Spatial rainfall variability in peninsular India: a nonlinear dynamic approach

  • R. Vignesh
  • V. Jothiprakash
  • B. SivakumarEmail author
Original Paper


This study examines the spatial rainfall variability in peninsular India from a nonlinear dynamic perspective. The rainfall variability is determined by employing the false nearest neighbor (FNN) method, a nonlinear dynamic dimensionality-based method, to rainfall data across peninsular India. The implementation of the FNN method involves phase space reconstruction of the rainfall time series using a delay embedding procedure and identification of the false nearest neighbors in the reconstructed phase space using a neighbor search approach. The method is applied to monthly rainfall data over a period of 35 years (January 1971–December 2005) at each of 367 grids (of 0.5° × 0.5° in size) across nine river basins in peninsular India, and the dimensionality (i.e. FNN dimension) is determined. The influence of the delay time (τ) on the FNN dimension estimation is also investigated by considering four different cases of separation of data in phase space reconstruction: monthly (τ = 1) and annual (τ = 12) as well as delay time values from the autocorrelation function method and the average mutual information method. The results indicate that, considering all the four τ values, the dimensionality of the 367 rainfall time series ranges from as low as 3 to as high as 20. The FNN dimension is found to generally vary, sometimes significantly, both for the same τ value for different grids and for the different τ values for the same grid. A basin-wise analysis of the dimensionality results also indicates that the Tapi basin (in the northwest part) exhibits the highest spatial rainfall variability (dimension ranging from 3 to 20), while the Vamsadhara basin (in the northeast part) exhibits the lowest rainfall variability (dimension ranging from 4 to 8). In general, rainfall in the southern and eastern basins have relatively lower dimensionality, while rainfall in the northern and western basins have relatively higher dimensionality. While there is almost no variability in the minimum FNN dimension of rainfall between the nine basins studied, there is significant variability in the maximum FNN dimension of rainfall between the basins. These results facilitate more reliable identification of rainfall spatial characteristics based on complexity and their classification and, thus, have important implications for the identification of the appropriate complexity of rainfall models for an area, rainfall interpolation/extrapolation, and basin water assessment and management.


Spatial rainfall variability Peninsular India Nonlinear dynamics Dimensionality False nearest neighbor method 


  1. Adarsh S, Janga Reddy M (2015) Trend analysis of rainfall in four meteorological subdivisions of southern India using nonparametric methods and discrete wavelet transforms. Int J Climatol 35:1107–1124Google Scholar
  2. Aittokallio T, Gyllenberg M, Hietarinta J, Kuusela T, Multamäki T (1999) Improving the false nearest neighbors method with graphical analysis. Phys Rev E 60:416–421Google Scholar
  3. Alvi SMA, Koteswaram P (1985) Time series analysis of annual rainfall over India. Mausam 36:479–490Google Scholar
  4. Ashok K, Guan Z, Saji N, Yamagata T (2004) Individual and combined influences of ENSO and the Indian Ocean dipole on the Indian summer monsoon. J Clim 17(16):3141–3155Google Scholar
  5. Berndtsson R, Jinno K, Kawamura A, Olsson J, Xu S (1994) Dynamical systems theory applied to long-term temperature and precipitation time series. Trends Hydrol 1:291–297Google Scholar
  6. Carrión IM, Antúnez EA, Castillo MMA, Canals JJM (2010) Parallel implementations of the False Nearest Neighbors method to study the behavior of dynamical models. Math Comput Model 52:1237–1242Google Scholar
  7. Carrión IM, Antúnez EA, Castillo MMA, Canals JJM (2011) A prediction method for nonlinear time series analysis by combining the false nearest neighbors and subspace identification methods. Int J Appl Math Inform 5:258–265Google Scholar
  8. Cherchi A, Navarra A (2013) Influence of ENSO and of the Indian Ocean Dipole on the Indian summer monsoon variability. Clim Dyn 41(1):81–103Google Scholar
  9. Dash SK, Jenamani RK, Kalsi SR, Panda SK (2007) Some evidence of climate change in twentieth-century India. Clim Change 85:299–321Google Scholar
  10. Dhanya CT, Nagesh Kumar D (2010) Nonlinear ensemble prediction of chaotic daily rainfall. Adv Water Resour 33:327–347Google Scholar
  11. Dhanya CT, Nagesh Kumar D (2011) Multivariate nonlinear ensemble prediction of daily chaotic rainfall with climate inputs. J Hydrol 403:292–306Google Scholar
  12. Drosdowsky W (1990) A simple index of the second POP component of Southern Oscillation. Trop Ocean Atmos Newsl (TOAN) 54:13–15Google Scholar
  13. Fathima TA, Jothiprakash V (2014) Behavioural analysis of a time series—a chaotic approach. Sadhana 39(3):659–676Google Scholar
  14. Fathima TA, Jothiprakash V (2016) Consequences of continuous zero values and constant values in time series modeling-understanding through chaotic approach. ASCE J Hydrol Eng 21(7):05016012-1–05016012-11Google Scholar
  15. Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33(2):1134–1140Google Scholar
  16. Fredkin DR, Rice JA (1995) Method of false nearest neighbors: a cautionary note. Phys Rev E 51(4):2950–2954Google Scholar
  17. Ghosh S, Das D, Kao S-C, Ganguly AR (2012) Lack of uniform trends but increasing spatial variability in observed Indian rainfall extremes. Nat Clim Change 2:86–91Google Scholar
  18. Gosain AK, Rao S, Basuray D (2006) Climate change impact assessment on hydrology of Indian river basins. Curr Sci 90(3):346–353Google Scholar
  19. Goswami BN, Krishnamurthy V, Annmalai H (1999) A broad-scale circulation index for the interannual variability of the Indian summer monsoon. Q J R Meteorol Soc 125:611–633Google Scholar
  20. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208Google Scholar
  21. Guhathakurta P, Rajeevan M (2008) Trends in the rainfall pattern over India. Int J Climatol 28:1453–1469Google Scholar
  22. Hastenrath S, Rosen A (1983) Patterns of Indian monsoon rainfall anomalies. Tellus A 35:324–331Google Scholar
  23. Hegger R, Kantz H (1999) Improved false nearest neighbor method to detect determinism in time series data. Phys Rev E 60(4):4970–4973Google Scholar
  24. Holzfuss J, Mayer-Kress G (1986) An approach to error-estimation in the application of dimension algorithms. In: Mayer-Kress G (ed) Dimensions and entropies in chaotic systems. Springer, New York, pp 114–122Google Scholar
  25. Islam MN, Sivakumar B (2002) Characterization and prediction of runoff dynamics: a nonlinear dynamical view. Adv Water Resour 25:179–190Google Scholar
  26. Joseph PV, Liebman B, Hendon HH (1991) Interannual variability of the Australian summer monsoon onset: possible influence of Indian summer monsoon and E1 Nino. J Clim 4:529–538Google Scholar
  27. Jothiprakash V, Fathima TA (2013) Chaotic analysis of daily rainfall series in Koyna reservoir catchment area. Stoch Environ Res Risk Assess 27(6):1371–1381Google Scholar
  28. Kantz H, Schreiber T (2004) Nonlinear time series analysis. Cambridge University Press, CambridgeGoogle Scholar
  29. Kennel MB, Abarbanel HDI (2002) False nearest neighbors and false strands: a reliable minimum embedding dimension algorithm. Phys Rev E 66(026209):1–19Google Scholar
  30. Kennel MB, Buhl M (2003) Estimating good discrete partitions from observed data: symbolic fast nearest neighbors. Phys Rev Lett 91(8):084102Google Scholar
  31. Kennel MB, Brown R, Abarbanel HDI (1992) Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 45(6):3403–3411Google Scholar
  32. Kim HS, Lee KH, Kyoung MS, Sivakumar B, Lee ET (2009) Measuring nonlinear dependence in hydrologic time series. Stoch Environ Res Risk Assess 23:907–916Google Scholar
  33. Köppen W (1936) Das geographisca System der Klimate. In: Koppen W, Geiger G (eds) Handbuch der Klimatologie, vol 1. Gebr, Borntraeger, Berlin, pp 1–44Google Scholar
  34. Kripalani RH, Singh SV, Arkin PA (1991) Large scale features of rainfall and outgoing long wave radiation over India and adjoining regions. Contrib Atmos Phys 64:159–168Google Scholar
  35. Kripalani RH, Inamdar S, Sontakke NA (1996) Rainfall variability over Bangladesh and Nepal: comparison and connections with features over India. Int J Climatol 16:689–703Google Scholar
  36. Kripalani RH, Kulkarni A, Sabade SS, Khandekar ML (2003) Indian monsoon variability in a global warming scenario. Nat Hazards 29:189–206Google Scholar
  37. Krishnamurthy V, Shukla J (2008) Seasonal persistence and propagation of intraseasonal patterns over the Indian summer monsoon region. Clim Dyn 30:353–369Google Scholar
  38. Kulkarni A, Kripalani RH, Singh SV (1992) Classification of summer monsoon rainfall patterns over India. Int J Climatol 12:269–280Google Scholar
  39. Kumar KR, Pant GB, Parthasarathy B, Sontakke NA (1992) Spatial and subseasonal patterns of the long-term trends on Indian summer monsoon rainfall. Int J Climatol 12:257–268Google Scholar
  40. Kumar V, Jain SK, Singh Y (2010) Analysis of long-term rainfall trends in India. Hydrol Sci J 55(4):484–496Google Scholar
  41. Kundu S, Khare D, Mondal A, Mishra PK (2015) Analysis of spatial and temporal variation in rainfall trend of Madhya Pradesh, India (1901–2011). Environ Earth Sci 73(12):8197–8216Google Scholar
  42. Kyoung MS, Kim HS, Sivakumar B, Singh VP, Ahn KS (2011) Dynamic characteristics of monthly rainfall in the Korean Peninsula under climate change. Stoch Environ Res Risk Assess 25:613–625Google Scholar
  43. Leibert W, Schuster HG (1989) Proper choice of the time delay for the analysis of chaotic time series. Phys Lett A 141:386–390Google Scholar
  44. Mitra AK, Das Gupta M, Singh SV, Krishnamurti TN (2003) Daily rainfall for the Indian monsoon region from merged satellite and raingauge values: large scale analysis from real time data. J Hydrometeorol 4:769–781Google Scholar
  45. Mujumdar PP, Ghosh S (2008) Modeling GCM and scenario uncertainty using a possibilistic approach: application to the Mahanadi River, India. Water Resour Res 44:W06407. Google Scholar
  46. New M, Hulme M, Jones PD (1999) Representing twentieth-century space-time variability. Part I: development of a 1961–1990 mean monthly terrestrial climatology. J Clim 12:829–856Google Scholar
  47. Niu J (2013) Precipitation in the Pearl River basin, South China: scaling, regional patterns, and influence of large-scale climate anomalies. Stoch Environ Res Risk Assess 27(5):1253–1268Google Scholar
  48. Packard NB, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from a time series. Phys Rev Lett 45(9):712–716Google Scholar
  49. Parthasarathy B, Munot AA, Kothawale DR (1994) All-India monthly and seasonal rainfall series: 1871–1993. Theor Appl Climatol 49:217–224Google Scholar
  50. Peel MC, Finlayson BL, McMahon TA (2007) Updated world map of the Koppen–Geiger climate classification. Hydrol Earth Syst Sci 11(5):1633–1644Google Scholar
  51. Prasad KD, Singh SV (1988) Large scale features of Indian summer monsoon rainfall and their association with some oceanic and atmospheric variables. Adv Atmos Sci 5:499–513Google Scholar
  52. Rajeevan M, Bhate J (2009) A high resolution daily gridded rainfall data set (1971–2005) for mesoscale meteorological studies. Curr Sci India 96(4):558–562Google Scholar
  53. Rajeevan M, Bhate J, Kale JD, Lal B (2006) High resolution daily gridded rainfall data for the Indian region: analysis of break and active monsoon spells. Curr Sci India 91(3):296–306Google Scholar
  54. Reshmidevi TV, Nagesh Kumar D, Mehrotra R, Sharma A (2018) Estimation of the climate change impact on a catchment water balance using an ensemble of GCMs. J Hydrol 556:1192–1204Google Scholar
  55. Rodriguez-Iturbe I, De Power FB, Sharifi MB, Georgakakos KP (1989) Chaos in rainfall. Water Resour Res 25(7):1667–1675Google Scholar
  56. Sahai AK, Grimm AM, Satyan V, Pant GB (2003) Long-lead prediction of Indian summer monsoon rainfall from global SST evolution. Clim Dyn 20:855–863Google Scholar
  57. Sangoyomi TB, Lall U, Abarbanel HDI (1996) Nonlinear dynamics of Great Salt lake: dimension estimation. Water Resour Res 32(1):149–159Google Scholar
  58. Seo Y, Kim S, Singh VP (2015) Estimating spatial precipitation using regression kriging and artificial neural network residual kriging (RKNNRK) hybrid approach. Water Resour Manage 29:2189–2204Google Scholar
  59. Sevruk B (1996) Adjustment of tipping-bucket precipitation gage measurement. Atmos Res 41:237–246Google Scholar
  60. Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data. In: Proceedings of the 1968 23rd ACM national conference, ACM’68, New York, pp 517–524Google Scholar
  61. Shi H, Chen J, Li T, Wang G (2017) A new method for estimation of spatially distributed rainfall through merging satellite observations, raingauge records, and terrain digital elevation model data. J Hydro-Environ Res.
  62. Singh N, Sontakke NA (2002) On climate fluctuations and environmental changes of the Indo-Gangetic Plains, India. Clim Change 52(3):287–313Google Scholar
  63. Sivakumar B (2001) Rainfall dynamics at different temporal scales: a chaotic perspective. Hydrol Earth Syst Sci 5(4):645–651Google Scholar
  64. Sivakumar B (2011) Global climate change and its impacts on water resources planning and management: assessment and challenges. Stoch Environ Res Risk Assess 25(4):583–600Google Scholar
  65. Sivakumar B (2017) Chaos in hydrology: bridging determinism and stochasticity. Springer Science + Business Media, DordrechtGoogle Scholar
  66. Sivakumar B, Liong SY, Liaw CY (1998) Evidence of chaotic behavior in Singapore rainfall. J Am Water Resour Assoc 34(2):301–310Google Scholar
  67. Sivakumar B, Liong SY, Liaw CY, Phoon KK (1999) Singapore rainfall behavior: chaotic? ASCE J Hydrol Eng 4(1):38–48Google Scholar
  68. Sivakumar B, Sorooshian S, Gupta HV, Gao X (2001) A chaotic approach to rainfall disaggregation. Water Resour Res 37(1):61–72Google Scholar
  69. Sivakumar B, Wallender WW, Horwath WR, Mitchell JP, Prentice SE, Joyce BA (2006) Nonlinear analysis of rainfall dynamics in California’s Sacramento Valley. Hydrol Process 20(8):1723–1736Google Scholar
  70. Sivakumar B, Woldemeskel FM, Puente CE (2014) Nonlinear analysis of rainfall variability in Australia. Stoch Environ Res Risk Assess 28(1):17–27Google Scholar
  71. Subash N, Sikka AK (2014) Trend analysis of rainfall and temperature and its relationship over India. Theor Appl Climatol 117(3):449–462Google Scholar
  72. Subbaramayya I, Naidu CV (1992) Spatial variations and trends in the Indian monsoon rainfall. Int J Climatol 12:597–609Google Scholar
  73. Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds) Dynamical systems and turbulence, vol 898. Lecture notes in Mathematics. Springer, Berlin, pp 366–381Google Scholar
  74. Taxak AK, Murumkar AR, Arya DS (2014) Long term spatial and temporal rainfall trends and homogeneity analysis in Wainganga basin, Central India. Weather Clim Extrem 4:50–61Google Scholar
  75. Unnikrishnan P, Jothiprakash V (2018) Daily rainfall forecasting for one year in a single run using singular spectrum analysis. J Hydrol 561:609–621Google Scholar
  76. Vignesh R, Jothiprakash V, Sivakumar B (2015) Streamflow variability and classification using false nearest neighbor method. J Hydrol 531:706–715Google Scholar
  77. Wahl ER, Morrill C (2010) Toward understanding and predicting monsoon patterns. Science 328:437–438Google Scholar
  78. Xie P, Yatagai A, Chen M, Hayasaka T, Fukushima Y, Liu C, Yang S (2007) A gauge-based analysis of daily precipitation over East Asia. J Hydrometeorol 8:607–627Google Scholar
  79. Yasunari T (1991) The monsoon year- A new concept of the climatic year in the tropics. Bull Am Meteor Soc 72:1331–1338Google Scholar
  80. Young PC, Parkinson SD, Lees M (1996) Simplicity out of complexity: occam’s razor revisited. J Appl Stat 23:165–210Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringIndian Institute of Technology BombayPowaiIndia
  2. 2.Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and TechnologyAvadi, ChennaiIndia
  3. 3.UNSW Water Research Centre, School of Civil and Environmental EngineeringThe University of New South WalesSydneyAustralia
  4. 4.State Key Laboratory of Hydroscience and EngineeringTsinghua UniversityBeijingChina

Personalised recommendations