Effect of resolution on the apparent fractal dimension of jagged force-displacement relationships and other irregular signatures

The apparent fractal dimension of computer simulated and experimental jagged force-displacement curves was determined using the Richardson and Kolmogorov algorithms. The curves analyzed had a wide range of amplitudes and a variety of temporal resolutions. Despite the fact that curves whose points are uniformly spaced (i.e. recorded at a constant sampling rate) are not truly fractal, both algorithms could be successfully employed, as judged by the linearity of their corresponding plots. The apparent Kolmogorov dimension of the simulated curves was generally smaller than that calculated from the Richardson plot. There was however a good agreement between the two in the experimental curves. Both dimensions increased in unison with the curve fluctuation's amplitude and resolution, indicating that the apparent fractal dimension is a consistent measure of jaggedness. In all the cases, the relationship between the apparent fractal dimension and the resolution could be described by simple empirical models which could be used interchangeably to calculate the asymptotic apparent fractal dimension at infinite resolution. Consequently, the record of real instruments with a finite resolution determined by technical and physical considerations could be used to estimate the 'ultimate jaggedness' of the force-displacement relationship or any other digitized signature. The same method can be used, through interpolation, to estimate the curve's jaggedness at any desired resolution for the sake of comparison. Without such an adjustment, the jaggedness of different curves when expressed in terms of their apparent dimension is meaningful only if they have been recorded at the same resolution. (C) 1997 Elsevier Science Limited.