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Regularization of the filter coefficients

The number of coefficients to estimate is usually much greater than the number of data points. This makes the problem very under-determined. A solution is to introduce more equations in equation (30) as follows:  
 \begin{displaymath}
\begin{array}
{rcccl}
 \bf{0} &\approx& {\bf r_y}& =& \bf{YKa}+\bf{y} \  \bf{0} &\approx& {\bf r_a}& =& \bf{Ra}.
 \end{array}\end{displaymath} (32)
The second term in equation (32) improves the conditioning of our problem and is called regularization. In the filter estimation problem, it is reasonable that ${\bf R}$ penalizes strong variations between filter coefficients. Hence, ${\bf R}$ is usually a gradient or a Laplacian. Crawley (2000) proposes smoothing the filter coefficients along radial directions. This proposal is valid for shot or common mid point gathers only where constant dips are roughly aligned along radial segments. For instance, if ${\bf R}$ is a gradient operator we have
\begin{displaymath}
\bf{R}=\left( \begin{array}
{c\vert c\vert c\vert c}
 \bf{I}...
 ...  \hline
 \vdots & \vdots & \vdots & \vdots
 \end{array}\right)\end{displaymath} (33)
with $\bf{I}$ the identity matrix. Equation (32) can be solved for ${\bf a}$ in a least-squares sense. We then want to minimize the objective function
\begin{displaymath}
f({\bf a})=\Vert{\bf r_y}\Vert^2+\epsilon^2\Vert{\bf r_a}\Vert^2\end{displaymath} (34)
which gives for ${\bf a}$ the least-squares estimate
\begin{displaymath}
\hat{\bf{a}}=-({\bf K'Y'YK}+\epsilon^2{\bf R'R})^{-1}\bf{K'Y'y}.\end{displaymath} (35)
Because of the number of unknowns and of the sparseness of the problem, we use a conjugate gradient method to estimate our PEFs.

 


next up previous print clean
Next: About this document ... Up: Estimation of nonstationary PEFs Previous: Filter estimation
Stanford Exploration Project
7/8/2003