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Next: Regularization of the filter Up: Estimation of nonstationary PEFs Previous: Definitions

Filter estimation

When PEFs are estimated, the matrix ${\bf A}$ is unknown. If ${\bf y}$is the data vector from which we want to estimate the filters, we minimize the vector ${\bf r_y}$ as follows:  
 \begin{displaymath}
{\bf 0 \approx r_y = Ay}\end{displaymath} (18)
which can be rewritten  
 \begin{displaymath}
{\bf 0 \approx r_y = Ya},\end{displaymath} (19)
where ${\bf Y}$ is the matrix representation of the non-stationary convolution or combination with the input vector ${\bf y}$.The transition between equations (18) and (19) is not simple. In particular, the shape of the matrix ${\bf Y}$ is quite different if we are doing non-stationary convolution or combination. For the convolution, we have  
 \begin{displaymath}
\bf{Y_{conv}}=\left( \begin{array}
{c\vert c\vert c\vert c}
...
 ...bf{Y^2_{conv}} & \cdots 
 \end{array} \right ),
\; \mbox{where}\end{displaymath} (20)

\begin{displaymath}
\bf{Y^0_{conv}} = \left( \begin{array}
{cccc}
 y_0 & 0 & 0 &...
 ... & \vdots & \vdots & \vdots 
 \end{array} \right )\mbox{etc...}\end{displaymath}

We see that for the convolution case, the ${\bf Y^k_{convo}}$ matrices are diagonal operators, translating the need for one filter to be applied to one input point. The size of the matrix $\bf{Y_{conv}}$ is $(n \times (m\times nf))$ where nf is the number of filter coefficients. Now, for the combination, we have  
 \begin{displaymath}
\bf{Y_{comb}}=\left( \begin{array}
{c\vert c\vert c\vert c}
...
 ...bf{Y^2_{comb}} & \cdots 
 \end{array} \right ),
\; \mbox{where}\end{displaymath} (21)

\begin{displaymath}
\bf{Y^0_{comb}} = \left( \begin{array}
{cccc}
 y_0 & 0 & 0 &...
 ... & \vdots & \vdots & \vdots 
 \end{array} \right )\mbox{etc...}\end{displaymath}

We see that for the combination case, the ${\bf Y^k_{combo}}$ matrices are row operators, translating the need for one filter to be constant for one output point. The size of $\bf{Y_{comb}}$ is equal to the size of $\bf{Y_{conv}}$.For the vector ${\bf a}$ in equation (19) we have  
 \begin{displaymath}
\bf{a}=\left( \begin{array}
{c}
 \bf{a_0} \  \hline
 \bf{a_...
 ...,k} \  a_{2,k} \  \vdots \  a_{nf-1,k}
 \end{array} \right )\end{displaymath} (22)
where nf is the number of coefficients per filter. This definition of ${\bf a}$ is independent of ${\bf Y}$.We might want to have one filter common to different input or output points instead of one filter per point. In that case, the matrix ${\bf Y}$ is obtained by adding successive ${\bf Y^k}$ matrices depending on how many points have a similar filter. Note that in the stationary case, for both the convolution and the combination case we have $\bf{a_0}=\bf{a_1}=\bf{a_2}=\cdots=\bf{a}$and
\begin{displaymath}
\bf{Ya}=
 \left( \bf{Y_0} + \bf{Y_1} + \bf{Y_2} + \cdots \right )\bf{a}. \end{displaymath} (23)
Therefore, for the matrix ${\bf Y}$, we have to add all the ${\bf Y^k}$ matrices together. If we take advantage of the special structure of $\bf{Y_k}$ for the convolution and the combination, we obtain for the stationary case  
 \begin{displaymath}
\bf{Ay}=\bf{Ya}= \left( \begin{array}
{cccccc}
 y_0 & 0 & 0 ...
 ...1 \  a_1 \  a_2 \  a_3 \  \vdots \  \end{array} 
 \right),\end{displaymath} (24)
which is the matrix formulation of the stationary convolution.

With the definitions given in equations (20), (21) and (22), the fitting goal in equation (19) can be rewritten  
 \begin{displaymath}
{\bf 0 \approx r_y = Y^0_{conv}a_0 + Y^1_{conv}a_1 + Y^2_{conv}a_2 +
 \cdots}\end{displaymath} (25)
or  
 \begin{displaymath}
{\bf 0 \approx r_y = Y^0_{comb}a_0 + Y^1_{comb}a_1 + Y^2_{comb}a_2 +
 \cdots}\end{displaymath} (26)
Each vector ${\bf a_k}$ has one constrained coefficient. We can then rewrite equations (25) and (26) as follows:  
 \begin{displaymath}
{\bf 0 \approx r_y = y+Y^0_{conv}Ma_0 + Y^1_{conv}Ma_1 + Y^2_{conv}Ma_2 +
 \cdots}\end{displaymath} (27)
and  
 \begin{displaymath}
{\bf 0 \approx r_y = y+Y^0_{comb}Ma_0 + Y^1_{comb}Ma_1 + Y^2_{comb}Ma_2 +
 \cdots}\end{displaymath} (28)
with
\begin{displaymath}
{\bf M}=
 \left( \begin{array}
{cccc}
 0 & 0 & 0 & \vdots \...
 ...ts \  \vdots & \vdots & \vdots & \vdots 
 \end{array} \right )\end{displaymath} (29)
The definition of ${\bf M}$ assumes that the first coefficient of each filter is known. Note that ${\bf M}$ is equal for both convolution and combination methods. Having defined the matrix ${\bf M}$, we can now rewrite equation (19) as follows:  
 \begin{displaymath}
\bf{0} \approx r_y = \bf{YKa}+\bf{y}\end{displaymath} (30)
where the square matrix ${\bf K}$ is
\begin{displaymath}
\bf{K}=\left( \begin{array}
{c\vert c\vert c\vert c}
 \bf{M}...
 ... \hline
 \vdots & \vdots & \vdots & \vdots
 \end{array}\right).\end{displaymath} (31)
The next step consists of estimating the filter coefficients in a least-squares sense.


next up previous print clean
Next: Regularization of the filter Up: Estimation of nonstationary PEFs Previous: Definitions
Stanford Exploration Project
7/8/2003