X (km)
12
27
Migration Image
Poststack Migration
0
Z (km)
Z (km)
Actual Reflectivity Model
0
3
3
6
-0.2
X (km)
0
12
0.2
6
12
X (km)
CPU
time =1800-4500
10 minutes m/s
Velocity:
Wavelength
: 70 -2.2
180
m
on a dual processor
GHz
28
Deblurring Filter with the Exact Model
Step1: Compute Matching Operator
Exact Model
Actual Migration Image
0
0
Z (km)
Z (km)
f
3
3
6
X (km)
12
6
X (km)
12
29
Deblurring Filter with the Exact Model
Step2: Apply the Operator
Actual Migration Image
0
Z (km)
Z (km)
f
Deblurred Migration Image
0
3
3
6
X (km)
12
6
X (km)
12
30
DLSM Convergence Curves
RDLSM
PDLSM
Residual
1
Residual
1
19
0
1
8
0
Iteration Number
30
Damping parameter:
Γ= 200000x0.5n-1, n=1,2,…,30
1
Iteration Number
30
31
DLSM Images
with the Exact Model
PDLSM after 8 Iterations
0
Z (km)
Z (km)
RDLSM after 19 Iterations
0
3
3
6
X (km)
12
6
X (km)
12
32
Model Sensitivity Test
• Exact model:
• Geological model:
– Skeletonized Migrated
Image
• Grid model:
– The region is divided
into sections; each
section has a point
scatterer in the center.
Zoom
View
Grid
Model
ExactofModel
Geological
Model
01
ZZ(km)
(km)
– the actual model
32
106
250 x 250 m
XX (km)
(km)
1211
33
Deblurring Filter with the Geological Model
Step1: Compute Matching Operator
Geological Model
0
Z (km)
Z (km)
f
Reference Migration Image
0
3
3
6
X (km)
12
6
X (km)
12
34
Deblurring Filter with the Geological Model
Step2: Apply the Operator
Actual Migration Image
0
Z (km)
Z (km)
f
Deblurred Migration Image
0
3
3
6
X (km)
12
6
X (km)
12
35
DLSM Convergence Curves
Regularized DLSM
Preconditioned DLSM
Residual
1
Residual
1
20
0
1
12
0
Iteration Number
30
Damping parameter:
Γ= 200000x0.5n-1, n=1,2,…,30
1
Iteration Number
30
36
DLSM Images
with the Geological Model
PDLSM after 12 Iterations
0
Z (km)
Z (km)
RDLSM after 20 Iterations
0
3
3
6
X (km)
12
6
X (km)
12
37
Deblurring Filter with the Grid Model
Step1: Compute Matching Operator
f
Reference Migration Image
0
Z (km)
Z (km)
Zoom View of Grid Model
1
2
10
3
X (km)
11
6
X (km)
12
38
Deblurring Filter with the Grid Model
Step2: Apply the Operator
Actual Migration Image
0
Z (km)
Z (km)
f
Deblurred Migration Image
0
3
3
6
X (km)
12
6
X (km)
12
39
DLSM Convergence Curves
Regularized DLSM
Preconditioned DLSM
Residual
1
Residual
1
20
0
1
10
0
Iteration Number
30
Damping parameter:
Γ= 200000x0.5n-1, n=1,2,…,30
1
Iteration Number
30
40
DLSM Images
with the Grid Model
PDLSM after 10 Iterations
0
Z (km)
Z (km)
RDLSM after 20 Iterations
0
3
3
6
X (km)
12
6
X (km)
12
41
Marmousi2 Test Summary (1)
• The deblurring filter can expedite the
computation of an LSM image.
– RDLSM and PDLSM provide acceptable LSM
images with about 2/3 and 1/3 the cost of standard
LSM, respectively.
• Controlling the model dependency is
required.
– RDLSM can control the model dependency with a
regularization parameter.
– In the PDLSM algorithm, not using a deblurring filter
after several iteration is recommended.
42
Marmousi2 Test Summary (2)
• DLSM with the geological model
– Computation of an LSM image can be expedited by a
human interpretation.
– A risk is an erroneous interpretation. The model
dependency should be carefully controlled.
• DLSM with the grid model
– The result is not good as that from a better geological
model.
– An advantage is that no expense of a human
interpretation is required for the model building.
43
The Gulf of Mexico Data Test
2D Poststack Marine Data
TWT(s)
0
4
8
18
X (km)
44
The Gulf of Mexico Data Test
• Both the regularization and preconditioning
schemes are employed in the DLSM.
• A geological model is created by the following
way:
1. A deblurred migration image is obtained with a grid
model.
2. A geological model is created by cosmetic filtering
and skeletonizing the deblurred migration image.
45
Zero-offset Data from
for a Grid Model
TWT(ｓ)
0
4
8
Scatterer Interval: 500 m x 500 m
X (km)
18
46
Zoom View of Reference Migration
Image for a Grid Model
Z (km)
0.4
1.2
8
10.5
X (km)
13
47
Kirchhoff Migration
Z (km)
0.5
1
1.5
8
10.5
X (km)
13
48
Deblurred Migration Image
Z (km)
0.5
1
1.5
8
10.5
X (km)
13
49
Geological Model
Reflectivity
0.1
Z (km)
0.5
0
1
1.5
8
10.5
X (km)
13
-0.1
50
Comparison of Imaging Results
Kirchhoff Migration
Z (km)
0.5
1.5
8
X (km)
13
51
Box A: Comparison of Images
Migration
LSM after 3 Iterations
Z (km)
0.5
Z (km)
0.5
0.7
0.7
9.6
X (km) 10.6
DLSM after 3 Iterations
0.5
9.6
X (km) 10.6
LSM after 10 Iterations
Z (km)
Z (km)
0.5
0.7
0.7
9.6
X (km)
10.6
52
9.6
X (km)
10.6
Box B: Comparison of Images
LSM after 3 Iterations
1
Z (km)
Z (km)
Migration
1
1.2
1.2
1
11
X (km) 12
LSM after 10 Iterations
Z (km)
Z (km)
11
X (km) 12
DLSM after 3 Iterations
1
1.2
1.2
53
11
X (km)
12
11
X (km)
12
Total Computational Cost
1
LSM after 3 Iterations
Z (km)
Z (km)
Migration
1
1
1.2
1.2
1
11
X (km) 12
LSM after 10 Iterations
Z (km)
11
X (km) 12
DLSM after 3 Iterations
Z (km)
9
19+
1
30
1.2
1.2
54
11
X (km)
12
11
X (km)
12
Total Computational Cost
•
•
•
•
Migration
LSM 3 Iterations
LSM 10 Iterations
DLSM 3 Iterations
– Deblurring with the grid model
– Deblurring with the geological model
– DLSM 3 Iterations
1
9
30
19+
3
4+
12
55
The GOM Data Test Summary
• DLSM can successfully provide an
improved LSM image with an affordable
computer expense.
56
Outlines
•
•
•
•
•
•
Motivation
Deblurring filter theory
A numerical results of the deblurring filter
Deblurred LSM theory
Numerical results of the deblurred LSM
Conclusions
57
Conclusions
• A deblurring filter provides a fine apriori
model for a regularized LSM, and it can
also be used as an effective
preconditioning filter.
• The DLSM algorithms provids acceptable
LSM images with 1/3 – 2/3 the cost of
standard LSM.
58
Future Works
• The deblurring filter requires some
improvement in quality and efficiency.
• A computer-aided skeletonization
method is required for reducing an
expense of a human interpretation.
59
Acknowledgements
• I would like to thank Prof. Gerard T. Schuster for
his encouragement throughout my stay at the
University of Utah.
• I also want to thank my group colleagues for
their academic discussions and personal help.
• I also thank JOGMEC and JAPEX for supporting
my study at the University of Utah.
60
Thanks
61