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. Objective. Simultaneous optimization for velocity and image Shot-record wave-equation migration. Theory. Nonlinear Local Optimization Objective function

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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


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

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

Gradient smoothing using spline evaluation

Vimage I

gimage

migration

differential migration*


Optimization

BFGS algorithm for nonlinear iteration

  • Objective function evaluation

  • Gradient calculation

loop

  • Update search direction

coutIout


Synthetic Examples

  • 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


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.


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