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, xl​, cumulative distributions of areas,
ξ, and perimeters, η, 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.Comment: 14 pages, 14 figures and 6 tables; V2: Added comments, references,
table and major correction