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

An automatic wave equation migration velocity analysis by differential semblance optimization

The Rice Inversion Project


Objective

Objective

  • Simultaneous optimization for velocity and image

  • Shot-record wave-equation migration.


Theory

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

Differential semblance criteria

z

x

offset image

angle image

z

z

h

h


Objective function

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

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


Gradient smoothing using spline evaluation

spline

Vmodel

gmodel

spline*

M : set of smooth velocity functions

Gradient smoothing using spline evaluation

Vimage I

gimage

migration

differential migration*


An automatic wave equation migration velocity analysis by differential semblance optimization

Optimization

BFGS algorithm for nonlinear iteration

  • Objective function evaluation

  • Gradient calculation

loop

  • Update search direction

coutIout


Synthetic examples

Synthetic Examples

  • Flat reflector, constant velocity

  • Marmousi data set


An automatic wave equation migration velocity analysis by differential semblance optimization

Experiment of flat reflector at constant velocity

x

Ccorrect = 2km/sec

z


An automatic wave equation migration velocity analysis by differential semblance optimization

Offset image

Angle image

Initial iterate:

Image (v0 = 1.8km/sec)

Image space: 401 by 80

Model space: 4 by 4


An automatic wave equation migration velocity analysis by differential semblance optimization

Offset image

Angle image

Iteration 5:

Image


An automatic wave equation migration velocity analysis by differential semblance optimization

Iterations

v5: Output velocity at

iteration 5

vbest - v5


An automatic wave equation migration velocity analysis by differential semblance optimization

Marmousi data set


Marmousi data set

Marmousi data set


An automatic wave equation migration velocity analysis by differential semblance optimization

V


An automatic wave equation migration velocity analysis by differential semblance optimization

Initial iterate:

Image (v0=1.8km/sec)

Image space: 921 by 60

Model space: 6 by 6

Offset image

Angle image


An automatic wave equation migration velocity analysis by differential semblance optimization

Iterate 5:

Image

Offset image

Angle image


An automatic wave equation migration velocity analysis by differential semblance optimization

v5: output velocity at iteration 5

vbest: best spline interpolated velocity

v5 - vbest

iterations


An automatic wave equation migration velocity analysis by differential semblance optimization

Low velocity lense + constant velocity background

Vbackground = 2 km/sec


An automatic wave equation migration velocity analysis by differential semblance optimization

Seismogram

Shot gathers far away from the low velocity lense

Shot gathers near the low velocity lense


An automatic wave equation migration velocity analysis by differential semblance optimization

Iteration 1

Start with v0 = 2km/sec

Iteration 2

Iteration 3

Iteration 4


An automatic wave equation migration velocity analysis by differential semblance optimization

1.0 1.5 2.0 2.5 3.0


An automatic wave equation migration velocity analysis by differential semblance optimization

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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