Fractal analysis of sampled profiles : systematic study

A quantitative evaluation of the influence of sampling on the numerical fractal analysis of experimental profiles is of critical importance. Although this aspect has been widely recognized, a systematic analysis of the sampling influence is still lacking. Here we present the results of a systematic analysis of synthetic self-affine profiles in order to clarify the consequences of the application of a poor sampling (up to 1000 points) typical of scanning probe microscopy for the characterization of real interfaces and surfaces. We interpret our results in terms of a deviation and a dispersion of the measured exponent with respect to the "true" one. Both the deviation and the dispersion have always been disregarded in the experimental literature, and this can be very misleading if results obtained from poorly sampled images are presented. We provide reasonable arguments to assess the universality of these effects and propose an empirical method to take them into account. We show that it is possible to correct the deviation of the measured Hurst exponent from the "true" one and give a reasonable estimate of the dispersion error. The last estimate is particularly important in the experimental results since it is an intrinsic error that depends only on the number of sampling points and can easily overwhelm the statistical error. Finally, we test our empirical method calculating the Hurst exponent for the well-known 1 + 1 dimensional directed percolation profiles, with a 512-point sampling.