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The Hurst phenomenon and fractional Gaussian noise made easy

Koutsoyiannis, D., The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences Journal, 47 (4), 573–595, 2002.

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[English]

The Hurst phenomenon, which characterises hydrological and other geophysical time series, is formulated and studied in an easy manner in terms of the variance and autocorrelation of a stochastic process on multiple temporal scales. In addition, a simple explanation of the Hurst phenomenon based on the fluctuation of a hydrologic process upon different temporal scales is presented. The stochastic process that was devised to represent the Hurst phenomenon, i.e. the fractional Gaussian noise, is also studied on the same grounds. Based on its studied properties, three simple and fast methods to generate fractional Gaussian noise or good approximations of it are proposed.

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See also: http://dx.doi.org/10.1080/02626660209492961

Remarks:

Alternative names for Hurst phenomenon are Hurst effect, Joseph effect, Long term persistence, Long range dependence, Scaling behaviour (in time), Multi-scale fluctuation, Hurst-Kolmogorov pragmaticity, etc.

Our works referenced by this work:

1. Koutsoyiannis, D., A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36 (6), 1519–1533, 2000.
2. Koutsoyiannis, D., Coupling stochastic models of different time scales, Water Resources Research, 37 (2), 379–392, 2001.

Our works that reference this work:

1. Koutsoyiannis, D., Climate change, the Hurst phenomenon, and hydrological statistics, Hydrological Sciences Journal, 48 (1), 3–24, 2003.
2. Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575–590, 2004.
3. Koutsoyiannis, D., Uncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence of hydrological processes and time scaling, Hydrological Sciences Journal, 50 (3), 405–426, 2005.
4. Langousis, A., and D. Koutsoyiannis, A stochastic methodology for generation of seasonal time series reproducing overyear scaling behaviour, Journal of Hydrology, 322, 138–154, 2006.
5. Koutsoyiannis, D., A toy model of climatic variability with scaling behaviour, Journal of Hydrology, 322, 25–48, 2006.
6. Koutsoyiannis, D., Nonstationarity versus scaling in hydrology, Journal of Hydrology, 324, 239–254, 2006.
7. Koutsoyiannis, D., An entropic-stochastic representation of rainfall intermittency: The origin of clustering and persistence, Water Resources Research, 42 (1), W01401, doi:10.1029/2005WR004175, 2006.
8. Koutsoyiannis, D., 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, 2007.
9. Koutsoyiannis, D., and A. Montanari, Statistical analysis of hydroclimatic time series: Uncertainty and insights, Water Resources Research, 43 (5), W05429, doi:10.1029/2006WR005592, 2007.
10. Koutsoyiannis, D., 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, 2008.
11. Koutsoyiannis, D., A. Efstratiadis, N. Mamassis, and A. Christofides, On the credibility of climate predictions, Hydrological Sciences Journal, 53 (4), 671–684, 2008.
12. Koutsoyiannis, D., Older and modern considerations in the design and management of reservoirs, dams and hydropower plants (Solicited), 1st Hellenic Conference on Large Dams, Larisa, Hellenic Commission on Large Dams, Technical Chamber of Greece, 2008.
13. Koutsoyiannis, D., A. Montanari, H. F. Lins, and T.A. Cohn, Climate, hydrology and freshwater: towards an interactive incorporation of hydrological experience into climate research—DISCUSSION of “The implications of projected climate change for freshwater resources and their management”, Hydrological Sciences Journal, 54 (2), 394–405, 2009.
14. Koutsoyiannis, D., and A. Langousis, Precipitation, Treatise on Water Science, edited by P. Wilderer and S. Uhlenbrook, 2, 27–78, Academic Press, Oxford, 2011.
15. Koutsoyiannis, D., Hurst-Kolmogorov dynamics and uncertainty, Journal of the American Water Resources Association, 47 (3), 481–495, 2011.
16. Koutsoyiannis, D., A. Paschalis, and N. Theodoratos, Two-dimensional Hurst-Kolmogorov process and its application to rainfall fields, Journal of Hydrology, 398 (1-2), 91–100, 2011.
17. Koutsoyiannis, D., Hurst-Kolmogorov dynamics as a result of extremal entropy production, Physica A: Statistical Mechanics and its Applications, 390 (8), 1424–1432, 2011.
18. Papalexiou, S.M., 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.
19. Lombardo, F., 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.
20. Markonis, Y., 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, 2013.
21. Koutsoyiannis, D., Hydrology and Change, Hydrological Sciences Journal, 58 (6), 1177–1197, 2013.
22. Lombardo, F., E. Volpi, D. Koutsoyiannis, and S.M. Papalexiou, Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology, Hydrology and Earth System Sciences, 18, 243–255, 2014.
23. Tsekouras, G., and D. Koutsoyiannis, Stochastic analysis and simulation of hydrometeorological processes associated with wind and solar energy, Renewable Energy, 63, 624–633, 2014.
24. Efstratiadis, A., 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, doi:10.1016/j.envsoft.2014.08.017, 2014, (in press).

Other works that reference this work:

1. Sakalauskiene, G., The Hurst Phenomenon in Hydrology, Environmental Research, Engineering and Management, 3(25), 16-20, 2003.
2. Green T.R., and R.H. Erskine, Measurement, scaling, and topographic analyses of spatial crop yield and soil water content, Hydrological Processes, 18 (8), 1447-1465, 2004.
3. Gebremeskel, S., Y.B. Liu, F. De Smedt, L. Hoffmann and L Pfister, Analysing the effect of climate changes on streamflow using statistically downscaled GCM scenarios, Intl. J. River Basin Management 2(4), 271-280, 2005.
4. #Gottschalk, L., Methods of analyzing variability, Encyclopedia of Hydrological Sciences, Part 1. Theory, Organization and Scale, DOI: 10.1002/0470848944.hsa006, Wiley, 2005.
5. Venema, V., S. Bachner, H.W. Rust and C. Simmer, Statistical characteristics of surrogate data based on geophysical measurements, Nonlinear Processes in Geophysics, 13(4), 449-466, 2006.
6. Wang, G., T. Jiang and G. Chen, Structure and long-term memory of discharge series in Yangtze River, Acta Geographica Sinica, 61(1), 47-56, 2006.
7. #Stockwell, D.R.B., Reconstruction of past climate using series with red noise, Australian Institute of Geoscientists News, 83, 14, March 2006.
8. Ochoa-Rivera, J.C., J. Andreu and R. Garcia-Bartual, Influence of inflows modeling on management simulation of water resources system, Journal of Water Resources Planning and Management, 133(2), 106-116, 2007.
9. Wang, W., P.H.A.J.M. Van Gelder, J.K. Vrijling and X. Chen, Detecting long-memory: Monte Carlo simulations and application to daily streamflow processes, Hydrology and Earth System Sciences, 11(2), 851-862, 2007.
10. #Stockwell, D., Niche Modeling: Predictions from Statistical Distributions, Chapman & Hall, Boka Raton, USA, 2007.
11. Cowpertwait, P., V. Isham and C. Onof, Point process models of rainfall: developments for fine-scale structure, Proceedings of the Royal Society, A-Mathematical Physical and Engineering Sciences 463(2086), 2569-2587, 2007.
12. Mackey, R., Rhodes Fairbridge and the idea that the solar system regulates the Earth's climate, Journal of Coastal Research, Special Issue 50, Proceedings ICS2007, 955- 968, 2007.
13. #Chen, Y.Q., R, Sun and A. Zhou, An overview of fractional order signal processing (FOSP) techniques, Proc. ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, 1205-1222, 2007.
14. #Eisler, Z., I. Bartos and J. Kertesz, Fluctuation scaling in complex systems: Taylor's law and beyond, arXiv:0708.2053v2 [physics.soc-ph], 2007.
15. Hamed, K.H., Trend detection in hydrologic data: The Mann-Kendall trend test under the scaling hypothesis, Journal of Hydrology, 349(3-4), 350-363, 2008.
16. Barnett, T.P., and D.W. Pierce, When will Lake Mead go dry?, Water Resources Research, 44, W03201, doi:10.1029/2007WR006704, 2008.
17. Jain, S.K. and P.K. Bhunya, Reliability, resilience and vulnerability of a multipurpose storage reservoir, Hydrological Sciences Journal, 53(2): 434-447, 2008.
18. Eisler, Z., I. Bartos and J. Kertesz, Fluctuation scaling in complex systems: Taylor's law and beyond, Advances in Physics, 57(1), 89-142, 2008.
19. #Padmanabhan, G., S. L. Shrestha and R. G. Kavasseri, Persistence in North American Palmer drought severity index data reconstructed from tree ring history, 13th World Water Congress, Montpellier, France, 2008.
20. Hrachowitz, M., C. Soulsby, D. Tetzlaff, J. Dawson, S. Dunn, and I. Malcolm, Using long-term data sets to understand transit times in contrasting headwater catchments, Journal of Hydrology, 367(3-4), 237-248, 2009.
21. #McKitrick, R., C. Essex, I. Clark, J. D'Aleo, O. Kärner, R. Willson, C. Idso, W. Kininmonth and M. Khandekar, Critical Topics in Global Warming, 124 pp., Fraser Institute, Calgary, Alberta, Canada, 2009.
22. He, W.P., Q. Wu, W. Zhang, O.G. Wang and Y. Zhang, Comparison of characteristics of moving detrended fluctuation analysis with that of approximate entropy method in detecting abrupt dynamic change, Acta Physica Sinica, 58 (4), 2862-2871, 2009.
23. Zhang, Q., C. Xu, and T. Yang, Scaling properties of the runoff variations in the arid and semi-arid regions of China: a case study of the Yellow River basin, Stochastic Environmental Research and Risk Assessment, 23 (8), 1103-1111, 2009.
24. #Stockwell, D. R. B., and A. Cox, Structural break models of climatic regime-shifts: claims and forecasts, arXiv:0907.1650, 2009.
25. Rupp, D. E., R. F. Keim, M. Ossiander, M. Brugnach and J. S. Selker, Time scale and intensity dependency in multiplicative cascades for temporal rainfall disaggregation, Water Resources Research, 45, W07409, doi:10.1029/2008WR007321, 2009.
26. Villarini, G., F. Serinaldi, J. A. Smith, and W. F. Krajewski, On the stationarity of annual flood peaks in the continental United States during the 20th century, Water Resour. Res., 45, W08417, doi:10.1029/2008WR007645, 2009.
27. Jain, S. K., Statistical performance indices for a hydropower reservoir, Hydrology Research, 40 (5), 454–464, 2009.
28. 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.
29. #Vivero, O., and W. P. Heath, On MLE methods for dynamical systems with fractionally differenced noise spectra, Proceedings of the IEEE Conference on Decision and Control, art. no. 5399549, 1842-1847, 2009.
30. Zhang, Z., A. D. Dehoff, R. D. Pody and J. W. Balay, Detection of Streamflow Change in the Susquehanna River Basin, Water Resources Management, 24 (10), 1947-1964, 2010.
31. Werner, G., Fractals in the nervous system: conceptual implications for theoretical neuroscience, Frontiers in Fractal Physiology, DOI: 10.3389/fphys.2010.00015, 2010.
32. Ehsanzadeh, E., and K. Adamowski, Trends in timing of low stream flows in Canada: impact of autocorrelation and long-term persistence, Hydrological Processes, 24, 970–980, 2010.
33. Blöschl, G., and A. Montanari, Climate change impacts - throwing the dice?, Hydrological Processes, DOI:10.1002/hyp.7574, 24(3), 374-381, 2010.
34. #De Domenico, M., and M. Ali Ghorbani, Chaos and scaling in daily river flow, arxiv, 2010.
35. 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.
36. Jain, S. K., Investigating the behavior of statistical indices for performance assessment of a reservoir, Journal of Hydrology, 391 (1-2), 90-96, 2010.
37. Molini, A., G. G. Katul, and A. Porporato, Causality across rainfall time scales revealed by continuous wavelet transforms, J. Geophys. Res., 115, D14123, doi:10.1029/2009JD013016, 2010.
38. Schertzer, D., I. Tchiguirinskaia, S. Lovejoy and P. Hubert, No monsters, no miracles: in nonlinear sciences hydrology is not an outlier! Hydrol. Sci. J., 55(6), 965–979, 2010.
39. #Mudelsee, M., Climate Time Series Analysis: Classical Statistical and Bootstrap Methods, 473 pp., Springer, Dordrecht, 2010.
40. 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.
41. Gürkan, M. A., Fractal geometry of angular momentum evolution in near-Keplerian systems, Mon. Not. R. Astron. Soc., 411 (1), L56–L60, 2011.
42. Ciflikli, C., and A. Gezer, Self similarity analysis via fractional Fourier transform, Simulation Modelling Practice and Theory, 19 (3), 986-995, 2011.
43. Villarini, G., J. A. Smith, M. L. Baeck, R. Vitolo, D. B. Stephenson and W. F. Krajewski, On the frequency of heavy rainfall for the midwest of the United States, Journal of Hydrology, 400 (1-2), 103-120, 2011.
44. Di Baldassarre, G., M. Elshamy, A. van Griensven, E. Soliman, M. Kigobe, P. Ndomba, J. Mutemi, F. Mutua, S. Moges, J.-Q. Xuan, D. Solomatine and S. Uhlenbrook, Future hydrology and climate in the River Nile basin: a review, Hydrol. Sci. J., 56(2), 199-211, 2011.
45. Frank, P. Imposed and neglected uncertainty in the global average surface air temperature index, Energy and Environment, 22 (4), 407-424, 2011.
46. Lee, T., and J. D. Salas, Copula-based stochastic simulation of hydrological data applied to Nile River flows, Hydrology Research, 42 (4), 318-330, 2011.
47. Hodgkins, G. A., and R. W. Dudley, Historical summer base flow and stormflow trends for New England rivers, Water Resour. Res., 47, W07528, doi: 10.1029/2010WR009109, 2011.
48. Johnson, F., S. Westra, A. Sharma and A. J. Pitman, An assessment of GCM skill in simulating persistence across multiple time scales, J. Climate, 24, 3609–3623, 2011.
49. Gudmundsson, L., L. M. Tallaksen, K. Stahl, and A. K. Fleig, Low-frequency variability of European runoff, Hydrol. Earth Syst. Sci., 15, 2853-2869, doi: 10.5194/hess-15-2853-2011, 2011.
50. Cowpertwait, P. S. P., G. Xie, V. Isham, C. Onof and D. C. I. Walsh, A fine-scale point process model of rainfall with dependent pulse depths within cells, Hydrol. Sci. J., 56 (7), 1110–1117, 2011.
51. #Liu, L. Z. Xu & J. Huang, Long-term trend of major climate variables in the Taihu basin during the last 53 years, Hydrological Cycle and Water Resources Sustainability in Changing Environments, Proceedings of the IWRM2010 Methodology in Hydrology Symposium, Nanjing, China (ed. by L. Ren, W. Wang and F. Yuan), IAHS Publ. 350, ISBN 978-1-907161-25-4, 18-27, 2011.
52. Ehsanzadeh, E., G.. van der Kamp and C. Spence, The impact of climatic variability and change in the hydroclimatology of Lake Winnipeg watershed, Hydrological Processes, 26 (18), 2802-2813, 2012.
53. Bakker, A. M. R., and B. J. J. M. van den Hurk, Estimation of persistence and trends in geostrophic wind speed for the assessment of wind energy yields in Northwest Europe, Climate Dynamics, 39 (3-4), 767-782, 2012.
54. Liu, L., Z.-X. Xu and J.-X. Huang, Spatio-temporal variation and abrupt changes for major climate variables in the Taihu Basin, China, Stochastic Environmental Research and Risk Assessment, 26 (6), 777-791, 2012.
55. Lee, K., S. Hong, S. J. Kim, I, Rhee and S. Chong, SLAW: Self-similar least-action human walk, IEEE/ACM Transactions on Networking, 20 (2), art. no. 6075290, 515-529, 2012.
56. #Das, S., and I. Pan, Fractional Order Signal Processing: Introductory Concepts and Applications, Springer, Heildelberg, 2012.
57. Lacombe, G., C. T. Hoanh and V. Smakhtin, Multi-year variability or unidirectional trends? Mapping long-term precipitation and temperature changes in continental Southeast Asia using PRECIS regional climate model, Climatic Change, Doi: 10.1007/s10584-011-0359-3, 2012.
58. Costa, A.C., A. Bronstert and D. Kneis, Probabilistic flood forecasting for a mountainous headwater catchment using a nonparametric stochastic dynamic approach, Hydrological Sciences Journal, 57 (1), 10–25, 2012.
59. Harvey, C. L., H. Dixon and J. Hannaford, An appraisal of the performance of data infilling methods for application to daily mean river flow records in the UK, Hydrology Research, 43 (5), 618–636, 2012.
60. #McKitrick, R., Adversarial versus consensus processes for assessing scientific evidence, Institutions and Incentives in Regulatory Science, Lexington Books, 55-74, 2012.
61. Mossberg, M., Analysis of moments based methods for fractional Gaussian noise estimation, IEEE Transactions on Signal Processing, 60 (7), 3823 – 3827, 2012.
62. 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.
63. Genz, F. and L. D. Luz, Distinguishing the effects of climate on discharge in a tropical river highly impacted by large dams, Hydrological Sciences Journal, 57 (5), 1020–1034, 2012.
64. Saraswat, P., and M. K. Sen, Pre-stack inversion of angle gathers using a hybrid evolutionary algorithm, Journal of Seismic Exploration, 21 (2), 177-200, 2012.
65. Meghea, I., M. Mihai, I. Lacatusu and T. Apostol, Time series model applied to environmental monitoring data analyses, Journal of Environmental Protection and Ecology, 13 (2), 426-434, 2012.
66. Goerg, G. M., Testing for white noise against locally stationary alternatives, Statistical Analysis and Data Mining, 5 (6), 478-492, 2012.
67. 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.
68. #Rianna, M., E. Ridolfi, L. Lorino, L. Alfonso, V. Montesarchio, G. Di Baldassarre, F. Russo and F. Napolitano, Definition of homogeneous regions through entropy theory, 3rd STAHY International Workshop on Statistical Methods for Hydrology and Water Resources Management, Tunis, Tunisia, 2012.
69. Pan, I., A. Korre, S. Das and S. Durucan, Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise, Nonlinear Dynamics, 70 (4), 2445-2461, 2012.
70. Todhunter, P. E., Uncertainty of the assumptions required for estimating the regulatory flood: Red River of the North, Journal of Hydrologic Engineering, 17 (9), 1011-1020, 2012.
71. Lacombe, G., V. Smakhtin and C. Hoanh, Wetting tendency in the Central Mekong Basin consistent with climate change-induced atmospheric disturbances already observed in East Asia, Theoretical and Applied Climatology, 111 (1-2), 251-263, 2013.
72. 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.
73. De Michele, C., and M. Ignaccolo, New perspectives on rainfall from a discrete view, Hydrological Processes, 10.1002/hyp.9782, 2013.
74. 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.
75. 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.
76. Hodgkins, G. A., The importance of record length in estimating the magnitude of climatic changes: an example using 175 years of lake ice-out dates in New England, Climatic Change, 10.1007/s10584-013-0766-8, 2013.
77. Witt, A., and B. D. Malamud, Quantification of long-range persistence in geophysical time series: conventional and benchmark-based improvement techniques, Surveys in Geophysics, 10.1007/s10712-012-9217-8, 2013.
78. #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.
79. #Ercan, A., M. L. Kavvas and R. Abbasov, Long-range dependence and ARFIMA models, Long-Range Dependence and Sea Level Forecasting, Springer International Publishing, 10.1007/978-3-319-01505-7_2, 2013.
80. Moschas, F., and E. Steirou, Statistical estimation of changes in the dominant frequencies of structures in long noisy series of monitoring data, Mathematical Problems in Engineering, 216860, 10.1155/2013/216860, 2013.
81. Liu, Z., J. Xu, Z, Chen, Q. Qin and C. Wei, Multifractal and long memory of humidity process in the Tarim River Basin, Stochastic Environmental Research and Risk Assessment, 28 (6), 1383-1400, 2014.
82. #Salas, J. D., R. S. Govindaraju, M. Anderson, M. Arabi, F. Francés, W. Suarez, W. S. Lavado-Casimiro and T. R. Green, Introduction to Hydrology, Modern Water Resources Engineering, Handbook of Environmental Engineering (ed. by L. K. Wang and C. T. Yang), v. 15, 1-126, 2014.
83. Sagarika, S., A. Kalra and S. Ahmad, Evaluating the effect of persistence on long-term trends and analyzing step changes in streamflows of the continental United States, Journal of Hydrology, 10.1016/j.jhydrol.2014.05.002, 2014.
84. Yang, G., and L. C. Bowling, Detection of changes in hydrologic system memory associated with urbanization in the Great Lakes region, Water Resources Research, 50 (5), 3750-3763, 2014.

Tagged under: Course bibliography: Stochastic methods, Hurst-Kolmogorov dynamics, Papers initially rejected, Stochastics