Deconvolution of seismic traces using homomorphic analysis and matching pursuit

We introduce a two-stage procedure to separate signals that were formed through convolution. A convolution may be represented as = The goal of deconvolution is to extract and from information recorded of For oil exploration applications, w is a seismic pressure wavelet, is the observed seismic response, and is the reflectivity of the earth. We first transform the seismic trace from a convolution domain to an additive domain (cepstral domain) via homomorphic analysis. Given the bandpass nature of the seismic trace, a constructive deconvolution procedure is applied so the signal can be properly transformed into the cepstral domain. We then perform a frequency-time (FT) decomposition to identify the characteristics of each signals so they can be properly separated through a linear lifter. Both signals can then be inverted, resulting in the separation of w(.) and T(.). Decomposition into the FT domain is performed using the matching pursuit algorithm in which reflectivity impulses and slowly-varying components are clearly recognized. The combination of homomorphic analysis, the constructive procedure and frequency-time decomposition offer novel opportunities for separating signals that are either formed through convolution or addition. We present examples that illustrate the procedure.