Least squares migration of elastic data Aaron Stanton and Mauricio Sacchi PIMS 2015

Outline Motivation Least squares migration of elastic data Adjoint (migration) operator – Wavefield de composition – Extrapolation – Imaging condition Forward (de-migration) operator – Adjoint of Imaging condition – Extrapolation – Wavefield re composition Preconditioning via Poynting vectors Example

Motivation To improve the imaging of converted wave data in the presence of noise, missing data, and poor illumination

Least squares migration with quadratic regularization L  extrapolates P & S potentials and recompose into data components

The forward operator Extrapolation and wavefield recomposition: Split-Step Padé Fourier propagator Blending of wavefields into data components

What is H -1 ? H -1 blends wavefield potentials into data components

What is H -1 ? H -1 blends wavefield potentials into data components

What is H -1 ? H -1 blends wavefield potentials into data components

What is H -1 ? H -1 blends wavefield potentials into data components If we assume isotropy we can use Helmholtz decomposition

Helmholtz decomposition

Etgen, 1988

Helmholtz recomposition Etgen, 1988

The forward operator Extrapolation and wavefield recomposition:

The adjoint operator wavefield decomposition and extrapolation: implies that the adjoint operator could result in some crosstalk artifacts

Quadratic Regularization

Angle domain regularization By a change of variables z = Dm we write Where D -1 is smoothing across angles within each angle gather

Poynting vector method Imaging is done without spatial lags Vectors are calculated from source and receiver side wavefields independently From the source and receiver side Poynting vectors the angle can be defined in many different ways Higginbotham et al, 2010

Poynting vectors Typically implemented in RTM: This approximation leads to a technique to calculate Poynting vectors in WEM (Dickens and Winbow, 2011) (Yoon and Marfurt, 2006)

Poynting vectors in WEM 1.calculate gradient components for the source side wavefield: 2.inverse Fourier transform over the spatial axes (x and z) 3.obtain that corresponds to the time of reflection by calculating the zero- lag cross correlation with the receiver wavefield: 4.normalize the elements of 5.repeat steps 1 to 4 for the z-component of the source side wavefield to obtain 6.repeat steps 1 to 5 for the receiver side wavefield

Unit vectors

Angle with respect to vertical: Angle with respect to reflector normal: Sign of angle (Duan and Sava 2014): Conversion from unit vectors to angle

Three Interfaces

m ps for 1 shot gather

Angles for 1 shot gather

Corrected gather

MARMOUSI 2 Synthetic data example

vpvp

vsvs

Shot gathers

Migration: m pp 25˚ incidence angle

LS Migration: m pp 25˚ incidence angle

Migration: m ps 25˚ incidence angle

LS Migration: m ps 25˚ incidence angle

ANTICLINE MODEL Synthetic data example

Simulated OBC Acquisition ★ osx = 100m ; dsx = 100m ; nsx = 69 ; sz = 10m ; dgx = 8m ; gz = 550m ; recording aperture = 6608m

X-Component

Z-Component

Angles for 1 migrated shot

Adjoint Mpp Constant Incidence angle of 10˚

Least Squares Mpp Constant Incidence angle of 10˚

Adjoint Mpp x = 1600m

Least Squares Mpp x = 1600m

Adjoint Mps Constant Incidence angle of 10˚

Least Squares Mps Constant Incidence angle of 10˚

Adjoint Mps x = 1600m

Least Squares Mps x = 1600m

Misfit Iteration number Relative misfit

Conclusions We implemented elastic least squares migration using the one way wave equation The forward operator consists of scalar extrapolation of P and S potentials followed by wavefield recomposition The method has application in imaging, regularization and wavefield separation of multicomponent data

Acknowledgements We gratefully acknowledge the sponsors of the Signal Analysis and Imaging Group for their generous support