Hurst-Kolmogorov dynamics and uncertainty

D. Koutsoyiannis, Hurst-Kolmogorov dynamics and uncertainty, Journal of the American Water Resources Association, 47 (3), 481–495, doi:10.1111/j.1752-1688.2011.00543.x, 2011.



The non-static, ever changing hydroclimatic processes are often described as nonstationary. However, revisiting the notions of stationarity and nonstationarity, defined within stochastics, suggests that claims of nonstationarity cannot stand unless the evolution in time of the statistical characteristics of the process is known in deterministic terms, particularly for the future. In reality, long-term deterministic predictions are difficult or impossible. Thus, change is not synonymous with nonstationarity, and even prominent change at a multitude of time scales, small and large, can be described satisfactorily by a stochastic approach admitting stationarity. This “novel” description does not depart from the 60- to 70-year old pioneering works of Hurst on natural processes and of Kolmogorov on turbulence. Contrasting stationary with nonstationary has important implications in engineering and management. The stationary description with Hurst-Kolmogorov (HK) stochastic dynamics demonstrates that nonstationary and classical stationary descriptions underestimate the uncertainty. This is illustrated using examples of hydrometeorological time series, which show the consistency of the HK approach with reality. One example demonstrates the implementation of this framework in the planning and management of the water supply system of Athens, Greece, also in comparison with alternative nonstationary approaches, including a trend-based and a climate-model-based approach.

Full text is only available to the NTUA network due to copyright restrictions

PDF Additional material:

See also:

Related works:


Blog posts and discussions: The Blackboard.

Our works referenced by this work:

1. I. Nalbantis, and D. Koutsoyiannis, A parametric rule for planning and management of multiple reservoir systems, Water Resources Research, 33 (9), 2165–2177, doi:10.1029/97WR01034, 1997.
2. D. Koutsoyiannis, A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36 (6), 1519–1533, 2000.
3. D. Koutsoyiannis, Coupling stochastic models of different time scales, Water Resources Research, 37 (2), 379–391, doi:10.1029/2000WR900200, 2001.
4. D. Koutsoyiannis, The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences Journal, 47 (4), 573–595, doi:10.1080/02626660209492961, 2002.
5. D. Koutsoyiannis, A. Efstratiadis, and G. Karavokiros, A decision support tool for the management of multi-reservoir systems, Journal of the American Water Resources Association, 38 (4), 945–958, doi:10.1111/j.1752-1688.2002.tb05536.x, 2002.
6. D. Koutsoyiannis, Climate change, the Hurst phenomenon, and hydrological statistics, Hydrological Sciences Journal, 48 (1), 3–24, doi:10.1623/hysj., 2003.
7. D. Koutsoyiannis, and A. Economou, Evaluation of the parameterization-simulation-optimization approach for the control of reservoir systems, Water Resources Research, 39 (6), 1170, doi:10.1029/2003WR002148, 2003.
8. D. Koutsoyiannis, G. Karavokiros, A. Efstratiadis, N. Mamassis, A. Koukouvinos, and A. Christofides, A decision support system for the management of the water resource system of Athens, Physics and Chemistry of the Earth, 28 (14-15), 599–609, doi:10.1016/S1474-7065(03)00106-2, 2003.
9. A. Efstratiadis, D. Koutsoyiannis, and D. Xenos, Minimising water cost in the water resource management of Athens, Urban Water Journal, 1 (1), 3–15, 2004.
10. D. Koutsoyiannis, Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575–590, 2004.
11. D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour, Journal of Hydrology, 322, 25–48, 2006.
12. D. Koutsoyiannis, Nonstationarity versus scaling in hydrology, Journal of Hydrology, 324, 239–254, 2006.
13. 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.
14. 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.
15. D. Koutsoyiannis, N. Zarkadoulas, A. N. Angelakis, and G. Tchobanoglous, Urban water management in Ancient Greece: Legacies and lessons, Journal of Water Resources Planning and Management - ASCE, 134 (1), 45–54, doi:10.1061/(ASCE)0733-9496(2008)134:1(45), 2008.
16. 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.
17. D. Koutsoyiannis, A. Efstratiadis, N. Mamassis, and A. Christofides, On the credibility of climate predictions, Hydrological Sciences Journal, 53 (4), 671–684, 2008.
18. D. Koutsoyiannis, A random walk on water, Hydrology and Earth System Sciences, 14, 585–601, doi:10.5194/hess-14-585-2010, 2010.
19. G. G. Anagnostopoulos, D. Koutsoyiannis, A. Christofides, A. Efstratiadis, and N. Mamassis, A comparison of local and aggregated climate model outputs with observed data, Hydrological Sciences Journal, 55 (7), 1094–1110, doi:10.1080/02626667.2010.513518, 2010.
20. H. Tyralis, and D. Koutsoyiannis, Simultaneous estimation of the parameters of the Hurst-Kolmogorov stochastic process, Stochastic Environmental Research & Risk Assessment, 25 (1), 21–33, 2011.

Our works that reference this work:

1. I. Nalbantis, A. Efstratiadis, E. Rozos, M. Kopsiafti, and D. Koutsoyiannis, Holistic versus monomeric strategies for hydrological modelling of human-modified hydrosystems, Hydrology and Earth System Sciences, 15, 743–758, doi:10.5194/hess-15-743-2011, 2011.
2. A. Montanari, and D. Koutsoyiannis, A blueprint for process-based modeling of uncertain hydrological systems, Water Resources Research, 48, W09555, doi:10.1029/2011WR011412, 2012.
3. Y. Markonis, and D. Koutsoyiannis, Climatic variability over time scales spanning nine orders of magnitude: Connecting Milankovitch cycles with Hurst–Kolmogorov dynamics, Surveys in Geophysics, 34 (2), 181–207, doi:10.1007/s10712-012-9208-9, 2013.
4. D. Koutsoyiannis, Hydrology and Change, Hydrological Sciences Journal, 58 (6), 1177–1197, doi:10.1080/02626667.2013.804626, 2013.
5. A. Efstratiadis, D. Bouziotas, and D. Koutsoyiannis, A decision support system for the management of hydropower systems – Application to the Acheloos-Thessaly hydrosystem, Proceedings of the 2nd Hellenic Concerence on Dams and Reservoirs, Athens, Zappeion, doi:10.13140/RG.2.1.1952.0244, Hellenic Commission on Large Dams, 2013.
6. C. Ioannou, G. Tsekouras, A. Efstratiadis, and D. Koutsoyiannis, Stochastic analysis and simulation of hydrometeorological processes for optimizing hybrid renewable energy systems, Proceedings of the 2nd Hellenic Concerence on Dams and Reservoirs, Athens, Zappeion, doi:10.13140/RG.2.1.3787.0327, Hellenic Commission on Large Dams, 2013.
7. G. Tsekouras, and D. Koutsoyiannis, Stochastic analysis and simulation of hydrometeorological processes associated with wind and solar energy, Renewable Energy, 63, 624–633, doi:10.1016/j.renene.2013.10.018, 2014.
8. D. Koutsoyiannis, Reconciling hydrology with engineering, Hydrology Research, 45 (1), 2–22, doi:10.2166/nh.2013.092, 2014.
9. D. Koutsoyiannis, Social vs. scientific perception of change in hydrology and climate — Reply to the Discussion by Arie Ben-Zvi on the Opinion Paper “Hydrology and Change”, Hydrological Sciences Journal, 59 (8), 1625–1626, doi:10.1080/02626667.2014.935382, 2014.
10. S. Ceola, A. Montanari, and D. Koutsoyiannis, Toward a theoretical framework for integrated modeling of hydrological change, WIREs Water, 1 (5), 427–438, doi:10.1002/wat2.1038, 2014.
11. A. Efstratiadis, Y. Dialynas, S. Kozanis, and D. Koutsoyiannis, A multivariate stochastic model for the generation of synthetic time series at multiple time scales reproducing long-term persistence, Environmental Modelling and Software, 62, 139–152, doi:10.1016/j.envsoft.2014.08.017, 2014.
12. A. Montanari, and D. Koutsoyiannis, Modeling and mitigating natural hazards: Stationarity is immortal!, Water Resources Research, 50 (12), 9748–9756, doi:10.1002/2014WR016092, 2014.
13. D. Koutsoyiannis, and A. Montanari, Negligent killing of scientific concepts: the stationarity case, Hydrological Sciences Journal, 60 (7-8), 1174–1183, doi:10.1080/02626667.2014.959959, 2015.
14. A. Efstratiadis, I. Nalbantis, and D. Koutsoyiannis, Hydrological modelling of temporally-varying catchments: Facets of change and the value of information, Hydrological Sciences Journal, 60 (7-8), 1438–1461, doi:10.1080/02626667.2014.982123, 2015.
15. I. Tsoukalas, and C. Makropoulos, Multiobjective optimisation on a budget: Exploring surrogate modelling for robust multi-reservoir rules generation under hydrological uncertainty, Environmental Modelling and Software, 69, 396–413, doi:10.1016/j.envsoft.2014.09.023, 2015.
16. I. Tsoukalas, and C. Makropoulos, A surrogate based optimization approach for the development of uncertainty-aware reservoir operational rules: the case of Nestos hydrosystem, Water Resources Management, 29 (13), 4719–4734, doi:10.1007/s11269-015-1086-8, 2015.
17. I. Tsoukalas, P. Dimas, and C. Makropoulos, Hydrosystem optimization on a budget: Investigating the potential of surrogate based optimization techniques, 14th International Conference on Environmental Science and Technology (CEST2015), Global Network on Environmental Science and Technology, University of the Aegean, 2015.
18. I. Tsoukalas, P. Kossieris, A. Efstratiadis, and C. Makropoulos, Surrogate-enhanced evolutionary annealing simplex algorithm for effective and efficient optimization of water resources problems on a budget, Environmental Modelling and Software, 77, 122–142, doi:10.1016/j.envsoft.2015.12.008, 2016.
19. P.E. O’Connell, D. Koutsoyiannis, H. F. Lins, Y. Markonis, A. Montanari, and T.A. Cohn, The scientific legacy of Harold Edwin Hurst (1880 – 1978), Hydrological Sciences Journal, 61 (9), 1571–1590, doi:10.1080/02626667.2015.1125998, 2016.
20. T. Iliopoulou, S.M. Papalexiou, Y. Markonis, and D. Koutsoyiannis, Revisiting long-range dependence in annual precipitation, Journal of Hydrology, doi:10.1016/j.jhydrol.2016.04.015, 2016.
21. P. Kossieris, C. Makropoulos, C. Onof, and D. Koutsoyiannis, A rainfall disaggregation scheme for sub-hourly time scales: Coupling a Bartlett-Lewis based model with adjusting procedures, Journal of Hydrology, doi:10.1016/j.jhydrol.2016.07.015, 2016.

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

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

1. Kiang, J. E., J. R. Olsen, and R. M. Waskom, Introduction to the featured collection on “Nonstationarity, Hydrologic Frequency Analysis, and Water Management.” Journal of the American Water Resources Association, 47(3), 433-435, 2011.
2. Stakhiv, E. Z., Pragmatic approaches for water management under climate change uncertainty, JAWRA Journal of the American Water Resources Association, 47(6), 1183-1196, 2011.
3. Beven, K., Causal models as multiple working hypotheses about environmental processes, Comptes Rendus Geoscience, 344 (2), 77-88, 2012.
4. Coron, L., V. Andréassian, C. Perrin, J. Lerat, J. Vaze, M. Bourqui, and F. Hendrickx, Crash testing hydrological models in contrasted climate conditions: An experiment on 216 Australian catchments, Water Resour. Res., 48, W05552, doi: 10.1029/2011WR011721, 2012.
5. #Schumann, A., Gumbel Distribution, ARMA, Copulas – The importance of stochastic tools for water management, 3rd STAHY International Workshop on Statistical Methods for Hydrology and Water Resources Management, Tunis, Tunisia, 2012.
6. Salas, J., B. Rajagopalan, L. Saito and C. Brown, Special Section on climate change and water resources: Climate nonstationarity and water resources management, J. Water Resour. Plann. Manage., 138(5), 385–388, 2012.
7. Kiparsky, M., A. Milman and S. Vicuña, Climate and water: knowledge of impacts to action on adaptation, Annual Review of Environment and Resources, 37, 163-194, 2012.
8. Merz, B., T. Maurer and K. Kaiser, Wie gut können wir vergangene und zukünftige Veränderungen des Wasserhaushalts quantifizieren? [How well can we quantify past and future changes of the water cycle?], Hydrologie und Wasserbewirtschaftung, 5, 244-256, DOI: 10.5675/HyWa_2012,5_1, 2012.
9. Resta, M., Hurst exponent and its applications in time-series analysis, Recent Patents on Computer Science, 5 (3), 211-219, 2012.
10. Schumann, A., Talsperrenbewirtschaftung unter veränderten gesellschaftlichen Anforderungen, Wasserbauliche Mitteilungen der TU Dresden, Heft 47, 35, Dresdner Wasserbaukolloquium 2012 “Staubauwerke - Planen, Bauen, Betreiben”, 3-12, 2012.
11. #Islam, S., and L.E. Susskind, Water diplomacy: A negotiated approach to managing complex water networks, Water Diplomacy: A Negotiated Approach to Managing Complex Water Networks, 1-342, 2012.
12. Serrat-Capdevila, A., J. B. Valdes, F. Dominguez, and S. Rajagopal, Characterizing the water extremes of the new century in the US South-west: a comprehensive assessment from state-of-the-art climate model projections, International Journal of Water Resources Development, 29 (2), 152-171, 2013.
13. Beven, K., So how much of your error is epistemic? Lessons from Japan and Italy, Hydrological Processes, 27 (11), 1677-168, 2013.
14. Jiang, P., M. R. Gautam, J. Zhu, and Z. Yu, How well do the GCMs/RCMs capture the multi-scale temporal variability of precipitation in the Southwestern United States?, Journal of Hydrology, 479, 13-23, 2013.
15. Salas, J. D., Discussion ‘‘Pragmatic Approaches for Water Management Under Climate Change Uncertainty’’ by E. Z. Stakhiv, Journal of the American Water Resources Association, 49 (2), 475-478, 2013.
16. Lisniak, D., J. Franke and C. Bernhofer, Circulation pattern based parameterization of a multiplicative random cascade for disaggregation of observed and projected daily rainfall time series, Hydrol. Earth Syst. Sci., 17, 2487-2500, 10.5194/hess-17-2487-2013, 2013.
17. #Ercan, A., M. L. Kavvas and R. Abbasov, Introduction, Long-Range Dependence and Sea Level Forecasting, Springer International Publishing, 10.1007/978-3-319-01505-7_1, 2013.
18. #Loukas, A., and L. Vasiliades, Review of applied methods for flood-frequency analysis in a changing environment in Greece, In: A review of applied methods in Europe for flood-frequency analysis in a changing environment, Floodfreq COST action ES0901: European procedures for flood frequency estimation (ed. by H. Madsen et al.), Centre for Ecology & Hydrology, Wallingford, UK, 2013.
19. Serinaldi, F., L. Zunino and O. Rosso, Complexity–entropy analysis of daily stream flow time series in the continental United States, Stochastic Environmental Research and Risk Assessment, 28 (7), 1685-1708, 2014.
20. Silva, A. T., M. M. Portela and M. Naghettini, On peaks-over-threshold modeling of floods with zero-inflated Poisson arrivals under stationarity and nonstationarity, Stochastic Environmental Research and Risk Assessment, 28 (6), 1587-1599, 2014.
21. Markovic, D., and M. Koch, Long-term variations and temporal scaling of hydroclimatic time series with focus on the German part of the Elbe River Basin, Hydrological Processes, 28 (4), 2202-2211, 2014.
22. Coron, L., V. Andréassian, C. Perrin, M. Bourqui, and F. Hendrickx, On the lack of robustness of hydrologic models regarding water balance simulation – a diagnostic approach on 20 mountainous catchments using three models of increasing complexity, Hydrology and Earth System Sciences, 18, 727-746, 2014.
23. Kundzewicz, Z.W., S. Kanae, S. I. Seneviratne, J. Handmer, N. Nicholls, P. Peduzzi, R. Mechler, L. M. Bouweri, N. Arnell, K. Mach, R. Muir-Wood, G. R. Brakenridge, W. Kron, G. Benito, Y. Honda, K. Takahashi, and B. Sherstyukov, Flood risk and climate change: global and regional perspectives, Hydrological Sciences Journal, 2014.
24. Panagoulia, D., P. Economou and C. Caroni, Stationary and nonstationary generalized extreme value modelling of extreme precipitation over a mountainous area under climate change, Environmetrics, 25 (1), 29-43, 2014.
25. O'Connell, P. E. and G. O'Donnell, Towards modelling flood protection investment as a coupled human and natural system, Hydrol. Earth Syst. Sci., 18, 155-171, 2014.
26. Panagoulia, D., and E. I. Vlahogianni, Non-linear dynamics and recurrence analysis of extreme precipitation for observed and general circulation model generated climates, Hydrological Processes, 28(4), 2281–2292, 2014.
27. Borwein, J., P. Howlett and J. Piantadosi, Modelling and simulation of seasonal rainfall using the principle of maximum entropy, Entropy, 16(2), 747-769, 2014.
28. Serinaldi, F., and C. G. Kilsby, Simulating daily rainfall fields over large areas for collective risk estimation, Journal of Hydrology, 10.1016/j.jhydrol.2014.02.043, 2014.
29. Turner, S. W. D., P. J. Jeffrey, D. Marlow, M. Ekström, B. G. Rhodes and U. Kularathna, Linking climate projections to performance: A yield-based decision scaling assessment of a large urban water resources system, Water Resources Research, 10.1002/2013WR015156, 2014.
30. Hughes, D. A., Simulating temporal variability in catchment response using a monthly rainfall-runoff model, Hydrological Sciences Journal, 10.1080/02626667.2014.909598, 2014.
31. Dong, Q., and Y. Zhang, Advances in research of hydrological serial variation under non-stationary conditions and their impacts on flood control of reservoirs, Advances in Science and Technology of Water Resources, 34 (2), 71-75, 2014.
32. #Beven, K., and R. Lamb, The uncertainty cascade in model fusion, Geological Society, London, Special Publications, 408, SP408-3, 10.1144/SP408.3, 2014
33. Kling, H., P. Stanzel and M. Preishuber, Impact modelling of water resources development and climate scenarios on Zambezi River discharge, Journal of Hydrology: Regional Studies, 1, 17-43, 2014.
34. Beven, K., and P. Smith, Concepts of information content and likelihood in parameter calibration for hydrological simulation models, Journal of Hydrologic Engineering, 20 (1), 10.1061/(ASCE)HE.1943-5584.0000991, art. no. A4014010, 2015.
35. Serinaldi, F., and C.G. Kilsby, Stationarity is undead: Uncertainty dominates the distribution of extremes, Advances in Water Resources, 77, 17-36, 2015.
36. 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.
37. Toledo, C., E. Muñoz and M. Zambrano-Bigiarini, Comparison of stationary and dynamic conceptual models in a mountainous and data-sparse catchment in the South-Central Chilean Andes, Advances in Meteorology, Art. ID 526158, 2015.
38. #Toledo, C., and E. Muñoz, Hydrological processes dynamics in a mountainous river basin in south-central Chile, E-proceedings of the 36th IAHR World Congress, The Hague, the Netherlands, 2015.

Tagged under: Course bibliography: Hydrometeorology, Course bibliography: Water Resources Management, Climate stochastics, Hurst-Kolmogorov dynamics, Hydrosystems