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.

[doc_id=642]

[English]

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 and lag-one autocorrelation. In the first part of the study, the marginal distributional properties of hydrological processes and the state scaling behaviour were investigated. This second part of the study is devoted to joint distributional properties of hydrological processes. Specifically, it investigates the time dependence structure that may result from the ME principle and shows that the time scaling behaviour (or the Hurst phenomenon) may be obtained by this principle under the additional general condition that all time scales are of equal importance for the application of the ME principle. The omnipresence of the time scaling behaviour in numerous long hydrological time series examined in the literature (one of which is used here as an example), validates the applicability of the ME principle, thus emphasizing the dominance of uncertainty in hydrological processes.

**See also:**
http://dx.doi.org/10.1623/hysj.50.3.405.65028

**Our works referenced by this work:**

1. | D. Koutsoyiannis, A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36 (6), 1519–1533, doi:10.1029/2000WR900044, 2000. |

2. | D. Koutsoyiannis, The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences Journal, 47 (4), 573–595, doi:10.1080/02626660209492961, 2002. |

3. | D. Koutsoyiannis, Climate change, the Hurst phenomenon, and hydrological statistics, Hydrological Sciences Journal, 48 (1), 3–24, doi:10.1623/hysj.48.1.3.43481, 2003. |

4. | D. Koutsoyiannis, Hydrological statistics for engineering design in a varying climate, EGS-AGU-EUG Joint Assembly, Geophysical Research Abstracts, Vol. 5, Nice, doi:10.13140/RG.2.2.16291.45602, European Geophysical Society, 2003. |

5. | D. Koutsoyiannis, Reliability concepts in reservoir design, Water Encyclopedia, Vol. 4, Surface and Agricultural Water, edited by J. H. Lehr and J. Keeley, 259–265, doi:10.1002/047147844X.sw776, Wiley, New York, 2005. |

6. | 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. |

7. | D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour, Journal of Hydrology, 322, 25–48, doi:10.1016/j.jhydrol.2005.02.030, 2006. |

**Our works that reference this work:**

1. | 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. |

2. | D. Koutsoyiannis, Nonstationarity versus scaling in hydrology, Journal of Hydrology, 324, 239–254, doi:10.1016/j.jhydrol.2005.09.022, 2006. |

3. | 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. |

4. | 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. |

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, A. Efstratiadis, N. Mamassis, and A. Christofides, On the credibility of climate predictions, Hydrological Sciences Journal, 53 (4), 671–684, doi:10.1623/hysj.53.4.671, 2008. |

8. | 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. |

9. | 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. |

10. | 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. |

11. | D. Koutsoyiannis, 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, 2011. |

12. | 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. |

13. | 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. |

14. | 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. |

15. | 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. |

16. | D. Koutsoyiannis, Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, Edition 3, ISBN: 978-618-85370-0-2, 391 pages, doi:10.57713/kallipos-1, Kallipos Open Academic Editions, Athens, 2023. |

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

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

1. | Markovic, D., and M. Koch, Sensitivity of Hurst parameter estimation to periodic signals in time series and filtering approaches, Geophysical Research Letters, 32(17), L17401, 2005. |

2. | 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. |

3. | Markovic, D., and M. Koch, Characteristic scales, temporal variability modes and simulation of monthly Elbe River flow time series at ungauged locations, Physics and Chemistry of the Earth, 31(18), 1262-1273, 2006. |

4. | 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. |

5. | Mudelsee, M., Long memory of rivers from spatial aggregation, Water Resources Research, 43(1), W01202, 2007. |

6. | #Stockwell, D., Niche Modeling: Predictions from Statistical Distributions, Chapman & Hall, Boka Raton, USA, 2007. |

7. | Conradt, T., Z.W. Kundzewicz, F. Hattermann and F. Wechsung, Measured effects of new lake surfaces on regional precipitation, Hydrological Sciences Journal 52(5), 936-955, 2007. |

8. | 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. |

9. | Ozger, M., Comparison of fuzzy inference systems for streamflow prediction, Hydrological Sciences Journal, 54(2), 261-273, 2009. |

10. | Mackey, R., The sun's role regulating the earth's climate dynamics, Energy and Environment, 20 (1-2), 25-73, 2009. |

11. | Fatichi, S., S. M. Barbosa, E. Caporali and M. E. Silva, Deterministic versus stochastic trends: Detection and challenges, Journal Of Geophysical Research-Atmospheres, 114, D18121, doi:10.1029/2009JD011960, 2009. |

12. | #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. |

13. | Singh, V. P., Entropy theory for derivation of infiltration equations, Water Resour. Res., 46, W03527, doi:10.1029/2009WR008193, 2010. |

14. | Singh, V. P., Entropy theory for movement of moisture in soils, Water Resour. Res., 46, W03516, doi:10.1029/2009WR008288, 2010. |

15. | 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. |

16. | #Mudelsee, M., Climate Time Series Analysis: Classical Statistical and Bootstrap Methods, 473 pp., Springer, Dordrecht, 2010. |

17. | Stockwell, D. R. B., Critique of drought models in the Australian Drought Exceptional Circumstances Report (DECR), Energy and Environment, 21(5), 425-436, 2010. |

18. | Poveda, G., Mixed memory, (non) Hurst effect, and maximum entropy of rainfall in the tropical Andes, Advances in Water Resources, 34 (2), 243-256, 2011. |

19. | Luo, H., and V. P. Singh, Entropy theory for two-dimensional velocity distribution, Journal of Hydrologic Engineering, 16 (4), 303-315, 2011. |

20. | Singh, V. P., Hydrologic synthesis using entropy theory: Review, Journal of Hydrologic Engineering, 16 (5), 421-433, 2011. |

21. | 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. |

22. | Hamed, K. H., A probabilistic approach to calculating the reliability of over-year storage reservoirs with persistent Gaussian inflow, Journal of Hydrology, 448-449, 93-99, 2012. |

23. | 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. |

24. | #Ignaccolo, M., and M. Marani, Metastatistics of extreme values and its application in hydrology, arXiv: 1211.3087, 2012. |

25. | 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. |

26. | Hrachowitz, M., H.H.G. Savenije, G. Blöschl, J.J. McDonnell, M. Sivapalan, J.W. Pomeroy, B. Arheimer, T. Blume, M.P. Clark, U. Ehret, F. Fenicia, J.E. Freer, A. Gelfan, H.V. Gupta, D.A. Hughes, R.W. Hut, A. Montanari, S. Pande, D. Tetzlaff, P.A. Troch, S. Uhlenbrook, T. Wagener, H.C. Winsemius, R.A. Woods, E. Zehe, and C. Cudennec, A decade of Predictions in Ungauged Basins (PUB) — a review, Hydrological Sciences Journal, 58(6), 1198-1255, 2013. |

27. | Cui, H., and V. Singh, Suspended sediment concentration in open channels using Tsallis entropy, J. Hydrol. Eng., 19 (5), 966-977, 2014. |

28. | 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. |

29. | Yuan, X., B. Ji, H. Tian and Y. Huang, Multiscaling analysis of monthly runoff series using improved MF-DFA approach, Water Resources Management, 10.1007/s11269-014-0715-y, 2014. |

30. | 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. |

31. | 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. |

32. | 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. |

33. | Marani, M., and M. Ignaccolo, A metastatistical approach to rainfall extremes, Advances in Water Resources, 79, 121-126, 2015. |

34. | Nicolis, O., and J. Mateu, 2D anisotropic wavelet entropy with an application to earthquakes in Chile, Entropy, 17 (6), 4155-4172, 2015. |

35. | 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,
Hurst-Kolmogorov dynamics,
Stochastics,
Uncertainty