Transcript Extended Diffraction-Slice Theorem for Wavepath Traveltime

Least Squares Datuming with
Wave Equation
Yanwei Xue
February 5, 2009
Outline
• Motivation for least squares interferometry
• Interferometric Transform: VSP to SWP
• Numerical test:
• Synthetic data test
• Field data test
• Conclusion & future work
• Progress on VSP to SSP
Motivation
VSP
s
VSP
s
g’
SWP
s
g’
g
g’
g
To get better imaging below salt
g
Outline
• Motivation for least squares interferometry
• Interferometric Transform: VSP to SWP
• Numerical Test:
• Synthetic data test
• Field data test
• Conclusion & Future Work
• Progress on VSP to SSP
VSP to SWP:
∫
G(A|x)
G(B|x) dx
G(A|B)
Workflow
Define d as the input upgoing
data, v as the output SWP and L
as crosscorrelation transform
matrix
Input csg separate csg into
downgoing, upgoing
waves
Redatum d to v by
Forward propagate v and
calculate the residual
Calculate the step length by CG
method, update v, calculate the
residual again
Output v
Yes
Δd is small enough?
No
Outline
• Motivation for least squares interferometry
• Interferometric Transform: VSP to SWP
• Numerical Test:
• Synthetic data test
• Field data test
• Conclusion & Future Work
• Progress on VSP to SSP
VSP CSG
0.0
0.0
Time (s)
LSD Result
Time (s)
Real Data
2.0
2.0
0
x (m)
6250
0
x (m)
6250
SWP CSG
0.0
0.0
Time (s)
Int. Result
Time (s)
Real Data
2.0
2.0
0
x (m)
6250
0
x (m)
6250
SWP CSG
0.0
0.0
Time (s)
LS Result
Time (s)
Real Data
2.0
2.0
0
x (m)
6250
0
x (m)
6250
SWP CSG Traces
Normalized RMS vs
Iteration Number
Normalized RMS
1
.15
0
0
10
Iteration Number
50
Outline
• Motivation for least squares interferometry
• Interferometric Transform: VSP to SWP
• Numerical Test:
• Synthetic data test
• Field data test
• Conclusion & Future Work
• Progress on VSP to SSP
VSP CSG
0
0
Time (s)
INT. CRG
Time (s)
Real CRG
4.0
4.0
0
4000
X (m)
0
4000
X (m)
VSP CSG
0
0
Time (s)
LS CRG
Time (s)
Real CRG
4.0
4.0
0
4000
X (m)
0
4000
X (m)
Normalized RMS vs Iteration Number
Normalized RMS
1
0.4
0
Iteration Number
500
S0
SWP CSG
0
0
Time (s)
LSD CRG
Time (s)
INT. CRG
4.0
4.0
0
4000
X (m)
S1
0
4000
X (m)
Outline
• Motivation for least squares interferometry
• Interferometric Transform: VSP to SWP
• Numerical Test:
• Synthetic data test
• Field data test
• Conclusion & Future Work
• Progress on VSP to SSP
Conclusion
• Least squares interferometric method improves the
datuming results and suppresses artifacts.
• Up/down going separation is required to get good
results
• Deconvolution with source wavelet is required for
convergence
Future work:
Preconditioning scheme to accelerate the convergence rate
Outline
• Motivation for least squares interferometry
• Interferometric Transform: VSP to SWP
• Numerical Test:
• Synthetic data test
• Field data test
• Conclusion & Future Work
• Progress on VSP to SSP
Motivation
VSP
VSP
SSP
Enlarge the illumination of the
VSP data
VSP to SSP
∫
G(A|x)
G(B|x) dx
G(A|B)
Direct Wave
25 ft
24 receivers
0
1.0
0
Time (s)
1.0
0
1000
Z (ft)
Downgoing Wave
0
Z (ft)
1000
Time (s)
10 ft
CRG
Time (s)
98 shots
0
1.0
0
Z (ft)
1000
SSP CSG
INT. Result
LSD Result
0
0.2
Time (s)
Time (s)
0
0
X (ft)
600
0.2
0
X (ft)
600
Normalized RMS vs Iteration Number
Normalized RMS
1
0.1
0
Iteration Number
20
Summary
Least squares method improves the
interferometric results
Next Steps:
Further synthetic tests are needed
Preconditioning scheme for upgoing VSP to SSP
transform.
Thanks