The deconvolution of seismic data as a fluctuation analysis

The seismic convolutional model states that a seismic record is the convolution of the earth’s reflectivity with the seismic wavelet. In seismic processing the purpose of deconvolution is to remove (or collapse) the seismic wavelet. In this way, the deconvolved seismic record provides an estimate of the reflectivity. Deconvolution is usually described in terms of the appearance of the deconvolved seismic record, as to whether the events have their durations significantly shortened or not. The change in the appearance of the deconvolved record as compared to the original record is normally understood as the justification for including the deconvolution step in the processing sequence. However, beyond the mere degree in which events are collapsed and located in the time direction, the statistical properties of the reflectivity should be represented in the result of a deconvolution. Using relations like the Wiener-Khinthchin theorem, and Birkhoff’s ergodic theorem as heuristic bases, we may presume that these statistical properties are completely given by the amplitude spectrum of the reflectivity.

The main obstacle to the determination of this amplitude spectrum is that it is not uniquely related to the data: In order to determine the amplitude spectrum of the reflectivity, it is necessary to know the spectrum of the seismic wavelet, or source signature, which is convolved with the reflectivity to produce the seismic records. There is the need of physical hypotheses to solve the problem, which may affect the statistics of the estimated reflectivity, as it is the case with the whiteness hypothesis in conventional deconvoltution. It is possible to be less strict when formulating these hypotheses that define the spectral model, trying to infer not only the positions of the reflectors, but also the statistics of the reflectivity. To do this, it is necessary to rely only on general properties of the reflectivity and seismic wavelets, as they are met in practice. In this paper we explore this possibility, which has also been explored by other researchers. We substitute the whiteness hypothesis by an hypothesis based on the regularity contrast between the reflectivity ϵ (which is a rough convolutive component of the seismic trace x), and the seismic wavelet b (which is a smooth convolutive component the seismic trace x). This physical hypothesis allows us to associate the reflectivity with the fluctuations χ ϵ of the quantity χ x ≡ ln | x ˆ | , thus interpreting the deterministic deconvolution as a detrending of this quantity, and the statistical analysis of reflectivity as a fluctuation analysis. Then we employ recently proposed fluctuation analysis tools based on discrete wavelet transforms to perform deterministic deconvolutions, and to characterize some statistical properties of reflectivity, without and with the presence of Gaussian white noise.