Uncertainty, entropy, scaling and hydrological stochastics, 1, Marginal distributional properties of hydrological processes and state scaling

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



The well-established physical and mathematical principle of maximum entropy (ME), is used to explain the distributional and autocorrelation properties of hydrological processes, including the scaling behaviour both in state and in time. In this context, maximum entropy is interpreted as maximum uncertainty. The conditions used for the maximization of entropy are as simple as possible, i.e. that hydrological processes are non-negative with specified coefficients of variation (CV) and lag one autocorrelation. In this first part of the study, the marginal distributional properties of hydrological variables and the state scaling behaviour are investigated. Application of the ME principle under these very simple conditions results in the truncated normal distribution for small values of CV and in a nonexponential type (Pareto) distribution for high values of CV. In addition, the normal and the exponential distributions appear as limiting cases of these two distributions. Testing of these theoretical results with numerous hydrological data sets on several scales validates the applicability of the ME principle, thus emphasizing the dominance of uncertainty in hydrological processes. Both theoretical and empirical results show that the state scaling is only an approximation for the high return periods, which is merely valid when processes have high variation on small time scales. In other cases the normal distributional behaviour, which does not have state scaling properties, is a more appropriate approximation. Interestingly however, as discussed in the second part of the study, the normal distribution combined with positive autocorrelation of a process, results in time scaling behaviour due to the ME principle.

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Our works referenced by this work:

1. D. Koutsoyiannis, Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575–590, doi:10.1623/hysj.49.4.575.54430, 2004.
2. D. Koutsoyiannis, Statistics of extremes and estimation of extreme rainfall, 2, Empirical investigation of long rainfall records, Hydrological Sciences Journal, 49 (4), 591–610, doi:10.1623/hysj.49.4.591.54424, 2004.
3. D. Koutsoyiannis, Uncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence of hydrological processes and time scaling, Hydrological Sciences Journal, 50 (3), 405–426, doi:10.1623/hysj.50.3.405.65028, 2005.

Our works that reference this work:

1. D. Koutsoyiannis, Uncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence of hydrological processes and time scaling, Hydrological Sciences Journal, 50 (3), 405–426, doi:10.1623/hysj.50.3.405.65028, 2005.
2. D. Koutsoyiannis, An entropic-stochastic representation of rainfall intermittency: The origin of clustering and persistence, Water Resources Research, 42 (1), W01401, doi:10.1029/2005WR004175, 2006.
3. D. Koutsoyiannis, A. Efstratiadis, and K. Georgakakos, Uncertainty assessment of future hydroclimatic predictions: A comparison of probabilistic and scenario-based approaches, Journal of Hydrometeorology, 8 (3), 261–281, doi:10.1175/JHM576.1, 2007.
4. D. Koutsoyiannis, and Z. W. Kundzewicz, Editorial - Quantifying the impact of hydrological studies, Hydrological Sciences Journal, 52 (1), 3–17, 2007.
5. D. Koutsoyiannis, and A. Montanari, Statistical analysis of hydroclimatic time series: Uncertainty and insights, Water Resources Research, 43 (5), W05429, doi:10.1029/2006WR005592, 2007.
6. D. Koutsoyiannis, H. Yao, and A. Georgakakos, Medium-range flow prediction for the Nile: a comparison of stochastic and deterministic methods, Hydrological Sciences Journal, 53 (1), 142–164, doi:10.1623/hysj.53.1.142, 2008.
7. D. Koutsoyiannis, C. Makropoulos, A. Langousis, S. Baki, A. Efstratiadis, A. Christofides, G. Karavokiros, and N. Mamassis, Climate, hydrology, energy, water: recognizing uncertainty and seeking sustainability, Hydrology and Earth System Sciences, 13, 247–257, doi:10.5194/hess-13-247-2009, 2009.
8. D. Koutsoyiannis, and A. Langousis, Precipitation, Treatise on Water Science, edited by P. Wilderer and S. Uhlenbrook, 2, 27–78, doi:10.1016/B978-0-444-53199-5.00027-0, Academic Press, Oxford, 2011.
9. D. Koutsoyiannis, A. Paschalis, and N. Theodoratos, Two-dimensional Hurst-Kolmogorov process and its application to rainfall fields, Journal of Hydrology, 398 (1-2), 91–100, doi:10.1016/j.jhydrol.2010.12.012, 2011.
10. S.M. Papalexiou, D. Koutsoyiannis, and A. Montanari, Can a simple stochastic model generate rich patterns of rainfall events?, Journal of Hydrology, 411 (3-4), 279–289, 2011.
11. S.M. Papalexiou, and D. Koutsoyiannis, 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, 2012.
12. F. Lombardo, E. Volpi, and D. Koutsoyiannis, Rainfall downscaling in time: Theoretical and empirical comparison between multifractal and Hurst-Kolmogorov discrete random cascades, Hydrological Sciences Journal, 57 (6), 1052–1066, 2012.
13. D. Koutsoyiannis, Physics of uncertainty, the Gibbs paradox and indistinguishable particles, Studies in History and Philosophy of Modern Physics, 44, 480–489, doi:10.1016/j.shpsb.2013.08.007, 2013.
14. D. Koutsoyiannis, Entropy: from thermodynamics to hydrology, Entropy, 16 (3), 1287–1314, doi:10.3390/e16031287, 2014.
15. D. Koutsoyiannis, Generic and parsimonious stochastic modelling for hydrology and beyond, Hydrological Sciences Journal, 61 (2), 225–244, doi:10.1080/02626667.2015.1016950, 2016.
16. D. Koutsoyiannis, and S.M. Papalexiou, Extreme rainfall: Global perspective, Handbook of Applied Hydrology, Second Edition, edited by V.P. Singh, 74.1–74.16, McGraw-Hill, New York, 2017.
17. P. Dimitriadis, and D. Koutsoyiannis, Stochastic synthesis approximating any process dependence and distribution, Stochastic Environmental Research & Risk Assessment, 32 (6), 1493–1515, doi:10.1007/s00477-018-1540-2, 2018.
18. I. Tsoukalas, A. Efstratiadis, and C. Makropoulos, Stochastic periodic autoregressive to anything (SPARTA): Modelling and simulation of cyclostationary processes with arbitrary marginal distributions, Water Resources Research, 54 (1), 161–185, WRCR23047, doi:10.1002/2017WR021394, 2018.
19. I. Tsoukalas, S.M. Papalexiou, A. Efstratiadis, and C. Makropoulos, A cautionary note on the reproduction of dependencies through linear stochastic models with non-Gaussian white noise, Water, 10 (6), 771, doi:10.3390/w10060771, 2018.
20. P. Dimitriadis, D. Koutsoyiannis, T. Iliopoulou, and P. Papanicolaou, A global-scale investigation of stochastic similarities in marginal distribution and dependence structure of key hydrological-cycle processes, Hydrology, 8 (2), 59, doi:10.3390/hydrology8020059, 2021.
21. D. Koutsoyiannis, Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, Edition 2, ISBN: 978-618-85370-0-2, 346 pages, doi:10.57713/kallipos-1, Kallipos Open Academic Editions, Athens, 2022.

Works that cite this document: View on Google Scholar or ResearchGate

Other works that reference this work (this list might be obsolete):

1. Jayawardena, A.W., P.C. Xu, F.L. Tsang and W.K. Li, Determining the structure of a radial basis function network for prediction of nonlinear hydrological time series, Hydrological Sciences Journal, 51(1), 21-44, 2006.
2. Ou, C.-P., J. Xia and G.H. Huang, Study of watershed hydrological spatio-temporal variability analysis based on information entropy, Journal of Dalian University of Technology, 46, 168-173, 2006.
3. Wang, G.J., B.D. Su, Z.W. Kundzewicz and T. Jiang, Linear and non-linear scaling of the Yangtze River flow, Hydrological Processes, 22(10), 1532-1536, 2008.
4. Ozger, M., Comparison of fuzzy inference systems for streamflow prediction, Hydrological Sciences Journal, 54(2), 261-273, 2009.
5. Mackey, R., The sun's role regulating the earth's climate dynamics, Energy and Environment, 20 (1-2), 25-73, 2009.
6. Muller, A., P. Arnaud, M. Lang and J. Lavabre, Uncertainties of extreme rainfall quantiles estimated by a stochastic rainfall model and by a generalized Pareto distribution, Hydrological Sciences Journal, 54(3), 417-429, 2009.
7. Wang, D., V. P. Singh, Y.-s. Zhu and J.-c. Wu, Stochastic observation error and uncertainty in water quality evaluation, Advances in Water Resources, 32 (10), 1526-1534, 2009.
8. #Kileshye Onema, J.-M., Z. Katambara and A. Taigbenu, Shuffled complex evolution and multi-linear approaches to flow prediction in the equatorial Nile basin, First Annual Nile Basin Research Conference, Dar Es Salaam, Tanzania, 2009.
9. Singh, V. P., Entropy theory for derivation of infiltration equations, Water Resour. Res., 46, W03527, doi:10.1029/2009WR008193, 2010.
10. Singh, V. P., Entropy theory for movement of moisture in soils, Water Resour. Res., 46, W03516, doi:10.1029/2009WR008288, 2010.
11. Wang, D., Accelerating entropy theory: New approach to the risks of risk analysis in water issues, Human and Ecological Risk Assessment, 16 (1), 4-9, 2010.
12. Andrés-Doménech, I., A. Montanari and J. B. Marco, Stochastic rainfall analysis for storm tank performance evaluation, Hydrol. Earth Syst. Sci., 14, 1221-1232, doi:10.5194/hess-14-1221-2010, 2010.
13. Baek, C., and V. Pipiras, Estimation of parameters in heavy-tailed distribution when its second order tail parameter is known, Journal of Statistical Planning and Inference, 140 (7), 1957-1967, 2010.
14. Singh, V. P., Tsallis entropy theory for derivation of infiltration equations, Transactions of the American Society of Agricultural and Biological Engineers, 53(2), 447-463, 2010.
15. Brunsell, N.A., A multiscale information theory approach to assess spatial-temporal variability of daily precipitation, Journal of Hydrology, 385 (1-4), 165-172, 2010.
16. Ben-Zvi, A., and B. Azmon, Direct relationships of discharges to the mean and standard deviation of the intervals between their exceedences, Hydrol. Sci. J., 55(4), 565–577, 2010.
17. Weijs, S. V., G. Schoups and N. van de Giesen, Why hydrological forecasts should be evaluated using information theory, Hydrol. Earth Syst. Sci., 14, 2545-2558, doi: 10.5194/hess-14-2545-2010, 2010.
18. #Pechlivanidis, I.G., B. Jackson and H. McMillan, The use of entropy as a model diagnostic in rainfall-runoff modeling, International Environmental Modelling and Software Society (iEMSs), 2010 International Congress on Environmental Modelling and Software, Modelling for Environment’s Sake, Fifth Biennial Meeting, Ottawa, Canada, D. A. Swayne, Wanhong Yang, A. A. Voinov, A. Rizzoli, T. Filatova (Eds.), 2010.
19. Viglione, A., Confidence intervals for the coefficient of L-variation in hydrological applications, Hydrol. Earth Syst. Sci., 14, 2229-2242, doi: 10.5194/hess-14-2229-2010, 2010.
20. Dupuis, D.J., Statistical modeling of the monthly Palmer drought severity index, Journal of Hydrologic Engineering, 15 (10), 796-807, art. no. 004010QHE, 2010.
21. Kerkhoven, E., and T. Y. Gan, Unconditional uncertainties of historical and simulated river flows subjected to climate change, Journal of Hydrology, 396 (1-2), 113-127, 2011.
22. Sang Y.-F., D. Wang, J.-C. Wu, Q.-P. Zhu and L. Wang, Wavelet-based analysis on the complexity of hydrologic series data under multi-temporal scales, Entropy, 13 (1), 195-210, 2011.
23. Zhao, C., Y. Ding, B. Ye, S. Yao, Q. Zhao, Z. Wang and Y. Wang, An analyses of long-term precipitation variability based on entropy over Xinjiang, northwestern China, Hydrol. Earth Syst. Sci. Discuss., 8, 2975-2999, doi: 10.5194/hessd-8-2975-2011, 2011.
24. Luo, H., and V. P. Singh, Entropy theory for two-dimensional velocity distribution, Journal of Hydrologic Engineering, 16 (4), 303-315, 2011.
25. Singh, V. P., Hydrologic synthesis using entropy theory: Review, Journal of Hydrologic Engineering, 16 (5), 421-433, 2011.
26. Ridolfi, E., V. Montesarchio, F. Russo and F. Napolitano, An entropy approach for evaluating the maximum information content achievable by an urban rainfall network, Nat. Hazards Earth Syst. Sci., 11, 2075-2083, 2011.
27. Singh, V. P., and H. Luo, Entropy theory for distribution of one-dimensional velocity in open channels, Journal of Hydrologic Engineering ASCE, 16, 725-735, 2011.
28. #Pechlivanidis, I. G., B. Jackson, H. Mcmillan and H. Gupta, Use of informational entropy- based metrics to drive model parameter identification, Proceedings of the 12th International Conference on Environmental Science and Technology, Rhodes, Greece, A-1476 – A1483, 2011.
29. Andrés-Doménech, I., A. Montanari and J. B. Marco, Efficiency of storm detention tanks for urban drainage systems under climate variability, Journal of Water Resources Planning and Management, 138 (1), 36-46, 2012.
30. Kileshye Onema, J.-M., A., Taigbenu and J. Ndiritu, J.: Classification and flow prediction in a data-scarce watershed of the Equatorial Nile region, Hydrol. Earth Syst. Sci., 16, 1435-1443, 2012.
31. Pechlivanidis, I. G., B. M. Jackson, H. K. Mcmillan and H. V. Gupta, Using an informational entropy-based metric as a diagnostic of flow duration to drive model parameter identification, Global NEST Journal, 14 (3), 325-334, 2012.
32. Sang, Y.-F., A review on the applications of wavelet transform in hydrology time series analysis, Atmospheric Research, 122, 8-15, 2013.
33. Sang, Y.-F., Wavelet entropy-based investigation into the daily precipitation variability in the Yangtze River Delta, China, with rapid urbanizations, Theoretical and Applied Climatology, 111 (3-4), 361-370, 2013.
34. Wrzesiński, D., Uncertainty of flow regime characteristics of rivers in Europe, Quaestiones Geographicae, 32(1), 43–53, 10.2478/quageo-2013-0006, 2013.
35. Fan, J., Q. Huang, J. Chang, D. Sun and S. Cui, Detecting Abrupt Change of Streamflow at Lintong Station of Wei River, Mathematical Problems in Engineering, 10.1155/2013/976591, 2013.
36. Yusof, F., I. L. Kane and Z. Yusop, Structural break or long memory: an empirical survey on daily rainfall data sets across Malaysia, Hydrol. Earth Syst. Sci., 17, 1311-1318, 2013.
37. Lin, P.-L., and N. Brunsell, Assessing regional climate and local landcover impacts on vegetation with remote sensing, Remote Sensing, 5 (9), 4347-4369, 2013.
38. Wang, X., J. Zhang, Z. Yang, S. Shahid, R. He, X. Xia and H. Liu, Historic water consumptions and future management strategies for Haihe River basin of Northern China, Mitigation and Adaptation Strategies for Global Change, 10.1007/s11027-013-9496-5, 2013.
39. Cui, H., and V. Singh, Suspended sediment concentration in open channels using Tsallis entropy, J. Hydrol. Eng., 19 (5), 966-977, 2014.
40. Ridolfi, E., L. Alfonso, G. Di Baldassarre, F. Dottori, F. Russo, and F. Napolitano, An entropy approach for the optimization of cross-section spacing for river modelling, Hydrological Sciences Journal, 59 (1), 126-137, 2014.
41. Pechlivanidis, I. G., B Jackson, H. McMillan and H. Gupta, Use of an entropy‐based metric in multiobjective calibration to improve model performance, Water Resources Research, 10.1002/2013WR014537, 2014.
42. Andrés-Doménech, I., R. García-Bartual, A. Montanari and J. B. Marco, Climate and hydrological variability: the catchment filtering role, Hydrol. Earth Syst. Sci., 19 (1), 379-387, 2015.
43. Singh, V.P., and J. Oh, A Tsallis entropy-based redundancy measure for water distribution networks, Physica A: Statistical Mechanics and its Applications, 421, 360-376, 2015.
44. Markovic, D., and M. Koch, Stream response to precipitation variability: A spectral view based on analysis and modelling of hydrological cycle components, Hydrological Processes, 29 (7), 1806-1816, 2015.
45. Sang, Y.F., V.P. Singh, J. Wen and C.M. Liu, Gradation of complexity and predictability of hydrological processes, Journal Of Geophysical Research-Atmospheres, 120 (11), 5334-5343, 10.1002/2014JD022844, 2015.
46. Pechlivanidis, I.G., B. Jackson, H. McMillan and H.V. Gupta, Robust informational entropy-based descriptors of flow in catchment hydrology, Hydrological Sciences Journal, 10.1080/02626667.2014.983516, 2015.

Tagged under: Entropy, Extremes, Stochastics, Uncertainty