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

# Filter estimation

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: (125)
which can be rewritten (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 (127) 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 (128) 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 (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 (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 (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 (132)
or (133)
Each vector has one constrained coefficient. We can then rewrite equations ( ) and ( ) as follows: (134)
and (135)
with (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: (137)
where the square matrix is (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