Tutorial: Linear inverse filters in orthogonal coordinates
E. J. Douze
Journal name: Geophysical Prospecting
Issue: Vol 33, No 8, December 1985 pp. 1093 - 1102
Info: Article, PDF ( 452.46Kb )
The linear filter is used extensively in exploration geophysics, and is usually computed using the least squares normal equations. In the general field of time series, the inverse problem is often solved through eigenvalue expansion solutions to integral equations.
The normal equations can be solved in terms of the eigenvalues and eigenvectors of the autocorrelation matrix. It has been suggested that a spectral expansion technique should be used which computes the inverse directly without explicit use of the normal equations. If all possible spiking positions are calculated using the normal equations, the spiking operator matrix is obtained. The matrix operator obtained from the spectral expansion is closely related to the spiking operator matrix. Thus, it is possible to compute the spectral expansion filter using the normal equations. Therefore, it is possible to use the best features of both methods, i.e. obtaining the optimum filter with the normal equations, and discarding the poorly determined parts of the solution based on spectral expansion.