Tempered Fractional Brownian Motion: Wavelet Estimation and Modeling of Turbulence in Geophysical Flows

Abstract

Fractional Brownian motion (fBm) is a Gaussian, stationary-increment process whose self-similarity property is governed by the so-named Hurst parameter H ∈ (0,1). FBm is one of the most widely used models of scale invariance, and its instance H = 1/3 corresponds to the classical Kolmogorov spectrum for the inertial range of turbulence. Tempered fractional Brownian motion (tfBm) was recently introduced as a new canonical model that displays the so-named Davenport spectrum, a model that also accounts for the low frequency behavior of turbulence. The autocorrelation of its increments displays semi-long range dependence, i.e., hyperbolic decay over moderate scales and quasi-exponential decay over large scales. The latter property has now been observed in many phenomena, from wind speed to geophysics to finance. This paper introduces a wavelet framework to construct the first estimation method for tfBm. The properties of the wavelet coefficients and spectrum of tfBm are studied, and the estimator’s performance is assessed by means of Monte Carlo experiments. We also use tfBm to model geophysical flow data in the wavelet domain and show that tfBm provides a closer fit than fBm.