     Next: Filter estimation Up: Estimation of nonstationary PEFs Previous: Estimation of nonstationary PEFs

# Definitions

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 (120)
and for the non-stationary combination we have (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 (122)
The helical boundary conditions allow to generalize this 1-D convolution to higher dimensions. We can rewrite equation ( ) for the convolution as follows: (123)
where nf is the number of filter coefficients, and for the combination (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