I show in Figure the result of the signal noise separation when no weight is applied, i.e, equation (1). Figures a and c show the estimated signal and noise, respectively. The noise attenuation clearly failed here. Figures b and d display the impulse responses of the signal PEF and the noise PEF, respectively. In theory, Figure b should look like the signal (or at least have the same spectrum) and Figure d should look like the noise. The signal PEF is clearly wrong here since we do not retrieve the signal spectrum very well. This mismatch comes from the high amplitudes at short offset in Figure a that bias the estimation of the filter.
To make things right, a weighting function is needed for the residual. My choice of a weighting function is based on statistical considerations. I apply a weighting function on the data and look at the histogram. If the histogram has a Gaussian distribution, I keep it and use it during the estimation of the noise and signal PEFs. I keep this weighting function unchanged during the iterations, although I could reestimate it as it is done with Iteratively Reweighted Least Squares methods. Figure shows four histograms corresponding to different weighting functions. The Amplitude Gain Control (AGC) produces the most satisfying result in term of distribution of the data. I then derive an appropriate weighting function from the AGC and estimated the noise and signal PEFs. To finish, I perform the signal separation on the raw data.
Figure displays the final noise attenuation result with the weighted PEF estimation. Figure a shows the estimated signal and Figure c the estimated noise. The noise has been successfully attenuated. Note that some signal has leaked in the noise in Figure c around 0.5 second. This is because I estimate only one noise and signal filter for the whole gather. But still, the noise is correctly attenuated. Figures b and d show a spike divided by the signal and noise PEFs respectively. Figure b demonstrates that the signal PEF correctly represents the quasi-flat signal. Figure d looks very similar to the noise with the correct amplitude behavior, as expected.
A last necessary comparison is by estimating the noise and signal PEFs from the weighted data [equation (5)] and look at the noise attenuation result. Figure displays the noise attenuation result with the weighted data for the PEF estimation. The estimated signal in Figure a is satisfying with artifacts above 0.5 second (also visible in the estimated noise in Figure c). Figure shows the difference between Figures a and a. In addition to the artifacts above 0.5 second, the noise is better removed at short offset in Figure a. The inverse PEF for the signal in Figure b is very similar to Figure b, but the amplitude decay is less important. The inverse PEF for the noise in Figure d is very different from Figure d. With the weighted residual, the filter is minimum phase whereas with the weighted data, it is not because the amplitude increases constantly with offset and time in Figure d.
Therefore, weighting the data or the residual for the PEF estimation produces very different filters affecting the final noise attenuation result. The best results are obtained when the residual is weighted to balance the relative importance of the regression equations. In the next section, I exemplify the weighted PEF estimation with a 3-D example and non-stationary filters.
Figure 6 Difference plot between Figures and . Both results are very similar but the weighted data strategy leads to a less efficient noise attenuation at short offset and leaves artifacts on top of the gather.