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Linear least-squares inversion

Linear least-squares inversion provides a theoretical approach to compensate for experimental deficiencies (e.g., limited acquisition geometry) and complexities of the overburden, while maintaining consistency with the acquired data. For seismic imaging, it can be summarized as follows. Given a linear modeling operator $ {\bf L}$, compute synthetic data d using $ {\bf d}={\bf L}{\bf m}$, where m is a reflectivity model.

The quadratic cost function,

$\displaystyle S({\bf m})=\Vert {\bf d} - {\bf d}_{obs} \Vert^2 =\Vert {\bf L}{\bf m} - {\bf d}_{obs} \Vert^2,$ (1)

is formed, where $ {\bf d}_{obs}$ denotes the recorded data.

The reflectivity model $ \hat{{\bf m}}$ that minimizes $ S({\bf m})$ is given by

$\displaystyle \hat{{\bf m}}=({\bf L}'{\bf L})^{-1}{\bf L}' {\bf d}_{obs} = {\bf H}^{-1} {\bf m}_{mig},$ (2)

where $ {\bf L}'$ (the migration operator) is the adjoint of the linear modeling operator $ {\bf L}$, $ {\bf m}_{mig}$ is the migrated image, $ {\bf m}_{mig}={\bf L}' {\bf d}_{obs}$, and $ {\bf H}={\bf L}'{\bf L}$ is the Hessian of $ S({\bf m})$.

Since the model space can be large, computing the inverse of the Hessian matrix is a big challenge for most geophysical imaging problems. For this reason, it is often more feasible to compute the inverse image as the solution of the linear system of equations,

$\displaystyle {\bf H} \hat{{\bf m}}={\bf m}_{mig},$ (3)

by using an iterative inversion algorithm. In this approach to the inverse problem, only one matrix-vector multiplication of the Hessian matrix with the model vector is necessary per iteration, in contrast with other methods (Clapp, 2005) that require one migration and one modeling every iteration. Still, explicit computation of the Hessian for the entire model space is too expensive in practice. Valenciano (2008) discusses how exploiting the structure of the Hessian matrix and the localization in the model space makes this problem tractable.

By using a priori information about the covariance of the model (model regularization), one can add regularization to solve the otherwise ill-posed inversion problem. A more customary regularization for the inversion in the poststack image domain is to add a damping factor that penalizes an increase of the values of the model. This regularization makes no use of any physical knowledge we might have about the seismic reflectors. It is implemented by adding a small value to the diagonal of $ {\bf H}({\bf x},{\bf x}')$ in equation 3:

$\displaystyle \left({\bf H+\varepsilon I}\right)\hat{{\bf m}} - {\bf m}_{mig} = {\bf r} \approx 0 ,$ (4)

where $ {\bf I}$ is the identity operator, $ {\bf r}$ is the residuals vector, and $ \varepsilon$ is an scalar parameter that governs the strength of the regularization.


next up previous [pdf]

Next: Noise Characterization Up: Reconciling processing and inversion: Previous: Introduction

2009-04-13