Week 4: Analysis of fractal geometries in the geosciences

Chapt. 1 and 2 of Turcotte, Fractals and chaos in geology and geophysics.p. 1-17. Read this carefully, and pay attention to the basic equation.

Marrett, R. and Allmendinger, 1992, Amount of extension on "small" faults: An example from the Viking graben; Geology, 20, p. 47-50. This is a good example of just one ways fractal analysis can be useful. Peruse it.

Note the geometry of the skyline profile here. How would you describe it?

What is a fractal?

There are so many ways to introduce fractals that one wonders why they weren't thought of much earlier. Perhaps it is because it takes some thinking outside the classic 3-D box. I will take a variant of a classic approach to understanding fractals. Think of the profile of a mountain ridge as a line. Pick two endpoints on the line. As a line most would think of it as a one dimensional feature, with a specific length. If you try to measure that length in a variety of ways (using strings, small rulers, boxes) an interesting thing happens as you try to get better estimates by using smaller rulers and/or more precise mappings of the line. You don't converge on or approach a specific number. A natural assumption would be that as you look at the line more carefully, as you measure it with more detail, with better vision and a smaller ruler you would converge on a true value. What happens with a fractal line is that as you look closer and closer, the same bumpiness or roughness is found again at the new scale. The bumps, have bumps, which have bumps and so on, theoretically forever. The length just continues to grow and not converge on a true value. So in theory this is a line with two well defined endpoints, but without a length. What dimension is it? It has a fractional dimension, somewhere in between a line and a surface, between 1 and 2 dimensions. So actually, an infinite number of dimensions can exist in our simple three dimensional static world, never minding the 4th 5th and 6th dimensions.

Fractals are entities that are scale invariant. For fractal patterns this means they look similar at a greater variety of scales, i.e. they are self similar. You may have seen pictures devoid of vegetation where it is difficult to tell how big something is. There is no feature with a characteristic scale or length. That landscape maybe fractal. Indeed, using fractal relationships people have been able to design very realistic looking computer landscapes. In addition, there are fractal relationships where the behavior is scale invariant. Fractals are often produced by reiterations or feedback loops where the results of the previous operation are fed into and influence the next operation.

It is important to remember that ultimately no natural feature is fundamentally fractal. At some point, clearly at the molecular level, the scale invariance breaks down and gives away to another world with its own characteristic and unique scale. So in real life the scale invariance only exists for a range of scales, from a scale which includes the entire feature being analyzed, down to a small enough scale where we enter new physical worlds, e.g. those of quantum mechanics. However, the important thing to remember is that fractal models can successfully describe natural features over many orders of magnitude. Consider a mountain ridge some 1 km long made primarily of a sandstone. By the time we get down to the grain size of several millimeters, we would leave the scale invariant behavior controlled by fracture planes, and individual sand grains will control the profile shape. The grains outlines themselves may have a different fractal relationship. So a fractal relationship might exist from 1 km to mms, or over 5-6 orders of magnitude. As with most mathematical descriptions of our world it only works for part of that world.

Why learn about fractals in geologic analysis?

A criticism is that fractals produce pretty images and are interesting, but not particularly useful. Your one reading indicates just one way in which this criticism is wrong. For phenomena like earthquakes that show a fractal distribution of frequency of occurrence this relationship can be used to infer the recurrence interval of a given size earthquake in an area. It may also give insight into the dynamics of earthquakes. You will explore this aspect in your exercise this week. Estimation of earthquake recurrence interval is of course of immense utility in seismic hazard assessment, in engineering, and hazard mitigation. Any place someone is interested in surface area or roughness, fractals provide a chance at a more accurate presentation. For example, they have been applied to studies of the effect of surface roughness on intergrain permeability. The literature is rich in examples of the utility of fractals in geology. Fractals provide a theoretical basis for extrapolation as you will see in your exercise. However, when extrapolating the question about the real world range of scale over which the fractal distribution applies is a an important consideration. You could extrapolate for the recurrence interval of a RM 12 fault earthquake, but there is no reason to believe they can occur.

Examples of phenomena with a fractal distribution or geometry in geology.

• frequency of earthquakes of various sizes.
• all sorts of fault parameters scale fractally.
• size of clasts in fragmented material.
• crater size (?).
• ore body size.
• styolite surfaces.
• dendritic branching.
• some geomorphic surfaces.

The basic equation and the fractal dimension.

Nn = C / rn**D, where is Nn the number of objects with a characteristic linear dimension rn. D is the fractal dimension and C is a proportionality constant. If we take the natural log of both sides and solve for D we get the equation below:

D = ln(Nn+1/Nn)/ln(rn/rn+1)

The log of Nn vs. rn plots as a line with a negative slope in log-log space. The slope yields the fractal dimension. See your reading for a more complete treatment.

Example of coastline "convoluteness".

Cubic island example - fractal in theory.

Fractal in practice? This is an in-class, data gathering part of your exercise. The class will measure the same section of coast line but with different size rulers. Then we will plot the values to see what, if any, fractal relationship exists. The literature suggests there should be one.

If you think about it, there is no reason you can't do the same exercise with contour lines off a topographic map (although at smaller scales they might not have the detail you need). There is also another method that can be used which uses different size boxes instead of different size rulers. Turcotte describes it in some detail.

Photo of Norwegian coast from http://visibleearth.nasa.gov/. Fractal?

Related examples.

Example of modeling restrained population growth and the Verhultz equation using Excel and an introduction to chaos.

Slider block models of earthquake movement.

Example of plotting of earthquake frequency. We will take a look at earthquake data from the Turkestan.

Exercise 4. Calculation of earthquake size-recurrence interval relationships.

To make comparison of results easier and more meaningful we will coordinate people's choice of where to conduct part 2 of exercise 4. The idea is to see how the fractal distribution and recurrence interval for a standard size area changes from place to place, and to consider what may be the reason(s) for the change.

Some references:

General information on fractals:

• Mandelbrot, Benoit, 1967, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, 156, 636-8.
• Mandelbrot, Benoit, 1982, The Fractal Geometry of Nature, Freeman. By the IBM research who literally created the mathematical subdiscipline of fractals.
• Briggs, J. & Peat, F. D., 1989, Turbulent Mirror, Harper Row, 222 p. This is a fun introduction into fractals and chaos and includes some of the elementary math.

Fractals and geology:

• Bak, P. & Chen, K., 1991, Self-organized Criticality, Scientific American, Jan, p. 46-53. A very interesting article that looks at the dynamics of earthquakes and sandpiles, among other things.
• Barton, Christopher C. and Paul R. La Pointe, eds., 1995, Fractals in the Earth Sciences, Plenum Press, 265 pp.
• Marrett, R. and Allmendinger, 1992, Amount of extension on "small" faults: An example from the Viking graben; Geology, 20, p. 47-50.
• Middleton, Gerard V., ed., 1991, Nonlinear Dynamics, Chaos and Fractals, with Applications to Geological Systems, Geol. Assoc. of Canada Short Course Notes, Vol. 9.
• Turcotte, Donald L., 1992, Fractals and Chaos in Geology and Geophysics, Cambridge Press, 221 pp. This will be our "text" for fractals.