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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 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 || ||: 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
spline Vmodel gmodel spline* M : set of smooth velocity functions Gradient smoothing using spline evaluation Vimage I gimage migration differential migration*
Optimization BFGS algorithm for nonlinear iteration • Objective function evaluation • Gradient calculation loop • Update search direction coutIout
Synthetic Examples • Flat reflector, constant velocity • Marmousi data set
Experiment of flat reflector at constant velocity x Ccorrect = 2km/sec z
Offset image Angle image Initial iterate: Image (v0 = 1.8km/sec) Image space: 401 by 80 Model space: 4 by 4
Offset image Angle image Iteration 5: Image
Iterations v5: Output velocity at iteration 5 vbest - v5
Initial iterate: Image (v0=1.8km/sec) Image space: 921 by 60 Model space: 6 by 6 Offset image Angle image
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
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
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.