Wave-equation migration velocity analysis Paul Sava* Stanford University Biondo Biondi Stanford University Sergey Fomel UT Austin

The problem Depth imaging –image: migration –velocity: migration velocity analysis Migration and MVA are inseparable “Everyhing depends on v(x,y,z)” »JF Claerbout, 1999

An approximation

A better approximation

In the “big picture” Kirchhoff migration traveltime tomography wavefronts wave-equation migration wave-equation MVA (WEMVA) wavefields

Agenda Theoretical background WEMVA methodology Scattering Imaging Non-linear operator Linear operator Image perturbation WEMVA applications

Wavefield scattering

Wavefield scattering

Scattered wavefield Medium perturbation Wavefield perturbation

Agenda Theoretical background WEMVA methodology Scattering Imaging Non-linear operator Linear operator Image perturbation WEMVA applications

Imaging: Correct velocity Background velocity Migrated image Reflectivity model What the data tell us...What migration does... location depth location depth

Imaging: Incorrect velocity Perturbed velocity Migrated image Reflectivity model What the data tell us...What migration does... location depth location depth

WEMVA objective Velocity perturbation Image perturbation slowness perturbation (unknown) WEMVA operator image perturbation (known) location depth location depth

Agenda Theoretical background WEMVA methodology Scattering Imaging Non-linear operator Linear operator Image perturbation WEMVA applications

Double Square-Root Equation Fourier Finite Difference Generalized Screen Propagator Wavefield extrapolation

Slowness perturbation

slowness perturbation background wavefield perturbation Wavefield perturbation

Agenda Theoretical background WEMVA methodology Scattering Imaging Non-linear operator Linear operator Image perturbation WEMVA applications

Linearizations Unit circle Born approximation

Linearizations Unit circle

Linearizations Unit circle

Linear WEMVA slowness perturbation (unknown) WEMVA operator image perturbation (known)

Agenda Theoretical background WEMVA methodology Scattering Imaging Non-linear operator Linear operator Image perturbation WEMVA applications

Correct velocity

Incorrect velocity

Image perturbation

Failure!

Small phase limitation

What can we do? Define another objective function –e.g. DSO Construct an image perturbation which obeys the Born approximation...

Residual migration

Analytical image perturbation Computed analytically Picked from data

Analytical image perturbation

Image perturbations comparison

Slowness perturbations

Migrated images

Migrated images: angle gathers

Agenda Theoretical background WEMVA methodology Scattering Imaging Non-linear operator Linear operator Image perturbation WEMVA applications

Other applications 4-D seismic monitoring –image perturbations over time –no need to construct Focusing MVA –zero offset data

4D seismic monitoring

4D seismic monitoring

4D seismic monitoring

Focusing MVA Incorrect image Correct image

Focusing MVA

Focusing MVA

Focusing MVA

Focusing MVA

Focusing MVA

Focusing MVA

Summary Wave-equation MVA wavefield extrapolation image space objective focusing and moveouts interpretation guided Linearization linear operator construct image perturbations