An Analysis of Deconvolution: Modeling Reflectivity by Fractionally Integrated Noise

Reflection coefficients are observed in nature to have stochastic behavior that departs significantly from the white noise model. Conventional deconvolution methods, however, assume reflectivity to be a white noise process. In this paper we analyze the deconvolution process, study the implications of the assumption of white noise, and show that the conventional operator can recover only the white component of reflectivity. A new stochastic model, fractionally integrated noise, is proposed for modeling reflectivity. This model more closely approximates its spectral character and that encompasses white noise as a special case. We discuss different techniques to generalize the conventional deconvolution method based on the new model in order to handle reflectivity that is not white, and compare the results of the conventional and generalized filters using data derived from well logs. INTRODUCTION Conventional deconvolution schemes assume that the earth's reflectivity has a white noise correlation structure. However, reflection coefficients in nature tend to behave in a different manner: generally their power spectra are proportional to frequency (Hosken, 1980; Walden and Hosken, 1985; Todoeschuck et al., 1990; Rosa and Ulrych, 1991). Figures 1a-f show the power spectra of typical reflectivity logs from four different wells. These were derived from sonic and density logs in various areas of the central and eastern regions of Saudi Arabia and were computed for a plane wave with normal incidence by r = (P2V2-PIVr)/(P2V2+PIVd, where Pi and Vi are the density and acoustic velocity in layer i, respectively. The spectra were calculated by FFT analysis on the