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EAS 434 Reflection Seismology HW 4 More Deconvolution Due 11/13/04 Consider our basic filter model. yt = xt * ft where yt = xt = ft = output of filter input to filter filter impulse response function Sometimes we need to compute the input xt from the output yt. This computation requires that we essential “reverse engineer” a new filter f-1t which undoes the affect of the original filter ft. This “deconvolution” is best viewed in the z- transform domain, i.e. if Y(z) = X(z) . F(z), Then X(z) = Y(z) / F(z) Where Y, X and F are the z transforms of yt, xt, and ft respectively. Given the “filter” : ft = 6, 5, 1 1) Compute the output yt when the input is xt = 4, 1, 8 2) Determine the first 3 terms of the stable inverse f-1t by polynomial long division (don’t use MATLAB) 3) Compute the output of the inverse filter f-1t when the input is yt from 1). 4) Plot the output in c) along with the the original input xt. How similar are they? 5) Is ft minimum or maximum delay? How do you know? Is the stable inverse in 2) causal or non-causal? 6) Use the deconv function in matlab to compute the first 15 terms of the inverse of ft. Compute and plot the output of the inverse filter using these 15 terms for input xt. How does this output compare to that in 4). The inverse filter computed above is an example of a “spiking” deconvolution, since we are trying to turn the original filter’s response into a unit response, e.g. 1 ft * f-1t ~ 1, 0, 0, 0 … An example of a common application of spiking deconvolution in seismic data is when we want to remove the “smearing” effect of a finite length seismic wavelet so we can “see” the corresponding reflection coefficients, i.e. st = et * wt, Where st = seismic trace et = the “earth” , as represented by its reflection coefficient series wt = the seismic wavelet emitted by the source, here playing the role of “filter” Deconvolution is then used to computer et from st and wt, that is we deconvolve the wavelet out of the seismogram to see the reflection coefficients. The inverse filter w-1t, essentially acts to “cancel” the effects of the original filter wt, i.e. st * w-1t = (et * wt )* w-1t = et * (wt *w-1t)= et * 1 = et In an ideal world, we get the “answer” and are done. In the real world, noise can sometimes lead to unsatisfactory results when try to replace the wavelet wt by a spike. Often we can achieve more satisfactory results by replacing the wavelet with another wavelet with a more desirable shape. This involves deconvolving wt out of the seismic trace, then refiltering with our new wavelet w2 to get a better seismic trace s2. st = et * wt => S(z) = E(z) . W(z), or E(z) = S(z)/W(z) s2t = et * w2t => S2(z) = E(z) . W2(z)= [S(z)/W(z)] .W2(z) = S(z) . [W2(z)/W(z)] The inverse wavelet corresponding to W2(z)/W(z) is a wavelet shaping filter, which converts the wavelet wt into the wavelet w2t. 7. Let the filter ft above be our original wavelet wt. Compute the wavelet shaping filter that converts wt into w2t = 4, 5, -3. 8. Compute the output of our wavelet shaping filter for the output yt in 1) above. Plot both the original st and the new s2t together. 2