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References on Entropy function in hydrology


This page is managed by Simon Michael Papalexiou (National Technical University of Athens) and by Federico Lombardo (Università degli Studi di Roma 3).


I update December 22, 2011 (62 references) thanks to Simon Michael Papalexiou

II update January 15, 2012 (4 references) thanks to Federico Lombardo

III update May 9, 2012 (2 references) thanks to Federico Lombardo

IV update June 18, 2012 (3 references) thanks to Simon Michael Papalexiou

V update November 9, 2012 (2 references) thanks to Federico Lombardo  

VI update November 26, 2012 (6 references) thanks to Federico Lombardo

VII update February 12, 2014 (11 references) thanks to Simon Michael Papalexiou

VIII update December 15, 2014 (3 references) thanks to Simon Michael Papalexiou


References with *star are the last update

Before 2003

Amorocho, J. & Espildora, B. (1973) Entropy in the assessment of uncertainty in hydrologic systems and models. Water Resources Research 9(6), PP. 1511-1522. doi:197310.1029/WR009i006p01511

Chapman, T. G. (1986) Entropy as a measure of hydrologic data uncertainty and model performance. Journal of Hydrology 85(1-2), 111-126. doi:10.1016/0022-1694(86)90079-X

Claps, P., Fiorentino, M. & Oliveto, G. (1996) Informational entropy of fractal river networks. Journal of Hydrology 187(1-2), 145-156. doi:10.1016/S0022-1694(96)03092-2

Dalezios, N. R. & Tyraskis, P. A. (1989) Maximum entropy spectra for regional precipitation analysis and forecasting. Journal of Hydrology 109(1-2), 25-42. doi:10.1016/0022-1694(89)90004-8

Havrda, J. & Charvát, F. (1967) Concept of structural a-entropy. Kybernetika 3, 30–35.

Jaynes, E. T. (1957) Information Theory and Statistical Mechanics. Physical review 106(4), 620. doi:10.1103/PhysRev.106.620

Jaynes, E. T. (1957) Information Theory and Statistical Mechanics. II. Physical review 108(2), 171. doi:10.1103/PhysRev.108.171

Jowitt, P. W. (1979) The extreme-value type-1 distribution and the principle of maximum entropy. Journal of Hydrology 42(1-2), 23-38. doi:10.1016/0022-1694(79)90004-0

Kawachi, T., Maruyama, T. & Singh, V. (2001) Rainfall entropy for delineation of water resources zones in Japan. Journal of Hydrology 246(1-4), 36-44. doi:10.1016/S0022-1694(01)00355-9

Krstanovic, P. F. & Singh, V. P. (1992) Evaluation of rainfall networks using entropy: I. Theoretical development. Water Resources Management 6(4), 279-293. doi:10.1007/BF00872281

Lind, N. C. & Hong, H. P. (1991) Entropy estimation of hydrological extremes. Stochastic Hydrology and Hydraulics 5(1), 77-87. doi:10.1007/BF01544180

Lind, N. C., Hong, H. P. & Solana, V. (1989) A cross entropy method for flood frequency analysis. Stochastic Hydrology and Hydraulics 3(3), 191-202. doi:10.1007/BF01543859

Padmanabhan, G. & Rao, A. R. (1988) Maximum entropy spectral analysis of hydrologic data. Water Resour. Res. 24(9), 1519-1533. doi:198810.1029/WR024i009p01519

Singh, V. (1997) Effect of class-interval size on entropy. Stochastic Hydrology and Hydraulics 11(5), 423-431. doi:10.1007/BF02427927

Singh, V. & Guo, H. (1997) Parameter estimation for 2-parameter generalized pareto distribution by POME. Stochastic Hydrology and Hydraulics 11(3), 211-227. doi:10.1007/BF02427916

Singh, V. P. (1987) On derivation of the extreme value (EV) type III distribution for low flows using entropy. Hydrological sciences journal 32(4), 521-533.

Singh, V. P. & Guo, H. (1995) Parameter estimation for 3-parameter generalized pareto distribution by the principle of maximum entropy (POME). Hydrological Sciences Journal 40(2), 165-181. doi:10.1080/02626669509491402

Singh, V. P., Guo, H. & Yu, F. X. (1993) Parameter estimation for 3-parameter log-logistic distribution (LLD3) by Pome. Stochastic Hydrology and Hydraulics 7(3), 163-177. doi:10.1007/BF01585596

Singh, V.P & Singh, K. (1985) Derivation of the Gamma Distribution by Using the Principle of Maximum Entropy (pome)1. JAWRA Journal of the American Water Resources Association 21(6), 941-952. doi:10.1111/j.1752-1688.1985.tb00189.x

Singh, V.P. (1987) On application of the Weibull distribution in hydrology. Water Resources Management 1(1), 33-43. doi:10.1007/BF00421796

Singh, V.P. (1997) The Use of Entropy in Hydrology and Water Resources. Hydrological Processes 11, 587–626.

Singh, Vijay P. & Singh, K. (1985) Derivation of the Pearson type (PT) III distribution by using the principle of maximum entropy (POME). Journal of Hydrology 80(3-4), 197-214. doi:10.1016/0022-1694(85)90117-9

Singh, Vijay P., Rajagopal, A. K. & Singh, K. (1986) Derivation of some frequency distributions using the principle of maximum entropy (POME). Advances in Water Resources 9(2), 91-106. doi:10.1016/0309-1708(86)90015-1

Solana-Ortega, A. & Solana, V. (2001) Entropy-based inference of simple physical models for regional flood analysis. Stochastic Environmental Research and Risk Assessment 15(6), 415-446. doi:10.1007/s004770100079

Sonuga, J. O. (1972) Principle of maximum entropy in hydrologic frequency analysis. Journal of Hydrology 17(3), 177-191. doi:10.1016/0022-1694(72)90003-0

Sonuga, J. O. (1976) Entropy principle applied to the rainfall-runoff process. Journal of Hydrology 30(1-2), 81-94. doi:10.1016/0022-1694(76)90090-1

Tsallis, C. (1988) Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics 52(1), 479-487. doi:10.1007/BF01016429

Ulrych, T., Velis, D., Woodbury, A. & Sacchi, R. (2000) L-moments and C-moments. Stochastic Environmental Research and Risk Assessment 14(1), 50-68. doi:10.1007/s004770050004


Markus, M., Knapp, H. & Tasker, G. (2003) Entropy and generalized least square methods in assessment of the regional value of streamgages. Journal of Hydrology 283(1-4), 107-121. doi:10.1016/S0022-1694(03)00244-0

Singh, V. & Deng, Z. (2003) Entropy-based parameter estimation for kappa distribution. Journal of Hydrologic Engineering 8(2), 81-92. doi:10.1061/(ASCE)1084-0699(2003)8:2(81)


Kottegoda, N., Natale, L. & Raiteri, E. (2004) Some considerations of periodicity and persistence in daily rainfalls. Journal of Hydrology 296(1-4), 23-37. doi:10.1016/j.jhydrol.2004.03.001

Solana-Ortega, A. & Solana, V. (2004) Entropy inference of recurrence models for variability analysis of extreme events. Stochastic Environmental Research and Risk Assessment 18(3), 167-187. doi:10.1007/s00477-003-0148-2


Agrawal, D., Singh, J. K. & Kumar, A. (2005) Maximum Entropy-based Conditional Probability Distribution Runoff Model. Biosystems Engineering 90(1), 103-113. doi:10.1016/j.biosystemseng.2004.08.003

Keylock, C. (2005) Describing the recurrence interval of extreme floods using nonextensive thermodynamics and Tsallis statistics. Advances in Water Resources 28(8), 773-778. doi:10.1016/j.advwatres.2005.02.011

Koutsoyiannis, D. (2005) Uncertainty, entropy, scaling and hydrological stochastics. 1. Marginal distributional properties of hydrological processes and state scaling. Hydrological Sciences Journal-Journal Des Sciences Hydrologiques 50(3), 381-404. doi:10.1623/hysj.50.3.381.65031

Koutsoyiannis, D. (2005) Uncertainty, entropy, scaling and hydrological stochastics. 2. Time dependence of hydrological processes and time scaling. Hydrological Sciences Journal-Journal Des Sciences Hydrologiques 50(3), 405-426. doi:10.1623/hysj.50.3.405.65028

Maruyama, T., Kawachi, T. & Singh, V. (2005) Entropy-based assessment and clustering of potential water resources availability. Journal of Hydrology 309(1-4), 104-113. doi:10.1016/j.jhydrol.2004.11.020


Koutsoyiannis, D. (2006) An entropic-stochastic representation of rainfall intermittency: The origin of clustering and persistence. Water Resources Research 42(1). doi:10.1029/2005WR00417



Hosking, J. R. M. (2007) Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics. Journal of Statistical Planning and Inference 137(9), 2870-2891. doi:10.1016/j.jspi.2006.10.010


Chen, Y.-C., Wei, C. & Yeh, H.-C. (2008) Rainfall network design using kriging and entropy. Hydrological Processes 22(3), 340-346. doi:10.1002/hyp.6292

Deng, J. & Pandey, M. D. (2008) Cross entropy quantile function estimation from censored samples using partial probability weighted moments. Journal of Hydrology 363(1-4), 18-31. doi:10.1016/j.jhydrol.2008.09.004

Hejazi, M. I., Cai, X. & Ruddell, B. L. (2008) The role of hydrologic information in reservoir operation - Learning from historical releases. Advances in Water Resources 31(12), 1636-1650. doi:10.1016/j.advwatres.2008.07.013

Koutsoyiannis, Demetris, Yao, H. & Georgakakos, A. (2008) Medium-range flow prediction for the Nile: a comparison of stochastic and deterministic methods. Hydrological Sciences Journal-Journal Des Sciences Hydrologiques 53(1), 142-164. doi:10.1623/hysj.53.1.142

Li, Z. & Zhang, Y.-K. (2008) Multi-scale entropy analysis of mississippi river flow. Stochastic Environmental Research and Risk Assessment 22(4), 507-512. doi:10.1007/s00477-007-0161-y

Liu, Q., Yang, Z. & Cui, B. (2008) Spatial and temporal variability of annual precipitation during 1961-2006 in Yellow River Basin, China. Journal of Hydrology 361(3-4), 330-338. doi:10.1016/j.jhydrol.2008.08.002

Perelman, L., Ostfeld, A. & Salomons, E. (2008) Cross Entropy multiobjective optimization for water distribution systems design. Water Resources Research 44(9). doi:10.1029/2007WR006248

Yoo, C., Jung, K. & Lee, J. (2008) Evaluation of rain gauge network using entropy theory: Comparison of mixed and continuous distribution function applications. Journal of Hydrologic Engineering 13(4), 226-235. doi:10.1061/(ASCE)1084-0699(2008)13:4(226)


Hao, Z. & Singh, V. P. (2009) Entropy-based parameter estimation for extended Burr XII distribution. Stochastic Environmental Research and Risk Assessment 23(8), 1113-1122. doi:10.1007/s00477-008-0286-7

Mishra, A. K., Ozger, M. & Singh, V. P. (2009) An entropy-based investigation into the variability of precipitation. Journal of Hydrology 370(1-4), 139-154. doi:10.1016/j.jhydrol.2009.03.006

Ruddell, B. L. & Kumar, P. (2009) Ecohydrologic process networks: 2. Analysis and characterization. Water Resour. Res. 45, 14 PP. doi:200910.1029/2008WR007280

Ruddell, B. L. & Kumar, P. (2009) Ecohydrologic process networks: 1. Identification. Water Resour. Res. 45, 22 PP. doi:200910.1029/2008WR007279


Alfonso, L., Lobbrecht, A. & Price, R. (2010) Information theory–based approach for location of monitoring water level gauges in polders. Water Resources Research 46(3), n/a–n/a. doi:10.1029/2009WR008101

Alfonso, L., Lobbrecht, A. & Price, R. (2010) Optimization of water level monitoring network in polder systems using information theory. Water Resources Research 46(12), n/a–n/a. doi:10.1029/2009WR008953 

Brunsell, N. A. (2010) A multiscale information theory approach to assess spatial-temporal variability of daily precipitation. Journal of Hydrology 385(1-4), 165-172. doi:10.1016/j.jhydrol.2010.02.016

Kumar, P. & Ruddell, B. L. (2010) Information Driven Ecohydrologic Self-Organization. Entropy 12(10), 2085–2096.

Rigby, J. R. & Porporato, A. (2010) Precipitation, dynamical intermittency, and sporadic randomness. Advances in Water Resources 33(8), 923-932. doi:10.1016/j.advwatres.2010.04.008

Rubio-Alvarez, E. & McPhee, J. (2010) Patterns of spatial and temporal variability in streamflow records in south central Chile in the period 1952-2003. Water Resour. Res. 46. doi:10.1029/2009WR007982

Singh, V. P. (2010) Derivation of the Singh-Yu Infiltration Equation Using Entropy Theory. Journal of Hydrologic Engineering 16(2), 187-191. doi:10.1061/(ASCE)HE.1943-5584.0000302

Singh, V. P. (2010) Entropy theory for movement of moisture in soils. Water Resources Research 46. doi:10.1029/2009WR008288

Singh, V. P. (2010) Entropy theory for derivation of infiltration equations. Water Resources Research 46. doi:10.1029/2009WR008193

Singh, V. P. (2010) Derivation of rating curves using entropy theory. Transactions of the ASABE 53(6), 1811–1821.

Singh, V. P. (2010) Tsallis entropy theory for derivation of infiltration equations. Transactions of the ASABE 53(2), 447–463.

Weijs, S. V., Schoups, G. & Giesen, N. van de. (2010) Why hydrological predictions should be evaluated using information theory. Hydrology and Earth System Sciences 14(12), 2545-2558. doi:10.5194/hess-14-2545-2010


Hao, Z. & Singh, V. P. (2011) Single-site monthly streamflow simulation using entropy theory. Water Resources Research 47. doi:10.1029/2010WR010208

Hasan, M. M. & Dunn, P. K. (2011) Entropy, consistency in rainfall distribution and potential water resource availability in Australia. Hydrological Processes 25(16), 2613-2622. doi:10.1002/hyp.8038

Khatibi, R. (2011) Evolutionary systemic modelling of practices on flood risk. Journal of Hydrology 401(1-2), 36-52. doi:10.1016/j.jhydrol.2011.02.006

Koutsoyiannis, D. (2011) Hurst-Kolmogorov dynamics as a result of extremal entropy production. Physica A: Statistical Mechanics and its Applications 390(8), 1424–1432. doi:10.1016/j.physa.2010.12.035

Montesarchio, V., Ridolfi, E., Russo, F. & Napolitano, F. (2011) Rainfall threshold definition using an entropy decision approach and radar data. Natural Hazards and Earth System Sciences 11, 2061–2074. doi:10.5194/nhess-11-2061-2011

Poveda, G. (2011) Mixed memory, (non) Hurst effect, and maximum entropy of rainfall in the tropical Andes. Advances in Water Resources 34(2), 243-256. doi:10.1016/j.advwatres.2010.11.007

Ridolfi, E., Montesarchio, V., Russo, F. & Napolitano F. (2011) An entropy approach for evaluating the maximum information content achievable by an urban rainfall network. Natural Hazards and Earth System Sciences 11, 2075-2083. doi:10.5194/nhess-11-2075-2011

Singh, V. P. (2011) Hydrologic Synthesis Using Entropy Theory: Review. Journal of Hydrologic Engineering 16(5), 421-433. doi:10.1061/(ASCE)HE.1943-5584.0000332

Singh, V. P. (2011) A Shannon entropy-based general derivation of infiltration equations. Transactions of the ASABE 54(1), 123–129.

Singh, V. P. (2011) An IUH equation based on entropy theory. Transactions of the ASABE 54(1), 131–140.

Wang, J. & Bras, R. L. (2011) A model of evapotranspiration based on the theory of maximum entropy production. Water Resources Research 47. doi:10.1029/2010WR009392 


Giannakis, D., Majda, A. J. & Horenko, I. (2012) Information theory, model error, and predictive skill of stochastic models for complex nonlinear systems. Physica D 241, 1735–1752. doi:10.1016/j.physd.2012.07.005 

Koutsoyiannis, D. (2012) Clausius-Clapeyron equation and saturation vapour pressure: simple theory reconciled with practice. European Journal of Physics 33(2), 295–305. doi:10.1088/0143-0807/33/2/295 

Papalexiou, S.M. & Koutsoyiannis, D. (2012) Entropy based derivation of probability distributions: A case study to daily rainfall. Advances in Water Resources 45, 51–57. doi:10.1016/j.advwatres.2011.11.007

Sang, Y.-F., Wang, Z. & Liu, C. (2012) Period identification in hydrologic time series using empirical mode decomposition and maximum entropy spectral analysis. Journal of Hydrology 424–425, 154–164. doi:10.1016/j.jhydrol.2011.12.044

Singh, V. P. (2012) Derivation of furrow geometry using entropy theory. Transactions of the ASABE 55(3), 987–993.

Zeng, X., Wang, D. & Wu, J. (2012) Sensitivity analysis of the probability distribution of groundwater level series based on information entropy. Stochastic Environmental Research and Risk Assessment 26(3), 345–356. doi:10.1007/s00477-012-0556-2

Zhang, L. & Singh, V. P. (2012) Bivariate rainfall and runoff analysis using entropy and copula theories, Entropy, 14, 1784–1812. doi:10.3390/e14091784

Zhou, Y., Zhang, Q., Li, K. & Chen, X. (2012) Hydrological effects of water reservoirs on hydrological processes in the East River (China) basin: complexity evaluations based on the multi-scale entropy analysis. Hydrological Processes 26(21), 3253–3262 . doi:10.1002/hyp.8406



Chen, L., Ye, L., Singh, V., Zhou, J. & Guo, S. (2013) Determination of Input for Artificial Neural Networks for Flood Forecasting Using the Copula Entropy Method. Journal of Hydrologic Engineering. doi:10.1061/(ASCE)HE.1943-5584.0000932

Liu, B., Chen, X., Lian, Y. & Wu, L. (2013) Entropy-based assessment and zoning of rainfall distribution. Journal of Hydrology 490, 32–40. doi:10.1016/j.jhydrol.2013.03.020

Moramarco, T., Corato, G., Melone, F. & Singh, V. P. (2013) An entropy-based method for determining the flow depth distribution in natural channels. Journal of Hydrology 497, 176–188. doi:10.1016/j.jhydrol.2013.06.002

Ridolfi, E., Alfonso, L., Baldassarre, G. Di, Dottori, F., Russo, F. & Napolitano, F. (2013) An entropy approach for the optimization of cross-section spacing for river modelling. Hydrological Sciences Journal 59(1), 1–12. doi:10.1080/02626667.2013.822640 

Singh, V., Byrd, A. & Cui, H. (2013) Flow Duration Curve Using Entropy Theory. Journal of Hydrologic Engineering. doi:10.1061/(ASCE)HE.1943-5584.0000930

Weijs, S. V., Giesen, N. van de & Parlange, M. B. (2013) HydroZIP: How Hydrological Knowledge can Be Used to Improve Compression of Hydrological Data. Entropy 15(4), 1289–1310. doi:10.3390/e15041289

Zhang, L. & Singh, V. (2013) Joint and Conditional Probability Distributions of Runoff Depth and Peak Discharge Using Entropy Theory. Journal of Hydrologic Engineering. doi:10.1061/(ASCE)HE.1943-5584.0000906


  AghaKouchak A., (2014), Entropy-Copula in Hydrology and Climatology, Journal of Hydrometeorology, 15, 2176-2189, doi: 10.1175/JHM-D-13-0207.1

Dirmeyer, P. A., Wei, J., Bosilovich, M. G. & Mocko, D. M. (2014) Comparing Evaporative Sources of Terrestrial Precipitation and Their Extremes in MERRA Using Relative Entropy. Journal of Hydrometeorology 15(1), 102–116. doi:10.1175/JHM-D-13-053.1

Westhoff, M. C., Zehe, E. & Schymanski, S. J. (2014) Importance of temporal variability for hydrological predictions based on the maximum entropy production principle. Geophysical Research Letters n/a–n/a. doi:10.1002/grl.51233

Pechlivanidis, I.G., B. Jackson, H. McMillan, and H. Gupta (2014), Use of an entropy-based metric in multiobjective calibration to improve model performance, Water Resour. Res., 50, 8066-8083, doi:10.1002/2013WR014537.

Pechlivanidis, I.G., B. Jackson, H. McMillan, and H. Gupta (2014), Robust informational entropy-based descriptors of flow in catchment hydrology, Hydrological Sciences Journal, doi:10.1080/02626667.2014.983516.





Ministero dell'Ambiente e della Tutela del Territorio e del Mare

Institute for Environmental Protection and Research

 Honors Center of Italian Universities - H2CU 
Sapienza University of Rome

University of Tuscia,
Viterbo, Italy


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