Transcript Wave equation migration velocity analysis by differential

An automatic wave equation
migration velocity analysis by
differential semblance
optimization
The Rice Inversion Project
Objective
• Simultaneous optimization for velocity and
image
• Shot-record wave-equation migration.
Theory
•
•
Nonlinear Local Optimization
–
Objective function
–
Gradient of the objective function
Remark:
–
–
Objective function requires to be smooth .
Differential semblance objective function is smooth.
Differential semblance criteria
z
x
offset image
angle image
z
z
h
h
Objective function
I : offset domain image
c : velocity
h : offset parameter
P : differential semblance operator
|| ||: L2 norm
M : set of smooth velocity functions
Gradient calculation
Definitions:
Downward continuation and upward continuation
S0
R0
down
SZ
down
RZ
gradient
derivative cross correlate*
DS*
DR*
cross correlate
up
image
S* z
up
R*z
cross correlate reference field
Gradient smoothing using spline
evaluation
Vmodel
spline
gmodel
spline*
Vimage
gimage
migration
differential migration*
M : set of smooth velocity functions
I
Optimization
BFGS algorithm for nonlinear iteration
• Objective function evaluation
• Gradient calculation
• Update search direction
cout
Iout
loop
Synthetic Examples
• Flat reflector, constant velocity
• Marmousi data set
Experiment of flat reflector at constant velocity
x
Ccorrect = 2km/sec
z
Initial iterate:
Image (v0 = 1.8km/sec)
Image space: 401 by 80
Model space: 4 by 4
Offset image
Angle image
Iteration 5:
Image
Offset image
Angle image
Iterations
v5: Output velocity at
iteration 5
vbest - v5
Marmousi data set
Marmousi data set
V
Initial iterate:
Image (v0=1.8km/sec)
Image space: 921 by 60
Model space: 6 by 6
Offset image
Angle image
Iterate 5:
Image
Offset image
Angle image
v5: output velocity at
iteration 5
vbest: best spline
interpolated velocity
v5 - vbest
iterations
Low velocity lense + constant velocity background
Vbackground = 2 km/sec
Seismogram
Shot gathers far away from the
low velocity lense
Shot gathers near the low
velocity lense
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Start with v0 = 2km/sec
1.0
1.5
2.0
2.5
3.0
Conclusions
• Offset domain DSO is a good substitute for angle domain
DSO.
• Image domain gradient needs to be properly smoothed.
• DSO is sensitive to the quality of the image.
• Differential semblance optimization by wave equation
migration is promising.