Next: Regularization of the filter
Up: Estimation of nonstationary PEFs
Previous: Definitions
When PEFs are estimated, the matrix
is unknown. If
is the data vector from which we want to estimate the filters, we
minimize the vector
as follows:
| ![\begin{displaymath}
{\bf 0 \approx r_y = Ay}\end{displaymath}](img332.gif) |
(125) |
which can be rewritten
| ![\begin{displaymath}
{\bf 0 \approx r_y = Ya},\end{displaymath}](img333.gif) |
(126) |
where
is the matrix representation of the non-stationary
convolution or combination with the input vector
.The transition between equations (
) and (
)
is not simple. In particular, the shape of the matrix
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}](img334.gif) |
(127) |
![\begin{displaymath}
\bf{Y^0_{conv}} = \left( \begin{array}
{cccc}
y_0 & 0 & 0 &...
... & \vdots & \vdots & \vdots
\end{array} \right )\mbox{etc...}\end{displaymath}](img335.gif)
We see that for the convolution case, the
matrices
are diagonal operators, translating the need for one filter to be
applied to one input point. The size of the matrix
is
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}](img339.gif) |
(128) |
![\begin{displaymath}
\bf{Y^0_{comb}} = \left( \begin{array}
{cccc}
y_0 & 0 & 0 &...
... & \vdots & \vdots & \vdots
\end{array} \right )\mbox{etc...}\end{displaymath}](img340.gif)
We see that for the combination case, the
matrices
are row operators, translating the need for one filter to be
constant for one output point. The size of
is equal to
the size of
.For the vector
in equation (
) 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}](img343.gif) |
(129) |
where nf is the number of coefficients per filter. This definition
of
is independent of
.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
is obtained by adding successive
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
and
| ![\begin{displaymath}
\bf{Ya}=
\left( \bf{Y_0} + \bf{Y_1} + \bf{Y_2} + \cdots \right )\bf{a}. \end{displaymath}](img346.gif) |
(130) |
Therefore, for the matrix
, we have to add all the
matrices together. If we take advantage of the
special structure of
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}](img348.gif) |
(131) |
which is the matrix formulation of the stationary convolution.
With the definitions given in equations
(
), (
) and (
), the fitting goal in
equation (
) 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}](img349.gif) |
(132) |
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}](img350.gif) |
(133) |
Each vector
has one constrained coefficient. We can then
rewrite equations (
) and (
) 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}](img352.gif) |
(134) |
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}](img353.gif) |
(135) |
with
| ![\begin{displaymath}
{\bf M}=
\left( \begin{array}
{cccc}
0 & 0 & 0 & \vdots \\...
...ts \\ \vdots & \vdots & \vdots & \vdots
\end{array} \right )\end{displaymath}](img354.gif) |
(136) |
The definition of
assumes that the first coefficient of each
filter is known. Note that
is equal for both convolution and
combination methods.
Having defined the matrix
, we can now rewrite
equation (
) as follows:
| ![\begin{displaymath}
\bf{0} \approx r_y = \bf{YKa}+\bf{y}\end{displaymath}](img355.gif) |
(137) |
where the square matrix
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}](img356.gif) |
(138) |
The next step consists of estimating the filter coefficients in a
least-squares sense.
Next: Regularization of the filter
Up: Estimation of nonstationary PEFs
Previous: Definitions
Stanford Exploration Project
5/5/2005