Practical application of fractional Brownian Motion and noise to synthetic hydrology

Abstract

A synopsis of self‐similar stochastic processes is presented with special emphasis on fractional Brownian motion and noises. Detailed analysis and discussion are on the approach of ‘fast fractional Gaussian noises’ by Mandelbrot and his collaborators. Modifications are incorporated and parameter selection criteria suggested so that the method becomes simpler, more flexible to use, and easier to follow. A step‐by‐step procedure is included for convenience of application.

Modeling and analyzing fractal point processes

“…[5]; the intensity process is then obtained by passing it through the appropriate transformation. The point process is generated using the method described in figure 1.1.…”

“…The point process is generated using the method described in § 1.1 and fGn is generated using the method of Chi et al [5). …”

“…This scheme is described in figure 2.4. Mandelbrot [22] has found that values of 2,3, or 4 to be appropriate for B and Chi et al [5] have determined values in the range 15-20 to be suitable for N.…”

“…We will employ the fast fractional Gaussian noise (ffGn) generator proposed by Mandelbrot [22]. Chi et al [5] proposed an implementation of this generator with simpler parameter selection criteria. We shall follow and extend both of these references in subsequent discussion.…”

“…Unequally-spaced sequences have been found to be most suitable for this purpose; Mandelbrot proposed with increasing values of n (in the range [0, oo) ); this implies that e-1 is the point of separation between high and low frequencies. After some manipulation, Chi et al [5) determined the covariance of XL(t) to be Figure 2.4 Fast fractional Gaussian noise is generated as a bank of N + 2 filters (see equation (2.20) …”