Greetings,
A question and two observation:

Question: What deformation processes are better understood by characterizing a phenomenon as quantitatively fractal or power-law (not exactly the same!)?  The question is offered not because improvement of understanding of process via fractal characterization has never been achieved, but rather, a number of cases exist where it has not.   Geology (not just structural geology!) is well populated with past efforts to quantify observations that did not improved the understanding of the underlying geological processes.  So, it might be better to refocus on the work that has used fractal characterization to successfully improve the understanding of process.

Observation: Consider the case of joint network with a fractal dimension of 1.57 (a different number can be picked).  Inspection of the literature will yield a number of networks of joints with this dimension.  Yet, because the fractal dimension is essentially a global characterization to do with network abundance and complexity, it is difficult (not possible?) to ascertain aspects of the network (e.g., number of sets, fracture tracelength size distributions, angular relationships between fractures) that they share for sharing the same dimension.  So, some thought needs to be given (and has been given by some workers!) to why one is ascertaining a fractal dimension.

Observation: Care needs to be taking when determining whether a data set is sufficient to determine whether a characteristic is fractal or not fractal.  Many folks have done thoughtful work on this matter, with the result that some analyses withstand inspection, but many do not.  My personal experience is with attributes of fractures such as tracelength and aperture.  Given censoring and truncation, the consensus seems to be that that three clear clean orders of size (so a data set spanning five to six orders of magnitude in size) is needed to reach a quality result.  That does happen in some analyses, but many others are typically working with only 1 or 1.5 orders of magnitude.

Good luck with folding the concept of similarity or disimilarity of form with changing scale into your second year course.  It is a good concept to include as it should cause students to have to think (Pumpelly's Rule!)
Bill
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William M. Dunne
Associate Dean & Professor
Research & Facilities
College of Engineering
University of Tennessee  Knoxville
Ph: 865-974-3608
Email: [log in to unmask]
Website: http://www.engr.utk.edu/ 
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-----Original Message-----
From: Tectonics & structural geology discussion list [mailto:[log in to unmask]] On Behalf Of Potts, Graham
Sent: Monday, January 20, 2014 8:28 AM
To: [log in to unmask]
Subject: fractal and non-fractal structures

Hi

I am in the process of "refreshing" the introductory lecture for my second year module in structural geology at Liverpool, U.K.

one of the components of the module is the interactions between similar and dissimilar structures

as a result of a somewhat cursory review I have been struck by a couple of issues

1.   the number of structures or their attributes (e.g. maximum displacement, length) documented as fractal, self-similar..... has changed little over the last ten years

        a.   Is this perception wrong?

        b.   Or do we, as a community, believe we have identified most those structures or their attributes that are fractal in nature

        c.   Or has this phase of documentation simply "run out of steam"

2.   with the exception of some attributes of some fracture (s.l.) systems the number of documented examples of structures or their attributes that are NOT fractal is much smaller than those that have been shown to be fractal in nature

       a.   Is this perception wrong?

       b.   Or is there a bias because many structures or their attributes not amenable to such detailed quantification?

       c.  Or is there a bias because non-fractal behaviours have been documented but the results have been declared not very interesting or important?

if I have missed your "favourite" paper I apologise in advance but I am buried by marking at the moment

Graham Potts