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Averaging Properties of Channel Networks Using Methods in Stochastic Branching Theory

  • Brent M. Troutman
  • Michael R. Karlinger
Chapter
Part of the Water Science and Technology Library book series (WSTL, volume 6)

Abstract

Methods in branching theory are used to average properties of channel networks, resulting in expressions for the instantaneous unit hydrograph (IUH) in terms of fundamental network characteristics (Z, α, β), where α parameterizes the link (channel segment) length distribution and β is a vector of hydraulic parameters. Several possibilities for Z are considered, including N, (N, D), (N, M), \(\tilde{D}\) , and (N, \(\tilde{D}\)), where N is magnitude (number of first-order streams), D is topological diameter, M is order, and \(\tilde{D}\) is mainstream length. Linear routing schemes, including translation, diffusion, and general linear routing, are used, and it is demonstrated that translation routing leads to an IUH identical to that obtained by use of the width function, where, for a given distance x, the width of a network is defined to be the number of links some point of which lies at channel distance x from the outlet (analogous to population size in branching theory).

The IUH is taken to be the conditional expectation of actual basin response given Z, and it is derived based on assumptions that the links are independent and identically distributed random variables and that the network is a member of a topologically random population. Uncertainty in use of this expectation to approximate actual basin response is given by the corresponding conditional variance.

Asymptotic (for large N,D, and \(\tilde{D}\)) results are available for several cases. For example, when Z = N asymptotic considerations lead to a Weibull probability density function for the IUH for all linear routing schemes, with only a single parameter depending on the particular routing method.

A simulation study compares different possibilities for Z in terms of ability to predict actual IUH characteristics. These characteristics include peak and time to peak.

Keywords

Conditional Expectation Hydraulic Parameter Water Resource Research Channel Network Link Length 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Athreya, K.B. and Ney, P.E., 1972. Branching Processes, Springer-Verlag, Berlin.Google Scholar
  2. Bailey, N.T.J., 1964. The Elements of Stochastic Processes with Applications to the Natural Sciences, John Wiley and Sons, NY.Google Scholar
  3. Dooge, J.C.I. and Harley, B.M., 1967. Linear Routing in Uniform Open Channels, Proc. Int’l. Hydrol Symp., v. 1, pp. 8. 1–8. 7.Google Scholar
  4. Gupta, V.K., Waymire, E., and Wang, C.T., 1980. A Representation of an Instantaneous Unit Hydrograph From Geomorphology, Water Resources Research, 16(5), pp. 855–862.CrossRefGoogle Scholar
  5. Gupta, V.K., and Waymire, E., 1983. On the Formulation of an Analytical Approach to Hydrologic Response and Similarity at the Basin Scale, J. Hydrology, 65, pp. 95–124.CrossRefGoogle Scholar
  6. Harley, B.M., 1967. Linear Routing in Uniform Open Channels, M.S. Thesis, National University of Ireland.Google Scholar
  7. Harris, T.E., 1963. The Theory of Branching Processes, Springer-Verlag, Berlin.Google Scholar
  8. Karlin, S., 1966. A First Course in Stochastic Processes, Academic Press, NY.Google Scholar
  9. Karlinger, M.R., and Troutman, B.M., 1985. An Assessment of the Instantaneous Unit Hydrograph Derived from the Theory of Topologically Random Networks, Water Resources Research, 21(11), pp. 1693–1702.CrossRefGoogle Scholar
  10. Keefer, T.N., and McQuivey, K.S., 1974. Multiple Linearization Flow Routing Model, J. Hydraul Div., ASCE, 100(HY7), pp. 1031–1046.Google Scholar
  11. Kirkby, M.J., 1976. Tests of the Random Network Model and Its Applications to Basin Hydrology, Earth Surf Proc., 1, pp. 197–212.CrossRefGoogle Scholar
  12. Kirshen, D.M., and Bras, R.L., 1983. The Linear Channel and its Effect on the Geomorphic IUH, J. Hydrology, 65, pp. 175–208.CrossRefGoogle Scholar
  13. Leopold, L.B., and Maddock, T., Jr., 1953. The Hydraulic Geometry of Stream Channels and Some Geomorphologic Implications, U.S. Geol. Survey Prof. Paper 252, 56 pp.Google Scholar
  14. Lienhard, J.H., 1964. A Statistical Mechanical Prediction of the Dimensionless Unit Hydrograph, J. Geophys. Res., 69(24), pp. 5231–5238.CrossRefGoogle Scholar
  15. Pilgrim, P.H., 1977. Isochrones of Travel Time and Distribution of Flood Storage from a Tracer Study on a Small Watershed, Water Resources Research, 13(3), pp. 587–595.CrossRefGoogle Scholar
  16. Rodríguez-Iturbe, I., and Valdés, J.B., 1979. The Geomorphologic Structure of Hydrologic Response, Water Resources Research, 15(6), pp. 1409–1420.CrossRefGoogle Scholar
  17. Rosenberger, J.L., and Gasko, M., 1983. Comparing Location Estimators: Trimmed Means, Medians, and Trimean, in Understanding Robust and Exploratory Data Analysis, edited by D.C. Hoaglin, F. Mosteller, and J.W. Tukey, John Wiley & Sons, NY.Google Scholar
  18. Shreve, R.L., 1966. Statistical Law of Stream Numbers, J. Geology, 74, pp. 17–37.CrossRefGoogle Scholar
  19. Shreve, R.L., 1967. Infinite Topologically Random Channel Networks, J. Geology, 75, pp. 178–186.CrossRefGoogle Scholar
  20. Shreve, R.L., 1969. Stream Lengths and Basin Areas in Topologically Random Channel Networks, J. Geology, 77, pp. 397–414.CrossRefGoogle Scholar
  21. Shreve, R.L., 1974. Variation of Mainstream Length with Basin Area in River Networks, Water Resources Research, 10, pp. 1167–1177.CrossRefGoogle Scholar
  22. Smart, J.S., 1968. Statistical Properties of Stream Lengths, Water Resources Research, 4, pp. 1001–1014.CrossRefGoogle Scholar
  23. Smart, J.S., 1972. Channel Networks, in Advances in Hydroscience, edited by Ven Te Chow, Academic Press, NY.Google Scholar
  24. Smart, J.S., 1978. Analysis of Drainage Network Composition, Earth Surf. Proc., 3, pp. 129–170.CrossRefGoogle Scholar
  25. Smart, J.S., and Werner, C, 1976. Analysis of the Random Model of Drainage Basin Composition, Earth Surf Proc., pp. 219–233.Google Scholar
  26. Troutman, B.M., and Karlinger, M.R., 1984. On the Expected Width Function for Topologically Random Channel Networks, J. of Applied Probability, 21, pp. 836–849.CrossRefGoogle Scholar
  27. Troutman, B.M., and Karlinger, M.R., 1985. Unit Hydrograph Approximations Assuming Linear Flow Through Topologically Random Channel Networks, Water Resources Research, 21(5), pp. 743–754.CrossRefGoogle Scholar
  28. Valdés, J.B., Fiallo, Y., and Rodríguez-Iturbe, I., 1979. A Rainfall-Runoff Analysis of the Geomorphologic IUH, Water Resources Research, 15(6), pp. 1421–1434.CrossRefGoogle Scholar
  29. Wang, C.T., Gupta, V.K., and Waymire, E., 1981. A Geomorphic Synthesis of Nonlinearity in Surface Runoff, Water Resources Research, 17(3), pp. 545–554.CrossRefGoogle Scholar
  30. Werner, C., and Smart, J.S., 1973. Some New Methods of Topologic Classification of Channel Networks, Geographical Analysis, 5, pp. 271–295.CrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • Brent M. Troutman
  • Michael R. Karlinger

There are no affiliations available

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