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*

slide9
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
slide12
Offset image

Angle image

Initial iterate:

Image (v0 = 1.8km/sec)

Image space: 401 by 80

Model space: 4 by 4

slide13
Offset image

Angle image

Iteration 5:

Image

slide14
Iterations

v5: Output velocity at

iteration 5

vbest - v5

slide18
Initial iterate:

Image (v0=1.8km/sec)

Image space: 921 by 60

Model space: 6 by 6

Offset image

Angle image

slide19
Iterate 5:

Image

Offset image

Angle image

slide20
v5: output velocity at iteration 5

vbest: best spline interpolated velocity

v5 - vbest

iterations

slide22
Seismogram

Shot gathers far away from the low velocity lense

Shot gathers near the low velocity lense

slide23
Iteration 1

Start with v0 = 2km/sec

Iteration 2

Iteration 3

Iteration 4

slide25
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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