SYNTHETIC EXAMPLE

I will illustrate the properties of the L¹ deconvolution on a synthetic trace, first without noise (``pure trace''), then with noise (several random spikes). Two kinds of deconvolution will be considered: deconvolution with a known wavelet, and predictive deconvolution. Though spiky noise is not usual in real data, it might be found for some kinds of acquisition (with a bad marine streamer for example) and is representative of the robustness of L¹-norm processes.

Figure  is a plot of all the synthetic inputs I used. The time sampling is dt=4 msec; the traces contain 512 samples. The first trace represents the synthetic wavelet; I chose it minimum-phase, in order to optimize the result of the standard L²-Wiener predictive deconvolution, which I will compare to the L¹ deconvolution. Its frequency band is 10-70 Hz. The second trace represents a spiky sequence of reflection coefficients. Next comes the convolution between the synthetic wavelet and this spiky sequence. Finally, the fourth trace represents the noisy trace, formed with five high-amplitude spikes added to the pure trace.