HESS Hydrology and Earth System Sciences HESS Hydrol. Earth Syst. Sci. 1607-7938 Copernicus Publications Göttingen, Germany 10.5194/hess-18-5331-2014 Imperfect scaling in distributions of radar-derived rainfall fields van den Berg M. J. 1 Delobbe L. 2 Verhoest N. E. C. https://orcid.org/0000-0003-4116-8881 1 Laboratory of Hydrology and Water Management, Ghent University, Coupure links 653, 9000 Ghent, Belgium Royal Meteorological Institute, Avenue Circulaire 3, 1180 Uccle, Belgium 19 12 2014 18 12 5331 5344 Copyright: © 2014 M. J. van den Berg et al. 2014 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://hess.copernicus.org/articles/18/5331/2014/hess-18-5331-2014.html The full text article is available as a PDF file from https://hess.copernicus.org/articles/18/5331/2014/hess-18-5331-2014.pdf

Fine-scale rainfall observations for modelling exercises are often not available, but rather coarser data derived from a variety of sources are used. Effectively using these data sources in models often requires the probability distribution of the data at the applicable scale. Although numerous models for scaling distributions exist, these are often based on theoretical developments, rather than on data. In this study, we develop a model based on the α-stable distribution of rainfall fields, and tested on 5 min radar data from a Belgian weather radar. We use these data to estimate functions that describe parameters of the distribution over various scales. Moreover, we study how the mean of the distribution and the intermittency change with scale, and validate and design functions to describe the shape parameter of the distribution. This information was combined into an effective model of the distribution.

 References Davis, A., Marshak, A., Wiscombe, W., and Cahalan, R.: Multifractal characterizations of nonstationarity and intermittency in geophysical fields: Observed, retrieved, or simulated, J. Geophys. Res., 99, 8055–8072, 1994. De Jongh, I. L. M., Verhoest, N. E. C., and De Troch, F. P.: Analysis of a 105-year time series of precipitation observed at Uccle, Belgium, Int. J. Climatol., 26, 2023–2039, 2006. Deidda, R.: Rainfall downscaling in a space-time multifractal framework, Water Resour. Res., 36, 1779–1794, 2000. Delobbe, L., Dehem, D., Dierickx, P., Roulin, E., Thunus, M., and Tricot, C.: Combined use of radar and gauge observations for hydrological applications in the Walloon region of Belgium, Proceedings of ERAD, 418–421, 2006. Ferraris, L., Rudari, R., and Siccardi, F.: The uncertainty in the prediction of flash floods in the northern Mediterranean environment, J. Hydrometeorol., 3, 714–727, 2002. Ferraris, L., Gabellani, S., Parodi, U., Rebora, N., Von Hardenberg, J., and Provenzale, A.: Revisiting multifractality in rainfall fields, J. Hydrometeorol, 4, 544–551, 2003. Garel, B. and Kodia, B.: Signed symmetric covariation coefficient for alpha-stable dependence modeling, CR Math., 347, 315–320, 2009. Gires, A., Onof, C., Maksimovic, C., Schertzer, D., Tchiguirinskaia, I., and Simoes, N.: Quantifying the impact of small scale unmeasured rainfall variability on urban runoff through multifractal downscaling: A case study, J. Hydrol., 442, 117–128, 2012a. Gires, A., Tchiguirinskaia, I., Schertzer, D., and Lovejoy, S.: Influence of the zero-rainfall on the assessment of the multifractal parameters, Adv. Water Res., 45, 13–25, 2012b. Goudenhoofdt, E. and Delobbe, L.: Evaluation of radar-gauge merging methods for quantitative precipitation estimates, Hydrol. Earth Syst. Sci., 13, 195–203, <a href="http://dx.doi.org/10.5194/hess-13-195-2009">https://doi.org/10.5194/hess-13-195-2009</a>, 2009. Gupta, V. K. and Waymire, E. C.: A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteorol., 32, 251–251, 1993. Harris, D. and Foufoula-Georgiou, E.: Subgrid variability and stochastic downscaling of modeled clouds: Effects on radiative transfer computations for rainfall retrieval (1984–2012), J. Geophys. Res.-Atmos., 106, 10349–10362, 2001. Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., and Stanley, H. E.: Multifractal detrended fluctuation analysis of nonstationary time series, Physica A, 316, 87–114, 2002. Kolmogorov, A. N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, in: Dokl. Akad. Nauk SSSR, Vol. 30, 299–303, 1941. Koutrouvelis, I.: Regression-type estimation of the parameters of stable laws, J. Am. Stat. Assoc., 75, 918–928, 1980. Koutsoyiannis, D., Paschalis, A., and Theodoratos, N.: Two-dimensional Hurst–Kolmogorov process and its application to rainfall fields, J. Hydrol., 398, 91–100, 2010. Lovejoy, S. and Schertzer, D.: The weather and climate: emergent laws and multifractal cascades, Cambridge University Press, 2013. McCulloch, J.: Simple consistent estimators of stable distribution parameters, Communications in Statistics-Simulation and Computation, 15, 1109–1136, 1986. Menabde, M. and Sivapalan, M.: Modeling of rainfall time series and extremes using bounded random cascades and Levy-stable distributions, Water Resour. Res., 36, 3293–3300, 2000. Menabde, M., Harris, D., Seed, A., Austin, G., and Stow, D.: Multiscaling properties of rainfall and bounded random cascades, Water Resour. Res., 33, 2823–2830, 1997. Nolan, J. P.: Parameter estimation and data analysis for stable distributions, in: Conference Record of the Thirty-First Asilomar Conference on Signals, Systems & Computers, 1997, vol. 1, IEEE, Pacific Grove(CA), USA, 443–447, 1997. Nolan, J. P.: Maximum likelihood estimation and diagnostics for stable distributions, in: Lévy processes, 379–400, Springer, 2001. Nolan, J. P.: Stable Distributions – Models for Heavy Tailed Data, Birkhauser, Boston, in progress, Chapter 1 available at: <a href="http://academic2.american.edu/ jpnolan">http://academic2.american.edu/ jpnolan</a> (last access: 20 August 2013), 2012. Olsson, J.: Limits and characteristics of the multifractal behaviour of a high-resolution rainfall time series, Nonl. Proc. Geophys., 2, 23–29, 1995. Olsson, J.: Evaluation of a scaling cascade model for temporal rain- fall disaggregation, Hydrol. Earth Syst. Sci., 2, 19–30, https://doi.org/10.5194/hess-2-19-1998, 1998. Over, T. and Gupta, V.: Statistical analysis of mesoscale rainfall: dependence of a random cascade generator on large-scale forcing, J. Appl. Meteorol., 33, 1526–1542, 1994. Parisi, G. and Frisch, U.: A multifractal model of intermittency, Turbulence and Predictability, in: Geophysical Fluid Dynamics, edited by: Ghil, M., Benzi, R., and Parisi, G., 84–88, 1985. Paulson, K. S. and Baxter, P. D.: Downscaling of rain gauge time series by multiplicative beta cascade, J. Geophys. Res., 112, D09105, <a href="http://dx.doi.org/10.1029/2006JD007333">https://doi.org/10.1029/2006JD007333</a>, 2007. Rebora, N., Ferraris, L., von Hardenberg, J., and Provenzale, A.: Rainfall downscaling and flood forecasting: a case study in the Mediterranean area, Nat. Hazards Earth Syst. Sci., 6, 611–619, <a href="http://dx.doi.org/10.5194/nhess-6-611-2006">https://doi.org/10.5194/nhess-6-611-2006</a>, 2006. Rupp, D. E., Keim, R. F., Ossiander, M., Brugnach, M., and Selker, J. S.: Time scale and intensity dependency in multiplicative cascades for temporal rainfall disaggregation, Water Resour. Res., 45, W07409, <a href="http://dx.doi.org/10.1029/2008WR007321">https://doi.org/10.1029/2008WR007321</a>, 2009. Schertzer, D. and Lovejoy, S.: Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes, J. Geophys. Res., 92, 9693–9714, 1987. Schertzer, D. and Lovejoy, S.: Universal multifractals do exist!: Comments on – A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteorol., 36, 1296–1303, 1997. Schertzer, D., Lovejoy, S., and Hubert, P.: An introduction to stochastic multifractal fields, in: ISFMA Symposium on Environmental Science and Engineering with related Mathematical Problems, 106–179, Higher Education Press, Beijing, 2002. Serinaldi, F.: Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models, Nonl. Proc. Geophys., 17, 697–714, 2010. Steiner, M., Houze Jr, R. A., and Yuter, S. E.: Climatological characterization of three-dimensional storm structure from operational radar and rain gauge data, J. Appl. Meteorol., 34, 1978–2007, 1995. Tchiguirinskaia, I., Lu, S., Molz, F., Williams, T., and Lavallée, D.: Multifractal versus monofractal analysis of wetland topography, Stochastic Environmental Research and Risk Assessment, 14, 8–32, 2000. Veneziano, D. and Furcolo, P.: A modified double trace moment method of multifractal analysis, Fractals, 7, 181–196, 1999. Veneziano, D., Furcolo, P., and Iacobellis, V.: Imperfect scaling of time and space-time rainfall, J. Hydrol., 322, 105–119, 2006. Venugopal, V., Roux, S. G., Foufoula-Georgiou, E., and Arneodo, A.: Revisiting multifractality of high-resolution temporal rainfall using a wavelet-based formalism, Water Resour. Res., 42, W06D14, <a href="http://dx.doi.org/10.1029/2005WR004489">https://doi.org/10.1029/2005WR004489</a>, 2006. Verrier, S., Mallet, C., and Barthès, L.: Multiscaling properties of rain in the time domain, taking into account rain support biases, J. Geophys. Res., 116, D20119, <a href="http://dx.doi.org/10.1029/2011JD015719">https://doi.org/10.1029/2011JD015719</a>, 2011. Wuertz, D., Maechler, M., and R metrics core team members: stabledist: Stable Distribution Functions, <a href="http://CRAN.R-project.org/package=stabledist">http://CRAN.R-project.org/package=stabledist</a>, r package version 0.6-5, 2012.