Transcript Least Squares Migration Combined with a Deblurring Filter

Iterative Migration Deconvolution (IMD)
with Migration Green’s Functions as
Preconditioners
Naoshi Aoki
Feb. 5, 2009
1
Outline
• Introduction
• Theory
– Inexpensive IMD
• Numerical results
– 2D model IMD test
– 3D model IMD test
• When should we use IMD and LSM ?
• Conclusions
2
Outline
• Introduction
• Theory
– Inexpensive IMD
• Numerical results
– 2D model IMD test
– 3D model IMD test
• When should we use IMD and LSM ?
• Conclusions
3
Deblurring Migration Image
• Migration
mmig = L d  = L Lm 
T
T
• Two methods to deblur the migration image
– Least Squares Migration (e.g., Nemeth et al.,1999)
( k 1)
m
(k )
=m
  L (Lm
(k )
T
(k )
- d),
– Migration Deconvolution (Hu and Schuster, 2001)
T
1
m = [L L] mmig
4
Migration Green’s Function or MGF
[L L] is known as the
migration Green’s function.
• It is an impulse response of
migration operator.
TWT (s)
•
Synthetic Data
T
Grid Reflectivity
MGFs Model
– acquisition geometry,
– and velocity distribution.
Z (km)
• MGF variation depends on:
5
X (km)
Outline
• Introduction
• Theory
– Inexpensive IMD
• Numerical results
– 2D model IMD test
– 3D model IMD test
• When should we use IMD and LSM ?
• Conclusions
6
Inexpensive IMD Theory
Expensive IMD


m ( k 1)  m ( k )   ( k ) [LT L] [LT L]m ( k ) - m mig ,
where m( k 1) and m( k ) represent the k+1 and k th
T
models,  ( k ) is a step length, and [L L] is
expensive MGF.
Inexpensive IMD with Preconditioned MGFs


ˆ T Gm
ˆ (k ) - m
ˆ mig ,
m( k 1)  m( k )   ( k ) G
ˆ  (LT L)1[LT L] represents a preconditioned
where G
xx
normal matrix that contains the preconditioned MGF in
ˆ mig  (LT L)xx1 LT d denotes amplitude
each subsection, m
compensated migration image.
Expensive and Inexpensive MGFs
T
[L L]
T
1
T
ˆ
G  (L L)xx [L L]
8
Outline
• Introduction
• Theory
– Inexpensive IMD
• Numerical results
– 2D model IMD test
– 3D model IMD test
• When should we use IMD and LSM ?
• Conclusions
9
Test Workflow
Data Preparation Part
MGF Computation Part
Point Scatterer Model
Reflectivity Model
Migration Image
Compute MGFs
IMD Computation Part
Compute IMD
Compare with LSM
10
Reflectivity Model
Data Preparation Part
Migration Image
2D Stick Model
Prestack Migration
Z (km)
0
Z (km)
0
1.8
0
1.8
X (km)
1.8
0
X (km)
1.811
Point Scatterer Model
MGF Computation Part
Compute MGFs
Scatterers
MGFs
0
Z (km)
Z (km)
0
1.8
1.8
0
X (km)
1.8
0
X (km)
1.812
Compute IMD
IMD Computation Part
IMD Image
after 43 Iterations
Z (km)
0
Z (km)
0
Prestack Migration
1.8
1.8
0
X (km)
1.8
0
X (km)
1.813
Compute IMD
IMD vs LSM
Compare with LSM
IMD Image
after 43 Iterations
Z (km)
0
Z (km)
0
1.8
1.8
0
X (km)
1.8
LSM Image
after 30 Iterations
0
X (km)
1.814
Model Residual
Residual
7
IMD Noise level
5
1
30
Iteration number
43
15
Computational Costs
Process
1 Migration
54 MGFs
Computational Costs
(CPU seconds)
9
640
43 IMD Iterations
55
30 LSM Iterations
860
16
Expensive and Inexpensive MGFs
Compute MGFs One by One
Clean MGFs without interference
Compute MGFs at Once
A possible problem is interference
17
from other MGFs.
Outline
• Introduction
• Theory
– Inexpensive IMD
• Numerical results
– 2D model IMD test
– 3D model IMD test
• When should we use IMD and LSM ?
• Conclusions
18
3D Model Test
Model
Model Description
● Source
● Receiver
0
• Model size:
– 1.8 x 1.8 x 1.8 km
Z (km)
• U shape reflectivity anomaly
2
0
0
X (km)
2
2
Depth (m)
Reflectivity
250
1
500
-1
750
1
1000
-1
1250
1
Y (km)
• Cross-spread geometry
– Source : 16 shots, 100 m int.
– Receiver : 16 receivers , 100 m
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int.
Test Workflow
Data Preparation Part
MGF Computation Part
Reflectivity Model
Point Scatterer Model
Migration Image
Compute MGFs
IMD Computation Part
Compute IMD
Compare with LSM
20
Reflectivity Model
Data Preparation
Migration Image
Prestack Migration
Y = 1 km
Prestack Migration
Z = 750 m
0
Z (km)
Y (km)
0
1.8
1.8
0
X (km)
1.8
0
X (km)
1.8
21
Point Scatterer Model
MGF Computation Part
Compute MGFs
MGF Image
Z = 750 m
MGF Image
Y = 1 km
0
Z (km)
Y (km)
0
1.8
1.8
0
X (km)
1.8
0
X (km)
1.8
22
Compute IMD
IMD Computation Part
IMD Image
after 30 Iterations
Prestack
Migration
Y = 1000 m
0
Z (km)
Z (km)
0
1.8
0
1.8
1.8
Z = 750 m
1.8
0
Y (km)
Y (km)
0
0
1.8
1.8
0
X (km)
1.8
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0
X (km)
1.8
IMD vs Prestack Migration
IMD after 30 Iterations
Z = 250 m
750 m
1000 m
1250 m
1500 m
Y (km)
0
1.8
0
1.8 0
1.8 0
1.8 0
X (km)
1.8 0
1.8
Prestack Migration
Y (km)
0
1.8 0
1.8 0
1.8 0
1.8 0
X (km)
1.8 0
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1.8
Compute IMD
IMD vs LSM
IMD image
after 30 Iterations
Compare with LSM
LSM Image
after 30 Iterations
Y = 1000 m
0
Z (km)
Z (km)
0
1.8
0
1.8
1.8
Z = 750 m
1.8
0
Y (km)
Y (km)
0
0
1.8
1.8
0
X (km)
1.8
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0
X (km)
1.8
IMD vs LSM
IMD Images after 30 Iterations
Z = 250 m
750 m
1000 m
1250 m
1500 m
Y (km)
0
1.8 0
1.8 0
1.8 0
1.8 0
X (km)
1.8 0
1.8
LSM Images after 30 Iterations
Y (km)
0
1.8 0
1.8 0
1.8 0
1.8 0
X (km)
1.8 0
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1.8
Model Residual
Residual
84
IMD Noise level
76
1
30
Iteration number
27
Computational Costs
Process
1 Migration
486 MGFs
Computational Costs
(CPU seconds)
190
25500
30 IMD Iterations
65400
30 LSM Iterations
15400
28
Why Is IMD So Slow?
• Computational cost of IMD is 6 times
higher than that of LSM because:
– the cross-spread geometry has a large MGF
variation,
– convolution / cross-correlation is used in the
space domain.
29
Outline
• Introduction
• Theory
– Inexpensive IMD
• Numerical results
– 2D model IMD test
– 3D model IMD test
• When should we use IMD and LSM ?
• Conclusions
30
Difference Between LSM and IMD
• Both methods minimize the misfit with the data.
LSM
IMD

m ( k 1)  m ( k )   ( k ) LT Lm ( k ) - d
d
m
(k )


ˆ T Gm
ˆ (k ) - m
ˆ mig
m( k 1)  m( k )   ( k ) G
ˆ mig
m
m
(k )
31

When Should We Use IMD and LSM ?
IMD
Larger amount of 5-D data
provides smaller MGF variation.
d
LSM
Smaller amount of 5-D data
provides larger MGF variation.
d
32
Suitable Application
IMD
LSM
Large amount of
data
Coarse acquisition
geometry
Complicated
geology
Target oriented
deblurring
33
Outline
• Introduction
• Theory
– Inexpensive IMD
• Numerical results
– 2D model IMD test
– 3D model IMD test
• When should we use IMD and LSM ?
• Conclusions
34
Conclusions
• Inexpensive IMD method with preconditioned
MGF is developed.
• 2D IMD achieves a quality almost equal to that
from LSM with cheaper computational cost.
• 3D IMD test suggests that IMD quality and cost
depend on required MGF density, and
investigation of the required MGF density is
important.
35
Continued Work
• An IMD test on PEMEX RTM image is
presented by Qiong Wu.
36
Thanks
37