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I call
the convolution or combination operator with a bank of
non-stationary filters. For the non-stationary convolution, the
filters are in the column of
(one filter corresponds to one
point in the input space) whereas for the non-stationary
combination, the filters lie in the rows of
(one filter corresponds
to one point in the output space). For the convolution matrix,
I define ai,j as the ith coefficient of the filter for the jth
data point in the input space. For the combination matrix, I define
ai,j as the jth coefficient of the filter for the ith
data point in the output space. Therefore, for the non-stationary convolution we have
| ![\begin{displaymath}
\bf{A}_{conv}=\left ( \begin{array}
{cccc}
1 & 0 & 0 & \vdo...
... \\ \vdots & \vdots & \vdots & \vdots
\end{array}
\right )\end{displaymath}](img325.gif) |
(120) |
and for the non-stationary combination we have
| ![\begin{displaymath}
\bf{A}_{comb}=\left ( \begin{array}
{cccc}
1 & 0 & 0 & \vdo...
... \\ \vdots & \vdots & \vdots & \vdots
\end{array}
\right )\end{displaymath}](img326.gif) |
(121) |
The size of both matrices is
where n is the size of
an output vector (
) and m the size of an input vector
(
) if
| ![\begin{displaymath}
{\bf Ay=x}\end{displaymath}](img329.gif) |
(122) |
The helical boundary conditions allow to generalize this
1-D convolution to higher dimensions. We can rewrite equation
(
) for the convolution as follows:
| ![\begin{displaymath}
y_k = x_k + \sum_{i=1}^{\min(nf-1, k-1)} a_{i,(k-i)} \; x_{k-i}\end{displaymath}](img330.gif) |
(123) |
where nf is the number of filter coefficients, and for the
combination
| ![\begin{displaymath}
y_k = x_k + \sum_{i=1}^{\min(nf-1, k-1)} a_{i,k} \; x_{k-i}.\end{displaymath}](img331.gif) |
(124) |
In the next section, I show how the non-stationary PEFs are estimated.
Next: Filter estimation
Up: Estimation of nonstationary PEFs
Previous: Estimation of nonstationary PEFs
Stanford Exploration Project
5/5/2005