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Least squares migration of elastic data Aaron Stanton and Mauricio Sacchi PIMS 2015
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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
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Motivation To improve the imaging of converted wave data in the presence of noise, missing data, and poor illumination
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Least squares migration with quadratic regularization L extrapolates P & S potentials and recompose into data components
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The forward operator Extrapolation and wavefield recomposition: Split-Step Padé Fourier propagator Blending of wavefields into data components
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What is H -1 ? H -1 blends wavefield potentials into data components
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What is H -1 ? H -1 blends wavefield potentials into data components
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What is H -1 ? H -1 blends wavefield potentials into data components
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What is H -1 ? H -1 blends wavefield potentials into data components If we assume isotropy we can use Helmholtz decomposition
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Helmholtz decomposition
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Etgen, 1988
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Helmholtz recomposition Etgen, 1988
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The forward operator Extrapolation and wavefield recomposition:
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The adjoint operator wavefield decomposition and extrapolation: implies that the adjoint operator could result in some crosstalk artifacts
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Quadratic Regularization
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Angle domain regularization By a change of variables z = Dm we write Where D -1 is smoothing across angles within each angle gather
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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
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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)
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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
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Unit vectors
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Angle with respect to vertical: Angle with respect to reflector normal: Sign of angle (Duan and Sava 2014): Conversion from unit vectors to angle
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Three Interfaces
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m ps for 1 shot gather
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Angles for 1 shot gather
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Corrected gather
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MARMOUSI 2 Synthetic data example
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vpvp
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vsvs
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Shot gathers
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Migration: m pp 25˚ incidence angle
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LS Migration: m pp 25˚ incidence angle
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Migration: m ps 25˚ incidence angle
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LS Migration: m ps 25˚ incidence angle
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ANTICLINE MODEL Synthetic data example
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Simulated OBC Acquisition ★ osx = 100m ; dsx = 100m ; nsx = 69 ; sz = 10m ; dgx = 8m ; gz = 550m ; recording aperture = 6608m
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X-Component
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Z-Component
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Angles for 1 migrated shot
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Adjoint Mpp Constant Incidence angle of 10˚
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Least Squares Mpp Constant Incidence angle of 10˚
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Adjoint Mpp CIG @ x = 1600m
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Least Squares Mpp CIG @ x = 1600m
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Adjoint Mps Constant Incidence angle of 10˚
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Least Squares Mps Constant Incidence angle of 10˚
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Adjoint Mps CIG @ x = 1600m
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Least Squares Mps CIG @ x = 1600m
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Misfit Iteration number Relative misfit
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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
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Acknowledgements We gratefully acknowledge the sponsors of the Signal Analysis and Imaging Group for their generous support
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