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The slow steepest-descent method

The steepest descent method requires the computation of the gradient. The model space is a vector of filter coefficients $ u(t)$ . Claerbout shows that the gradient $ du(t)$ of the sparse log-decon method corresponds to the crosscorrelation of the residual (the reflectivity series) with the soft-clipped residual (see Claerbout et al. (2012) for a generalization with a variable gain). The pseudo-code below shows the steepest descent algorithm.

Once $ u(t)$ is estimated, we obtain the wavelet $ w(t)=FT^{-1} \left ( e^{-U(\omega)} \right )$ and the sparse decon output $ r(t)=FT^{-1} \left ( D(\omega)e^{U(\omega)} \right ) $ , where $ D(\omega)$ is the Fourier transformed input data.