ANALYSIS OF FRACTAL SURFACES USING SCANNING PROBE MICROSCOPY AND MULTIPLE-IMAGE VARIOGRAPHY .1. SOME GENERAL-CONSIDERATIONS

Even when fractal models describe well the topography of a physical surface, the geometry of scanning probe microscopy will generally introduce artifacts into single images that can seriously bias estimates of the fractal dimension D from most techniques of analysis (e.g., power spectrum, correlation function, structure function). Other methods that remain unbiased (e.g., lake/island perimeter-area analysis) cannot extend to crossovers into nonfractal regimes of scale, which are also important to detect. This paper demonstrates that for multiple images fitted to their least-squares planes the topographic variance as a function of image size is a convenient statistic that retains the self-affine scaling form of the original fractal, allows unbiased estimates of D, and also detects crossover scales into nonfractal regimes.