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 differential semblance optimization

  • Simultaneous optimization for velocity and image

  • Shot-record wave-equation migration.


Theory
Theory differential semblance optimization

  • 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 differential semblance optimization

z

x

offset image

angle image

z

z

h

h


Objective function
Objective function differential semblance optimization

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 differential semblance optimization

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 differential semblance optimization

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 differential semblance optimization

BFGS algorithm for nonlinear iteration

  • Objective function evaluation

  • Gradient calculation

loop

  • Update search direction

coutIout


Synthetic examples
Synthetic Examples differential semblance optimization

  • 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 differential semblance optimization

x

Ccorrect = 2km/sec

z


An automatic wave equation migration velocity analysis by differential semblance optimization

Offset image differential semblance optimization

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 differential semblance optimization

Angle image

Iteration 5:

Image


An automatic wave equation migration velocity analysis by differential semblance optimization

Iterations differential semblance optimization

v5: Output velocity at

iteration 5

vbest - v5


An automatic wave equation migration velocity analysis by differential semblance optimization

Marmousi data set differential semblance optimization


Marmousi data set
Marmousi data set differential semblance optimization


An automatic wave equation migration velocity analysis by differential semblance optimization

V differential semblance optimization


An automatic wave equation migration velocity analysis by differential semblance optimization

Initial iterate: differential semblance optimization

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: differential semblance optimization

Image

Offset image

Angle image


An automatic wave equation migration velocity analysis by differential semblance optimization

v differential semblance optimization5: 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 differential semblance optimization

Vbackground = 2 km/sec


An automatic wave equation migration velocity analysis by differential semblance optimization

Seismogram differential semblance optimization

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 differential semblance optimization

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 differential semblance optimization


An automatic wave equation migration velocity analysis by differential semblance optimization

Conclusions differential semblance optimization

  • 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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