The Rice Inversion Project 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 Remark: 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 || ||: L2 norm M : set of smooth velocity functions

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

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

Synthetic Examples Flat reflector, constant velocity Marmousi data set

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

Initial iterate: Image (v0 = 1.8km/sec) Offset image Angle image Image space: 401 by 80 Model space: 4 by 4 Offset image Angle image

Iteration 5: Image Offset image Angle image

Iterations v5: Output velocity at iteration 5 vbest - v5

Marmousi data set

Marmousi data set

V

Initial iterate: Image (v0=1.8km/sec) Offset image Angle image Image space: 921 by 60 Model space: 6 by 6 Offset image Angle image

Iterate 5: Image Offset image Angle image

v5 - vbest iterations v5: output velocity at iteration 5 vbest: best spline interpolated velocity v5 - 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 Start with v0 = 2km/sec Iteration 2 Iteration 3 Iteration 4

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.