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Regularization and Preconditioning

Seismic inversion is an ill-posed problem. Therefore, regularization operators are required to stabilize the inversion and to prevent divergence to unrealistic solutions. A regularized least squares solution $ {\bf\hat m}$ is obtained by minimizing a modified objective function:

\begin{displaymath}\begin{array}{cc} S({\bf m})=\Vert {\bf\widetilde L}{\bf m} -...
...t^2 + \epsilon^{2}\Vert {\bf R}{\bf m} \Vert^{2},\\ \end{array}\end{displaymath} (13)

where $ \epsilon$ is a damping factor that determines the strength of the regularization operator $ {\bf R}$. In this paper, we consider a fixed, heuristically determined, damping factor computed as a function of the data as follows:

$\displaystyle \epsilon = \frac{max \bf \vert \widetilde d \vert }{100}.$ (14)

Relevant examples of regularization criteria for geophysical inverse problems include model smoothness (Tikhonov and Arsenin, 1977), temporal smoothness (Ajo-Franklin et al., 2005), and horizontally smooth angle gathers (Clapp, 2005).

Minimizing equation (13) is equivalent to solving the problem

$\displaystyle \left [ \begin{array}{cc} {\bf\widetilde L} \\ {\bf\epsilon R} \e...
...} = \left [ \begin{array}{cc} {\bf\widetilde d} \\ {\bf0} \end{array} \right ].$ (15)

Fast iterative convergence can be obtained by preconditioning the regularization (Claerbout and Fomel, 2008). This is equivalent to making the variable substitution

$\displaystyle {\bf m=R^{-1} p= A p},$ (16)

so that equation (15) becomes

$\displaystyle \left [ \begin{array}{cc} {\bf\widetilde LA} \\ {\bf\epsilon I} \...
...} = \left [ \begin{array}{cc} {\bf\widetilde d} \\ {\bf0} \end{array} \right ],$ (17)

where $ {\bf A}$ is the preconditioner, and $ {\bf p}$ is the preconditioned variable. By selecting an invertible regularization operator $ {\bf R=A^{-1}}$, we can solve the preconditioned problem (equation (17)) at fewer iterations than the regularized problem (equation (15)).

For the current problem, we require two regularization constraints (spatial and temporal). The spatial regularization operator is a system of non-stationary dip filters applied on the helix (Claerbout and Fomel, 2008). These symmetric filters, built from puck filters (Hale, 2007; Claerbout and Fomel, 2008), are then factored into causal dip filters using Wilson-Burg factorization (Fomel et al., 2003). The preconditioner, implemented as a helical polynomial division, uses dips estimated from plane-wave destruction (Fomel, 2002) to determine the appropriate filters for each model point. The temporal preconditioner is a bi-directional leaky integration operator which penalizes sudden changes over time.

The preconditioned joint inverse problem is

$\displaystyle \left [ \begin{array}{cc} {\bf\L\AA } \\ {\bf\epsilon I } \end{ar...
...\end{array} = \left [ \begin{array}{cc} {\bf d} \\ {\bf0} \end{array} \right ],$ (18)

where

$\displaystyle {\bf\L } \\ = \left [ \begin{array}{cc} {\bf\widetilde L}_{0} & {\bf0} \\ {\bf0 } & {\bf\widetilde L}_{1} \end{array} \right ],$ (19)

$\displaystyle {\bf\AA } \\ = {\bf AT},$ (20)

with

$\displaystyle {\bf A } \\ = \left [ \begin{array}{cc} {\bf A}_{0} & {\bf0} \\ {\bf0 } & {\bf A}_{1} \end{array} \right ],$ (21)

and

$\displaystyle {\bf T } \\ = \left [ \begin{array}{cc} {\bf I } & {\Lambda } \\ {\Lambda } & {\bf I } \\ \end{array} \right ].$ (22)

The operators $ {\bf A}_{0}$ and $ {\bf A}_{1}$ are preconditioners for the baseline and monitor images, respectively, while $ {\bf I}$ is identity and $ {\Lambda}$ is a diagonal operator containing the leak rates $ {\lambda}$. Equation (18) is directly extendable to an arbitrary number of surveys. The proposed method, joint preconditioned least squares inversion (J-PLSI ) refers to the definition in equation (18). We solve equation (18) using a conjugate gradient algorithm.


next up previous [pdf]

Next: Cascaded covariance-based preconditioning Up: Linear least-squares migration/inversion Previous: Joint-inversion

2009-09-25