Measuring Hurst Exponents With the First Return Method

The First Return method has proven to be an efficient method for determining the Hurst exponent, H, of self-affine surfaces. We discuss its foundations and some corrections to scaling which must be taken into account for an adequate estimation of H. Using this method, we analyze a set of artificially generated curves with known Hurst exponent and compare the result with Fourier analysis. We also discuss the case where the surface to be analyzed has a curved bias with a maximum or minimum. In this case, the relation between the scaling exponents associated with the first-return histograms and the Hurst exponent H is different from the unbiased case.