fast least squares migration with a deblurring filter
DESCRIPTION
Fast Least Squares Migration with a Deblurring Filter. 30 October 2008 Naoshi Aoki. Outlines. Motivation Deblurring filter theory A numerical result of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions. Outlines. Motivation - PowerPoint PPT PresentationTRANSCRIPT
Fast Least SquaresFast Least Squares MigrationMigrationwith a Deblurring Filterwith a Deblurring Filter
30 October 2008
Naoshi Aoki
1
OutlinesOutlines
• Motivation
• Deblurring filter theory
• A numerical result of the deblurring filter
• Deblurred LSM theory
• Numerical results of the deblurred LSM
• Conclusions
2
OutlinesOutlines
• 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 ProblemsForward and Inverse Problemsfor Acoustic Wavefieldfor Acoustic Wavefield
• Forward problem:
where d is data, L is forward modeling operator, and m is reflectivity model.
• Inverse problem:
where LT is an adjoint of forward modeling operator, and [LTL]-1 is the inverse of Hessian.
,d Lm
, -1T Tm L L L d
4
Alternatives to Direct InversionAlternatives to Direct Inversion
• Migration
• LSM (e.g., Nemeth, Wu and Schuster,1999)
where
Tmigm = L d
1 ,n n n m =m g
,n Tng L (Lm -d)
,
,n n
nn n
g g
Lg Lg
T= L Lm
5
The The UU Model Test Model Test
3D U Model Model Description• Model size:
– 1.8 x 1.8 x 1.8 km
• U shape reflectivity anomaly
• Cross-spread geometry– Source : 16 shots, 100 m int.– Receiver : 16 receivers , 100 m int.
Depth (m) Reflectivity
250 1
500 -1
750 1
1000 -1
1250 1
● Source● Receiver
U model is designed for testing Prestack 3D LSM with arbitrary 3D survey geometry.
Data0
5
TW
T (
s)
0 1.8X (m)
6
Depth Slices fromDepth Slices fromMigration and LSMMigration and LSM
(c) Z = 250 m (e) Z = 750 m (g) Z=1250m(a) Actual Reflectivity
Kirchhoff Migration Images
(b) Test geometry(d) Z=250m
LSM Images after 30 Iterations(f) Z=750m (h) Z=1250m
● Source● Receiver
7
Challenges in LSM ProcessingChallenges 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
OutlinesOutlines
• 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 LSMAn 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 TheoryDeblurring Filter Theory• Actual Migration Image:
• Compute a reference migration image from a reference model m’:
• Find a deblurring operator with a matching filter (He, 2003) :
• Apply the operator to the actual migration image
T TL d = L Lm
' Td'F L =m
'T TL d' = L Lm
TdF L m
-1TLF L
The computational cost is about one iteration of LSM
11
OutlinesOutlines
• Motivation
• Deblurring filter theory
• A numerical result of the deblurring filter
• Deblurred LSM theory
• Numerical results of the deblurred LSM
• Conclusions
12
0
2.5
Z (
km)
0 2.5X (km)
0.1-0.1 0
Actual Reflectivity Model
Point Scatterer Model TestPoint Scatterer Model Test
TW
T (
sec)
X (km)0.5 1.5
1.8
2.8
CSG Example
Fdominant = 5 Hz; λ=200 m
Scatterer:50 m x 50 m
V=1000 m/s
▼▼▼▼▼▼▼▼▼▼▼▼▼
13
Migration ImageMigration Image
0
2.5
Z (
km)
0 2.5X (km)
Actual Reflectivity Image
Z (
km)
0
2.50 2.5
X (km)
Migration Image
0.1-0.1 0The Rayleigh resolution limit = 200 m 14
Deblurred Migration ImageDeblurred Migration Image
0
2.5
Z (
km)
0 2.5X (km)
Actual Reflectivity Image
0
2.5
Z (
km)
0 2.5X (km)
Deblurred Migration Image
0.1-0.1 015
LSM ImageLSM Image
0
2.5
Z (
km)
0 2.5X (km)
Actual Reflectivity Image
0.1-0.1 0
0
2.5
Z (
km)
0 2.5X (km)
LSM Image after 30 Iterations
16
Horizontal Image of the ScattererHorizontal Image of the Scatterer
0.1
0
0.5 1.5
Ref
lect
ivity
X(km)17
Migration Deblurring Test Summary 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
OutlinesOutlines
• Motivation
• Deblurring filter theory
• A numerical results of the deblurring filter
• Deblurred LSM theory
• Numerical results of the deblurred LSM
• Conclusions
19
Deblurred LSM TheoryDeblurred LSM Theory
• DLSM is a fast LSM with a deblurring filter.• 2 types of DLSM algorithms are proposed:
1. Regularized DLSM (or RDLSM)
where mapri is a skeletonized version of ,
and γ is a regularization parameter.
2. Preconditioned DLSM (or PDLSM)
1 ,n n n m =m g
,n Tn aprig L (Lm -d) mm -
TFL d
1 ,n n nm =m Fg 2
,.n n
n
n
g g
gFL
F
2
2 2 ,n
n
n
g
Lg g
20
OutlinesOutlines
• Motivation
• Deblurring filter theory
• A numerical results of the deblurring filter
• Deblurred LSM theory
• Numerical results of the deblurred LSM
• Conclusions
21
Numerical ResultsNumerical Results
• A synthetic data set from the Marmousi2 model.
• A 2D marine data set from the Gulf of Mexico.
22
Marmousi2 ModelMarmousi2 ModelGeological Cross SectionGeological Cross Section
(Martin et. al., 2006)(Martin et. al., 2006) 23
Velocity and Density ModelsVelocity and Density Models
0
3
0 15
Z (
km)
X (km)
P wave Velocity Model
4.51.5Velocity (km/s)
0
3
0 15Z
(km
)
X (km)
Density Model
2.61Density (g/cc) 24
Traveltime Field ComputationTraveltime Field Computation
0
3
0 15
Z (
km)
X (km)
P wave Velocity Model
4.51.5Velocity (km/s)
0
3
0 15Z
(km
)
X (km)
Traveltime Field Example
41 Velocity (km/s)
(UTAM ray- tracing code written by He, 2002)25
Reflectivity Model and DataReflectivity Model and Data
0 300Time (msec)
0
2000
-2000A
mpl
itude
Source WaveletReflectivity Model
0
3
Z (
km)
6 12X (km)
0.2-0.2 0
Fdom = 25 Hz
26
Reflectivity Model and DataReflectivity Model and Data
Zero-offset Data
0
3T
WT
(s)
6 12X (km)
Reflectivity Model
0
3
Z (
km)
6 12X (km)
0.2-0.2 0 27
Migration ImageMigration Image
Poststack Migration
0
3Z
(km
)
6 12X (km)
Actual Reflectivity Model
0
3
Z (
km)
6 12X (km)
0.2-0.2 0
CPU time = 10 minutes
on a dual processor 2.2 GHz
Velocity: 1800-4500 m/sWavelength : 70 - 180 m
28
Deblurring Filter with the Exact Model Deblurring Filter with the Exact Model Step1: Compute Matching OperatorStep1: Compute Matching Operator
Actual Migration Image
0
3Z
(km
)
6 12X (km)
Exact Model
0
3
Z (
km)
6 12X (km)
f
29
Deblurring Filter with the Exact Model Deblurring Filter with the Exact Model Step2: Apply the OperatorStep2: Apply the Operator
Deblurred Migration Image
0
3Z
(km
)
6 12X (km)
Actual Migration Image
0
3
Z (
km)
6 12X (km)
f
30
DLSM Convergence CurvesDLSM Convergence Curves
PDLSMPDLSM1
01 30
Iteration Number
Res
idua
l
1
01 30
Iteration Number
Res
idua
l
819
Damping parameter: Γ= 200000x0.5n-1, n=1,2,…,30
RDLSMRDLSM
31
DLSM ImagesDLSM Images with the Exact Model with the Exact Model
0
3Z
(km
)
6 12X (km)
PDLSM after 8 Iterations0
3
Z (
km)
6 12X (km)
RDLSM after 19 Iterations
32
Model Sensitivity TestModel Sensitivity Test• Exact model:
– the actual model
• Geological model:– Skeletonized Migrated
Image
• Grid model:– The region is divided into
sections; each section has a point scatterer in the center.
Exact Model
0
3Z
(km
)
6 12X (km)
Geological Model
0
3Z
(km
)
6 12X (km)
Zoom View of Grid Model
1
2Z
(km
)
10 11X (km)
250 x 250 m
33
Deblurring Filter with the Geological Model Deblurring Filter with the Geological Model Step1: Compute Matching OperatorStep1: Compute Matching Operator
Reference Migration Image
0
3Z
(km
)
6 12X (km)
Geological Model
0
3
Z (
km)
6 12X (km)
f
34
Deblurring Filter with the Geological Model Deblurring Filter with the Geological Model Step2: Apply the OperatorStep2: Apply the Operator
Deblurred Migration Image
0
3Z
(km
)
6 12X (km)
Actual Migration Image
0
3
Z (
km)
6 12X (km)
f
35
DLSM Convergence CurvesDLSM Convergence Curves
Preconditioned DLSMPreconditioned DLSM
1
01 30
Iteration Number
Res
idua
l
Regularized DLSMRegularized DLSM
1
01 30
Iteration Number
Res
idua
l
20 12
Damping parameter: Γ= 200000x0.5n-1, n=1,2,…,30 36
DLSM ImagesDLSM Images with the Geological Model with the Geological Model
0
3Z
(km
)
6 12X (km)
PDLSM after 12 Iterations0
3
Z (
km)
6 12X (km)
RDLSM after 20 Iterations
37
Zoom View of Grid Model
1
2
Z (
km)
10 11X (km)
Deblurring Filter with the Grid Model Deblurring Filter with the Grid Model Step1: Compute Matching OperatorStep1: Compute Matching Operator
Reference Migration Image0
3Z
(km
)
6 12X (km)
f
38
Deblurring Filter with the Grid Model Deblurring Filter with the Grid Model Step2: Apply the OperatorStep2: Apply the Operator
Deblurred Migration Image
0
3Z
(km
)
6 12X (km)
Actual Migration Image
0
3
Z (
km)
6 12X (km)
f
39
Regularized DLSMRegularized DLSM
1
01 30
Iteration Number
Res
idua
l
Damping parameter: Γ= 200000x0.5n-1, n=1,2,…,30
DLSM Convergence CurvesDLSM Convergence Curves
Preconditioned DLSMPreconditioned DLSM
1
01 30
Iteration Number
Res
idua
l20 10
40
DLSM ImagesDLSM Images with the Grid Model with the Grid Model
0
3
Z (
km)
6 12X (km)
RDLSM after 20 Iterations0
3Z
(km
)
6 12X (km)
PDLSM after 10 Iterations
41
Marmousi2 Test Summary (1)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)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 TestThe Gulf of Mexico Data Test
84
TW
T(s
)
X (km)18
02D Poststack Marine Data
44
The Gulf of Mexico Data TestThe 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 Zero-offset Data from for a Grid Modelfor a Grid Model
84
TW
T(s
)
X (km)18
0
Scatterer Interval: 500 m x 500 m
46
Zoom View of Reference Migration Zoom View of Reference Migration Image for a Grid ModelImage for a Grid Model
8
1.2
Z (
km)
X (km)
1310.5
0.4
47
Kirchhoff MigrationKirchhoff Migration
8
1
1.5
Z (
km)
X (km)
1310.5
0.5
48
Deblurred Migration ImageDeblurred Migration ImageZ
(km
)
X (km)
8
1
1.51310.5
0.5
49
Geological ModelGeological Model
8
1
1.5
Z (
km)
X (km)
1310.5
0.5
0
0.1
-0.1
Reflectivity
50
Comparison of Imaging ResultsComparison of Imaging Results
0.5
1.5
Z (
km)
8 13X (km)
Kirchhoff Migration
51
Box A: Comparison of ImagesBox A: Comparison of Images
0.5
0.7
Z (
km)
9.6 10.6X (km)
Migration
0.5
0.7
Z (
km)
9.6 10.6X (km)
LSM after 3 Iterations
0.5
0.7
Z (
km)
9.6 10.6X (km)
DLSM after 3 Iterations
0.5
0.7
Z (
km)
9.6 10.6X (km)
LSM after 10 Iterations
52
Box B: Comparison of ImagesBox B: Comparison of ImagesMigration
1
1.2
Z (
km)
11 12X (km)
LSM after 3 Iterations
1.2
Z (
km)
11 12X (km)
1
DLSM after 3 Iterations
1.2Z (
km)
11 12X (km)
1
LSM after 10 Iterations
1.2
Z (
km)
11 12X (km)
1
53
Total Computational CostTotal Computational CostMigration
1
1.2
Z (
km)
11 12X (km)
LSM after 3 Iterations
1.2
Z (
km)
11 12X (km)
1
DLSM after 3 Iterations
1.2Z (
km)
11 12X (km)
1
LSM after 10 Iterations
1.2
Z (
km)
11 12X (km)
1
1 9
19+ 3054
Total Computational CostTotal Computational Cost
• Migration 1• LSM 3 Iterations 9• LSM 10 Iterations 30• DLSM 3 Iterations 19+
– Deblurring with the grid model 3– Deblurring with the geological model 4+– DLSM 3 Iterations 12
55
The GOM Data Test SummaryThe GOM Data Test Summary
• DLSM can successfully provide an improved LSM image with an affordable computer expense.
56
OutlinesOutlines
• Motivation
• Deblurring filter theory
• A numerical results of the deblurring filter
• Deblurred LSM theory
• Numerical results of the deblurred LSM
• Conclusions
57
ConclusionsConclusions
• 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 WorksFuture 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
AcknowledgementsAcknowledgements• 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
ThanksThanks
61