AN ALTERNATIVE STRATEGY FOR NON‐LINEAR INVERSION OF SEISMIC WAVEFORMS
Identifieur interne : 00DC92 ( Main/Exploration ); précédent : 00DC91; suivant : 00DC93AN ALTERNATIVE STRATEGY FOR NON‐LINEAR INVERSION OF SEISMIC WAVEFORMS
Auteurs : M. S. Sambridge [Royaume-Uni] ; A. Tarantola [France] ; B. L. N. Kennett [Australie]Source :
- Geophysical Prospecting [ 0016-8025 ] ; 1991-08.
Descripteurs français
- Wicri :
- topic : Sciences de la terre.
English descriptors
- KwdEn :
- Algorithm, Alternative strategy, Conjugate gradient method, Conjugate gradient methods, Covariance, Covariance operator, Covariance operators, Descent direction, Descent directions, Different dimension, Different parameter types, Duality product, Duality products, Earth model, Earth sciences, Elastic wave equation, Full hessian, Functional space, Geophysical journal, Geophysical prospecting, Gradient methods, Gradient vector, Greatest reduction, Impedance, Inverse problem, Inverse problems, Inversion, Inversion algorithms, Iteration, Kennett, Local linearization, Matrix, Matrix inversion, Misfit, Misfit function, Model parameter type, Model parametrization, Model space, Model vector, Numerical solution, Objective function, Optimization, Optimization problem, Other hand, Overall descent direction, Parameter, Parameter type, Parameter types, Previous descent directions, Receiver positions, Reflection data, Reflection problem, Reflection seismology, Sambridge, Second term, Seismic, Seismic reflection data, Seismic reflection waveforms, Seismic waveforms, Single gradient method, Source function, Steepest, Steepest descent algorithm, Steepest descent direction, Steepest descent vector, Step lengths, Subspace, Subspace approach, Subspace directions, Subspace method, Subspace methods, Subspace scheme, Subspace technique, Surface displacements, Tarantola, Theoretical physics, Tomographic inversion, Waveform, Waveform inversion, Weighted residuals, Weighting functions, Williamson.
- Teeft :
- Algorithm, Alternative strategy, Conjugate gradient method, Conjugate gradient methods, Covariance, Covariance operator, Covariance operators, Descent direction, Descent directions, Different dimension, Different parameter types, Duality product, Duality products, Earth model, Earth sciences, Elastic wave equation, Full hessian, Functional space, Geophysical journal, Geophysical prospecting, Gradient methods, Gradient vector, Greatest reduction, Impedance, Inverse problem, Inverse problems, Inversion, Inversion algorithms, Iteration, Kennett, Local linearization, Matrix, Matrix inversion, Misfit, Misfit function, Model parameter type, Model parametrization, Model space, Model vector, Numerical solution, Objective function, Optimization, Optimization problem, Other hand, Overall descent direction, Parameter, Parameter type, Parameter types, Previous descent directions, Receiver positions, Reflection data, Reflection problem, Reflection seismology, Sambridge, Second term, Seismic, Seismic reflection data, Seismic reflection waveforms, Seismic waveforms, Single gradient method, Source function, Steepest, Steepest descent algorithm, Steepest descent direction, Steepest descent vector, Step lengths, Subspace, Subspace approach, Subspace directions, Subspace method, Subspace methods, Subspace scheme, Subspace technique, Surface displacements, Tarantola, Theoretical physics, Tomographic inversion, Waveform, Waveform inversion, Weighted residuals, Weighting functions, Williamson.
Abstract
A common example of a large‐scale non‐linear inverse problem is the inversion of seismic waveforms. Techniques used to solve this type of problem usually involve finding the minimum of some misfit function between observations and theoretical predictions. As the size of the problem increases, techniques requiring the inversion of large matrices become very cumbersome. Considerable storage and computational effort are required to perform the inversion and to avoid stability problems. Consequently methods which do not require any large‐scale matrix inversion have proved to be very popular. Currently, descent type algorithms are in widespread use. Usually at each iteration a descent direction is derived from the gradient of the misfit function and an improvement is made to an existing model based on this, and perhaps previous descent directions. A common feature in nearly all geophysically relevant problems is the existence of separate parameter types in the inversion, i.e. unknowns of different dimension and character. However, this fundamental difference in parameter types is not reflected in the inversion algorithms used. Usually gradient methods either mix parameter types together and take little notice of the individual character or assume some knowledge of their relative importance within the inversion process. We propose a new strategy for the non‐linear inversion of multi‐offset reflection data. The paper is entirely theoretical and its aim is to show how a technique which has been applied in reflection tomography and to the inversion of arrival times for 3D structure, may be used in the waveform case. Specifically we show how to extend the algorithm presented by Tarantola to incorporate the subspace scheme. The proposed strategy involves no large‐scale matrix inversion but pays particular attention to different parameter types in the inversion. We use the formulae of Tarantola to state the problem as one of optimization and derive the same descent vectors. The new technique splits the descent vector so that each part depends on a different parameter type, and proceeds to minimize the misfit function within the sub‐space defined by these individual descent vectors. In this way, optimal use is made of the descent vector components, i.e. one finds the combination which produces the greatest reduction in the misfit function based on a local linearization of the problem within the subspace. This is not the case with other gradient methods. By solving a linearized problem in the chosen subspace, at each iteration one need only invert a small well‐conditioned matrix (the projection of the full Hessian on to the subspace). The method is a hybrid between gradient and matrix inversion methods. The proposed algorithm requires the same gradient vectors to be determined as in the algorithm of Tarantola, although its primary aim is to make better use of those calculations in minimizing the objective function.
Url:
DOI: 10.1111/j.1365-2478.1991.tb00341.x
Affiliations:
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<term>Descent directions</term>
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<term>Steepest descent vector</term>
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<term>Subspace approach</term>
<term>Subspace directions</term>
<term>Subspace method</term>
<term>Subspace methods</term>
<term>Subspace scheme</term>
<term>Subspace technique</term>
<term>Surface displacements</term>
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<term>Theoretical physics</term>
<term>Tomographic inversion</term>
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<term>Different parameter types</term>
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<term>Earth sciences</term>
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<front><div type="abstract" xml:lang="en">A common example of a large‐scale non‐linear inverse problem is the inversion of seismic waveforms. Techniques used to solve this type of problem usually involve finding the minimum of some misfit function between observations and theoretical predictions. As the size of the problem increases, techniques requiring the inversion of large matrices become very cumbersome. Considerable storage and computational effort are required to perform the inversion and to avoid stability problems. Consequently methods which do not require any large‐scale matrix inversion have proved to be very popular. Currently, descent type algorithms are in widespread use. Usually at each iteration a descent direction is derived from the gradient of the misfit function and an improvement is made to an existing model based on this, and perhaps previous descent directions. A common feature in nearly all geophysically relevant problems is the existence of separate parameter types in the inversion, i.e. unknowns of different dimension and character. However, this fundamental difference in parameter types is not reflected in the inversion algorithms used. Usually gradient methods either mix parameter types together and take little notice of the individual character or assume some knowledge of their relative importance within the inversion process. We propose a new strategy for the non‐linear inversion of multi‐offset reflection data. The paper is entirely theoretical and its aim is to show how a technique which has been applied in reflection tomography and to the inversion of arrival times for 3D structure, may be used in the waveform case. Specifically we show how to extend the algorithm presented by Tarantola to incorporate the subspace scheme. The proposed strategy involves no large‐scale matrix inversion but pays particular attention to different parameter types in the inversion. We use the formulae of Tarantola to state the problem as one of optimization and derive the same descent vectors. The new technique splits the descent vector so that each part depends on a different parameter type, and proceeds to minimize the misfit function within the sub‐space defined by these individual descent vectors. In this way, optimal use is made of the descent vector components, i.e. one finds the combination which produces the greatest reduction in the misfit function based on a local linearization of the problem within the subspace. This is not the case with other gradient methods. By solving a linearized problem in the chosen subspace, at each iteration one need only invert a small well‐conditioned matrix (the projection of the full Hessian on to the subspace). The method is a hybrid between gradient and matrix inversion methods. The proposed algorithm requires the same gradient vectors to be determined as in the algorithm of Tarantola, although its primary aim is to make better use of those calculations in minimizing the objective function.</div>
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