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Hi, Good discussion - just catching up. > Question: What deformation processes are better understood by > characterizing a phenomenon as quantitatively fractal or power-law > (not exactly the same!)? Yes, a power law exponent is not strictly a fractal dimension (a la Mandelbrot), since it does not account for spatial variability or clustering. It is however often an attribute of fractal (scale-invariant or self-similar) systems, since a power-law has no characteristic scale. As a result some authors include it as a 'fractal dimension' (e.g. in Don Turcotte's book on 'Fractals and Chaos in geophysics'). > .. it might > be better to refocus on the work that has used fractal > characterization to successfully improve the understanding of > process. From my perspective a key outcome is in understanding brittle deformation in rocks and the Earth's crust as near-critical phenomena. In rock samples and simulations of brittle failure we do see systematic changes in correlation dimension (fractal clustering) and broad-band power-law exponents with ongoing deformation, more consistent with an approach to the critical point. http://pre.aps.org/abstract/PRE/v88/i6/e062207 but Nature seems already to have evolved to a dynamic equilibrium or steady-state where systematic variability in fractal dimension is much less than we see in the lab, making identification of changes, and associated prediction in space and/or time much harder. http://www.geos.ed.ac.uk/homes/imain/igmpapers/rg1996.pdf Even this statistically-stationary state can have significant *structural* switches - changing not the fractal dimension but the location of deformation as in the slip switching anticipated in http://www.geos.ed.ac.uk/~cowie/PAPERS/cowie_statphys_JGR_1993.pdf and now shown also to be consistent with power-law creep processes in the lower crust http://www.nature.com/ngeo/journal/v6/n12/full/ngeo1991.html This coupling I think is a major frontier in explaining surface structures, fractal or not. Actually fractals have more often been used in anger for pragmatic reasons, e.g. quantitative calculations of geo-hazards, or to characterise geo-statistical properties for calculating fluid flow properties at different scales etc. The main problem is that fractals inherently contain variability, so it's like having the amplitude but not the phase in a signal processing terms, making it hard to predict exactly where the systematic variations are, unless the structures are accessible at the surface. > ...many are typically working with only 1 or 1.5 orders of magnitude. Yes, a 'narrow-band fractal' is an oxymoron. Unfortunately all too common in the literature - see the review and discussion in http://www.geos.ed.ac.uk/homes/imain/igmpapers/rg2001.pdf Cheers, Ian. -- Ian Main FRSE Professor of Seismology and Rock Physics http://www.geos.ed.ac.uk/homes/imain The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336.