Scale Independence – Aha! Moment
I had an Aha! moment this afternoon while trawling through my draft posts for edmedley.com . I came across the title Soul-Soaring (Fract)Elation for an unwritten draft, the provenance of the title being from the Q&A session after one of my Jahns Lectures at California State University at Fresno last year. A student asked me if I had had a break-through moment in my PhD research. I told him I had - it had been a moment of "soul-soaring elation". That was such an odd, poetic expression to tumble from me, that the geologist audience and I laughed. But I shall never forget the joy, the soul-soaring elation, I felt during my breakthrough research moment in 1994. See if you can feel it too... I was measuring the sizes of blocks in Franciscan Complex melanges from photographs and maps then plotting histograms of the sizes. Histograms are graphs of the numbers of somethings of certain sizes: in this case the number of blocks that fell into "bins" or "classes" of ranges. My measurements ranged form millimeters to kilometers - seven orders of magnitude (or the largest measurements were 10 million times larger than the smallest measurements.) It is really impossible to show the data on conventional graphs so I switched to logarithmic plots. The plots thus became "log-histograms". When plotted as log-histograms much of the data assumed neat order - steeply peaked curves marched nicely across the graph paper, looking very similar. Indeed: the block size plots (we call them block size distributions) are "self-similar" meaning that if you do not have some way to know the scale there is no real way to tell them apart. The plots are also "fractal" which basically means the same thing. Fractal mathematics is pretty cool stuff and very sexy in geology. But I am not a mathematician and you likeley are not either, so let's skip ahead... The plots were so invitingly similar that I gnawed at finding some way to collapse the plots some more. I eventually tried dividing all the block size data by the square root of the area of the photo or map containing the individual data sets. Doing that was pretty easy- I just worked my way through the exercise using Excel. When I hit the Excel graphing button , up came all the data, collapsed into one fluffy big plot, with many colored symbols, organized into a constellation of datapoints that grouped together looked just like the individual plots. That was a thrilling moment, seeing the constellation on my screen; an amazed moment of "Holy shit! - look at that!" - which is what "soul-soaring elation" is, expressed in (my) words. I had stumbled upon the "scale-independence" of the block-size distributions of the Franciscan Complex melanges I had measured. It did not matter at what scale, it seemed that the melanges would always have a similar-looking block size distribution. In practice that meant many, many small blocks and a decreasing number of larger blocks. The collapsed plot gave me the insight to come up with a way to answer the pesky question "If there are seven orders of magintitude of block sizes, then what is block and what is matrix?" The answer: "It depends on your scale of interest". That is not a throw-away answer; it has significant implications in engineering - I have written about that significance. The findings also provided a solid justification for the work of my research partner, now-Dr. Eric Lindquist. Eric was working on relatively small physical models of melanges in the laboratory. Since our joint problem was the characterization of melange under a dam, one valid critique could be: "What is the validity of laboratory results based on small models to the real-world behavior of a full-scale dam and its foundations?" The answer was that the small-scale laboratory melanges were actually much more like scale models of real conditions than is generally found in geotechnical engineering. Not all PhD researchers have the same soul-soaring elation, the same "Holy shit!" moment that I had. I consider myself lucky and privileged, and I hope one day you have one too.