Generating two-dimensional fractional Brownian motion using the fractional Gaussian process (FGp) algorithm

Abstract

Fractional Brownian motion (FBM) is a random fractal that has been used to model many one-, two- and multi-dimensional natural phenomena. The increments process of FBM has a Gaussian distribution and a stationary correlation function. The fractional Gaussian process (FGp) algorithm is an exact algorithm to simulate Gaussian processes that have stationary correlation functions. The approximate second partial derivative of two-dimensional FBM, called 2D fractional Gaussian noise, is found to be a stationary isotropic Gaussian process. In this paper, the expected correlation function for 2D fractional Gaussian noise is derived. The 2D FGp algorithm is used to simulate the approximate second partial derivative of 2D FBM (FBM2) which is then numerically integrated to generate 2D fractional Brownian motion (FBM2). Ensemble averages of surfaces simulated by the FGp2 algorithm show that the correlation function and power spectral density have the desired properties of 2D fractional Brownian motion.