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

offset image

z

x

z z

angle image

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

derivative cross correlate*

cross correlate reference field

cross correlate

R0S0

image

DS* DR*

gradient

down down

up up

S*z R*

z

Downward continuation and upward continuationDefinitions:

SZ RZ

Gradient smoothing using spline evaluation

Vimage I

gimage

migration

differential migration*

spline

spline*

Vmodel

gmodel

M : set of smooth velocity functions

Optimization

• Objective function evaluation

• Gradient calculation

BFGS algorithm for nonlinear iteration

• Update search direction

loop

cout Iout

Synthetic Examples

• Flat reflector, constant velocity

• Marmousi data set

Experiment of flat reflector at constant velocity

x

z Ccorrect = 2km/sec

Initial iterate:

Image (v0 = 1.8km/sec)

Image space: 401 by 80

Model space: 4 by 4

Offset image Angle image

Offset image Angle image

Iteration 5:

Image

Iterations

v5: Output velocity at

iteration 5

vbest - v5

Marmousi data set

Marmousi data set

V

Offset image Angle image

Initial iterate:

Image (v0=1.8km/sec)

Image space: 921 by 60

Model space: 6 by 6

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

Shot gathers far away from the low velocity lense

Shot gathers near the low velocity lense

Seismogram

Iteration 1

Iteration 2

Iteration 3

Iteration 4

Start with v0 = 2km/sec

1.0 1.5 2.0 2.5 3.0

• 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.

Conclusions