Deconvolution with a known wavelet

Figure  shows the results of the L² and L¹ deconvolutions for the pure trace. Because the original wavelet lacked very low and very high frequencies, I used a damping factor for both deconvolutions: 1/1000 of the autocorrelation $\sum_i\vert w(i)\vert^2$ in the L² algorithm, and 1/1000 of the weighted autocorrelation $\sum_iW(i)\vert w(i)\vert^2$ at each iteration of the L¹ algorithm. The results are similar, as no noise was introduced. The ringing around the main peaks comes from the lack of high frequencies in the data, which forces the output to be convolved with a ``sinc'' function (impulse response of a high-cut filter).

Then I did the same process with the noisy trace. The results are presented on Figure . Even with a damping factor, the L² deconvolution cannot avoid the influence of the noise, because a damping factor is adapted to gaussian noise. By giving the same weights to all the residuals (W=Identity matrix), it overestimates the importance of the noise bursts, and damages the output around these bursts. On the contrary, as expected, the L¹ deconvolution is insensitive to this noise.