An automatic wave equation migration velocity analysis by differential semblance optimization

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An automatic wave equation migration velocity analysis by differential semblance optimization

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An automatic wave equation migration velocity analysis by differential semblance optimization

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An automatic wave equation migration velocity analysis by differential semblance optimization

The Rice Inversion Project

- Simultaneous optimization for velocity and image
- Shot-record wave-equation migration.

- Nonlinear Local Optimization
- Objective function
- Gradient of the objective function

- Remark:
- Objective function requires to be smooth .
- Differential semblance objective function is smooth.

z

x

offset image

angle image

z

z

h

h

I : offset domain image

c : velocity

h : offset parameter

P : differential semblance operator

|| ||: L2 norm

M : set of smooth velocity functions

Definitions:

Downward continuation and upward continuation

S0

R0

gradient

derivative cross correlate*

down

down

SZ

RZ

DS*

DR*

cross correlate

up

up

S*z

R*z

image

cross correlate reference field

spline

Vmodel

gmodel

spline*

M : set of smooth velocity functions

Vimage I

gimage

migration

differential migration*

Optimization

BFGS algorithm for nonlinear iteration

- Objective function evaluation

- Gradient calculation

loop

- Update search direction

coutIout

- Flat reflector, constant velocity
- Marmousi data set

Experiment of flat reflector at constant velocity

x

Ccorrect = 2km/sec

z

Offset image

Angle image

Initial iterate:

Image (v0 = 1.8km/sec)

Image space: 401 by 80

Model space: 4 by 4

Offset image

Angle image

Iteration 5:

Image

Iterations

v5: Output velocity at

iteration 5

vbest - v5

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

Start with v0 = 2km/sec

Iteration 2

Iteration 3

Iteration 4

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.