Fractal analysis in geology. Week 4: Analysis of fractal geometries. Lecture index: What is a. Calculation. of earthquake size- recurrence interval relationships. Turcotte, Fractals and chaos. Read this carefully, and pay. Marrett, R. This is a good example of. Peruse it. Note the geometry of the skyline. How would you describe it? What is a fractal? There are so many ways to introduce fractals. Perhaps. it is because it takes some thinking outside the classic 3- D box. Pick two endpoints. As a line most would think of it as a one dimensional. If you try to measure that length. You don't converge. A natural assumption would be. What happens with a fractal line is. The bumps, have bumps, which. The Chaos Hypertextbook. An introductory primer on chaos and fractals. Fractals, in Layman's Terms; Fractals, fractal dimension, chaos, plane filling curves at cut. Get this from a library! Fractals and hyperspaces. The length just continues. So in theory this is. It has a fractional dimension, somewhere. So. actually, an infinite number of dimensions can exist in our simple. Fractals are entities that are scale invariant. You may. have seen pictures devoid of vegetation where it is difficult. The Application of Fractal Geometry to Ecology New insights into the natural world are just a few of the results from the use of fractal geometry. The online version of Chaos, Solitons & Fractals at ScienceDirect.com, the world's leading platform for high quality peer-reviewed full-text journals. Explores the subject of fractal geometry focusing on the occurrence of fractal-like shapes in the natural world. Topics include iterated functions, chaos theory, the. The Mandelbrot set is the set of complex numbers c for which the function () = + does not diverge when iterated from =, i.e., for which the sequence (), (()), etc. There is no feature with a characteristic. That landscape maybe fractal. Indeed, using fractal. In addition, there are fractal relationships. Fractals are often produced. It is important to remember that ultimately. 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. By Author Abraham, Ralph H., and Christopher D. Dynamics: The Geometry of Behavior. Fractals For the Classroom. Few of us have escaped the ubiquitous fractals that appear on screen savers or coffee table picture books, but what bearing do these. At some point, clearly. However, the. important thing to remember is that fractal models can successfully. Consider. a mountain ridge some 1 km long made primarily of a sandstone. As with most mathematical descriptions. Why learn about fractals in geologic analysis? A criticism is that fractals produce pretty. Your. one reading indicates just one way in which this criticism is. For phenomena like earthquakes that show a fractal distribution. You. will explore this aspect in your exercise this week. Estimation. of earthquake recurrence interval is of course of immense utility. For. example, they have been applied to studies of the effect of surface. The literature is rich in. Fractals provide. 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 1. Examples of phenomena with a fractal distribution. The basic equation and the fractal dimension. Nn. = C / rn**D, where is Nn. 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. D = ln(Nn+1/Nn)/ln(rn/rn+1)The log of Nn vs. The slope yields the fractal dimension. See your reading for a more complete treatment. Example of coastline ? This. is an in- class, data gathering part of your exercise. The class. will measure the same section of coast line but with different. Then we will plot the values to see what, if any. The literature suggests there should. If you think about it, there is no reason you. There is also another method that can be used which. Turcotte. describes it in some detail. Photo of Norwegian coast from http: //visibleearth. 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. 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 links for further exploration: Some references: General information on fractals: Mandelbrot, Benoit, 1. How long is the. coast of Britain? Statistical self- similarity and fractional. Science, 1. 56, 6. Mandelbrot, Benoit, 1. The Fractal Geometry. Nature, Freeman. By the IBM research who literally created. Briggs, J. D., 1. Turbulent. Mirror, Harper Row, 2. This is a fun introduction into fractals. Fractals and geology: Bak, P. A very interesting. Barton, Christopher C. La Pointe. eds., 1. Fractals in the Earth Sciences, Plenum Press, 2. Marrett, R. 4. 7- 5. Middleton, Gerard V., ed., 1. Nonlinear. Dynamics, Chaos and Fractals, with Applications to Geological. Systems, Geol. 9. Turcotte, Donald L., 1. Fractals and Chaos. Geology and Geophysics, Cambridge Press, 2. This material. may be used for non- profit educational purposes if proper attribution. Otherwise please contact Harmon D.
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