Skip to main content
SpringerLink
Log in
Book cover

Mathematics of Planet Earth pp 55–74Cite as

Mathematical Challenges in Measuring Variability Patterns for Precipitation Analysis

  • Chapter
  • First Online:

Part of the book series: Mathematics of Planet Earth ((MPE,volume 5))

Abstract

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  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). https://doi.org/10.1007/BF02065878

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  5. Bacchi, B., Kottegoda, N.: Identification and calibration of spatial correlation patterns of rainfall. J. Hydrol. 165, 311–348 (1995)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  11. Bras, R.L., Rodriguez-Iturbe, I.: Random Functions and Hydrology. Courier Corporation, Chelmsford (1985)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  15. Cheng, L.: Nonstationary Extreme Value Analysis (NEVA) software package, version 2.0. http://amir.eng.uci.edu/neva.php (2014)

  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). https://doi.org/10.1007/s10584-014-1254-5

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  18. Ciach, G.: Local random errors in tipping-bucket rain gauge measurements. J. Atmos. Ocean. Technol. 20(5), 752–759 (2003)

    Article  Google Scholar 

  19. Ciach, G.J., Krajewski, W.F.: On the estimation of radar rainfall error variance. Adv. Water Resour. 22(6), 585–595 (1999)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  21. Cressie, N.A.C.: Statistics for Spatial Data. John Wiley and Sons, Hoboken (1993)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  25. Di, Z., Maggioni, V., Mei Y., Vazquez M., Houser P., Emelianenko M., 2019, arXiv, arXiv:1908.10403

    Google Scholar 

  26. Dommenget, D., Latif, M.: A cautionary note on the interpretation of EOFs. J. Climate 15, 216–225 (2001)

    Article  Google Scholar 

  27. Duan, J., Goldys, B.: Ergodicity of stochastically forced large scale geophysical flows. J. Math. Math. Sci. 28, 313–320 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Du, Q., Gunzburger, M.: Grid generation and optimization based on centroidal Voronoi tessellations. Appl. Math. Comput. 133, 591–607 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. SIAM Review 41, 637–676 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  32. Emelianenko, M.: Fast multilevel CVT-based adaptive data visualization algorithm. Numer. Math. Theor. Meth. Appl. 3(2), 195–211 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  34. Hateley, J.C., Wei, H., Chen, L.: Fast methods for computing centroidal Voronoi tessellations. J. Sci. Comput. 63(1), 185–212 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hirsch, R.M.: A perspective on nonstationarity and water management. J. Amer. Water Resources Assoc. (JAWRA) 47(3), 436–446 (2011)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Hossain, F., Anagnostou, E.N.: A two-dimensional satellite rainfall error model. IEEE Trans. Geosci. Remote Sens. 44(6), 1511–1522 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. ftp://arthurhou.pps.eosdis.nasa.gov/gpmdata/

  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)

    Article  Google Scholar 

  43. Kidd, C., Bauer, P., Turk, J., et al.: Intercomparison of high-resolution precipitation products over northwest Europe. J. Hydrometeorl. 13, 67–83 (2012)

    Article  Google Scholar 

  44. Kottegoda, N.T.: Stochastic Water Resources Technology. Palgrave, Macmillan (1980). https://books.google.com/books?id=3SiuCwAAQBAJ

    Book  Google Scholar 

  45. Koutsoyiannis, D.: Stochastic simulation of hydrosystems. Water Encyclopedia 3, 421–430 (2005)

    Google Scholar 

  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)

    Article  Google Scholar 

  47. Krauth, W.: Statistical Mechanics: Algorithms and Computations. Oxford Master Series in Physics. Oxford University Press, UK (2006). https://books.google.com/books?id=B3koVucDyKUC

    MATH  Google Scholar 

  48. Kummerow, C.: Beamfilling errors in passive microwave rainfall retrievals. J. Appl. Meteorol. 37(4), 356–370 (1998)

    Article  Google Scholar 

  49. Lins, H.F.: A note on stationarity and non-stationarity. 14th Session of the Commission for Hydrology (2012)

    Google Scholar 

  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)

    Article  Google Scholar 

  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. Michaelides, S., Levizzani, V., Anagnostou, E.N., et al.: Precipitation science: measurement, remote sensing, climatology and modeling. Atmos. Res. 94, 512–533 (2009)

    Article  Google Scholar 

  53. Milly, P.C.D., Betancourt, J., Fallkenmark, M., et al.: Stationarity is dead: whither water management? Science 319, 573–574 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Schwarzl, M., Godec, A., Metzler, R.: Quantifying non-ergodicity of anomalous diffusion with higher order moments. Sci. Rep. 7, 3878 (2017)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  65. von Storch, H., Navarra, A.: Analysis of Climate Variability Applications of Statistical Techniques. Springer, Berlin (1999)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  67. Wang, H., Wang, C., Zhao, Y., et al.: Toward a practical approach for ergodicity analysis. Nonlin. Processes Geophys. Discuss. 2, 1425–1446 (2015)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Maria Emelianenko

  2. Sid and Reva Dewberry Department of Civil, Environmental, and Infrastructure Engineering, George Mason University, Fairfax, VA, USA

    Viviana Maggioni

Authors
  1. Maria Emelianenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viviana Maggioni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Emelianenko .

Editor information

Editors and Affiliations

  1. Mathematics and Statistics, Georgetown University, Washington, DC, USA

    Hans G. Kaper

  2. DIMACS Center, Rutgers University, Piscataway, NJ, USA

    Fred S. Roberts

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Emelianenko, M., Maggioni, V. (2019). Mathematical Challenges in Measuring Variability Patterns for Precipitation Analysis. In: Kaper, H., Roberts, F. (eds) Mathematics of Planet Earth. Mathematics of Planet Earth, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-22044-0_3

Download citation

Publish with us

Policies and ethics

Search

Navigation