In problems of large dimensional complexities, matrix methods are frequently the favored mathematical tools. In this paper some extensions of matrix methods to calculus operations are introduced. Consistent array structural definitions are given for derivatives of matrix-valued functions with respect to matrices, for matrix differentials, and for matrix integrals, and some operational properties arising therefrom are detailed. Novel structures are developed for Taylor expansions of a matrix-valued function, which have some attractive features both for manipulative and for computational purposes.

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ConclusionsAccurate seismic porosity and formation pressure logs can be derived from seismic logs estimated from wavelet deconvolved data through seismic inversion. The key steps of wavelet deconvolution deal with the removal of the system wavelet, wavelet spatial distortions (migration), time variant Q and dispersion effects, and residual wavelet. Careful application of this sequence yields a good estimate of the band-limited version of the reflectivity function.Seismic inversion by modeling provides accurate seismic logs with the incorporation of an iterative recovery of the low frequency interval velocity component. This feature leads to a more accurate depth estimation of the anomalies and higher vertical resolution. Consequently, seismic porosity and formation pressure are estimated more accurately.

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