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Definitions

I call ${\bf A}$ the convolution or combination operator with a bank of non-stationary filters. For the non-stationary convolution, the filters are in the column of ${\bf A_{conv}}$ (one filter corresponds to one point in the input space) whereas for the non-stationary combination, the filters lie in the rows of ${\bf A_{comb}}$ (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} (13)
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} (14)
The size of both matrices is $(n\times m )$ where n is the size of an output vector (${\bf x}$) and m the size of an input vector (${\bf y}$) if  
 \begin{displaymath}
{\bf Ay=x}\end{displaymath} (15)
The helical boundary conditions allow to generalize this 1-D convolution to higher dimensions. We can rewrite equation (15) 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} (16)
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} (17)
In the next section, I show how the non-stationary PEFs are estimated.
next up previous print clean
Next: Filter estimation Up: Estimation of nonstationary PEFs Previous: Estimation of nonstationary PEFs
Stanford Exploration Project
7/8/2003