Wave-equation migration velocity analysis beyond the Born approximation Paul Sava* Stanford University Sergey Fomel UT Austin (UC Berkeley)

Imaging=MVA+Migration Migration wavefield based Migration velocity analysis (MVA) traveltime based Compatible migration and MVA methods

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

Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

Wavefields or traveltimes?

Wavefields or traveltimes?

Scattered wavefield Medium perturbation Wavefield perturbation

Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

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

Wave-equation MVA: Objective Velocity perturbation Image perturbation slowness perturbation (unknown) WEMVA operator image perturbation (known) location depth location depth

–migrated images –moveout and focusing –use amplitudes –parabolic wave equation –multipathing –slow –picked traveltimes –moveout –ignore amplitudes –eikonal equation –fast Comparison of MVA methods Wave-equation MVATraveltime tomography

Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

What is the image perturbation? FocusingFlatness Residual process: moveout migration focusing slowness perturbation (unknown) WEMVA operator image perturbation (known) location depth angle

Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

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

“Wave-equation” migration

Slowness perturbation

slowness perturbation background wavefield perturbation Wavefield perturbation

Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

Born approximation Small perturbations! Born linearization Non-linear WEMVA slowness perturbation (unknown) WEMVA operator image perturbation (known) Unit circle

Does it work? What if the perturbations are not small? Location [km] Depth [km]

Synthetic example

Born approximation 1%10%

Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

Wavefield continuation Bilinear Implicit Explicit(Born approximation)

Exponential approximations Unit circle

A family of linearizations Linear WEMVA slowness perturbation (unknown) WEMVA operator image perturbation (known)

Improved linearizations 1%10% 40%

Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

Summary Wave-equation MVA wavefield-continuation improved focusing image space (improve the image) interpretation guided Improved WEMVA better approximations no additional cost further refinement