| This dissertation mainly deals with two separate but closely related topics. The first part is focused on solving current problems in monoscaling analysis on well logging data and provided an acceptable framework for more accurate analysis. My arguments were made based on theoretical considerations, analyses from simulations of a group of fGn/fBm, as well as studies on real data examples. It is argued that raw well data should be considered as fBm type of time series that have spectrum power β ≥ 1. Consequently, for fBm-like data, in order to make consistent and comparable results from rescaled-range and power spectrum techniques, and to make meaningful estimates of fractal dimension, we should use their incremental series, rather than the raw time series themselves for R/S analysis. R/S analyses applied directly on raw well data always give erroneously high estimates of Hu (>0.85). On the other hand, R/S analyses on the incremental series give estimates of Hu close to H from power spectrum analyses on the raw data.; Monoscaling analysis assumes inherently that the scaling parameter is constant along the data trajectory. In this sense Fourier transform is well suited for the analysis because no time/space localization is necessary. For more complicated data, however, we have to introduce wavelet transform and utilize its ability of localization in time/space. In the second part of this dissertation I introduced wavelet-based singularity analysis and demonstrated that Hölder exponent from this analysis can serve as a good seismic attribute for more detailed stratigraphic interpretation. Hölder exponent gives a close link between acoustic impedance and seismograms due to the nature of physics and the property of Hölder exponent. Geological information that would be obscured on original seismic amplitude display can stand out more clearly on Hölder attribute. The Hölder exponent can also help for acoustic impedance inversion.; The application of wavelet transform in geosciences has become a rapidly growing research field. As a contribution to this trend, I included a section of discussion in the epilog on seismic deconvolution in wavelet domain, which may circumvent many strict assumptions in traditional techniques and therefore provide more accurate imaging. |
