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.
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