Simulating rainfall time-series: how to account for statistical variability at multiple scales?

  • Fabio Oriani
  • Raj Mehrotra
  • Grégoire Mariethoz
  • Julien Straubhaar
  • Ashish Sharma
  • Philippe Renard
Original Paper
  • 147 Downloads

Abstract

Daily rainfall is a complex signal exhibiting alternation of dry and wet states, seasonal fluctuations and an irregular behavior at multiple scales that cannot be preserved by stationary stochastic simulation models. In this paper, we try to investigate some of the strategies devoted to preserve these features by comparing two recent algorithms for stochastic rainfall simulation: the first one is the modified Markov model, belonging to the family of Markov-chain based techniques, which introduces non-stationarity in the chain parameters to preserve the long-term behavior of rainfall. The second technique is direct sampling, based on multiple-point statistics, which aims at simulating a complex statistical structure by reproducing the same data patterns found in a training data set. The two techniques are compared by first simulating a synthetic daily rainfall time-series showing a highly irregular alternation of two regimes and then a real rainfall data set. This comparison allows analyzing the efficiency of different elements characterizing the two techniques, such as the application of a variable time dependence, the adaptive kernel smoothing or the use of low-frequency rainfall covariates. The results suggest, under different data availability scenarios, which of these elements are more appropriate to represent the rainfall amount probability distribution at different scales, the annual seasonality, the dry-wet temporal pattern, and the persistence of the rainfall events.

Keywords

Rainfall Simulation Markov chain Multiple point statistics Long-term Time-series 

Notes

Acknowledgements

This research was funded by the Swiss National Science Foundation (Project No. 134614) and the National Centre for Groundwater Research and Training (Australia). We thank Prof. Geoffrey G.S. Pegram for his review and suggested modifications prior to the submission of the final version of this paper. The data used to produce the results of this paper are freely available upon request to the corresponding author.

References

  1. Andrade C, Trigo RM, Freitas MC, Gallego MC, Borges P, Ramos AM (2008) Comparing historic records of storm frequency and the north atlantic oscillation (nao) chronology for the azores region. Holocene 18(5):745–754. doi: 10.1177/0959683608091794 CrossRefGoogle Scholar
  2. Arpat G, Caers J (2007) Conditional simulation with patterns. Math Geol 39(2):177–203CrossRefGoogle Scholar
  3. Bardossy A, Plate EJ (1992) Space–time model for daily rainfall using atmospheric circulation patterns. Water Resour Res 28(5):1247–1259. doi: 10.1029/91WR02589 CrossRefGoogle Scholar
  4. Basu S, Andharia HI (1992) The chaotic time-series of indian monsoon rainfall and its prediction. Proc Indian Acad Sci Earth Planet Sci 101(1):27–34Google Scholar
  5. Briggs WM, Wilks DS (1996) Estimating monthly and seasonal distributions of temperature and precipitation using the new cpc long-range forecasts. J Clim 9(4):818–826. doi: 10.1175/1520-0442(1996)009<0818:EMASDO>2.0.CO;2 CrossRefGoogle Scholar
  6. Buishand T (1978) Some remarks on the use of daily rainfall models. J Hydrol 36(3–4):295–308CrossRefGoogle Scholar
  7. Buishand TA, Brandsma T (2001) Multisite simulation of daily precipitation and temperature in the rhine basin by nearest-neighbor resampling. Water Resour Res 37(11):2761–2776. doi: 10.1029/2001WR000291 CrossRefGoogle Scholar
  8. Chipperfield A, Fleming P, Fonseca C (1994) Genetic algorithm tools for control systems engineering. In: Proceedings of adaptive computing in engineering design and control. Citeseer, pp 128–133Google Scholar
  9. Chou C, Tu JY, Yu JY (2003) Interannual variability of the western north pacific summer monsoon: differences between enso and non-enso years. J Clim 16(13):2275–2287. doi: 10.1175/2761.1 CrossRefGoogle Scholar
  10. Elsanabary MH, Gan TY, Mwale D (2014) Application of wavelet empirical orthogonal function analysis to investigate the nonstationary character of ethiopian rainfall and its teleconnection to nonstationary global sea surface temperature variations for 1900–1998. Int J Climatol 34(6):1798–1813. doi: 10.1002/joc.3802 CrossRefGoogle Scholar
  11. Feng X, ChangZheng L (2008) The influence of moderate enso on summer rainfall in eastern china and its comparison with strong enso. Chin Sci Bull 53(5):791–800. doi: 10.1007/s11434-008-0002-5 CrossRefGoogle Scholar
  12. Gabriel K, Neumann J (1962) A markov chain model for daily rainfall occurrence at tel aviv. Q J R Meteorol Soc 88(375):90–95CrossRefGoogle Scholar
  13. Garcia-Barron L, Aguilar M, Sousa A (2011) Evolution of annual rainfall irregularity in the southwest of the iberian peninsula. Theor Appl Climatol 103(1–2):13–26. doi: 10.1007/s00704-010-0280-0 CrossRefGoogle Scholar
  14. Guardiano F, Srivastava R (1993) Multivariate geostatistics: beyond bivariate moments. Geostat Troia 1:133–144Google Scholar
  15. Harrold TI, Sharma A, Sheather SJ (2003a) A nonparametric model for stochastic generation of daily rainfall amounts. Water Resour Res 39(12):1343. doi: 10.1029/2003WR002570 CrossRefGoogle Scholar
  16. Harrold TI, Sharma A, Sheather SJ (2003b) A nonparametric model for stochastic generation of daily rainfall occurrence. Water Resour Res 39(10):1300. doi: 10.1029/2003WR002182 CrossRefGoogle Scholar
  17. Hay LE, Mccabe GJ, Wolock DM, Ayers MA (1991) Simulation of precipitation by weather type analysis. Water Resour Res 27(4):493–501. doi: 10.1029/90WR02650 CrossRefGoogle Scholar
  18. Honarkhah M, Caers J (2010) Stochastic simulation of patterns using distance-based pattern modeling. Math Geosci 42(5):487–517CrossRefGoogle Scholar
  19. Hughes J, Guttorp P (1994) A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena. Water Resour Res 30(5):1535–1546CrossRefGoogle Scholar
  20. Hughes J, Guttorp P, Charles S (1999) A non-homogeneous hidden markov model for precipitation occurrence. J Roy Stat Soc: Ser C (Appl Stat) 48(1):15–30CrossRefGoogle Scholar
  21. Jayawardena AW, Lai FZ (1994) Analysis and prediction of chaos in rainfall and stream-flow time-series. J Hydrol 153(1–4):23–52. doi: 10.1016/0022-1694(94)90185-6 CrossRefGoogle Scholar
  22. Jones PG, Thornton PK (1997) Spatial and temporal variability of rainfall related to a third-order markov model. Agric For Meteorol 86(1–2):127–138. doi: 10.1016/S0168-1923(96)02399-4 CrossRefGoogle Scholar
  23. Jothiprakash V, Fathima TA (2013) Chaotic analysis of daily rainfall series in koyna reservoir catchment area, india. Stoch Env Res Risk Assess 27(6):1371–1381. doi: 10.1007/s00477-012-0673-y CrossRefGoogle Scholar
  24. Katz R, Parlange M (1998) Overdispersion phenomenon in stochastic modeling of precipitation. J Clim 11(4):591–601CrossRefGoogle Scholar
  25. Katz RW, Parlange MB (1993) Effects of an index of atmospheric circulation on stochastic properties of precipitation. Water Resour Res 29(7):2335–2344. doi: 10.1029/93WR00569 CrossRefGoogle Scholar
  26. Katz RW, Zheng XG (1999) Mixture model for overdispersion of precipitation. J Clim 12(8):2528–2537. doi: 10.1175/1520-0442(1999)012<2528:MMFOOP>2.0.CO;2 CrossRefGoogle Scholar
  27. Kelley OA (2014) Where the least rainfall occurs in the sahara desert, the trmm radar reveals a different pattern of rainfall each season. J Clim 27(18):6919–6939. doi: 10.1175/JCLI-D-14-00145.1 CrossRefGoogle Scholar
  28. Khan S, Ganguly AR, Saigal S (2005) Detection and predictive modeling of chaos in finite hydrological time series. Nonlinear Process Geophys 12(1):41–53CrossRefGoogle Scholar
  29. Kiely G, Albertson JD, Parlange MB, Katz RW (1998) Conditioning stochastic properties of daily precipitation on indices of atmospheric circulation. Meteorol Appl 5(1):75–87. doi: 10.1017/S1350482798000656 CrossRefGoogle Scholar
  30. Lall U, Sharma A (1996) A nearest neighbor bootstrap for resampling hydrologic time series. Water Resour Res 32(3):679–693. doi: 10.1029/95WR02966 CrossRefGoogle Scholar
  31. Mariethoz G, Renard P, Straubhaar J (2010) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46(11):W11,536. doi: 10.1029/2008WR007621 Google Scholar
  32. Mehrotra R, Li JW, Westra S, Sharma A (2015) A programming tool to generate multi-site daily rainfall using a two-stage semi parametric model. Environ Model Softw 63:230–239. doi: 10.1016/j.envsoft.2014.10.016 CrossRefGoogle Scholar
  33. Mehrotra R, Sharma A (2007a) Preserving low-frequency variability in generated daily rainfall sequences. J Hydrol 345(1–2):102–120. doi: 10.1016/j.jhydrol.2007.08.003 CrossRefGoogle Scholar
  34. Mehrotra R, Sharma A (2007b) A semi-parametric model for stochastic generation of multi-site daily rainfall exhibiting low-frequency variability. J Hydrol 335(1–2):180–193. doi: 10.1016/j.jhydrol.2006.11.011 CrossRefGoogle Scholar
  35. Millan H, Rodriguez J, Ghanbarian-Alavijeh B, Biondi R, Llerena G (2011) Temporal complexity of daily precipitation records from different atmospheric environments: Chaotic and levy stable parameters. Atmos Res 101(4):879–892. doi: 10.1016/j.atmosres.2011.05.021 CrossRefGoogle Scholar
  36. Munoz-Diaz D, Rodrigo FS (2003) Effects of the north atlantic oscillation on the probability for climatic categories of local monthly rainfall in southern spain. Int J Climatol 23(4):381–397. doi: 10.1002/joc.886 CrossRefGoogle Scholar
  37. Oriani F, Straubhaar J, Renard P, Mariethoz G (2014) Simulation of rainfall time series from different climatic regions using the direct sampling technique. Hydrol Earth Syst Sci 18(8):3015–3031. doi:  10.5194/hess-18-3015-2014 http://www.hydrol-earth-syst-sci.net/18/3015/2014/
  38. Prairie JR, Rajagopalan B, Fulp TJ, Zagona EA (2006) Modified k-nn model for stochastic streamflow simulation. J Hydrol Eng 11(4):371–378. doi: 10.1061/(ASCE)1084-0699(2006)11:4(371) CrossRefGoogle Scholar
  39. Rajagopalan B, Lall U (1999) A k-nearest-neighhor simulator for daily precipitation and other weather variables. Water Resour Res 35(10):3089–3101. doi: 10.1029/1999WR900028 CrossRefGoogle Scholar
  40. Schertzer D, Tchiguirinskaia I, Lovejoy S, Hubert P, Bendjoudi H, Larcheveque M (2002) Discussion of “evidence of chaos in the rainfall-runoff process”—which chaos in the rainfall-runoff process? Hydrol Sci J 47(1):139–148. doi: 10.1080/02626660209492913 CrossRefGoogle Scholar
  41. Scott DW (1992) Multivariate density estimation: theory, practice, and visualization. Wiley, New YorkCrossRefGoogle Scholar
  42. Sharif M, Burn DH (2007) Improved k-nearest neighbor weather generating model. J Hydrol Eng 12(1):42–51. doi: 10.1061/(ASCE)1084-0699(2007)12:1(42) CrossRefGoogle Scholar
  43. Sharma A, Tarboton D, Lall U (1997) Streamflow simulation: a nonparametric approach. Water Resour Res 33(2):291–308. doi: 10.1029/96WR02839 CrossRefGoogle Scholar
  44. Singhrattna N, Rajagopalan B, Clark M, Kumar KK (2005) Seasonal forecasting of thailand summer monsoon rainfall. Int J Climatol 25(5):649–664. doi: 10.1002/joc.1144 CrossRefGoogle Scholar
  45. Sivakumar B, Berndtsson R, Olsson J, Jinno K (2001) Evidence of chaos in the rainfall-runoff process. Hydrol Sci J 46(1):131–145. doi: 10.1080/02626660109492805 CrossRefGoogle Scholar
  46. Sivakumar B, Liong S, Liaw C (1998) Evidence of chotic behaviour in singapore rainfall. JAWRA J Am Water Resour Assoc 34(2):301–310CrossRefGoogle Scholar
  47. Sivakumar B, Woldemeskel FM, Puente CE (2014) Nonlinear analysis of rainfall variability in australia. Stoch Env Res Risk Assess 28(1):17–27. doi: 10.1007/s00477-013-0689-y CrossRefGoogle Scholar
  48. Srikanthan R (2004) Stochastic generation of daily rainfall data using a nested model. In: 57th Canadian water resources association annual congress. pp 16–18Google Scholar
  49. Srikanthan R (2005) Stochastic generation of daily rainfall data using a nested transition probability matrix model. In: 29th hydrology and water resources symposium: water capital, 20–23 February 2005, Rydges Lakeside, Canberra. Engineers Australia, p 26Google Scholar
  50. Srikanthan R, Pegram GGS (2009) A nested multisite daily rainfall stochastic generation model. J Hydrol 371(1–4):142–153. doi: 10.1016/j.jhydrol.2009.03.025 CrossRefGoogle Scholar
  51. Straubhaar J (2011) MPDS technical reference guide. Centre d’hydrogeologie et geothermie, University of Neuchâtel, NeuenburgGoogle Scholar
  52. Straubhaar J, Renard P, Mariethoz G (2016) Conditioning multiple-point statistics simulations to block data. Spat Stat 16:53–71CrossRefGoogle Scholar
  53. Straubhaar J, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43(3):305–328CrossRefGoogle Scholar
  54. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21CrossRefGoogle Scholar
  55. Tahmasebi P, Hezarkhani A, Sahimi M (2012) Multiple-point geostatistical modeling based on the cross-correlation functions. Comput Geosci 16(3):779–797CrossRefGoogle Scholar
  56. Trigo R, Zezere JL, Rodrigues ML, Trigo IF (2005) The influence of the north atlantic oscillation on rainfall triggering of landslides near lisbon. Nat Hazards 36(3):331–354. doi: 10.1007/s11069-005-1709-0 CrossRefGoogle Scholar
  57. Wallis TWR, Griffiths JF (1997) Simulated meteorological input for agricultural models. Agric For Meteorol 88(1–4):241–258. doi: 10.1016/S0168-1923(97)00035-X CrossRefGoogle Scholar
  58. Wang QJ, Nathan RJ (2002) A daily and monthly mixed algorithm for stochastic generation of rainfall time series. In: Water challenge: balancing the risks: hydrology and water resources symposium 2002. Institution of Engineers, Australia, p 698Google Scholar
  59. Wilby RL (1998) Modelling low-frequency rainfall events using airflow indices, weather patterns and frontal frequencies. J Hydrol 212(1–4):380–392. doi: 10.1016/S0022-1694(98)00218-2 CrossRefGoogle Scholar
  60. Wilks DS (1989) Conditioning stochastic daily precipitation models on total monthly precipitation. Water Resour Res 25(6):1429–1439. doi: 10.1029/WR025i006p01429 CrossRefGoogle Scholar
  61. Wojcik R, McLaughlin D, Konings A, Entekhabi D (2009) Conditioning stochastic rainfall replicates on remote sensing data. IEEE Trans Geosci Remote Sens 47(8):2436–2449CrossRefGoogle Scholar
  62. Woolhiser DA, Keefer TO, Redmond KT (1993) Southern oscillation effects on daily precipitation in the southwestern united-states. Water Resour Res 29(4):1287–1295. doi: 10.1029/92WR02536 CrossRefGoogle Scholar
  63. Zanchettin D, Franks SW, Traverso P, Tomasino M (2008) On enso impacts on european wintertime rainfalls and their modulation by the nao and the pacific multi-decadal variability described through the pdo index. Int J Climatol 28(8):995–1006. doi: 10.1002/joc.1601 CrossRefGoogle Scholar
  64. Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38(1):63–80CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of HydrologyGeological Survey of Denmark and GreenlandCopenhagen KDenmark
  2. 2.School of Civil and Environmental EngineeringUniversity of New South WalesSydneyAustralia
  3. 3.Institute of Earth Surface DynamicsUniversité de LausanneLausanneSwitzerland
  4. 4.Centre for Hydrogeology and GeothermicsUniversité de NeuchâtelNeuchâtelSwitzerland

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