# Spectral Deconvolution Lab

Figure 1. The a) input time series and b) output after application of spectral deconvolution.

Theory: Given an input signal x, find the deconvolution filter f by spectral division f=x*/(|x|2+ eps), where eps is a small positive number known as the damping parameter (see Yilmaz, 2001).

Objective: Practice the craft of spectral deconvolution of a signal.

Procedure:

1. Execute the command deconsimp.m.
2. Change values of damping parameter eps1. Explain what happens to deconvolved wavelet if eps1 is to small. Explain what happens if eps is too large. Note, as eps becomes really large the inverse filter f is a constant for a zero-phase x and is an all pass filter. Thus, larger values of eps lead to deconvolved results with fatter main lobes and smaller side lobes. As eps becomes smaller the main lobe thins and the side lobes become more pronounced.
3. Add noise to wavelet by the command x1=3*x./max(x)+rand(1,nt)-.5; plot(x1); and repeat exercise. What is the effect of random noise?
4. Download the shot gather CSG and type "load CSG". To display the shot gather type "CSG=CSG/max(CSG(:)); clim=[-.03 .03];imagesc(CSG',clim);". Extract one of the traces and autocorrelate it, then truncate the autocorrelation after the 2nd zero crossing. This truncated wavelet will now be your estimated input wavelet. Find the inverse filter f to this correlation wavelet and use it to deconvolve the autocorrelation trace. Compare the deconvolved autocorrelation trace to original autocorrelation trace.
5. A more robust estimate of the decon filter might be obtained by averaging truncated autocorrelation signals for a number of different traces. Use averaging to construct your new inverse filter and deconvolve the autocorrelated traces. Compare the new results to the results from the previous question.
6. A trick that is almost always used after any deconvolution process is to bandpass filter the deconvolved results, where the passband is over the frequency range of the estimated signal.