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Multi-source Least-squares Migration with Topography Dongliang Zhang and Gerard Schuster King Abdullah University of Science and Technology

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Outline Summary Theory Use ghost extrapolation to reduce stair-step diffractions from irregular surfaces Numerical Example Tests on Marmousi model and Foothills model Motivation Irregular surface problems

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Outline Summary Theory Use ghost extrapolation to reduce stair-step diffractions from irregular surfaces Numerical Example Tests on Marmousi model and Foothills model Motivation Irregular surface problems

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Irregular Surface Problems Datuming the data from irregular surface to flat surface Motivation

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Problem: Irregular Surface Using Ghost extrapolation Motivation RTM migrates directly from the irregular surface Air Surface Stair step Subsurface Solution: Ghost RTM

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Outline Summary Theory Use ghost extrapolation to reduce stair-step diffractions from irregular surfaces Numerical Example Tests on Marmousi model and Foothills model Motivation Irregular surface problems

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Least-squares Migration

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Workflow of Multisource LSM with Topography 1.Forward modeling with topography to calculate the data residual 3. Update the reflectivity using the conjugate gradient method 2. Calculate gradient (RTM image) of data residual with topography Blended encoded shot gathers

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Forward Modeling with Topography Difficulty : Implement free surface boundary condition Calculate the pressure on the points near by the free surface Acoustic equation: Ghost point

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Ghost Extrapolation Z i,j Z i-1,j Z i-2,j Z i+1,j Z i+2,j Surface ZbZb 0 Taylor Series

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Extrapolation in z directionExtrapolation in x direction Ghost Extrapolation

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Example of Dipping Surface Surface Air Surface Stair step Subsurface 0 X (km) 2 ZoomModel 0 1.5 Z (km)

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Mirror image Common Shot Gather P i-1,j P i-2,j P i+2,j =- P i-2,j P i+1,j =-P i-1,j Air Zero velocity layer V=0 Subsurface Air Ghost extrapolation 0 X (km) 2 0 1.5 Z (km)

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Zoom Views Conventional method New method

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Outline Summary Theory Use ghost extrapolation to reduce stair-step diffractions from irregular surfaces Numerical Example Tests on Marmousi model and Foothills model Motivation Irregular surface problems

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0 X (km) 2 Grids size: 201 x 400 dx=dz=5 m Peak Freq.: 25 Hz Shots: 200 Receiver: 400 Max difference of elevation: 180 m Marmousi Model 0 1 Z (km) 0 1 V (km/s)

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Migration Velocity Reflectivity Model Marmousi Model 0 X (km) 2 0 1 Z (km) 0 1 Ghost FD 0 X (km) 2 Common Shot Gather 0 2 T (s)

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0 X (km) 2 Ghost LSRTM Image Ghost FD Marmousi Model Ghost FD Conventional FD LSRTM Image RTM ImageGhost RTM Image 0 X (km) 2 0 1 Z (km) 0 1

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Zoom Views Ghost FD Ghost LSRTM Image Ghost FDConventional FD LSRTM Image RTM ImageGhost RTM Image Conventional FD

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0 X (km) 8 Grids size: 333 x 833 dx=dz=10 m Peak Freq.: 15 Hz Shots: 208 Receiver: 833 Max difference of elevation: 500 m Foothills Model 0 3 Z (km) 0 6 V (km/s)

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Migration Velocity Reflectivity Model 0 X (km) 8 0 X (km) 2 Common Shot Gather Ghost FD Foothills Model 0 3 Z (km) 0 3 0 2 T (s)

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0 X (km) 8 Ghost LSRTM Image 0 X (km) 8 LSRTM Image Ghost FD Ghost RTM Image Conventional FD RTM Image Conventional FD Foothills Model 0 3 Z (km) 0 3

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Ghost LSRTM Image LSRTM Image Ghost FD Ghost RTM Image Conventional FD RTM Image Conventional FD Zoom Views

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Summary MLSM can produce high quality images efficiently: MLSM with topography produces high quality image, multi-source saves the computational time Ghost extrapolation can reduce stair-step diffraction artifacts Future work: Using 2D ghost extrapolation Test on field data High accuracy for the free surface boundary condition Elastic

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Thank you!

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