Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Fighting ambiguity of inverse problems in seismic imaging Jörg Schleicher University of Campinas & INCT-GP Colóquio Brasileiro de Matemática Rio de Janeiro, 30/07/2013 Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Reflector-oriented regularization in slope tomography Decomposition of sensitivity kernels in full-waveform inversion Conclusions Contents Introduction 1 Reflector-oriented regularization in slope tomography 2 Decomposition of sensitivity kernels in full-waveform 3 inversion Conclusions 4 Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Contents Introduction 1 Reflector-oriented regularization in slope tomography 2 Decomposition of sensitivity kernels in full-waveform 3 inversion Conclusions 4 Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Motivation Velocity Model 0 0.5 1 Depth (km) 1.5 2 2.5 3 4 5 6 7 8 9 Distance (km) 1.5 2 2.5 3 3.5 4 4.5 km/s Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Motivation Velocity Model 0 0.5 1 Depth (km) 1.5 2 2.5 3 4 5 6 7 8 9 Distance (km) 1.5 2 2.5 3 3.5 4 4.5 km/s Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Motivation Velocity Model 0 0.5 1 Depth (km) 1.5 2 2.5 3 4 5 6 7 8 9 Distance (km) 1.5 2 2.5 3 3.5 4 4.5 km/s Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Objectives Slope tomography: 1 Introduce geologically meaningful constraints Improve velocity model building for depth migration Better recover large scale structural features Improve convergence of layer and grid-based tomography Full-waveform inversion: 2 Decompose sensitivity kernels Understand contributions Invert only important ones Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Objectives Slope tomography: 1 Introduce geologically meaningful constraints Improve velocity model building for depth migration Better recover large scale structural features Improve convergence of layer and grid-based tomography Full-waveform inversion: 2 Decompose sensitivity kernels Understand contributions Invert only important ones Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Inversion Problem: invert Nonlinear relationship between data and parameters d = F ( m ) m ≡ model parameters d ≡ data parameters F ≡ nonlinear functional (wave propagation) Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction Motivation Reflector-oriented regularization in slope tomography Objective Decomposition of sensitivity kernels in full-waveform inversion Inversion Conclusions Frechèt derivatives Solution: Linear iterations: the Frechét derivative δ d = D F ( m 0 ) δ m m 0 ≡ reference model parameters δ d ≡ data perturbation δ m ≡ model parameters perturbations around m 0 D F ≡ Frechét derivative of F Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Contents Introduction 1 Reflector-oriented regularization in slope tomography 2 Decomposition of sensitivity kernels in full-waveform 3 inversion Conclusions 4 Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion What is slope tomography? Data space: d = { ( x s , x r , T sr , s s , s r ) n } for n = 1 , . . . , N Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion What is slope tomography? Model space: m = { p , ( X , τ s , τ r , θ s , θ r ) n } for n = 1 , . . . , N Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Frechét derivatives computation: dynamic ray tracing The Hamiltonian (eikonal equation): H ( x , s ) = 1 2 ( p ( x ) s · s − 1 ) = 0 x - position along the ray s - slowness vector along the ray p ( x ) - velocity square field Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Frechét derivatives computation: dynamic ray tracing x ∇ x H d = d τ s −∇ s H ∇ s ∇ T ∇ s ∇ T δ x x H s H δ x d = d τ −∇ x ∇ T −∇ x ∇ T δ s x H s H δ s ∇ s ∇ T p H δ p . + −∇ x ( ∇ T p H δ p ) Reference ray Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Frechét derivatives computation: dynamic ray tracing x ∇ x H d = d τ s −∇ s H ∇ s ∇ T ∇ s ∇ T δ x x H s H δ x d = d τ −∇ x ∇ T −∇ x ∇ T δ s x H s H δ s ∇ s ∇ T p H δ p . + −∇ x ( ∇ T p H δ p ) Paraxial rays Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Initial Conditions Slowness Direction: � � d n I − n ∇ T s H δ x = 0 and δ s = s d θ δθ ∇ T s H n Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Initial Conditions Scattering point position: ∇ T and δ s = − ∇ s H x H δ X δ x = δ X �∇ s H� �∇ s H� Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Initial Conditions Velocity model parameters: ∇ T p H δ p and δ s = − ∇ s H δ x = 0 �∇ s H� �∇ s H� Jörg Schleicher Fighting ambiguity in seismic inverse problems
Introduction What is slope tomography? Reflector-oriented regularization in slope tomography Smoothness constraints Decomposition of sensitivity kernels in full-waveform inversion Numerical Experiments Conclusions Discussion Linear iterations 1- reference model m 0 2- ray tracing is perfomed to calculate synthetic data ( d c ), δ d = d OBS − d c 3- compute Frechét derivatives D F ( m 0 ) 4- solve for model perturbations δ m 5- Update reference model m 0 ← m 0 + δ m 6- If updated model fits the data within a specified tolerance stop; otherwise, iterate Jörg Schleicher Fighting ambiguity in seismic inverse problems
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