Efficient inversion of near singular geophysical signals

Abstract

In a method for inverting a seismic signal, an input seismic signal is represented as an input vector i(t) of length m+n−1, an output seismic signal is represented as an output vector o(t) of length m, and a finite impulse response filter is represented as a solution vector u(t) of length n, satisfying a convolutional equation i(t){circle around (x)}u(t)=o(t). An m×, n Toeplitz matrix T(t,τ) corresponding to i(t) is calculated, satisfying a matrix equation T(t,τ)·u(t)=o(t). Both sides of the matrix equation are transformed to a transform domain, to generate a transformed equation. Both sides of the transformed equation are pruned, to generate a pruned equation. The pruned equation is then solved. In one embodiment of the invention, the transformed domain is the frequency domain.

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