Mathematical Challenges in Measuring Variability Patterns for Precipitation Analysis

  • Maria EmelianenkoEmail author
  • Viviana Maggioni
Part of the Mathematics of Planet Earth book series (MPE, volume 5)


This chapter addresses some of the mathematical challenges associated with current experimental and computational methods to analyze spatiotemporal precipitation patterns. After a brief overview of the various methods to measure precipitation from in situ observations, satellite platforms, and via model simulations, the chapter focuses on the statistical assumptions underlying the most common spatiotemporal and pattern-recognition techniques: stationarity, isotropy, and ergodicity. As the variability of Earth’s climate increases and the volume of observational data keeps growing, these assumptions may no longer be satisfied, and new mathematical methodologies may be required. The chapter discusses spatiotemporal decorrelation measures, a nonstationary intensity-duration-function, and 2-dimension reduction methodologies to address these challenges.


Centroidal Voronoi tessellation Data reduction Decorrelation Empirical orthogonality functions Ergodicity Isotropy Precipitation patterns Stationarity Statistical assumptions 



This work was instigated at the Mason Modeling Days workshop held at George Mason University, generously supported by the National Science Foundation grant DMS-1056821. The authors are grateful to Paul Houser for stimulating discussions at the initial stages of this collaboration. ME also wishes to thank Hans Engler and Hans Kaper for their encouragement over the years, and for introducing this research group to the MPE community.


  1. 1.
    Ababou, R., Bagtzoglou, A.C., Wood, E.F.: On the condition number of covariance matrices in kriging, estimation, and simulation of random fields. Math. Geol. 26(1), 99–133 (1994). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Agilan, V., Umamahesh, N.V.: What are the best covariates for developing non-stationary rainfall intensity-duration-frequency relationship? Adv. Water Resources 101, 11–22 (2017)CrossRefGoogle Scholar
  3. 3.
    Artan, G., Gadain, H., Smith, J.L., et al.: Adequacy of satellite derived rainfall data for streamflow modeling. Nat. Hazards 43, 167–185 (2007)CrossRefGoogle Scholar
  4. 4.
    Atencia, A., Mediero, L., Llasat, M.C., et al.: Effect of radar rainfall time resolution on predictive capability of a distributed hydrological model. Hydrol. Earth Syst. Sci. 15, 3809–3827 (2011)CrossRefGoogle Scholar
  5. 5.
    Bacchi, B., Kottegoda, N.: Identification and calibration of spatial correlation patterns of rainfall. J. Hydrol. 165, 311–348 (1995)CrossRefGoogle Scholar
  6. 6.
    Bauer, P., Lopez, P., Benedetti, A., et al.: Implementation of 1D +  4D-Var assimilation of precipitation-affected microwave radiances at ECMWF. I: 1D-Var. Q. J. Roy. Meteorol. Soc. 132(620), 2277–2306 (2006)Google Scholar
  7. 7.
    Bell, T.L., Kundu, P.K.: Dependence of satellite sampling error on monthly averaged rain rates: comparison of simple models and recent studies. J. Climate 13(2), 449–462 (2000)CrossRefGoogle Scholar
  8. 8.
    Berne, A., Delrieu, G., Creutin, J.D., et al.: Temporal and spatial resolution of rainfall measurements required for urban hydrology. J. Hydrol. 299, 166–179 (2004)CrossRefGoogle Scholar
  9. 9.
    Bonnin, G.M., Maitaria, K., Yekta, M.: Trends in rainfall exceedances in the observed record in selected areas of the United States 1. J. Am. Water Resour. Assoc. 47(6), 1173–1182 (2011)CrossRefGoogle Scholar
  10. 10.
    Borga, M., Anagnostou, E.N., Frank, E.: On the use of real-time radar rainfall estimates for flood prediction in mountainous basins. J. Geophys. Res. 105(D2), 2269–2280 (2000)CrossRefGoogle Scholar
  11. 11.
    Bras, R.L., Rodriguez-Iturbe, I.: Random Functions and Hydrology. Courier Corporation, Chelmsford (1985)Google Scholar
  12. 12.
    Brown, P.E., Diggle, P.J., Lord, M.E., et al.: Space-time calibration of radar rainfall data. J. Royal Statistical Society: Series C (Applied Statistics) 50(2), 221–241 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Burkardt, J., Gunzburger, M., Lee, H.C.: Centroidal Voronoi tessellation-based reduced order modeling of complex systems. SIAM J. Sci. Comput. 28(2), 459–484 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chang, A.T., Chiu, L.S.: Nonsystematic errors of monthly oceanic rainfall derived from SSM/I. Mon. Weather Rev. 127(7), 1630–1638 (1999)CrossRefGoogle Scholar
  15. 15.
    Cheng, L.: Nonstationary Extreme Value Analysis (NEVA) software package, version 2.0. (2014)
  16. 16.
    Cheng, L., AghaKouchak, A., Gilleland, E., et al.: Non-stationary extreme value analysis in a changing climate. Clim. Chang. 127(2), 353–369 (2014). CrossRefGoogle Scholar
  17. 17.
    Chumchean, S., Sharma, A., Seed, A.: Radar rainfall error variance and its impact on radar rainfall calibration. Phys. Chem. Earth, Parts A/B/C 28(1–3), 27–39 (2003)CrossRefGoogle Scholar
  18. 18.
    Ciach, G.: Local random errors in tipping-bucket rain gauge measurements. J. Atmos. Ocean. Technol. 20(5), 752–759 (2003)CrossRefGoogle Scholar
  19. 19.
    Ciach, G.J., Krajewski, W.F.: On the estimation of radar rainfall error variance. Adv. Water Resour. 22(6), 585–595 (1999)CrossRefGoogle Scholar
  20. 20.
    Ciach, G.J., Krajewski, W.F.: Analysis and modeling of spatial correlation structure in small-scale rainfall in Central Oklahoma. Adv. Water Resour. 29(10), 1450–1463 (2006)CrossRefGoogle Scholar
  21. 21.
    Cressie, N.A.C.: Statistics for Spatial Data. John Wiley and Sons, Hoboken (1993)zbMATHCrossRefGoogle Scholar
  22. 22.
    Cristiano, E., Ten Veldhuis, M.C., van de Giesen, N.: Spatial and temporal variability of rainfall and their effects on hydrological response in urban areas – a review. Hydrol. Earth Syst. Sci. 21, 3859–3878 (2017)CrossRefGoogle Scholar
  23. 23.
    Curriero, F.C., Hohn, M.E., Liebhold, A.M.: A statistical evaluation of non-ergodic variogram estimators. Environ. Ecol. Stat. 9, 89–110 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    DeGaetano, A.T.: Time-dependent changes in extreme-precipitation return-period amounts in the continental united states. J. Appl. Meteor. Climatol. 48, 2086–2099 (2009)CrossRefGoogle Scholar
  25. 25.
    Di, Z., Maggioni, V., Mei Y., Vazquez M., Houser P., Emelianenko M., 2019, arXiv, arXiv:1908.10403Google Scholar
  26. 26.
    Dommenget, D., Latif, M.: A cautionary note on the interpretation of EOFs. J. Climate 15, 216–225 (2001)CrossRefGoogle Scholar
  27. 27.
    Duan, J., Goldys, B.: Ergodicity of stochastically forced large scale geophysical flows. J. Math. Math. Sci. 28, 313–320 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Du, Q., Gunzburger, M.: Grid generation and optimization based on centroidal Voronoi tessellations. Appl. Math. Comput. 133, 591–607 (2002)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. SIAM Review 41, 637–676 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Du, Q., Emelianenko, M., Ju, L.: Convergence of the Lloyd algorithm for computing centroidal Voronoi tessellations. SIAM J. Num. Anal. 44, 102–119 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Ebert, E.E., Janowiak, J.E., Kidd, C.: Comparison of near-real-time precipitation estimates from satellite observations and numerical models. Bull. Amer. Meteor. Soc. 88, 47–64 (2007)CrossRefGoogle Scholar
  32. 32.
    Emelianenko, M.: Fast multilevel CVT-based adaptive data visualization algorithm. Numer. Math. Theor. Meth. Appl. 3(2), 195–211 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Gottschalck, J., Meng, J., Rodell, M., et al.: Analysis of multiple precipitation products and preliminary assessment of their impact on global land data assimilation system land surface states. J. Hydrometeorl. 6, 573–598 (2005)CrossRefGoogle Scholar
  34. 34.
    Hateley, J.C., Wei, H., Chen, L.: Fast methods for computing centroidal Voronoi tessellations. J. Sci. Comput. 63(1), 185–212 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Hirsch, R.M.: A perspective on nonstationarity and water management. J. Amer. Water Resources Assoc. (JAWRA) 47(3), 436–446 (2011)CrossRefGoogle Scholar
  36. 36.
    Hodgkins, G.A., Dudley, R.W.: Changes in the timing of winter–spring streamflows in eastern North America. Geophys. Res. Lett. 33, 1913–2002 (2006)CrossRefGoogle Scholar
  37. 37.
    Hossain, F., Anagnostou, E.N.: Assessment of current passive-microwave- and infrared-based satellite rainfall remote sensing for flood prediction. J. Geophys. Res. 109 (2004)Google Scholar
  38. 38.
    Hossain, F., Anagnostou, E.N.: A two-dimensional satellite rainfall error model. IEEE Trans. Geosci. Remote Sens. 44(6), 1511–1522 (2006)CrossRefGoogle Scholar
  39. 39.
    Hsu, K., Gao, X., Sorooshian, S., et al.: Precipitation estimation from remotely sensed information using artificial neural networks. J. Appl. Meteor. 36, 1176–1190 (1997)CrossRefGoogle Scholar
  40. 40.
    Huffman, G.J., Bolvin, D.T., Nelkin, E.J., et al.: The TRMM multisatellite precipitation analysis (TMPA): Quasi-global, multiyear, combined-sensor precipitation estimates at fine scales. J. Hydrometeorol. 8(1), 38–55 (2007)CrossRefGoogle Scholar
  41. 41.
    Huffman, G.J., Bolvin, D., Braithwaite, D., et al.: Integrated Multi-satellite Retrievals for GPM (IMERG), version 4.4. NASA’s Precipitation Processing Center. Accessed 31 March 2015.
  42. 42.
    Joyce, R.J., Janowiak, J.E., Arkin, P.A., et al.: Cmorph: a method that produces global precipitation estimates from passive microwave and infrared data at high spatial and temporal resolution. J. Hydrometeorl. 5, 487–503 (2004)CrossRefGoogle Scholar
  43. 43.
    Kidd, C., Bauer, P., Turk, J., et al.: Intercomparison of high-resolution precipitation products over northwest Europe. J. Hydrometeorl. 13, 67–83 (2012)CrossRefGoogle Scholar
  44. 44.
    Kottegoda, N.T.: Stochastic Water Resources Technology. Palgrave, Macmillan (1980). CrossRefGoogle Scholar
  45. 45.
    Koutsoyiannis, D.: Stochastic simulation of hydrosystems. Water Encyclopedia 3, 421–430 (2005)Google Scholar
  46. 46.
    Krajewski, W.F., Anderson, M.C., Eichinger, W.E., et al.: A remote sensing observatory for hydrologic sciences: a genesis for scaling to continental hydrology. Water Resour. Res. 42(7), W07,301 (2006)CrossRefGoogle Scholar
  47. 47.
    Krauth, W.: Statistical Mechanics: Algorithms and Computations. Oxford Master Series in Physics. Oxford University Press, UK (2006). zbMATHGoogle Scholar
  48. 48.
    Kummerow, C.: Beamfilling errors in passive microwave rainfall retrievals. J. Appl. Meteorol. 37(4), 356–370 (1998)CrossRefGoogle Scholar
  49. 49.
    Lins, H.F.: A note on stationarity and non-stationarity. 14th Session of the Commission for Hydrology (2012)Google Scholar
  50. 50.
    Lorenc, A.C.: The potential of the ensemble Kalman filter for NWP—a comparison with 4D-Var. Q. J. R. Meteorol. Soc. 129(595), 3183–3203 (2003)CrossRefGoogle Scholar
  51. 51.
    Marzano, F.S., Picciotti, E., Vulpiani, G.: Rain field and reflectivity vertical profile reconstruction from c-band radar volumetric data. IEEE Trans. Geosci. Remote Sens. 42(4), 1033–1046 (2004)Google Scholar
  52. 52.
    Michaelides, S., Levizzani, V., Anagnostou, E.N., et al.: Precipitation science: measurement, remote sensing, climatology and modeling. Atmos. Res. 94, 512–533 (2009)CrossRefGoogle Scholar
  53. 53.
    Milly, P.C.D., Betancourt, J., Fallkenmark, M., et al.: Stationarity is dead: whither water management? Science 319, 573–574 (2008)CrossRefGoogle Scholar
  54. 54.
    Nikolopoulos, E., Borga, M., Zoccatelli, D., et al.: Catchment scale storm velocity: quantification, scale dependence and effect on flood response. Hydrol. Sci. J. 59, 1363–1376 (2014)CrossRefGoogle Scholar
  55. 55.
    Ochoa-Rodriguez, S., Wang, L., Gires, A., et al.: Impact of spatial and temporal resolution of rainfall inputs on urban hydrodynamic modelling outputs: a multi-catchment investigation. J. Hydrol. 531, 389–407 (2015)CrossRefGoogle Scholar
  56. 56.
    Oliveira, T.F., Cunha, F.R., Bobenrieth, R.F.M.: A stochastic analysis of a nonlinear flow response. Probab. Eng. Mech. 21, 377–383 (2006)CrossRefGoogle Scholar
  57. 57.
    Oliveira, R., Maggioni, V., Vila, D., et al.: Characteristics and diurnal cycle of GPM rainfall estimates over the Central Amazon Region. Remote Sens. 8(7), 544 (2016)CrossRefGoogle Scholar
  58. 58.
    Rafieeinasab, A., Norouzi, A., Kim, S., et al.: Toward high-resolution flash flood prediction in large urban areas: analysis of sensitivity to spatiotemporal resolution of rainfall input and hydrologic modeling. J. Hydrol. 531, 370–388 (2015)CrossRefGoogle Scholar
  59. 59.
    Ringler, T., Ju, L., Gunzburger, M.: A multiresolution method for climate system modeling: application of spherical centroidal Voronoi tessellations. Ocean Dyn. 58, 475–498 (2008)CrossRefGoogle Scholar
  60. 60.
    Rodriguez-Iturbe, I., Isham, V.: Some models for rainfall based on stochastic point processes. Proc. R. Soc. Lond. A 410(1839), 269–288 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Schneider, U., Fuchs, T., Meyer-Christoffer, A., et al.: Global precipitation analysis products of the GPCC. Global Precipitation Climatology Centre (GPCC), DWD, Internet Publication 112 (2008)Google Scholar
  62. 62.
    Schwarzl, M., Godec, A., Metzler, R.: Quantifying non-ergodicity of anomalous diffusion with higher order moments. Sci. Rep. 7, 3878 (2017)CrossRefGoogle Scholar
  63. 63.
    Scofield, R.A., Kuligowski, R.J.: Status and outlook of operational satellite precipitation algorithms for extreme-precipitation events. Weather Forecast. 18, 1037–1051 (2003)CrossRefGoogle Scholar
  64. 64.
    Serrat-Capdevila, A., Valdes, J.B., Stakhiv, E.: Water management applications for satellite precipitation products: synthesis and recommendations. J. Am. Water Resour. Assoc. 50, 509–525 (2014)CrossRefGoogle Scholar
  65. 65.
    von Storch, H., Navarra, A.: Analysis of Climate Variability Applications of Statistical Techniques. Springer, Berlin (1999)CrossRefGoogle Scholar
  66. 66.
    Tian, Y., Peters-Lidard, C.D., Choudhury, B.J., et al.: Multitemporal analysis of TRMM-based satellite precipitation products for land data assimilation applications. J. Hydrometeorol. 8, 1165–1183 (2007)CrossRefGoogle Scholar
  67. 67.
    Wang, H., Wang, C., Zhao, Y., et al.: Toward a practical approach for ergodicity analysis. Nonlin. Processes Geophys. Discuss. 2, 1425–1446 (2015)CrossRefGoogle Scholar
  68. 68.
    Wood, E., Roundy, J.K., Troy, T.J., et al.: Hyper-resolution global land surface modeling: meeting a grand challenge for monitoring Earth’s terrestrial water. Water Resour. Res. 47, W05,301 (2011)CrossRefGoogle Scholar
  69. 69.
    Zhang, Q., Sun, P., Singh, V.P., et al.: Spatial-temporal precipitation changes (1956–2000) and their implications for agriculture in China. Global Planet. Change 82, 86–95 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  2. 2.Sid and Reva Dewberry Department of Civil, Environmental, and Infrastructure EngineeringGeorge Mason UniversityFairfaxUSA

Personalised recommendations