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We develop an approach for characterizing the morphology of rough surfaces based on the analysis of the scaling properties of contour loops, i.e., loops of constant height. Given a height profile of the surface we perform independent measurements of the fractal dimension of contour loops, and the exponent that characterizes their size distribution. Scaling formulas are derived, and used to relate these two geometrical exponents to the roughness exponent of a self-affine surface, thus providing independent measurements of this important quantity. Furthermore, we define the scale-dependent curvature, and demonstrate that by measuring its third moment departures of the height fluctuations from Gaussian behavior can be ascertained. These nonlinear measures are used to characterize the morphology of computer generated Gaussian rough surfaces, surfaces obtained in numerical simulations of a simple growth model, and surfaces observed by scanning-tunneling microscopes. For experimentally realized surfaces the self-affine scaling is cut off by a correlation length, and we generalize our theory of contour loops to take this into account.

Nonlinear measures for characterizing rough surface morphologies