Hydrologic Synthesis Using Entropy Theory: Review
Publication: Journal of Hydrologic Engineering
Volume 16, Issue 5
Abstract
An entropy theory, comprising the Shannon entropy, the principle of maximum entropy, and the concentration theorem, has been applied in recent years to a wide range of problems in hydrology. From a hydrologic point of view, the applications can be organized into three classes: (1) physical, (2) statistical, and (3) mixed. This study focuses on the physical applications of the entropy theory, wherein the theory is coupled with the laws of mathematical physics and solutions are derived either in a time or space domain rather than the frequency domain. It is shown that a general framework can be developed to derive solutions to a wide range of seemingly disparate problems. The theory seems to have much potential that remains yet to be fully exploited.
Get full access to this article
View all available purchase options and get full access to this article.
References
Al-Hamdan, O. Z., and Cruise, J. F. (2010). “Soil moisture profile development from surface observations by principle of maximum entropy.” J. Hydrol. Eng., 15(5), 327–337.
Amorocho, J., and Espildora, B. (1973). “Entropy in the assessment of uncertainty in hydrologic systems and models.” Water Resour. Res., 9(6), 1551–1522.
Barbe, D. E., Cruise, J. F., and Singh, V. P. (1991). “Derivation of a velocity distribution using the principle of maximum entropy.” J. Hydraul. Eng., 117(10), 1389–1396.
Barbe, D. E., Cruise, J. F., and Singh, V. P. (1994). “Derivation of a distribution for the piezometric head in groundwater flow using entropy.” Stochastic and statistical methods in hydrology and environmental engineering, K. W. Hipel, ed., Vol. 2, Kluwer, Dordrecht, Netherlands, 151–161.
Batty, M. (1974). “Spatial entropy.” Geogr. Anal., 6, 1–31.
Batty, M. (1976). “Entropy in spatial aggregation.” Geogr. Anal., 8, 1–21.
Cao, S. Y., and Chang, H. H. (1988). “Entropy as a probability concept in energy gradient distribution.” Proc., 1988 National Conf. on Hydraulic Engineering, 1013–1018.
Cao, S., and Knight, D. W. (1995a). “Design of threshold channels. Hydra 2000.” Proc., 26th IAHR Congress, London.
Cao, S., and Knight, D. W. (1995b). “New concept of hydraulic geometry of threshold channels.” Proc., 2nd Symp. on the Basic Theory of Sedimentation, Beijing.
Cao, S., and Knight, D. W. (1997). “Entropy-based approach of threshold alluvial channels.” J. Hydraul. Res., 35(4), 505–524.
Chiu, C. L. (1987). “Entropy and probability concepts in hydraulics.” J. Hydraul. Eng., 113(5), 583–600.
Chiu, C. L. (1988). “Entropy and 2D velocity distribution in open channels.” J. Hydraul. Eng., 114(7), 738–756.
Chiu, C. L. (1989). “Velocity distribution in open channel flow.” J. Hydraul. Eng., 115(5), 576–594.
Chiu, C. L. (1991). “Application of entropy concept in open channel flow study.” J. Hydraul. Eng., 117(5), 615–628.
Chiu, C. L., and Chen, Y. C. (2003). “An efficient method of discharge estimation based on probability concept.” J. Hydraul. Res., 41(6), 589–596.
Chiu, C. L., Hsu, S. M., and Tung, N. C. (2005). “Efficient methods of discharge measurements in rivers and streams based on the probability concept.” Hydrol. Processes, 19, 3935–3946.
Chiu, C. L., Jin, W., and Chen, Y. C. (2000). “Mathematical models of distribution of sediment concentration.” J. Hydraul. Eng., 126(1), 16–23.
Chiu, C. L., and Murray, D. W. (1992). “Variation of velocity distribution along nonuniform open channel flow.” J. Hydraul. Eng., 118(7), 989–1001.
Chiu, C. L., and Said, C. A. A. (1995). “Maximum and mean velocities and entropy in open-channel flow.” J. Hydraul. Eng., 121(1), 26–35.
Chiu, C. L., and Tung, N. C. (2002). “Maximum velocity and regularities in open channel flow.” J. Hydraul. Eng., 128(4), 390–398.
Choo, T. H. (2000). “An efficient method of the suspended sediment-discharge measurement using entropy concept.” Water Eng. Res., 1(2), 95–105.
de Araujo, J. C. (2007). “Entropy-based equation to assess hillslope sediment production.” Earth Surf. Processes Landforms, 32, 2005–2018.
Deng, Z. Q., and Zhang, K. Q. (1994). “Morphologic equations based on the principle of maximum entropy.” Int. J. Sediment Res., 9(1), 31–46.
Diplas, P., and Vigilar, G. (1992). “Hydraulic geometry of threshold channels.” J. Hydraul. Eng., 118(4), 597–614.
Fiorentino, M., Claps, P., and Singh, V. P. (1993). “An entropy-based morphological analysis of river-basin networks.” Water Resour. Res., 29(4), 1215–1224.
Green, W. H., and Ampt, C. A. (1911). “Studies on soil physics, I. Flow of air and water through soils.” J. Agric. Sci., 4, 1–24.
Harmancioglu, N. B., Singh, V. P., and Alpaslan, N. (1992). “Versatile uses of the entropy concept in water resources.” Entropy and energy dissipation in water resources, V. P. Singh and M. Fiorentino, ed., Kluwer, Dordrecht, Netherlands, 91–118.
Hartley, R. V. L. (1928). “Transmission of information.” Bell Syst. Tech. J., 7(3), 535–563.
Henderson, F. M. (1966). Open channel flow, Macmillan, New York.
Holtan, H. N. (1961). “A concept of infiltration estimates in watershed engineering.” ARS41-51, U.S. Dept. of Agriculture, Agricultural Research Service, Washington, DC.
Horton, R. I. (1938). “The interpretation and application of runoff plot experiments with reference to soil erosion problems.” Proc., Soil Science Society of America, 3, 340–349.
Jaynes, E. T. (1957a). “Information theory and statistical mechanics, I.” Phys. Rev., 106, 620–630.
Jaynes, E. T. (1957b). “Information theory and statistical mechanics, II.” Phys. Rev., 108, 171–190.
Jaynes, E. T. (1958). “Probability theory in science and engineering.” Colloquium lectures in pure and applied science, No. 4, Socony Mobil Oil Company, Dallas.
Jaynes, E. T. (1982). “On the rationale of maximum entropy methods.” Proc., IEEE, 70, 939–952.
Jaynes, E. T. (2003). Probability theory, Cambridge University Press, Cambridge, UK, 727.
Jowitt, P. W. (1991). “A maximum entropy view of probability-distributed catchment models.” Hydrol. Sci. J., 36(2), 123–134.
Kapur, J. N. (1989). “Maximum entropy models in science and engineering,” Wiley, New Delhi, India.
Kostiakov, A. N. (1932). “On the dynamics of the coefficient of water percolations in soils.” 6th Commission, Int. Society of Soil Science, Part A, 15–21.
Koutsoyiannis, D. (2005a). “Uncertainty, entropy, scaling and hydrological stochastics, 1. Marginal distributional properties of hydrological processes and state scaling.” Hydrol. Sci. J., 50(3), 381–404.
Koutsoyiannis, D. (2005b). “Uncertainty, entropy, scaling and hydrological stochastics. 1. Time dependence of hydrological processes and time scaling.” Hydrol. Sci. J., 50(3), 405–426.
Krasovskaia, I. (1997). “Entropy-based grouping of river flow regimes.” J. Hydrol., 202, 173–191.
Krasovskaia, I., and Gottschalk, L. (1992). “Stability of river flow regimes.” Nord. Hydrol., 23, 137–154.
Krstanovic, P. F., and Singh, V. P. (1991a). “A univariate model for long-term streamflow forecasting: I. Development.” Stochastic Hydrol. Hydraul., 5, 173–188.
Krstanovic, P. F., and Singh, V. P. (1991b). “A univariate model for long-term streamflow forecasting: II. Application.” Stochastic Hydrol. Hydraul., 5, 189–205.
Krstanovic, P. F., and Singh, V. P. (1993a). “A real-time flood forecasting model based on maximum entropy spectral analysis: I. Development.” Water Resour. Manage., 7, 109–129.
Krstanovic, P. F., and Singh, V. P. (1993b). “A real-time flood forecasting model based on maximum entropy spectral analysis: II. Application.” Water Resour. Manage., 7, 131–151.
Langbein, W. B. (1964). “Geometry of river channels.” J. Hydrau. Div., 90(HY2), 301–311.
Leopold, L. B., and Langbein, W. B. (1962). “The concept of entropy in landscape evolution.” Geological Survey Professional Paper 500-A,, USGS, U.S. Department of the Interior, Washington, DC, 1–55.
Leopold, L. B., and Maddock, T. J. (1953). “Hydraulic geometry of stream channels and some physiographic implications.” U. S. Geological Survey Professional Paper 252, 55.
Lienhard, J. H. (1964). “A statistical mechanical prediction of the dimensionless unit hydrograph.” J. Geophys. Res., 69(24), 5231–5238.
Moramarco, T., Barbetta, S., Melone, F., and Singh, V. P. (2005). “Relating local stage and remote discharge with significant lateral inflow.” J. Hydrol. Eng., 10(1), 58–69.
Moramarco, T., Saltalippi, C., and Singh, V. P. (2002). “Estimating the cross-sectional mean velocity in natural channels by the entropy approach.” Water resources planning and management, M. Al-Rashid, V. P. Singh, and M. M. Sherif, eds., A. A. Balkema, Rotterdam, Netherlands, 435–449.
Moramarco, T., Saltalippi, C., and Singh, V. P. (2004). “Estimating the cross-sectional mean velocity in natural channels using Chiu’s velocity distribution.” J. Hydrol. Eng., 9(1), 42–50.
Moramarco, T., Saltalippi, C., and Singh, V. P. (2009). “Velocity profiles assessment in natural channels during high floods.” Water, environment, energy and society, statistical and systems analysis techniques, Proc., WEES-09, Vol. 2, S. K. Jain, V. P. Singh, V. Kumar, R. Kumar, R. D. Singh, and K. D. Sharma, eds., Allied Publishers, New Delhi, India, 780–786.
Moramarco, T., and Singh, V. P. (2001). “Simple method for relating local stage and remote discharge.” J. Hydrol. Eng., 6(1), 78–81.
Nash, J. E. (1957). “The form of the instantaneous unit hydrograph.” Int. Assoc. Sci. Hydrol., 45(3), 114–121.
Overton, D. E. (1964). “Mathematical refinement of an infiltration equation for watershed engineering.” ARS 41-99, U.S. Department of Agriculture, Agricultural Research Service, Washington, DC.
Pal, N. R., and Pal, S. K. (1991). “Entropy: A new definition and its applications.” IEEE: Transactions on Systems, Man, and Cybernetics, 21(5), 1260–1270.
Philip, J. R. (1957). “Theory of infiltration, Parts 1 and 4.” Soil Science, 85(5), 345–357.
Rai, R., Sarkar, S., and Singh, V. P. (2009). “Evaluation of the adequacy of statistical distribution functions for deriving unit hydrograph.” Water Resour. Manage., 23, 899–929.
Rai, R., Sarkar, S., Upadhyay, A., and Singh, V. P. (2010). “Efficacy of Nakagami-m distribution function for deriving unit hydrograph.” Water Resour. Manage., 24, 563–575.
Renyi, A. (1961). “On measures of entropy and information.” Proc., 4th Berkeley Symposium on Mathematics, Statistics and Probability, Vol. 1, Berkeley, CA, 547–561.
Rosenthal, H., and Binia, J. (1988). “On the epsilon entropy of mixed random variables.” IEEE Trans. Inf. Theory, 34(5), 1110–1114.
Shannon, C. E. (1948). “The mathematical theory of communications, I and II.” Bell Syst. Tech. J., 27, 379–423.
Shannon, C. E., and Weaver, W. (1949). The mathematical theory of communication, Univ. of Illinois Press, Urbana, IL.
Singh, V. P. (1993). Elementary hydrology, Prentice Hall, Englewood Cliffs, NJ.
Singh, V. P. (1997a). “The use of entropy in hydrology and water resources.” Hydrol. Processes, 11, 587–626.
Singh, V. P. (1997b). “Effect of class interval size on entropy.” Stochastic Hydrol. Hydraul., 11, 423–431.
Singh, V. P. (1998). Entropy-based parameter estimation in hydrology, Kluwer, Boston, 365.
Singh, V. P. (2010a). “Entropy theory for derivation of infiltration equations.” Water Resour. Res., 46, W03527.
Singh, V. P. (2010b). “Entropy theory for movement of moisture in soils.” Water Resour. Res., 46, W03516.
Singh, V. P. (2010c). “Tsallis entropy theory for derivation of infiltration equations.” Trans. ASABE, 53(2), 447–463.
Singh, V. P., and Fiorentino, M. (1992). “A historical perspective of entropy applications in water resources.” Entropy and energy dissipation in water resources, V. P. Singh and M. Fiorentino, eds., Kluwer, Dordrecht, Netherlands, 21–61.
Singh, V. P., and Rajagopal, A. K. (l987 ). “A new method of parameter estimation for hydrologic frequency analysis.” Hydrol. Sci. Technol., 2(3), 33–40.
Singh, V. P., Yang, C. T., and Deng, Z. Q. (2003a). “Downstream hydraulic geometry relations: 1. Theoretical development.” Water Resour. Res., 39(12), 1–15.
Singh, V. P., Yang, C. T., and Deng, Z. Q. (2003b). “Downstream hydraulic geometry relations: 2. Calibration and testing.” Water Resour. Res., 39(12), 1–10.
Singh, V. P., and Zhang, L. (2008a). “At-a-station hydraulic geometry: I. Theoretical development.” Hydrol. Processes, 22, 189–215.
Singh, V. P., and Zhang, L. (2008b). “At-a-station hydraulic geometry: II. Calibration and testing.” Hydrol. Processes, 22, 216–228.
Sonuga, J. O. (1972). “Principle of maximum entropy in hydrologic frequency analysis.” J. Hydrol., 17(3), 177–219.
Sonuga, J. O. (1976). “Entropy principle applied to the rainfall-runoff process.” J. Hydrol., 30, 81–94.
Tsallis, C. (1988). “Possible generalization of Boltzmann-Gibbs statistics.” J. Stat. Phys., 52(1—2), 479–487.
Williams, G. P. (1978). “Hydraulic geometry of river cross-sections—Theory of minimum variance.” Professional paper 1029, U.S. Geological Survey, Washington, DC.
Wolman, M. G. (1955). “The natural channel of Brandywine Creek, PA.” U. S. Geol. Survey Professional Paper 271, Washington, DC.
Xia, R. (1997). “Relation between mean and maximum velocities in a natural river.” J. Hydraul. Eng., 123(8), 720–723.
Yang, C. T. (1971). “Potential energy and stream morphology.” Water Resour. Res., 2(2), 311–322.
Yang, C. T., Song, C. C., and Woldenberg, M. T. (1981). “Hydraulic geometry and minimum rate of energy dissipation.” Water Resour. Res., 17, 877–896.
Zurek, W. H. (1989). “Algorithmic randomness and physical entropy.” Phys. Rev. A, 40(8), 4731–4751.
Information & Authors
Information
Published In
Copyright
© 2011 American Society of Civil Engineers.
History
Received: Jun 3, 2010
Accepted: Sep 1, 2010
Published online: Apr 15, 2011
Published in print: May 1, 2011
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.