A non-linear and non-stationary perspective for downscaling mean monthly temperature: a wavelet coupled second order Volterra model

  • Anchit Lakhanpal
  • Vinit Sehgal
  • R. Maheswaran
  • R. Khosa
  • Venkataramana Sridhar
Original Paper


This study presents a multiscale framework for downscaling of the General Circulation Model (GCM) outputs to the mean monthly temperature at regional scale using a wavelet based Second order Voltera (SoV) model. The models are developed using the reanalysis climatic data from the National Centers for Environmental Prediction (NCEP) and are validated using the simulated climatic dataset from the Can CM4 GCM for five locations in the Krishna river basin, India. K-means clustering, based on the multiscale wavelet entropy of the predictors, is used for obtaining the clusters of the input climatic variables. Principal component analysis (PCA) is used to obtain the representative variables from each cluster. These input variables are then used to develop a wavelet based multiscale model using Second order Volterra approach to simulate observed mean monthly temperature for the selected locations in the basin. These models are called W-P-SoV models in this paper. For the purpose of comparison, linear multi-resolution models are developed using Multiple Linear regression (MLR) and are called W-P MLR models. The performance of the models is further compared with other Wavelet-PCA based models coupled with Multiple linear regression models (P-MLR) and Artificial Neural Networks (P-ANN), and, stand-alone MLR and ANN to establish the superiority of the proposed approach. The results indicate that the performance of the wavelet based models is superior in terms of downscaling accuracy when compared with the other models used.


Downscaling Global circulation models Wavelet analysis Principal component analysis Entropy Volterra 



This research was funded by Department of Science and Technology, India through the INSPIRE Faculty Fellowship held by Dr. Maheswaran Rathinasamy. Figures 4, 6 and 7 are reprinted from Sehgal et. al 2016, with permission from Elsevier.


  1. Adamowski JF (2008) River flow forecasting using wavelet and cross-wavelet transform models. Hydrol Process 22:4877–4891CrossRefGoogle Scholar
  2. Agarwal A, Maheswaran R, Sehgal V, Khosa R, Sivakumar B, Bernhofer C (2016) Hydrologic regionalization using wavelet-based multiscale entropy method. J Hydrol 538:22–32CrossRefGoogle Scholar
  3. Ajaaj AA, Mishra AK, Khan AA (2015) Comparison of bias correction techniques for GPCC rainfall data in semi-arid climate. Stoch Environ Res Risk Assess 30:1–17Google Scholar
  4. Aksornsingchai P, Srinilta C (2011) Statistical downscaling for rainfall and temperature prediction in Thailand. In: Proceedings of the international multiconference of engineers and computer scientistsGoogle Scholar
  5. Anandhi A, Srinivas V, Nanjundiah RS, Nagesh Kumar D (2008) Downscaling precipitation to river basin in India for IPCC SRES scenarios using support vector machine. Int J Climatol 28:401–420CrossRefGoogle Scholar
  6. Anandhi A, Srinivas V, Kumar DN, Nanjundiah RS (2009) Role of predictors in downscaling surface temperature to river basin in India for IPCC SRES scenarios using support vector machine. Int J Climatol 29:583–603CrossRefGoogle Scholar
  7. Ball GH, Hall DJ (1967) A clustering technique for summarizing multivariate data. Behav Sci 12:153–155CrossRefGoogle Scholar
  8. Beecham S, Rashid M, Chowdhury RK (2014) Statistical downscaling of multi-site daily rainfall in a South Australian catchment using a generalized linear model. Int J Climatol 34:3654–3670CrossRefGoogle Scholar
  9. Bolshakova N, Azuaje F (2003) Machaon CVE: cluster validation for gene expression data. Bioinformatics 19:2494–2495CrossRefGoogle Scholar
  10. Cai X, Wang D, Zhu T, Ringler C (2009) Assessing the regional variability of GCM simulations. Geophys Res Lett. doi: 10.1029/2008GL036443 Google Scholar
  11. Cannon AJ, Whitfield PH (2002) Downscaling recent streamflow conditions in British Columbia, Canada using ensemble neural network models. J Hydrol 259:136–151CrossRefGoogle Scholar
  12. Carter TR, Kenkyū KKKCK (1994) IPCC technical guidelines for assessing climate change impacts and adaptations: part of the IPCC special report to the first session of the conference of the parties to the UN framework convention on climate change. LondonGoogle Scholar
  13. Cawley GC, Haylock MR, Dorling SR, Goodess C, Jones PD (2003) Statistical downscaling with artificial neural networks. In: ESANN, pp 167–172Google Scholar
  14. Cek ME, Ozgoren M, Savaci FA (2010) Continuous time wavelet entropy of auditory evoked potentials. Comput Biol Med 40:90–96CrossRefGoogle Scholar
  15. Chadwick R, Coppola E, Giorgi F (2011) An artificial neural network technique for downscaling GCM outputs to RCM spatial scale. Nonlinear Process Geophys 18(6):1013–1028CrossRefGoogle Scholar
  16. Chen S, Billings SA, Luo W (1989) Orthogonal least squares methods and their application to non-linear system identification. Int J Control 50:1873–1896CrossRefGoogle Scholar
  17. Chou C-M, Wang R-Y (2002) On-line estimation of unit hydrographs using the wavelet-based LMS algorithm/Estimation en ligne des hydrogrammes unitaires grâce à l’algorithme des moindres carrés moyens à base d’ondelettes. Hydrol Sci J 47:721–738CrossRefGoogle Scholar
  18. Coulibaly P, Burn DH (2004) Wavelet analysis of variability in annual Canadian streamflows. Water Resour Res 40Google Scholar
  19. Coulibaly P, Dibike YB, Anctil F (2005) Downscaling precipitation and temperature with temporal neural networks. J Hydrometeorol 6:483–496CrossRefGoogle Scholar
  20. Dai X, Wang P, Chou J (2003) Multiscale characteristics of the rainy season rainfall and interdecadal decaying of summer monsoon in North China. Chin Sci Bull 48:2730–2734CrossRefGoogle Scholar
  21. Davies DL, Bouldin DW (1979) A cluster separation measure. IEEE Trans Pattern Anal Mach Intell 1(2):224–227CrossRefGoogle Scholar
  22. Devak M, Dhanya C (2014) Downscaling of precipitation in Mahanadi basin, India. Int J Civil Eng Res 5:111–120Google Scholar
  23. Devak M, Dhanya C (2016) Downscaling of precipitation in mahanadi basin, india using support vector machine, K-nearest neighbour and hybrid of support vector machine with K-nearest neighbour. In: Geostatistical and geospatial approaches for the characterization of natural resources in the environment, Springer, pp 657–663Google Scholar
  24. Devak M, Dhanya C, Gosain A (2015) Dynamic coupling of support vector machine and K-nearest neighbour for downscaling daily rainfall. J Hydrol 525:286–301CrossRefGoogle Scholar
  25. Duhan D, Pandey A (2015) Statistical downscaling of temperature using three techniques in the Tons River basin in Central India. Theor Appl Climatol 121:605–622CrossRefGoogle Scholar
  26. Dunn JC (1973) A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clustersGoogle Scholar
  27. Fistikoglu O, Okkan U (2010) Statistical downscaling of monthly precipitation using NCEP/NCAR reanalysis data for Tahtali River basin in Turkey. J Hydrol Eng 16:157–164CrossRefGoogle Scholar
  28. Foufoula-Georgiou E, Ebtehaj M (2013) Variational data assimilation via sparse regularization. In: EGU general assembly conference abstracts, 14147Google Scholar
  29. Ghosh S, Mujumdar P (2006) Future rainfall scenario over Orissa with GCM projections by statistical downscaling. Curr Sci 90:396–404Google Scholar
  30. Ghosh S, Mujumdar P (2008) Statistical downscaling of GCM simulations to streamflow using relevance vector machine. Adv Water Resour 31:132–146CrossRefGoogle Scholar
  31. Govindaraju RS (2005) Bayesian learning and relevance vector machines for hydrologic applications. In: 2nd Indian international conference on artificial intelligence (IICAI-05), Pune, IndiaGoogle Scholar
  32. Goyal MK, Ojha C (2012) Downscaling of surface temperature for lake catchment in an arid region in India using linear multiple regression and neural networks. Int J Climatol 32:552–566CrossRefGoogle Scholar
  33. Grinsted A, Moore JC, Jevrejeva S (2004) Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Process Geophys 11:561–566CrossRefGoogle Scholar
  34. Gudmundsson L, Bremnes J, Haugen J, Engen-Skaugen T (2012) Technical note: downscaling RCM precipitation to the station scale using statistical transformations—a comparison of methods. Hydrol Earth Syst Sci 16:3383–3390CrossRefGoogle Scholar
  35. Halkidi M, Batistakis Y, Vazirgiannis M (2001) On clustering validation techniques. J Intell Inf Syst 17:107–145CrossRefGoogle Scholar
  36. Hertig E, Jacobeit J (2013) A novel approach to statistical downscaling considering nonstationarities: application to daily precipitation in the Mediterranean area. J Geophys Res Atmos 118:520–533CrossRefGoogle Scholar
  37. Hessami M, Gachon P, Ouarda TB, St-Hilaire A (2008) Automated regression-based statistical downscaling tool. Environ Model Softw 23:813–834CrossRefGoogle Scholar
  38. Huang J, Tao H, Fischer T, Wang X (2015) Simulated and projected climate extremes in the Tarim River Basin using the regional climate model CCLM. Stoch Environ Res Risk Assess 29:2061–2071CrossRefGoogle Scholar
  39. Jeong D, St-Hilaire A, Ouarda T, Gachon P (2012) Comparison of transfer functions in statistical downscaling models for daily temperature and precipitation over Canada. Stoch Environ Res Risk Assess 26:633–653CrossRefGoogle Scholar
  40. Kannan S, Ghosh S (2013) A nonparametric kernel regression model for downscaling multisite daily precipitation in the Mahanadi basin. Water Resour Res 49:1360–1385CrossRefGoogle Scholar
  41. Kanungo T, Mount DM, Netanyahu NS, Piatko CD, Silverman R, Wu AY (2002) An efficient k-means clustering algorithm: analysis and implementation. IEEE Trans Pattern Anal Mach Intell 24:881–892CrossRefGoogle Scholar
  42. Kasturi J, Acharya R, Ramanathan M (2003) An information theoretic approach for analyzing temporal patterns of gene expression. Bioinformatics 19:449–458CrossRefGoogle Scholar
  43. Kim S (2004) Wavelet analysis of precipitation variability in northern California, USA. KSCE J Civil Eng 8:471–477CrossRefGoogle Scholar
  44. Labat D, Ababou R, Mangin A (2000) Rainfall–runoff relations for karstic springs. Part II: continuous wavelet and discrete orthogonal multiresolution analyses. J Hydrol 238:149–178CrossRefGoogle Scholar
  45. Li H, Sheffield J, Wood EF (2010) Bias correction of monthly precipitation and temperature fields from Intergovernmental Panel on Climate Change AR4 models using equidistant quantile matching. J Geophys Res Atmos. doi: 10.1029/2009JD012882
  46. MacQueen J (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol 14. Oakland, CA, pp 281–297Google Scholar
  47. Maheswaran R, Khosa R (2012a) Comparative study of different wavelets for hydrologic forecasting. Comput Geosci 46:284–295CrossRefGoogle Scholar
  48. Maheswaran R, Khosa R (2012b) Wavelet-Volterra coupled model for monthly stream flow forecasting. J Hydrol 450:320–335CrossRefGoogle Scholar
  49. Maheswaran R, Khosa R (2013) Wavelets-based non-linear model for real-time daily flow forecasting in Krishna River. J Hydroinf 15:1022–1041CrossRefGoogle Scholar
  50. Mahmood R, Babel MS (2014) Future changes in extreme temperature events using the statistical downscaling model (SDSM) in the trans-boundary region of the Jhelum river basin. Weather Clim Extrem 5:56–66CrossRefGoogle Scholar
  51. Maimon O, Rokach L (2005) Data mining and knowledge discovery handbook, vol 2. Springer, New YorkGoogle Scholar
  52. Maini P, Kumar A, Singh S, Rathore L (2004) Operational model for forecasting location specific quantitative precipitation and probability of precipitation over India. J Hydrol 288:170–188CrossRefGoogle Scholar
  53. Maraun D (2013) Bias correction, quantile mapping, and downscaling: revisiting the inflation issue. J Clim 26:2137–2143CrossRefGoogle Scholar
  54. Najafi MR, Moradkhani H, Wherry SA (2010) Statistical downscaling of precipitation using machine learning with optimal predictor selection. J Hydrol Eng 16:650–664CrossRefGoogle Scholar
  55. Nourani V, Komasi M, Mano A (2009) A multivariate ANN-wavelet approach for rainfall–runoff modeling. Water Resour Manag 23:2877–2894CrossRefGoogle Scholar
  56. Partal T, Kişi Ö (2007) Wavelet and neuro-fuzzy conjunction model for precipitation forecasting. J Hydrol 342:199–212CrossRefGoogle Scholar
  57. Perica S, Foufoula-Georgiou E (1996) Model for multiscale disaggregation of spatial rainfall based on coupling meteorological and scaling. J Geophys Res 101:26–347Google Scholar
  58. Piani C, Haerter J, Coppola E (2010a) Statistical bias correction for daily precipitation in regional climate models over Europe. Theor Appl Climatol 99:187–192CrossRefGoogle Scholar
  59. Piani C, Weedon G, Best M, Gomes S, Viterbo P, Hagemann S, Haerter J (2010b) Statistical bias correction of global simulated daily precipitation and temperature for the application of hydrological models. J Hydrol 395:199–215CrossRefGoogle Scholar
  60. Rajaee T, Mirbagheri S, Nourani V, Alikhani A (2010) Prediction of daily suspended sediment load using wavelet and neurofuzzy combined model. Int J Environ Sci Technol 7:93–110CrossRefGoogle Scholar
  61. Rashid MM, Beecham S, Chowdhury RK (2016) Statistical downscaling of rainfall: a non-stationary and multi-resolution approach. Theor Appl Climatol 124:919–933CrossRefGoogle Scholar
  62. Sachindra D, Perera B (2016) Statistical downscaling of general circulation model outputs to precipitation accounting for non-stationarities in predictor-predictand relationships. PLoS ONE 11:e0168701CrossRefGoogle Scholar
  63. Sachindra D, Huang F, Barton A, Perera B (2011) Statistical downscaling of general circulation model outputs to catchment streamflows. 19th International Congress on Modelling and Simulation (Modsim2011):2810–2816Google Scholar
  64. Sachindra D, Huang F, Barton A, Perera B (2014) Statistical downscaling of general circulation model outputs to precipitation—part 2: bias-correction and future projections. Int J Climatol 34:3282–3303CrossRefGoogle Scholar
  65. Sachindra D, Ng A, Muthukumaran S, Perera B (2016) Impact of climate change on urban heat island effect and extreme temperatures: a case-study. Q J R Meteorol Soc 142:172–186CrossRefGoogle Scholar
  66. Sahay R, Sehgal V (2013) Wavelet regression models for predicting flood stages in rivers: a case study in Eastern India. J Flood Risk Manag 6:146–155CrossRefGoogle Scholar
  67. Salvi K, Ghosh S (2013) High-resolution multisite daily rainfall projections in India with statistical downscaling for climate change impacts assessment. J Geophys Res Atmos 118:3557–3578CrossRefGoogle Scholar
  68. Sang Y-F (2013) A review on the applications of wavelet transform in hydrology time series analysis. Atmos Res 122:8–15CrossRefGoogle Scholar
  69. Sang Y-F, Wang D, Wu J-C, Zhu Q-P, Wang L (2011) Wavelet-based analysis on the complexity of hydrologic series data under multi-temporal scales. Entropy 13:195–210CrossRefGoogle Scholar
  70. Saraf VR, Regulwar DG (2016) Assessment of climate change for precipitation and temperature using statistical downscaling methods in Upper Godavari River basin, India. J Water Resour Prot 8:31CrossRefGoogle Scholar
  71. Schoof JT, Pryor S (2001) Downscaling temperature and precipitation: a comparison of regression-based methods and artificial neural networks. Int J Climatol 21:773–790CrossRefGoogle Scholar
  72. Sehgal V, Sahay RR, Chatterjee C (2014a) Effect of utilization of discrete wavelet components on flood forecasting performance of wavelet based ANFIS models. Water Resour Manag 28:1733–1749CrossRefGoogle Scholar
  73. Sehgal V, Tiwari MK, Chatterjee C (2014b) Wavelet bootstrap multiple linear regression based hybrid modeling for daily river discharge forecasting. Water Resour Manag 28:2793–2811CrossRefGoogle Scholar
  74. Sehgal V, Lakhanpal A, Maheswaran R, Khosa R, Sridhar V (2016) Application of multi-scale wavelet entropy and multi-resolution Volterra models for climatic downscaling. J Hydrol. doi: 10.1016/j.jhydrol.2016.10.048
  75. Sehgal V, Sridhar V, Tyagi A (2017) Stratified drought analysis using a stochastic ensemble of simulated and in situ soil moisture observations. J Hydrol 545:226–250CrossRefGoogle Scholar
  76. Shannon CE (1948) A note on the concept of entropy Bell System. Tech J 27:379–423Google Scholar
  77. Srinivas V, Basu B, Nagesh Kumar D, Jain SK (2014) Multi-site downscaling of maximum and minimum daily temperature using support vector machine. Int J Climatol 34:1538–1560CrossRefGoogle Scholar
  78. Suykens JA (2001) Nonlinear modelling and support vector machines. In: Proceedings of the 18th IEEE instrumentation and measurement technology conference. IMTC 2001, pp 287–294Google Scholar
  79. Torrence C, Compo G (1998) A practical guide to wavelet analysis. Bull Am Meteorol Soc 79:61–78. doi: 10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2 CrossRefGoogle Scholar
  80. Tripathi S, Srinivas V, Nanjundiah RS (2006) Downscaling of precipitation for climate change scenarios: a support vector machine approach. J Hydrol 330:621–640CrossRefGoogle Scholar
  81. White R, Toumi R (2013) The limitations of bias correcting regional climate model inputs. Geophys Res Lett 40:2907–2912CrossRefGoogle Scholar
  82. White MA, Schmidt JC, Topping DJ (2005) Application of wavelet analysis for monitoring the hydrologic effects of dam operation: Glen Canyon Dam and the Colorado River at Lees Ferry, Arizona. River Res Appl 21:551–565CrossRefGoogle Scholar
  83. Wigley T, Jones P, Briffa K, Smith G (1990) Obtaining sub-grid-scale information from coarse-resolution general circulation model output. J Geophys Res Atmos 95:1943–1953CrossRefGoogle Scholar
  84. Wilby R, Charles S, Zorita E, Timbal B, Whetton P, Mearns L (2004) Guidelines for use of climate scenarios developed from statistical downscaling methods. Supporting material of the intergovernmental panel on climate change, available from the DDC of IPCC TGCIA 27Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Civil EngineeringIndian Institute of TechnologyDelhiIndia
  2. 2.Department of Biological Systems EngineeringVirginia TechBlacksburgUSA
  3. 3.Department of Civil EngineeringMVGR College of EngineeringVizianagaramIndia

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