Two Methods of Deconvolution: Power Spectrum Smoothing and Parsimonious Deconvolution
, by Bob Godfrey and Jon F. Claerbout

Two independent methods of deconvolving seismic data are presented. The first section gives the theoretical development of each method and the second section follows with some examples.
Power spectrum smoothing consists of estimating the bubble spectrum by dividing a smoothed raw data spectrum by a super-smoothed raw data spectrum. The time domain bubble is computed using the minimum phase factorization of the resultant spectral ratio.
Parsimonius deconvolution is an iterative gradient descent algorithm. A norm measuring the parsimony of spikiness of a trace provides the gradient direction along which descent is made. The optimation process is subject to the constraint that the observed seismic trace is composed of random noise plus the convolution of a reflectivity series with a casual waveform.