Analytical image perturbations for wave-equation migration velocity analysis Paul Sava & Biondo Biondi Stanford University

Wave-equation MVA (WEMVA) Wavefield-based MVA method Closely related to –Wave-equation migration –Wave-equation tomography Benefits –Finite-frequency –Multipathing –Hi resolution

A tomography problem Traveltime tomography/MVA Wave-equation tomography Wave-equation MVA qq  t traveltime  d data  R image L ray fieldwavefield

Outline 1.WEMVA review 2.Image perturbation 3.Field data example

WEMVA: main idea

Born approximation

WEMVA: objective function slowness perturbation image perturbation slowness perturbation (unknown) Linear WEMVA operator image perturbation (known)

Slowness backprojection slowness perturbation image perturbation slowness perturbation image perturbation

MVA information Traveltime MVAWave-equation MVA Offset focusing (flat gathers) Spatial focusing Frequency redundancy  z  z xx

Outline 1.WEMVA review 2.Image perturbation 3.Field data example

“Data” estimate Traveltime MVA Wave-equation tomography Wave-equation MVA tt dd RR ray tracing data modeling residual migration

Prestack Stolt residual migration Background image R 1 Velocity ratio  Image perturbation  RR

Incorrect velocity Correct velocity Zero offset image Angle gathers Synthetic model

Residual migration: the problem

Differential image perturbation Image difference Image differential ComputedMeasured

Background image Zero offset image Angle gathers Background image

Differential image Zero offset image Angle gathers

Image to slowness perturbation Slowness perturbation Image perturbation

Image comparison Updated slowness Correct slowness Zero offset image slowness

Outline 1.WEMVA review 2.Image perturbation 3.Field data example

Field data example North Sea –Salt environment –One non-linear iteration Migration (background image) Residual migration (image perturbation) Slowness inversion (slowness perturbation) Slowness update (updated slowness) Re-migration (updated image) location depth

locationdepth Zero offset image Angle gathers Background slowness Background image

depth velocity ratio Semblance Angle-gathers

locationdepth Zero offset image Background image location “Ratio” map

locationdepthlocation Zero offset image Background image Image perturbation

locationdepthlocation Zero offset image Image perturbation Slowness perturbation

locationdepth Zero offset image Angle gathers Background slowness Background image

locationdepth Zero offset image Angle gathers Updated slowness Updated image

depth location Angle gathers “Correct” slowness Zero offset image “Correct” image

Summary Wave-equation MVA –Finite frequency –Multipathing –Hi resolution –Image space objective function Image perturbation –From prestack Stolt residual migration –Differential method –Compliant with the Born approximation