on the application of fractal geometry to stream networks€¦ · fractals and fractal dimensions...

On the Application of Fractal Geometry to

Stream NetworksCalculating and Comparing Fractal Dimensions

Adam ZhangGeorgia Institute of Technology

Mathematics of Stream NetworksQuantitative framework for stream networks

Horton-Strahler Stream Ordering System

Streams as fractals

Fractals and Fractal DimensionsSelf-similarity and Heavy Tails

Fractal Dimension

Complexity and Space-filling

Fractal methods in streams

Different values

Validity

Divider (Richardson) Method

Functional-Box Counting

As a function of bifurcation and length

H.J. Andrews

Bifurcation jpg

ImplicationsGenerally fractal

Similar to values across watersheds

Techniques create varying ranges of dimension

Only divider method illustrates sinuosity vs branching

Higher fractal dimensions have higher drainage densities

More space-filling

Further StudySimulate Tree Networks of Similar Dimension

Probability Distribution in Statistical Fractals

Catchment Area vs Bifurcation Ratio

AcknowledgementsMentors and Facilitators-

Dr. Desiree Tullos, Dr. Julia Jones, Dr. Fred Swanson, Dr. Catalina Segura, Cara Walter

EISI Group-

Sashka, Brent, Pete, Malia, Josh, Jane, Lydia, Elaina, Andrew, Stephanie

H.J. Andrews and Oregon State University

ReferencesB. B. Mandelbrot, “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,” Science, New Series, Vol. 156, No. 3775, 1967, pp. 636-638.

La Barbera, P., and R. Rosso. 1990. On fractal dimension of stream networks, Reply to Tarboton et al., Water. Resour. Res., 26(9), 2245-2248.

Lovejoy, S., D. Schertzer, and A. A. Tsonis. 1987. Functional box counting and multiple elliptical dimensions in rain, Science, 235, 1036-1038.

Horton, R. E. 1932. Drainage-basin characteristics, EOS Trans. AGU, 13, 350-361.

I. Rodriguez and A. Rinaldo, “Fractal River Basins (Chance and Self-Organization),” Cambridge University Press, Cambridge, 1997.

Strahler, A. N. 1952. Hypsometric (area altitude) analysis of erosional topography, Geol. Soc. Am. Bull., 63, 1117-1142.

Tarboton, D. G., R. L. Bras, and I. Rodgriquez-Iturbe. 1988. The fractal nature of river networks, Water. Resour. Res., 24(8), 1317-1322

Z. Khanbabaei, A. Karam, and G. Rostamizad, "Studying relationships between the fractal dimension of the drainage basins and some of their geomorphological characteristics", Int. J. Geosci., vol. 4, pp. 636-642, 2013.

DiscussionAdam Zhang

School of Mathematics

Georgia Institute of Technology, c/o 2018

azhang47@gatech.edu