Filtering method

The following results illustrate the first strategy. The left panel of Figure displays the input CMP gather for the velocity inversion. These data have been pre-whitened with a 1-D PEF (deconvolution). Notice that this CMP is made up of nearly horizontal events, hyperbolas and a slow velocity, low frequency event crossing the gather. The later is the coherent noise we want to get rid of. I iterated 30 times to obtain the final model m (Figure ). The residual PEF is 2-D with 25 coefficients on the time axis and 2 coefficients on the space axis. This PEF is re-estimated every ten iterations (see algorithm 1).

In Figure , I compare the input data with the remodeled data (${\bf \tilde{d}=Hm}$) after least-squares inversion with (Equation 11) and without PEF (Equation 9). Notice how close the two results are (the velocity of the linear event is not scanned in the velocity inversion). Figure shows a comparison of the residuals (${\bf r = \tilde{d} - d}$). As expected, the residual of the least-squares inversion with PEF estimation gives a white residual (right panel) as opposed to the ``simplest'' inversion residual contaminated with the linear noise (left panel). Notice that the use of the helical boundary conditions for the PEF estimation has left its footprint on the edges of the residual panel. Figure displays the two spectra for the two residuals.

A comparison of the two model space (Figure ) shows that (1) both results are difficult to interpret and (2) the inversion scheme with PEF gives a more satisfying panel. As a more striking comparison, Figure shows the output of the least-squares inversion with or without PEF as a function of the number of iterations. After 100 iterations, the ``simplest'' inversion (Equation 9) gives a velocity panel infested with artifacts, for it tries to fit the linear event left in the residual. In contrast, with the proposed scheme, the change in the number of iteration does not affect the final result: the inversion becomes stable. Figure displays the convolution of one of the inverse PEF estimated during the iterations with a panel of white noise. It demonstrates that the PEF is effectively after the linear event we want to attenuate.