Geometrical exponents of contour loops on synthetic multifractal rough surfaces: multiplicative hierarchical cascade p-model

TL;DR

This study investigates the geometrical properties of contour loops on synthetic multifractal rough surfaces generated by hierarchical cascade models, revealing that certain exponents depend on multifractal parameters and Hurst exponent, with implications for understanding surface morphology.

Contribution

It introduces a detailed analysis of contour loop exponents on synthetic multifractal surfaces, highlighting the roles of multifractal parameters and Hurst exponent in their geometrical properties.

Findings

Hyperscaling relations hold for both classes of surfaces.

H* influences geometrical exponents in smoothened surfaces.

Exponents depend on p-values and Hurst exponent in multifractal surfaces.

Abstract

In this paper, we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces, which are generated by multiplicative hierarchical cascading processes. To this end, two different classes of multifractal rough surfaces are numerically simulated. As the first group, singular measure multifractal rough surfaces are generated by using the $p$ model. The smoothened multifractal rough surface then is simulated by convolving the first group with a so-called Hurst exponent, $H^*$ . The generalized multifractal dimension of isoheight lines (contours), $D(q)$, correlation exponent of contours, $x_l$, cumulative distributions of areas, $\xi$, and perimeters, $\eta$, are calculated for both synthetic multifractal rough surfaces. Our results show that for both mentioned classes, hyperscaling relations for contour loops are the same as that of monofractal systems. In contrast to singular measure multifractal rough surfaces, $H^*$ plays a leading role in smoothened multifractal rough surfaces. All computed geometrical exponents for the first class depend not only on its Hurst exponent but also on the set of $p$ values. But in spite of multifractal nature of smoothened surfaces (second class), the corresponding geometrical exponents are controlled by $H^*$, the same as what happens for monofractal rough surfaces.