An automatic wave equation migration velocity analysis by

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 || ||: L 2 norm M : set of smooth velocity functions

Gradient calculation Definitions: Downward continuation and upward continuation S 0 R 0 down SZ down RZ gradient derivative cross correlate* DS* DR* cross correlate up image S* z up R*z cross correlate reference field

Gradient smoothing using spline evaluation Vmodel spline gmodel spline* Vimage gimage migration differential migration* M : set of smooth velocity functions I

Optimization BFGS algorithm for nonlinear iteration • Objective function evaluation • Gradient calculation • Update search direction cout Iout loop

Synthetic Examples • Flat reflector, constant velocity • Marmousi data set

Experiment of flat reflector at constant velocity x Ccorrect = 2 km/sec z

Initial iterate: Image (v 0 = 1. 8 km/sec) Image space: 401 by 80 Model space: 4 by 4 Offset image Angle image

Iteration 5: Image Offset image Angle image

Iterations v 5: Output velocity at iteration 5 vbest - v 5

Marmousi data set

V

Initial iterate: Image (v 0=1. 8 km/sec) Image space: 921 by 60 Model space: 6 by 6 Offset image Angle image

Iterate 5: Image Offset image Angle image

v 5: output velocity at iteration 5 vbest: best spline interpolated velocity v 5 - 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 Iteration 2 Iteration 3 Iteration 4 Start with v 0 = 2 km/sec

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.