Next: EXAMPLE ON REAL DATA Up: SYNTHETIC EXAMPLE Previous: Other families of noise

## L1 norm and non minimum-phase wavelets

A predictive L2 deconvolution produces a minimum-phase inverse filter, independently of the phase of the wavelet w. Thus it would be interesting to study the effect of the L1 deconvolution on non minimum-phase inputs. I used the same input reflection sequence as before, but changed the input wavelet into a constant phase (60o) wavelet. I applied L2 and L1 deconvolution on the new synthetic trace. I initialized the L1 algorithm first with the L2 filter, then without the input filter. The results are shown on Figure .

It appears that the L1 deconvolution does not improve the phase properties of the output. In one case (starting from L2 filter), the filter is from the start close to the L2 filter; in the other case, the first iteration of the IRLS algorithm forces the filter to be close to the L2 filter. In the first case, the L1 deconvolution results in a sharpening of the L2 result, again because the residuals have an a priori exponential distribution.

The L1 deconvolution can be related to the variable norm deconvolution (Gray, 1979), in the particular case when the processing tries to draw the statistical distribution of the time series from a gaussian one (seismic trace) to an exponential one (residuals). However, L1 and Gray's deconvolutions are slightly different, because the IRLS algorithm does not suppose any a-priori distribution for the inputs; the normal equations of the variable norm deconvolution are in fact a linear combination of the L2 normal equations and of the L1 normal equations.

In conclusion, the IRLS algorithm in predictive deconvolution essentially sharpens the residuals. If an initial estimate is used (Wiener filter, Burg filter, or any minimum-entropy deconvolution filter), the residuals of this initial filter will be sharpened; otherwise, the result will be a sharpened version of the L2 residuals. It would be interesting in fact to modify the first iteration of the algorithm, but would it still guarantee the convergence?

Next: EXAMPLE ON REAL DATA Up: SYNTHETIC EXAMPLE Previous: Other families of noise
Stanford Exploration Project
1/13/1998