A covariance function with fractal, Hurst, and scale-bridging effects for random surfaces and turbulence

We examine a novel three-parameter covariance function class, designed to independently control fractal and Hurst effects, extending the capabilities of the two-parameter generalized Cauchy and Dagum models. Our analysis reveals that this class effectively decouples the behavior of the spectral density at comparatively low and high frequencies. A distinct parameter is used to fully control the transition between these frequency ranges, characterized by Hurst effects at low frequencies and fractal behaviors at high frequencies. We demonstrate the utility of this new model through its application to multiscale data from various processes, including the surface height distribution of rough surfaces and turbulent flows—such as wind tunnel boundary layer data, field data, and isotropic turbulence in clay–water mixtures. The model exhibits robust performance, capturing long-range dependencies and fractal behaviors not addressed by traditional models.