Sensitivity kernels for shear wave splitting in transverse isotropic media

Fractal theory and related developments have significantly enlightened our understanding of many natural phenomena and led to an upsurge in research efforts in many different disciplines. This has unfortunately caused some ambiguities, misunderstandings, and even misuses in the published literature concerning some new concepts in fractal theory and their actual implementations. As new applications and ideas arise almost daily, it is important to present the methods in a common framework to minimize further confusion. In this paper, we scrutinize and compare two of the most frequently applied methods for characterizing fractal time-series, namely the rescaled-range (R/S) and the power spectrum method, and describe their proper applications in wire line well-logging data. We outline the problems related to applications of these two methods. Based on numerical analysis on fractional Gaussian noise (fGn) and fractional Brownian motion (fBm), we demonstrate the clear necessity to distinguish between fGn-like and fBm-like data series. For fBm-like data series, the use of their successive increments is proposed against using various other alternatives in R/S analysis in order to make the estimated Hurst exponent (Hu) comparable with the global scaling exponent (H) from power spectrum analysis. We argue that well-logging data are generally analogous to fBm, and thus we use their incremental series rather than raw well data themselves to estimate Hu when applying R/S. We also reveal a connection between the transient zone length in the R/S technique and the shortest-wavelength that can be included for a linear regression in the power spectrum method.