An Investigation of Seismic Reflectivity Spectra, Moments, and Fractal Dimensions

Statistical models for seismic reflectivity spectra provide means of generating numerical reflection sequences as a priori information for seismic inversion, to test geophysical data processing techniques, and to parameterize real reflection sequences for classification purposes. I tested the effectiveness of using parametric and non-parametric statistical methods to estimate optimized reflectivity spectra and to identify the accurate form of the reflectivity distribution of seismic and well log data that best simulates observations. I show that a sum of generalized Gaussian distributions with variable mean, variance, skewness, and kurtosis characterizes measured distributions well. Reflectivity series distributions derived from sonic and density logs in 180 wells from 8 basins worldwide at forty sampling intervals between 0.1 milliseconds and 4 milliseconds show that the reflectivity exhibits a range of spectral shapes, from white to blue. The variation is strongly dependent on the sampling rate which has a remarkable effect on the shape of reflectivity distributions, spectra, and designed deconvolution filters. The sample rate dependence of spectral shape and the earth impulse response is greatly affected by internal multiples and transmission losses. Reflectivity moments and spectra vary with digital sampling interval and thus do not characterize a continuous earth well. I develop a novel technique whereby moments and spectral parameters are extrapolated to zero sample rate.