• Anselmet, F., Antonia R. A. , and Danaila L. , 2001: Turbulent flows and intermittency in laboratory experiments. Planet. Space Sci., 49, 11771191, doi:10.1016/S0032-0633(01)00059-9.

    • Search Google Scholar
    • Export Citation
  • Battaglia, A., Rustemeier E. , Tokay A. , Blahak U. , and Simmer C. , 2010: PARSIVEL snow observations: A critical assessment. J. Atmos. Oceanic Technol., 27, 333344, doi:10.1175/2009JTECHA1332.1.

    • Search Google Scholar
    • Export Citation
  • Cao, Q., and Coauthors, 2008: Analysis of video disdrometer and polarimetric radar data to characterize rain microphysics in Oklahoma. J. Appl. Meteor. Climatol., 47, 22382255, doi:10.1175/2008JAMC1732.1.

    • Search Google Scholar
    • Export Citation
  • Cotton, W. R., and Gokhale N. R. , 1967: Collision, coalescence, and breakup of large water drops in a vertical wind tunnel. J. Geophys. Res., 72, 40414049, doi:10.1029/JZ072i016p04041.

    • Search Google Scholar
    • Export Citation
  • de Lima, M. I. P., and Grasman J. , 1999: Multifractal analysis of 15-min and daily rainfall from a semi-arid region in Portugal. J. Hydrol., 220, 111, doi:10.1016/S0022-1694(99)00053-0.

    • Search Google Scholar
    • Export Citation
  • de Lima, M. I. P., and de Lima J. , 2009: Investigating the multifractality of point precipitation in the Madeira archipelago. Nonlinear Processes Geophys., 16, 299311, doi:10.5194/npg-16-299-2009.

    • Search Google Scholar
    • Export Citation
  • de Montera, L., Barthes L. , Mallet C. , and Gole P. , 2009: The effect of rain–no rain intermittency on the estimation of the universal multifractals model parameters. J. Hydrometeor., 10, 493506, doi:10.1175/2008JHM1040.1.

    • Search Google Scholar
    • Export Citation
  • Desaulniers-Soucy, N., Lovejoy S. , and Schertzer D. , 2001: The continuum limit in rain and the HYDROP experiment. Atmos. Res., 59–60, 163197, doi:10.1016/S0169-8095(01)00115-6.

    • Search Google Scholar
    • Export Citation
  • Ducrocq, V., and Coauthors, 2014: HyMeX-SOP1: The field campaign dedicated to heavy precipitation and flash flooding in the northwestern Mediterranean. Bull. Amer. Meteor. Soc., 95, 1083–1100, doi:10.1175/BAMS-D-12-00244.1.

    • Search Google Scholar
    • Export Citation
  • Fitton, G., Tchiguirinskaia I. , Schertzer D. , and Lovejoy S. , 2011: Scaling of turbulence in the atmospheric surface-layer: Which anisotropy? J. Phys. Conf. Ser., 318, 072008, doi:10.1088/1742-6596/318/7/072008.

    • Search Google Scholar
    • Export Citation
  • Fraedrich, K., and Larnder C. , 1993: Scaling regimes of composite rainfall time series. Tellus,45A, 289–298, doi:10.1034/j.1600-0870.1993.t01-3-00004.x.

  • Frasson, R. P. M., da Cunha L. K. , and Krajewski W. F. , 2011: Assessment of the Thies optical disdrometer performance. Atmos. Res., 101, 237255, doi:10.1016/j.atmosres.2011.02.014.

    • Search Google Scholar
    • Export Citation
  • Gabella, M., Pavone S. , and Perona G. , 2001: Errors in the estimate of the fractal correlation dimension of raindrop spatial distribution. J. Appl. Meteor., 40, 664668, doi:10.1175/1520-0450(2001)040<0664:EITEOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Tchiguirinskaia I. , Schertzer D. , and Lovejoy S. , 2011: Analyses multifractales et spatio-temporelles des précipitations du modèle Méso-NH et des données radar. Hydrol. Sci. J., 56, 380396, doi:10.1080/02626667.2011.564174.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Onof C. , Maksimovic C. , Schertzer D. , Tchiguirinskaia I. , and Simoes N. , 2012: Quantifying the impact of small scale unmeasured rainfall variability on urban runoff through multifractal downscaling: A case study. J. Hydrol., 442-443, 117128, doi:10.1016/j.jhydrol.2012.04.005.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Tchiguirinskaia I. , Schertzer D. , and Lovejoy S. , 2013: Development and analysis of a simple model to represent the zero rainfall in a universal multifractal framework. Nonlinear Processes Geophys., 20, 343356, doi:10.5194/npg-20-343-2013.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Tchiguirinskaia I. , Schertzer D. , Schellart A. , Berne A. , and Lovejoy S. , 2014: Influence of small scale rainfall variability on standard comparison tools between radar and rain gauge data. Atmos. Res., 138, 125138, doi:10.1016/j.atmosres.2013.11.008.

    • Search Google Scholar
    • Export Citation
  • Hubert, P., and Coauthors, 1993: Multifractals and extreme rainfall events. Geophys. Res. Lett., 20, 931934, doi:10.1029/93GL01245.

  • Jaffrain, J., and Berne A. , 2011: Experimental quantification of the sampling uncertainty associated with measurements from PARSIVEL disdrometers. J. Hydrometeor., 12, 352370, doi:10.1175/2010JHM1244.1.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., and Kostinski A. B. , 1998: Fluctuation properties of precipitation. Part II: Reconsideration of the meaning and measurement of raindrop size distributions. J. Atmos. Sci., 55, 283294, doi:10.1175/1520-0469(1998)055<0283:FPOPPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., Kostinski A. B. , and Kruger A. , 1999: Fluctuation properties of precipitation. Part IV: Finescale clustering of drops in variable rain. J. Atmos. Sci., 56, 8291, doi:10.1175/1520-0469(1999)056<0082:FPOPPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Joss, J., and Waldvogel A. , 1967: Ein Spektrograph für Niederschlagstropfen mit automatischer Auswertung (A spectrograph for raindrops with automatic interpretation). Pure Appl. Geophys., 68, 240246, doi:10.1007/BF00874898.

    • Search Google Scholar
    • Export Citation
  • Kostinski, A. B., and Jameson A. R. , 1997: Fluctuation properties of precipitation. Part I: On deviations of single-size drop counts from the Poisson distribution. J. Atmos. Sci., 54, 21742186, doi:10.1175/1520-0469(1997)054<2174:FPOPPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Krajewski, W. F., and Coauthors, 2006: DEVEX-disdrometer evaluation experiment: Basic results and implications for hydrologic studies. Adv. Water Resour., 29, 311325, doi:10.1016/j.advwatres.2005.03.018.

    • Search Google Scholar
    • Export Citation
  • Kruger, A., and Krajewski W. F. , 2002: Two-dimensional video disdrometer: A description. J. Atmos. Oceanic Technol., 19, 602617, doi:10.1175/1520-0426(2002)019<0602:TDVDAD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ladoy, P., Schmitt F. , Schertzer D. , and Lovejoy S. , 1993: The multifractal temporal variability of Nimes rainfall data. C. R. Acad. Sci. Ser. II, 317 (6), 775782.

    • Search Google Scholar
    • Export Citation
  • Lavergnat, J., and Golé P. , 1998: A stochastic raindrop time distribution model. J. Appl. Meteor., 37, 805818, doi:10.1175/1520-0450(1998)037<0805:ASRTDM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lazarev, A., Schertzer D. , Lovejoy S. , and Chigirinskaya Y. , 1994: Unified multifractal atmospheric dynamics tested in the tropics: Part II, vertical scaling and generalized scale invariance. Nonlinear Processes Geophys., 1, 115123, doi:10.5194/npg-1-115-1994.

    • Search Google Scholar
    • Export Citation
  • Lilley, M., Lovejoy S. , Desaulniers-Soucy N. , and Schertzer D. , 2006: Multifractal large number of drops limit in rain. J. Hydrol., 328, 2037, doi:10.1016/j.jhydrol.2005.11.063.

    • Search Google Scholar
    • Export Citation
  • Lovejoy, S., and Schertzer D. , 1990: Fractals, raindrops and resolution dependence of rain measurements. J. Appl. Meteor., 29, 11671170, doi:10.1175/1520-0450(1990)029<1167:FRARDO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lovejoy, S., and Schertzer D. , 1995: Multifractals and rain. New Uncertainty Concepts in Hydrology and Hydrological Modelling, A.W. Kundzewicz, Ed., Cambridge Press, 62–103.

  • Lovejoy, S., and Schertzer D. , 2008: Turbulence, rain drops and the l1/2 number density law. New J. Phys.,10, 075017, doi:10.1088/1367-2630/10/7/075017.

  • Lovejoy, S., Duncan M. R. , and Schertzer D. , 1996: Scalar multifractal radar observer’s problem. J. Geophys. Res., 101, 26 47926 492, doi:10.1029/96JD02208.

    • Search Google Scholar
    • Export Citation
  • Lovejoy, S., Schertzer D. , and Allaire V. , 2008: The remarkable wide range spatial scaling of TRMM precipitation. Atmos. Res., 90, 1032, doi:10.1016/j.atmosres.2008.02.016.

    • Search Google Scholar
    • Export Citation
  • Mandapaka, P. V., Lewandowski P. , Eichinger W. E. , and Krajewski W. F. , 2009: Multiscaling analysis of high resolution space–time lidar-rainfall. Nonlinear Processes Geophys., 16, 579586, doi:10.5194/npg-16-579-2009.

    • Search Google Scholar
    • Export Citation
  • Nykanen, D. K., and Harris D. , 2003: Orographic influences on the multiscale statistical properties of precipitation. J. Geophys. Res., 108, 8381, doi:10.1029/2001JD001518.

    • Search Google Scholar
    • Export Citation
  • Olsson, J., 1995: Limits and characteristics of the multifractal behavior of a high-resolution rainfall time series. Nonlinear Processes Geophys., 2, 2329, doi:10.5194/npg-2-23-1995.

    • Search Google Scholar
    • Export Citation
  • Royer, J.-F., Biaou A. , Chauvin F. , Schertzer D. , and Lovejoy S. , 2008: Multifractal analysis of the evolution of simulated precipitation over France in a climate scenario. C. R. Geosci., 340, 431440, doi:10.1016/j.crte.2008.05.002.

    • Search Google Scholar
    • Export Citation
  • Salles, C., Creutin J. D. , and Sempere-Torres D. , 1998: The optical spectropluviometer revisited. J. Atmos. Oceanic Technol., 15, 12151222, doi:10.1175/1520-0426(1998)015<1215:TOSR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., and Lovejoy S. , 1987: Physical modelling and analysis of rain and clouds by anisotropic scaling and multiplicative processes. J. Geophys. Res., 92, 96939714, doi:10.1029/JD092iD08p09693.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., and Lovejoy S. , 1997: Universal multifractals do exist!: Comments on “A Statistical analysis of mesoscale rainfall as a random cascade.” J. Appl. Meteor., 36, 12961303, doi:10.1175/1520-0450(1997)036<1296:UMDECO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., and Lovejoy S. , 2011: Multifractals, generalized scale invariance and complexity in geophysics. Int. J. Bifurcation Chaos, 21, 34173456, doi:10.1142/S0218127411030647.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., Tchiguirinskaia I. , Lovejoy S. , and Hubert P. , 2010: No monsters, no miracles: In nonlinear sciences hydrology is not an outlier! Hydrol. Sci. J., 55, 965979, doi:10.1080/02626667.2010.505173.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., Tchiguirinskaia I. , and Lovejoy S. , 2012: Getting higher resolution rainfall estimates: X-band radar technology and multifractal drop distribution. IAHS Publ.,351, 105–110.

  • Tchiguirinskaia, I., Salles C. , Hubert P. , Schertzer D. , Lovejoy S. , Creutin J. D. , and Bendjoudi H. , 2003: Multifractal analysis of the OSP measured rain rates over time scales from millisecond to day. IUGG 2003, Sapporo, Japan, IUGG, JSM18/01A/B20-003.

  • Tessier, Y., Lovejoy S. , and Schertzer D. , 1993: Universal multifractals: Theory and observations for rain and clouds. J. Appl. Meteor., 32, 223250, doi:10.1175/1520-0450(1993)032<0223:UMTAOF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tessier, Y., Lovejoy S. , Hubert P. , Schertzer D. , and Pecknold S. , 1996: Multifractal analysis and modeling of rainfall and river flows and scaling, causal transfer functions. J. Geophys. Res., 101, 26 42726 440, doi:10.1029/96JD01799.

    • Search Google Scholar
    • Export Citation
  • Thurai, M., and Bringi V. N. , 2005: Drop axis ratios from a 2D video disdrometer. J. Atmos. Oceanic Technol., 22, 966978, doi:10.1175/JTECH1767.1.

    • Search Google Scholar
    • Export Citation
  • Thurai, M., Peterson W. A. , Tokay A. , Schutz C. , and Gatlin P. , 2011: Drop size distribution comparisons between Parsivel and 2-D video disdrometers. Adv. Geosci., 30, 39, doi:10.5194/adgeo-30-3-2011.

    • Search Google Scholar
    • Export Citation
  • Tokay, A., Petersen W. A. , Gatlin P. , and Wingo M. , 2013: Comparison of raindrop size distribution measurements by collocated disdrometers. J. Atmos. Oceanic Technol., 30, 16721690, doi:10.1175/JTECH-D-12-00163.1.

    • Search Google Scholar
    • Export Citation
  • Uijlenhoet, R., Stricker J. N. M. , Torfs P. J. J. F. , and Creutin J. D. , 1999: Towards a stochastic model of rainfall for radar hydrology: Testing the Poisson homogeneity hypothesis. Phys. Chem. Earth, Part B, 24, 747755, doi:10.1016/S1464-1909(99)00076-3.

    • Search Google Scholar
    • Export Citation
  • Verrier, S., de Montera L. , Barthes L. , and Mallet C. , 2010: Multifractal analysis of African monsoon rain fields, taking into account the zero rain-rate problem. J. Hydrol., 389, 111120, doi:10.1016/j.jhydrol.2010.05.035.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Picture of the installed 2DVD in Le Pradel, Ardèche, France.

  • View in gallery

    (a) Example of a reconstruction for a 26-cm vertical column; dimensions are in millimeters, drops have been colored according to size, and their diameter has been multiplied by 4 to improve visibility. (b) Illustration of the 36 m × 11 cm × 11 cm reconstructed column divided into smaller boxes. (c) Vertical evolution of the LWC (g m−3) within the vertical column for the reconstructed field (black) and simulated assuming homogenous distribution of drop positions (red) for an instant of the 24 Sep 2012 event. The horizontal axis corresponds to the box number (0 is ground level and 8192 is the top of the reconstructed column).

  • View in gallery

    (a) Spectral analysis of the vertical column snapshots for 60 s starting at 0217 UTC 24 Sep 2012. (b) TM analysis of the same data (points) and the corresponding synthetic field–same drops (size and velocity), but their position is randomly (uniformly) assigned (solid, curved lines). The linear regressions (straight lines on the right) are only performed for large scales, that is, 0.5–36 m. (c) As in (b), but zoomed in on the large scales. (d) Spectral analysis of the corresponding synthetic field.

  • View in gallery

    Illustration of the absence of scaling in columns with a too low number of drops. (a) As in Fig. 3b, but for 60 time steps starting at 0223 UTC. (b) Scatterplot of r2 for q = 1.5 in the TM analysis vs the average number of drops in the studied columns. (c) Scatterplot of the average number of drops in the studied columns vs the indicative rain rate computed at ground level.

  • View in gallery

    Figures are plotted for the 24 Sep 2012 event. (a) Temporal evolution of the number of drops passing through the sampling area with 1-ms time steps. (b)–(d) Temporal evolution of the rain rate during the same event with time steps of 1 ms, 1 s, and 1 min, respectively. (e) Temporal evolution of the average mass weighted diameter. (f)–(h) As in (b)–(d), but for logλ.

  • View in gallery

    (a) Spectral analysis and (b) TM analysis data corresponding to 140 min of the 23 Oct 2013 event with 1-ms time steps (time series of length 223).

  • View in gallery

    (top) Natural drop accumulation: (left) rain accumulation map (mm × 10−3) and (right) occurrence map for 150 consecutive drops during the 23 Oct 2013 event. (bottom) As in (top), but for a homogenized version of the drop accumulation (see text for details).

  • View in gallery

    Fractal analysis [Eq. (1) in log–log plot] of the drop position for the 23 Oct 2013 event with 150 drops per drop accumulation map (ensemble analysis on the 1179 pictures recorded on a total duration of 12 h). (a) On the drop centers. (b) Considering all the pixels occluded by drops.

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    (a) Duration needed to record pictures with 350 drops during the 27 Oct 2013 event. (b) The exceedance probability of the distribution of the durations in a log–log plot (Δt and x are in seconds).

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2DVD Data Revisited: Multifractal Insights into Cuts of the Spatiotemporal Rainfall Process

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  • 1 Laboratoire Eau, Environnement, Systèmes Urbains, École des Ponts ParisTech, Université Paris-Est, Marne-la-Vallée, France
  • | 2 Laboratoire de Télédétection Environnementale, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
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Abstract

Data collected during four heavy rainfall events that occurred in Ardèche (France) with the help of a 2D video disdrometer (2DVD) are used to investigate the structure of the raindrop distribution in both space and time. A first type of analysis is based on the reconstruction of 36-m-height vertical rainfall columns above the measuring device. This reconstruction is obtained with the help of a ballistic hypothesis applied to 1-ms time step series. The corresponding snapshots are analyzed with the help of universal multifractals. For comparison, a similar analysis is performed on the time series with 1-ms time steps, as well as on time series of accumulation maps of N consecutive recorded drops (therefore with variable time steps). It turns out that the drop distribution exhibits a good scaling behavior in the range 0.5–36 m during the heaviest portion of the events, confirming the lack of empirical evidence of the widely used homogenous assumption for drop distribution. For smaller scales, drop positions seem to be homogeneously distributed. The notion of multifractal singularity is well illustrated by the very high-resolution time series.

Corresponding author address: Auguste Gires, École des Ponts ParisTech, LEESU, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-La-Vallée CEDEX 2, France. E-mail: auguste.gires@leesu.enpc.fr

Abstract

Data collected during four heavy rainfall events that occurred in Ardèche (France) with the help of a 2D video disdrometer (2DVD) are used to investigate the structure of the raindrop distribution in both space and time. A first type of analysis is based on the reconstruction of 36-m-height vertical rainfall columns above the measuring device. This reconstruction is obtained with the help of a ballistic hypothesis applied to 1-ms time step series. The corresponding snapshots are analyzed with the help of universal multifractals. For comparison, a similar analysis is performed on the time series with 1-ms time steps, as well as on time series of accumulation maps of N consecutive recorded drops (therefore with variable time steps). It turns out that the drop distribution exhibits a good scaling behavior in the range 0.5–36 m during the heaviest portion of the events, confirming the lack of empirical evidence of the widely used homogenous assumption for drop distribution. For smaller scales, drop positions seem to be homogeneously distributed. The notion of multifractal singularity is well illustrated by the very high-resolution time series.

Corresponding author address: Auguste Gires, École des Ponts ParisTech, LEESU, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-La-Vallée CEDEX 2, France. E-mail: auguste.gires@leesu.enpc.fr

1. Introduction

Rainfall is extremely variable over a wide range of scales both in space and time. It has become rather usual to characterize this behavior with the help of scaling properties and, more recently, multifractals [see Lovejoy and Schertzer (1995) and Schertzer et al. (2010) for reviews]. This framework is physically based in the sense that it is more than a simple tailored statistical framework. Indeed, it relies on the cascade process concept that was introduced to reflect the scale invariance properties of the Navier–Stokes equations that govern atmospheric dynamics. It is assumed that the unknown equations governing rainfall inherit these properties of scale invariance (Schertzer and Lovejoy 1987; Hubert et al. 1993). Since the advent of this framework in the 1980s, ample empirical evidence has established its relevancy on scales ranging from a few minutes to decades in time and hundreds of meters to planetary size in space (Lovejoy et al. 2008). Some authors reported not a single scaling regime but various ones separated by breaks, typically at a few minutes and a few days in time and few kilometers in space. Various types of data have been used: rain gauges (de Lima and de Lima 2009; de Lima and Grasman 1999; Fraedrich and Larnder 1993; Ladoy et al. 1993; Olsson 1995; Tessier et al. 1996), disdrometers (de Montera et al. 2009; Gires et al. 2014), weather radars (Gires et al. 2011, 2012; Nykanen and Harris 2003; Tessier et al. 1993; Verrier et al. 2010), satellite with TRMM data (Lovejoy et al. 2008), and even numerical outputs of climate simulations (Royer et al. 2008) or mesoscale models (Gires et al. 2011).

The highest resolutions mentioned before are rather coarse with regard to the millimeter scale down to which atmospheric turbulence—the embedding field of rainfall—is known to exhibit scaling behavior [see Anselmet et al. (2001) for a review]. Hence, there is a need to investigate more precisely the minimum spatiotemporal scale down to which scaling is observed in rainfall fields. Very few studies have analyzed this, mainly because of the lack of rainfall data at these scales. Mandapaka et al. (2009) reported scaling in the range 1–512 s in time and 2.5–320 m in space on rainfall data obtained with the help of a lidar. Lilley et al. (2006) and Lovejoy and Schertzer (2008), using a few reconstructed 3D snapshots of an 8-m3 volume with most of its drops (Desaulniers-Soucy et al. 2001), reported a scaling behavior down to a few tens of centimeters with a dependency on the turbulence intensity and drop size. Lovejoy and Schertzer (1990) and Gabella et al. (2001) using large (approximately 1 m) sheets of chemically blotted paper also found a scaling behavior down to almost drop scale, but these results have been disputed (Jameson and Kostinski 1998). In this paper we suggest investigating more in depth the spatiotemporal (three dimensions in space and one dimension in time) features of rainfall fields down to drop scale using data collected by a 2D video disdrometer (2DVD) [see Kruger and Krajewski (2002) for a precise description of the device’s functioning], deployed in the Ardèche region (southeastern France) in the framework of the Hydrological Cycle in Mediterranean Experiment (HyMeX; Ducrocq et al. 2014), in an innovative way. Indeed, this device has been extensively used as a reference in comparison with other rainfall-measuring ones (Krajewski et al. 2006; Tokay et al. 2013), or to investigate drops and, more generally, hydrometeor shape (Battaglia et al. 2010; Cao et al. 2008; Thurai and Bringi 2005) and size (Thurai et al. 2011) distribution features, but not to address this issue of the spatiotemporal structure of rainfall process at drop scale. More precisely, vertical, horizontal, and temporal cuts of rainfall fields are either obtained or reconstructed with the data provided by a 2DVD and investigated with the help of multifractal techniques.

A related issue is whether drops are homogeneously distributed in space and time. Scaling laws are incompatible with a homogenous distribution (Poisson statistics), but it remains a debated topic. It has mainly been discussed in time by analyzing drop counts over various time steps. Most authors have reported deviations from Poisson statistics also, not necessarily attributing them to an underlying scaling behavior; Kostinski and Jameson (1997) used 1-min drop counts by the Joss–Waldvogel disdrometer (Joss and Waldvogel 1967) to most likely invalidate a Poisson framework during heavy rainfall periods (more than 27 drops per minute) and confirmed this with 1-s (corresponding to few meters) counts computed with 15 min of 2DVD data (Jameson et al. 1999), noticing that larger drops are more correlated over longer coherence time. Uijlenhoet et al. (1999) also observed deviations from Poisson statistics with 35 min of 10-s time step disdrometer data (~6500 drops), but they found that the discrepancies are only due to small drops (with a diameter smaller than 1.1 mm) that basically do not influence rain rates or reflectivity that are related to higher-order moments of the drop-size distribution. At the interevent scale (4 months of data), Lavergnat and Golé (1998) observed a power-law decrease of the duration between the observations of two consecutive drops with a disdrometer, which is incompatible with Poisson statistics. Hence, in this paper, we suggest to also tackle this issue by systematically comparing (when possible) our results with those that would have been obtained with homogeneously distributed drops. Such methodology was already employed in Lilley et al. (2006) and Jameson and Kostinski (1998).

The collected data are presented in section 2, along with the basic ideas underlying the multifractal framework. Section 3 describes the analysis of snapshots of reconstructed 36-m-high vertical columns of air above the device with all its drops. The high temporal resolution of the measuring device is used in section 4 to study 1-ms time step series (over the 11 × 11 cm2 sampling area of the device). Drop accumulation maps (with varying time steps, but for the same sampling area) are analyzed in section 5.

2. Data description and multifractal framework

a. Data description

The data used in this paper were collected with the help of a 2DVD (Kruger and Krajewski 2002). The 2DVD provides detailed information about the geometry and the fall velocity of the particles falling through its sampling area of about 11 × 11 cm2, by means of two perpendicular high-speed line cameras (with a pixel size of about 0.2 mm and a temporal resolution of about 1 ms). From the two reconstructed side views of each raindrop, the shape and equivolume diameter are retrieved (assuming the raindrops are oblate spheroids). With the two cameras being shifted in the vertical by about 6.5 mm, the fall velocity of each particle is directly measured. The same raw data will be used to generate three types of representation and analysis in the following sections: vertical reconstructed columns, high-resolution time series, and small-scale accumulation maps. Methods used to obtain each cut of the underlying spatiotemporal rainfall representation are included in separate, dedicated sections to facilitate the reading of the paper.

The device was installed in Le Pradel, Ardèche, France (see Fig. 1), during the fall 2012 and 2013 in the framework of HyMeX (Ducrocq et al. 2014). The considered 2DVD is the low version (with reduced wind disturbance). A number of rainfall events were collected, and the two heaviest ones (in terms of 5-min rain rate) for each fall have been selected for the present study. Their main features are summarized in Table 1.

Fig. 1.
Fig. 1.

Picture of the installed 2DVD in Le Pradel, Ardèche, France.

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

Table 1.

Basic features of the studied events.

Table 1.

b. Multifractal framework

The basic principle of the multifractal framework is briefly recapped. For more details, refer to the recent review by Schertzer and Lovejoy (2011). Multifractals are used to characterize and simulate geophysical fields that are extremely variable over a wide range of spatiotemporal scales and that exhibit long-range correlation. It basically relies on the concept of multiplicative cascades. Let us denote ελ as a field at resolution λ (= L/l) defined as the ratio between the outer scale L of the phenomenon and the observation scale l. The field at a given resolution λ is obtained from the field at the maximum resolution Λ by averaging it over pixels of resolution λ.

Let us first introduce in a rather intuitive (but less mathematically rigorous) way the notion of multifractal fields with the help of the concept of fractal dimension. The variable A is a geometrical set embedded in a space of dimension d (for instance d = 1 for time series and d = 2 for maps), and Nλ is the number of nonoverlapping d-dimension boxes of size l needed to cover A at resolution λ. If the set is fractal, then we have
e1
where DF is the fractal dimension of A.

The geometrical set corresponding to the portion of a field ελ above a given threshold exhibits fractal features that can be quantified with the help of a fractal dimension. The decreasing dependency of this fractal dimension with regard to the considered threshold reflects the need of multiple fractal dimensions to characterize the field. In a more rigorous mathematical way, the threshold is replaced by the scale invariant notion of singularity, but the underlying idea remains the same.

More precisely, if a field is scaling, then its power spectra E is a power law with respect to the wavenumber k:
e2
where β is the spectral slope. A larger β reflects a weaker correlation.
Spectral analysis basically corresponds to a statistical analysis of the moment 2 of the field. In the multifractal framework, all of the moments are used to fully characterize the variability across scale. The statistical moment of order q at a given resolution scales with the resolution:
e3
where K(q) is the moment scaling function. In the specific framework of universal multifractals (UM), which correspond to the stable and attractive limits of nonlinearly interacting multifractal processes (i.e., a multiplicative generalization of the central limit theorem), K(q) has the following analytical expression:
e4
which only depends on three parameters having a strong physical meaning (Schertzer and Lovejoy 1987, 1997):
  • The degree of nonconservation H, which measures the scale dependency of the mean field H = 0 for a conservative field and can be either positive or negative corresponding to a fractional integration or differentiation, respectively, of a conserved field.

  • The mean intermittency codimension C1, which measures the mean sparseness of the field, that is, how concentrated is the mean field. A homogeneous field fills the embedding space and has C1 = 0, , where d is the dimension of the embedding space (a greater C1 could theoretically exist, but it would correspond to fields almost surely null everywhere).

  • The multifractality index , which measures the variability of the intermittency, that is, its dependence with respect to the considered level of activity. When α = 0, it means that all activity levels exhibit the same intermittency reflecting a fractal field.

The various parameters are related by the following equation from which H is usually estimated:
e5

3. Analysis of reconstructed vertical rainfall columns

a. Ballistic reconstruction of a column

As already explained, the 2DVD provides for each drop a direct measurement of the fall velocity and of the horizontal position within the 11 × 11 cm2 sampling area of the device. Assuming the validity of the hypothesis of vertical ballistic trajectories, that is, that both the position and the velocity remain constant during the last seconds of fall and equal to the ones measured near the ground surface, it is possible to reconstruct the trajectory of each drop. Achieving this for all the drops enables us to reconstruct the whole rainfall field drop by drop on a column above the device (Fig. 2a). The height of the column studied in this paper is 36 m.

Fig. 2.
Fig. 2.

(a) Example of a reconstruction for a 26-cm vertical column; dimensions are in millimeters, drops have been colored according to size, and their diameter has been multiplied by 4 to improve visibility. (b) Illustration of the 36 m × 11 cm × 11 cm reconstructed column divided into smaller boxes. (c) Vertical evolution of the LWC (g m−3) within the vertical column for the reconstructed field (black) and simulated assuming homogenous distribution of drop positions (red) for an instant of the 24 Sep 2012 event. The horizontal axis corresponds to the box number (0 is ground level and 8192 is the top of the reconstructed column).

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

The ballistic assumption is extremely coarse. Indeed, by assuming an underlying laminar flow, we neglect all interactions with the turbulent wind field and notably shear effects, which have an influence on the drops’ velocity, especially for small ones. Neglecting these effects tend to worsen the quality of the scaling we can observe with the help of this reconstruction. Drop population dynamics (e.g., coalescence in case of collisions or breakups) is also ignored, although wind tunnel studies have shown that raindrop interactions can occur on scales below 36 m (Cotton and Gokhale 1967). Possible effects on the tendency of small drops to collide or not, due to small-scale turbulence self-induced by the aerodynamic forces exerted on the larger falling raindrop (for large raindrops, the Reynolds number is on the order of a few thousand), is also not taken into account. However, despite its limitations, this reconstruction can yield some preliminary insight before considering more complex reconstruction, for example, with randomized trajectories taking into account turbulence effects as well as coalescence and breakup.

Finally, the column is divided into 8192 boxes of 4.3-mm height, the two horizontal dimensions being the horizontal sampling area ones, that is, ~11 cm (Fig. 2b). Although the minimum interdrop distance (defined as the drop concentration to the power −⅓) observed for the studied events is roughly 4 cm (and usually much larger), the chosen height of the boxes is much smaller so that results can be compared in the vertical and in the horizontal dimensions (see section 5). For each box, we consider the sum of the pth power of the volumes Vi of the drops (computed with help of the equivolume diameter Di estimated by the 2DVD device) contained in this box:
e6
As suggested by Lilley et al. (2006), by varying p, various physical quantities are represented; for instance, χp for p equal to 0, 1, , and 2 is proportional to the drop concentration, liquid water content (LWC), rain rate (assuming a fall velocity proportional to the square root of the drop diameter), and radar reflectivity (which is nearly proportional to the 6th power of the diameter in the Rayleigh scattering regime), respectively. This yields a spatial 1D (vertical) field. An example of vertical evolution of the LWC (basically χp for p = 1) within the column during the heaviest portion of the 24 September 2012 event is displayed in Fig. 2c. A reconstructed snapshot of the vertical column above the 2DVD is done every second. Before going on, it should be mentioned that small drops are not fully represented with the 2DVD since measurements for drops with an equivolume diameter smaller than 0.3 mm are unreliable (Tokay et al. 2013). This means that small moments are likely to be biased. Nevertheless, the quantities of interest studied in this paper are for p > 1 (moment of the diameter greater than 3), meaning that this limitation does not have a strong influence on the discussed results.

To compare the observed properties of the fields with the ones that would be obtained if drops were homogeneously distributed, pseudosynthetic fields are also generated. To obtain a realization for a given column, the drop centers are reassigned with the help of a random, uniform distribution, whereas the drop sizes are unchanged. The vertical evolution of the LWC obtained for a realization is displayed in Fig. 2c.

b. Scaling behavior

Statistical scaling properties are obtained by ensemble averaging [each field sample is first independently upscaled, i.e., its resolution is degraded by averaging over adjacent pixels, then raised to various powers q, and finally the ensemble average is performed to obtain an estimate of the theoretical moments and its scaling behavior; Eq. (3)] over 60 consecutive column snapshots (1 min in physical time). The spectral analysis of the field [Eq. (2) in log–log plot] is displayed in Fig. 3a for the 60 time steps starting at 0217 UTC 24 September 2012 with p = 1 (i.e., LWC estimate). This is a very intense period of the storm where there are 5573 drops in the column on average (which corresponds to roughly 13 000 drops per cubic meter) and the rain rate is approximately 180 mm h−1. The rain rate is computed as the average rain rate measured at the level of the 2DVD device (at the bottom of the column). It is only an indication of the current rainfall intensity because some of the drops present in the studied 60 consecutive snapshots do not reach the device during this minute, meaning that consistent comparison is not possible. We mention it simply because rain rates are much more often used in hydrometeorology than drop concentrations.

Fig. 3.
Fig. 3.

(a) Spectral analysis of the vertical column snapshots for 60 s starting at 0217 UTC 24 Sep 2012. (b) TM analysis of the same data (points) and the corresponding synthetic field–same drops (size and velocity), but their position is randomly (uniformly) assigned (solid, curved lines). The linear regressions (straight lines on the right) are only performed for large scales, that is, 0.5–36 m. (c) As in (b), but zoomed in on the large scales. (d) Spectral analysis of the corresponding synthetic field.

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

The reconstructed fields exhibit a scaling behavior from 36 m down to roughly 1 m [log (k/k0) ~ 3 with k0 = 1 Hz]. For smaller scales, the spectra are flat, as they would be for white noise resulting from randomly homogeneously distributed drops. These findings are confirmed by the trace moment (TM) analysis [Eq. (3) in a log–log plot] displayed in Fig. 3b for the whole range of available scales and in Fig. 3c, zoomed in on larger scales. It appears that the reconstructed fields exhibit a scaling behavior (the coefficients of determination r2 of the linear portions are all greater than 0.99) up to approximately λ = 64, which corresponds roughly to 0.5 m, a value rather similar to the one found in the spectral analysis. This confirms, on an extended range of scales and using more data, the results of Lilley et al. (2006), who observed scaling from 0.4 to 2 m on a few snapshots of 8-m3 volume with most of its drops. This suggests that drops are distributed in a scaling manner down to 0.5–1 m, and below they are homogeneously distributed, which is in agreement with the findings of Lovejoy and Schertzer (2008).

Similar analyses are also performed on the synthetic fields with homogeneously distributed drop positions. The spectra remain flat on the whole range of scales (Fig. 3d), and the TM analysis curves (solid lines in Figs. 3b,c) do not exhibit any scaling regime (i.e., linear portion). More quantitatively, over the scaling regime identified for reconstructed fields, the r2 for the homogeneous fields are typically equal to 0.80. The curves for the smallest scales (roughly from 4 mm to a few centimeters) are very similar between reconstructed and homogeneous fields. This confirms the results of the spectral analysis for this range of scales. For larger scales, deviations between reconstructed and homogeneous fields are visible with a transition regime before the scaling regime of the reconstructed fields (0.5–36 m). Very similar results are found for other realizations of homogeneously distributed fields, showing that these results are statistically meaningful, that is, the Poisson hypothesis of a homogeneous drop distribution, which is commonly used to model rainfall, is not correct for this minute of the storm. Indeed, it does not enable us to reproduce the observed structure of drop distributions.

An analysis of the other minutes of this storm and the others leads us to qualify this statement. Indeed, it appears that a scaling behavior is only retrieved when there are enough drops in the column (typically more than 2000). To illustrate this, Fig. 4a displays a TM analysis for 60 s starting at 0223 UTC for the same 24 September 2012 event (415 drops on average in the column, approximately 30 mm h−1). In this case, with fewer drops the linear portions on the TM curves tend to disappear, and furthermore, the spectra for large scales flatten with β estimates closer to zero, suggesting a behavior more in agreement with a homogeneous distribution of drops. To get an idea of the number of drops needed in the vertical columns to observe scaling, Fig. 4b displays a scatterplot of the r2 of the TM curve for q = 1.5, which is taken as an indication of the quality of the scaling, versus the average number of drops in the studied vertical columns for the 24 September 2012 event (60 consecutive snapshots are used for each point). Similar plots are obtained for the other events. It appears that scaling is only visible when there are more than 2000–3000 drops in the column, which is the case for roughly 500 snapshots (8 min) during the studied events. To get a similar threshold with rain rates, Fig. 4c displays the rain rate computed versus the average number of drops in the studied vertical columns for the same minute. The limitations of the comparison between these two quantities are visible on this figure, especially for extreme minutes for which the relation is not linear. The indicative rain-rate threshold to observe scaling in the vertical columns would be roughly 75 mm h−1.

Fig. 4.
Fig. 4.

Illustration of the absence of scaling in columns with a too low number of drops. (a) As in Fig. 3b, but for 60 time steps starting at 0223 UTC. (b) Scatterplot of r2 for q = 1.5 in the TM analysis vs the average number of drops in the studied columns. (c) Scatterplot of the average number of drops in the studied columns vs the indicative rain rate computed at ground level.

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

For the vertical columns exhibiting a scaling behavior, we find for scales ranging from 35 to 0.5 m that α ~ 1.8–2, C1 ~ 0.005–0.01, and β ~ 1–1.4 [slightly smaller than the values found by Lovejoy and Schertzer (2008)]. With the help of Eq. (5), it leads to H ~ 0–0.2. The degree of nonconservation is small enough that it is relevant to directly implement the TM analysis on the vertical series and not on their fluctuations. It is timely to mention that only a limited portion of the scaling regime is observed with this dataset since there is only a ratio of 64 between the maximum and the minimum resolutions of the scaling regime. This means that the reliability of the multifractal exponent estimates is not very high and their values should not be overinterpreted. Therefore, we will only briefly discuss them. The UM parameter estimates for the actual field are slightly different from those found by Lilley et al. (2006), with a greater α and smaller C1, and the storm-to-storm variability is also less pronounced, but it could be due to the fact that here the estimates are only discussed for the most intense portion of the events, whereas Lilley et al. only performed their analysis on a limited number of volume snapshots from various storms and not necessarily obtained during the peak in rain intensity.

The UM parameter estimates are slightly different, with especially a C1 smaller than those usually reported at coarser resolution (α ~ 1.7–1.9, C1 ~ 0.05–0.2) by authors who studied rainfall field in space (Mandapaka et al. 2009; Verrier et al. 2010; Gires et al. 2013). Although these authors studied rainfall fields horizontally, which makes a direct comparison harder (because of a possible anisotropy between the vertical and horizontal directions), this hints at a possible break between two scaling regimes, a small-scale one and a large-scale one, located at a few tens or a few hundreds of meters. Very high-resolution data on larger areas are needed to confirm this.

The UM estimates are also different from those found for wind turbulence, again with greater values of α and smaller values of C1 (Lazarev et al. 1994; Fitton et al. 2011). It would mean that there is no trivial link between the drop distributions and the wind turbulence field in which they are embedded and from which they presumably inherit the scaling behavior up to a given (high) degree, contrary to the ballistic assumption that was used for the column reconstruction and could partially explain the observed differences with wind turbulence parameters. Development of new instruments providing the size and 3D velocity of each raindrop over a few tens of cubic meters are required to properly address this issue.

With the UM parameter estimates found, one can expect sampling limitation and divergence of moment [see Schertzer and Lovejoy (2011) for more details] to only affect estimates of K(q) for q greater than 10, meaning that for lower statistical order the observed scaling is not spurious.

The same analyses were performed with other values of p or only considering drops with an equivolume diameter D belonging to a given interval. Taking a greater p or only larger drops yields similar results as expected, since greater moments enhance the influence of large drops in Eq. (6). There is no significant difference in the scaling for drops up to 2.5 mm (the average number of drops per column remains significant) or p smaller than 3. A tendency of α to decrease and C1 to increase with larger drops is noticed. For drops with D > 2.5 mm or p > 3, the scaling is lost and the discrepancies with the Poisson framework are much less pronounced. An explanation (Lilley et al. 2006) is that large drops decouple from atmospheric turbulence, from which they inherit the scaling behavior, at a larger scale than small drops. It means that, although the ballistic assumption on the studied scales is likely to be more valid for larger drops, on the limited range of available scales (the maximum range in 35 m), it will not be possible to observe scaling behavior. Another limitation to the study of large drops with this dataset is the very low number of drops in the reconstructed columns (typically less than 80 for the period with the heaviest rainfall available), which leads to less reliable statistics.

4. Rain-rate time series with 1-ms time steps

a. Description of the data

The data provided by the 2DVD enable us to compute a rain rate with time steps of 1 ms, given by
e7
where the sum over the Vi of the drops is now preformed over all the drops that passed through the sampling area S during (1 ms here). Note that we consider a constant sampling area (the maximum one) and do not take into account refinements with regard to edge effects, which can be assumed of second order. This very high-resolution rain rate was computed for 35 min (221 time steps) of the 24 September 2012 event and for 140 min (223 time steps) of the other events. The portion with the greatest cumulative depth was selected for each event. Before going on, we should mention that authors are not advocating for the use of 1-ms time step series for routine hydrological applications. Unfortunately, these series are likely to suffer from strong sampling errors; see Frasson et al. (2011) or Jaffrain and Berne (2011) for an illustration of this issue with standard occlusion optical disdrometers having much smaller sampling area. Here we use this series to demonstrate that the multifractal notion of singularity, mostly perceived at small scales, also has consequences at large scales.

The 1-ms rain-rate time series is plotted in Fig. 5b for the 24 September 2012 event along with the same series with 1-s time steps (Fig. 5c) and the more classical 1-min time step (i.e., as the average over 60 000 consecutive time steps; Fig. 5d). The temporal evolution of the number of drops recorded (Fig. 5a) and of the average mass weighted diameter (Fig. 5e) per millisecond is also displayed. Note that because of the high temporal resolution, the “average” is in fact most of the time steps performed over a unique drop. Two striking features should be noted: 96% of zeros occur while the series was taken from what is commonly considered an extreme rainfall event (this percentage ranges from 97% to 99% for the other events), and the maximum rain rate (over 1 ms) is 60 000 mm h−1, which is a much greater value than what hydrometeorologists are used to (this maximum ranges from 30 000 to 60 000 mm h−1 for the other events). These extremely high rain rates are due to the passing of several drops during the same millisecond, up to 14 during the 24 September event. Obviously, the diameter of the drops is also important, as can be seen for the second peak (between minutes 7 and 10) of the event in Fig. 5c, which is not visible on the drop counts time series. From Fig. 5e, it appears that this second peak is due to larger drops during this period. More standard values of rain rate are retrieved when averaging to the 1-min time steps.

Fig. 5.
Fig. 5.

Figures are plotted for the 24 Sep 2012 event. (a) Temporal evolution of the number of drops passing through the sampling area with 1-ms time steps. (b)–(d) Temporal evolution of the rain rate during the same event with time steps of 1 ms, 1 s, and 1 min, respectively. (e) Temporal evolution of the average mass weighted diameter. (f)–(h) As in (b)–(d), but for logλ.

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

These two features illustrate both the intermittency of the rainfall field and the notion of multifractal singularity, that is, the fact that the rain rate is not pointwise defined because it depends on the time interval over which it is estimated: it usually diverges when considering it for smaller and smaller intervals. Mathematically, it means that the rain accumulation is a singular measure with respect to the usual measure of time (i.e., the one-dimensional Lebesgue measure). The need to properly deal with these two complex properties led to the development of the theoretical framework of multifractals [see Schertzer and Lovejoy (2011) for more details]. In this framework, the rain rate is expected to behave as
e8
When positive, the singularity γ is the (algebraic) rate of divergence of the rain rate with the resolution. Here the singularity γ corresponding to the maximum peak occurring at about 5 min (Fig. 5) is equal to 0.8, 0.8, and 1.5 for time steps of 1 ms (λ = 2.1 × 106), 1 s (λ = 2.1 × 103), and 1 min (λ = 35), respectively. The differences are due to the fact that there are several scaling regimes over these scales, as will be shown in the next section. To emphasize this point on the whole time series, they are also displayed in Fig. 5 in a logλ scale.

b. Temporal multifractal analysis

Figure 6 displays the spectral analysis and the TM analysis for the 23 October 2013 event, for which the total duration taken into account is 140 min, enabling us to study a series that comprises 223 1-ms time steps! This duration was chosen because it is the longest one enabling us to remain in the event and to have a series whose length is a power of 2 as it is needed for the simplest TM analysis. Similar curves are obtained for the other events. The spectra do not reflect a very good scaling behavior, and it seems that there is a scaling breakdown for the frequency roughly equal to 70 [log(k/k0) ~ 4.2 with k0 = 1 Hz], which corresponds to 2 min−1 (the breakdown is also at roughly 2 min−1 for the 24 September 2012 event, the series of which is shorter). The coefficient of determination of the linear portion for large scales is weak (equal to 0.68); hence, the estimates of the spectral slope are not reliable and will therefore not be discussed. No spectral slope can be derived for smaller scales. Through the TM analysis, three scaling regimes can be identified: from 140 min to a few minutes (from 2 to 8 min according the event) with a rather good scaling (r2 = 0.97–0.9; blue lines in Fig. 6b), a transition regime ranging from a few minutes to 32 ms with a rather bad scaling (red lines in Fig. 6b), and a very small-scale regime from 32 to 1 ms with a good scaling (r2 = 0.99; green lines in Fig. 6b). The poor quality of the scaling observed through the spectral analysis means that the results of the TM analysis should not be overinterpreted. We observe similar scaling regimes to those reported and discussed by Schertzer et al. (2012), that is, the multifractal regime from 1 day to 7 min and the fractal regime from 2 s to 1 ms, obtained with the help of a multifractal analysis by Tchiguirinskaia et al. (2003) of an infrared optical spectropluviometer time series (Salles et al. 1998).

Fig. 6.
Fig. 6.

(a) Spectral analysis and (b) TM analysis data corresponding to 140 min of the 23 Oct 2013 event with 1-ms time steps (time series of length 223).

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

For the small scales (32–1 ms), we indeed find an index of multifractality roughly equal to zero, meaning that the observed behavior is monofractal. Values of C1 are in the interval 0.7–0.85 and estimates are roughly equal to the fractal codimension of the time series, which means that this regime simply reflects the passing of individual drops through the sampling area. Given the poor quality of the scaling, UM estimates are not computed for the medium scales corresponding to the transition regime. With regard to large scales, the UM parameter estimates for all the events are reported in Table 2. The values of α are no longer equal to zero, meaning that an actual multifractal behavior coming from the common influence of several drops is retrieved (contrary to what was observed for small scales). The strong variations according to the event are much greater, especially for α, than those expected if various realizations of the same process were analyzed. It means that the UM parameters seem to depend on the event. The standard drop-size distribution [DSD; N(D)] was computed for each event on average (not shown here), and it appears that the observed differences in the UM parameters are not related to DSD differences (especially to the thickness of the tail). There is for now no clear explanation for the physical process to which the UM parameter estimate differences could be attributed, and further investigations are required to clarify this.

Table 2.

UM parameter estimates for large scales in the temporal analysis of 1-ms resolution time series for the four studied events.

Table 2.

To compare these results in time with those obtained on the vertical column in the previous section, one has to consider similar time scales. According to its size, the duration taken by a drop to fall through the 36-m vertical column is between a few seconds and a few tens of seconds (observed drop velocity is between 0.5 and 10 m s−1). This range of scales corresponds to the middle regime for which there is no clear scaling behavior. It might be due to the fact that the data are considered for the whole event and not only for the most intense portions, as is done for the spatial 1D analysis. To test this hypothesis, a TM analysis was carried out only considering the 1-min-long time series of the 24 September 2012 event. Successive minutes starting every 15 s (moving window) were considered during the event and the scaling was quantified in the range 2–33 s, which would correspond to the regime observed in the vertical column. It appears that some minutes exhibit good scaling (with r2 greater than 0.98) and others do not (with r2 smaller than 0.90), without any direct link with the corresponding rainfall intensities. This confirms the bad scaling observed in this range of scales with longer time series and invalidates the hypothesis that it could be because the whole event was taken into account in the previous analysis. It means that more investigations on the 3D and 1D structure of rainfall are needed to properly link the results between the two types of analyses, which are furthermore affected by the bias associated with the coarse ballistic assumption in this paper.

5. Raindrop accumulation analysis

a. 2D raindrop accumulation for a given number of consecutive drops

The aim of this section is to study the spatial distribution of the accumulation of a given number of consecutive drops at ground level (more precisely at the height of the 2DVD) over the sampling area of the device (11 × 11 cm2). A 634 × 634 matrix corresponding to pixels seen by the 2DVD is created. To compute the pixel by pixel accumulation, the volume of a drop passing through the 2DVD is evenly distributed between all the pixels that are partially or totally obscured (information provided by the 2DVD), hence the unrealistic square shape of drops visible on the related figures. The drop accumulation is obtained by summing the contribution of N consecutive drops to yield either a 2D rain accumulation map or a 2D drop occurrence map. Various N values (from 50 to 2000) were tested and yielded similar results. Finally, a 512 × 512 pixel portion (a 9 × 9 cm2 area in the middle of the domain that enables us to remove potential bias due to instrumental side effects) is extracted to carry out fractal and multifractal analysis. Figure 7 displays an example of a raindrop accumulation of 150 consecutive drops during the 23 October 2013 event. As in section 3, each (natural) drop accumulation map is compared with a homogenized version, that is, a synthetic drop accumulation map. This synthetic map is obtained by keeping the same drops with their size and volume, but randomly homogeneously distributing their position over the sampling area with the help of a uniform law. A natural drop accumulation and one of its homogeneous versions are displayed in Fig. 7. Finally, the duration needed to record N consecutive drops is also retrieved and will be analyzed.

Fig. 7.
Fig. 7.

(top) Natural drop accumulation: (left) rain accumulation map (mm × 10−3) and (right) occurrence map for 150 consecutive drops during the 23 Oct 2013 event. (bottom) As in (top), but for a homogenized version of the drop accumulation (see text for details).

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

b. Fractal and multifractal analysis

The box-counting technique [i.e., Eq. (1) in log–log plot] was applied to drop occurrence accumulation maps to determine their fractal dimension. It is first applied to the drop centers, and then to the pixels occluded by drops. Both are displayed in Figs. 8a and 8b for the 23 October 2013 event with 150 drops per map. The numbers of selected pixels N(λ) were averaged over the ensemble of 1179 pictures recorded on a total duration of 12 h for each λ [Eq. (1)]. Very similar results are observed for the other four events and therefore not shown here. Figure 8 displays a plateau for scales smaller than approximately 12 mm (λ ≥ 8), because the resolution of pixels is too high with respect to the number of drop centers. For larger scales (23–92 mm, 4 ≥ λ ≥ 1), the box-counting technique yields a fractal dimension 2, meaning that the drop centers are homogeneously distributed and fill the whole embedding space. The disagreement with the result of Lovejoy and Schertzer (1990), who recorded the position of 452 drops on 128 × 128 cm2 chemically treated blotting paper, discussed and finally confirmed by Gabella et al. (2001), seems to be merely due to the fact that the scale of the sampling area is too small to reach one of the inhomogeneous regimes. This is in agreement with the fact that the rainfall field exhibited a scaling structure down to 0.5 m and a homogeneous distribution for smaller scales (see section 3).

Fig. 8.
Fig. 8.

Fractal analysis [Eq. (1) in log–log plot] of the drop position for the 23 Oct 2013 event with 150 drops per drop accumulation map (ensemble analysis on the 1179 pictures recorded on a total duration of 12 h). (a) On the drop centers. (b) Considering all the pixels occluded by drops.

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

When analyzing the pixels occluded by drops and not only the distribution of drop centers, similar features are retrieved for large scales, but for small scales two fractal dimensions are obtained: 1.1 in the range 23–1 mm (4 ≤ λ ≤ 128; red line in Fig. 8b) and 1.65 for scales smaller than 0.7 mm (128 ≤ λ; green line in Fig. 8b). This disagreement with the homogeneity of the drop centers might be due to the variability of the rain volumes (due to the drops’ sizes) that occurs over smaller scales.

A TM analysis was carried out on rain accumulation maps (not shown here). The main results are that the observed scaling is very bad, and that the curves obtained by analyzing the random-position pictures are very similar. This is simply a confirmation that there is no clear scaling regime for scales smaller than 9 cm, which, as already mentioned, was expected from the results of section 3.

c. Analysis of the distribution of duration needed for each picture

In this section, the durations needed to obtain N consecutive drops through the sampling area are analyzed. Figure 9a displays the strong fluctuations of the series of these durations for the 27 October 2013 event and N = 350. The event lasted 9.3 h with a total number of series of N drops equal to 302. Figure 9b displays the exceedance probability distribution of the durations in a log–log plot, which exhibits a power-law tail (linear slope in the log–log plot). This feature is opposed to Poisson statistics that yield exponential falloff of the probability distributions. For this event the power law is visible for scales ranging from 30 s to 10 min. Larger scales are not taken into account since they correspond to interevent features rather than the intraevent ones, which are accessible with the available dataset. For a more extensive study on interevent analysis, refer to Lavergnat and Golé (1998), who also observed power-law behavior in a slightly different context since they considered much longer time series and N = 1. The exponent of the power law is found roughly equal to 0.8–0.9 for N = 350 and similar values are found for other N values with a slight tendency to decrease with increasing N (N values ranging from 50 to 500, with steps of 50, were tested). The same power-law behavior is also observed for the 23 October 2012 event. For the 26 October 2013 and 24 September 2012 events, the range of available durations does not enable us to see the power law (especially true for the 24 September 2012 event). For the 20 October 2012 event, the range of available duration is similar to the one for the 27 and 23 October 2013 events, but the scaling is not very good, as was already noticed while analyzing the 1-ms time series.

Fig. 9.
Fig. 9.

(a) Duration needed to record pictures with 350 drops during the 27 Oct 2013 event. (b) The exceedance probability of the distribution of the durations in a log–log plot (Δt and x are in seconds).

Citation: Journal of Hydrometeorology 16, 2; 10.1175/JHM-D-14-0127.1

6. Conclusions

In this paper, 2DVD data of four rainfall events were analyzed to grasp some insight into the 3D and 1D structure of the rainfall field at the drop scale. First, based on a ballistic assumption, vertical 36-m-high rainfall columns above the measuring device are reconstructed. It appears that during the most intense portions of the events, a good scaling behavior is retrieved on scales ranging from 0.5 to 36 m. This inner scale seems to depend on the drop-size distribution, and larger-scale data are needed to investigate this dependency in greater depth. These observations are incompatible with a homogeneous distribution of drops. On the other hand, for smaller scales the observations are in agreement with the hypothesis of a homogeneous distribution of drops. Results show that integrated values (such as the number of drops or LWC) are well represented by UM simulations for the heaviest rainfall period on scales ranging from 0.5 to 36 m. An interesting future perspective, which would require investigations that are beyond the scope of this paper, would be to actually generate a rainfall simulator at the drop scale that enables us to reproduce observations. This simulator would need to be capable of integrating breakup and coalescence as well as small-scale turbulence wind effect in order to overcome the limitations of the ballistic assumption used here. Results show that using a multifractal framework would be relevant to design this rainfall simulator.

Second, 1-ms rain-rate time series are analyzed. With such a resolution it is possible to actually grasp the underlying assumption of the multifractal framework that rainfall is extremely concentrated on small portions of time or space. The mathematical interpretation is that rain accumulation is actually a measure that is singular with respect to the usual space–time measures (Lebesgue measures). Two scaling regimes are confirmed with a transition in between. The first one for very small scales (from 1 ms to a few tens of milliseconds) is actually fractal and corresponds to rather individual drops. The large-scale multifractal regime (from a few minutes to a few tens of minutes) corresponds to a kind of collective regime of drops, and UM parameter values depend on the event. We found that multifractality index α ranges between 1 and 2 and the codimension of the mean intermittency C1 ranges between 0.2 and 0.5.

Third, N consecutive drop accumulation maps observed by the measuring device are analyzed. Fractal spatial analyses were carried out on the actual drop accumulation, as well as homogenized versions. Both yield very similar results and do not exhibit scaling features. This is due to the fact that the size of the sampling area (11 × 11 cm2) is smaller than the inner scale of the scaling regime identified with the analysis of the vertical columns. Nevertheless, the distribution of the durations needed to record N drops exhibits a power-law tail, which invalidates the usual hypothesis of a Poisson distribution.

The spatiotemporal structure of the rainfall field is investigated through the analysis of 1D (vertical column, time series) or 2D (drop accumulations) cuts of the spatiotemporal field. Further investigations are needed to establish a rigorous link between the full underlying process and these cuts. This would enable us to improve our representation of the full 3D and 1D structures of the field with all its drops, which is the ultimate goal. We showed that during the heaviest rainfall period the commonly used Poisson hypothesis is not valid and results suggest that a framework relying on multifractals may help overcome some of the current discrepancies. Finally, it should be mentioned that this work has some strong consequences for instance on the remote sensing of rainfall with weather radar. Indeed, radar rainfall retrieval algorithms usually assume that drops are homogeneously distributed within the scanned volume, which is not necessarily the case, especially during the most intense portions of rainfall events. This issue of nonuniform beam filling is experienced by radar analysts. Given the scale gap between the volume scanned by radar and the studied volume of the vertical column (at least a ratio of 106 between the volumes), it is difficult to be more affirmative, but it converges with studies of the speckle effect, due to coherent backscattering of inhomogeneously distributed drops, which underpinned strong biases in the rainfall rate estimates. Some theoretical investigations of this effect with the help of multifractals have already been carried out (Lovejoy et al. 1996; Schertzer et al. 2012), and further empirical analysis with the reconstructed columns could help to better quantify the actual consequences of this effect. The variability observed between consecutive snapshots of vertical columns (done every second in this paper) also suggests that the sampling uncertainty of radar data due to limited revisit time in the scanning strategies should be investigated more in depth. These two issues will be analyzed in future work.

Acknowledgments

The authors greatly acknowledge partial financial support from the Chair “Hydrology for Resilient Cities” (sponsored by Veolia) of École des Ponts ParisTech, EU NEW-INTERREG IV RainGain Project (www.raingain.eu), and EU Climate KIC Blue Green Dream project (www.bgd.org.uk). Tim Raupach is also acknowledged for having selected the studied events and for providing Fig. 1.

REFERENCES

  • Anselmet, F., Antonia R. A. , and Danaila L. , 2001: Turbulent flows and intermittency in laboratory experiments. Planet. Space Sci., 49, 11771191, doi:10.1016/S0032-0633(01)00059-9.

    • Search Google Scholar
    • Export Citation
  • Battaglia, A., Rustemeier E. , Tokay A. , Blahak U. , and Simmer C. , 2010: PARSIVEL snow observations: A critical assessment. J. Atmos. Oceanic Technol., 27, 333344, doi:10.1175/2009JTECHA1332.1.

    • Search Google Scholar
    • Export Citation
  • Cao, Q., and Coauthors, 2008: Analysis of video disdrometer and polarimetric radar data to characterize rain microphysics in Oklahoma. J. Appl. Meteor. Climatol., 47, 22382255, doi:10.1175/2008JAMC1732.1.

    • Search Google Scholar
    • Export Citation
  • Cotton, W. R., and Gokhale N. R. , 1967: Collision, coalescence, and breakup of large water drops in a vertical wind tunnel. J. Geophys. Res., 72, 40414049, doi:10.1029/JZ072i016p04041.

    • Search Google Scholar
    • Export Citation
  • de Lima, M. I. P., and Grasman J. , 1999: Multifractal analysis of 15-min and daily rainfall from a semi-arid region in Portugal. J. Hydrol., 220, 111, doi:10.1016/S0022-1694(99)00053-0.

    • Search Google Scholar
    • Export Citation
  • de Lima, M. I. P., and de Lima J. , 2009: Investigating the multifractality of point precipitation in the Madeira archipelago. Nonlinear Processes Geophys., 16, 299311, doi:10.5194/npg-16-299-2009.

    • Search Google Scholar
    • Export Citation
  • de Montera, L., Barthes L. , Mallet C. , and Gole P. , 2009: The effect of rain–no rain intermittency on the estimation of the universal multifractals model parameters. J. Hydrometeor., 10, 493506, doi:10.1175/2008JHM1040.1.

    • Search Google Scholar
    • Export Citation
  • Desaulniers-Soucy, N., Lovejoy S. , and Schertzer D. , 2001: The continuum limit in rain and the HYDROP experiment. Atmos. Res., 59–60, 163197, doi:10.1016/S0169-8095(01)00115-6.

    • Search Google Scholar
    • Export Citation
  • Ducrocq, V., and Coauthors, 2014: HyMeX-SOP1: The field campaign dedicated to heavy precipitation and flash flooding in the northwestern Mediterranean. Bull. Amer. Meteor. Soc., 95, 1083–1100, doi:10.1175/BAMS-D-12-00244.1.

    • Search Google Scholar
    • Export Citation
  • Fitton, G., Tchiguirinskaia I. , Schertzer D. , and Lovejoy S. , 2011: Scaling of turbulence in the atmospheric surface-layer: Which anisotropy? J. Phys. Conf. Ser., 318, 072008, doi:10.1088/1742-6596/318/7/072008.

    • Search Google Scholar
    • Export Citation
  • Fraedrich, K., and Larnder C. , 1993: Scaling regimes of composite rainfall time series. Tellus,45A, 289–298, doi:10.1034/j.1600-0870.1993.t01-3-00004.x.

  • Frasson, R. P. M., da Cunha L. K. , and Krajewski W. F. , 2011: Assessment of the Thies optical disdrometer performance. Atmos. Res., 101, 237255, doi:10.1016/j.atmosres.2011.02.014.

    • Search Google Scholar
    • Export Citation
  • Gabella, M., Pavone S. , and Perona G. , 2001: Errors in the estimate of the fractal correlation dimension of raindrop spatial distribution. J. Appl. Meteor., 40, 664668, doi:10.1175/1520-0450(2001)040<0664:EITEOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Tchiguirinskaia I. , Schertzer D. , and Lovejoy S. , 2011: Analyses multifractales et spatio-temporelles des précipitations du modèle Méso-NH et des données radar. Hydrol. Sci. J., 56, 380396, doi:10.1080/02626667.2011.564174.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Onof C. , Maksimovic C. , Schertzer D. , Tchiguirinskaia I. , and Simoes N. , 2012: Quantifying the impact of small scale unmeasured rainfall variability on urban runoff through multifractal downscaling: A case study. J. Hydrol., 442-443, 117128, doi:10.1016/j.jhydrol.2012.04.005.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Tchiguirinskaia I. , Schertzer D. , and Lovejoy S. , 2013: Development and analysis of a simple model to represent the zero rainfall in a universal multifractal framework. Nonlinear Processes Geophys., 20, 343356, doi:10.5194/npg-20-343-2013.

    • Search Google Scholar
    • Export Citation
  • Gires, A., Tchiguirinskaia I. , Schertzer D. , Schellart A. , Berne A. , and Lovejoy S. , 2014: Influence of small scale rainfall variability on standard comparison tools between radar and rain gauge data. Atmos. Res., 138, 125138, doi:10.1016/j.atmosres.2013.11.008.

    • Search Google Scholar
    • Export Citation
  • Hubert, P., and Coauthors, 1993: Multifractals and extreme rainfall events. Geophys. Res. Lett., 20, 931934, doi:10.1029/93GL01245.

  • Jaffrain, J., and Berne A. , 2011: Experimental quantification of the sampling uncertainty associated with measurements from PARSIVEL disdrometers. J. Hydrometeor., 12, 352370, doi:10.1175/2010JHM1244.1.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., and Kostinski A. B. , 1998: Fluctuation properties of precipitation. Part II: Reconsideration of the meaning and measurement of raindrop size distributions. J. Atmos. Sci., 55, 283294, doi:10.1175/1520-0469(1998)055<0283:FPOPPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., Kostinski A. B. , and Kruger A. , 1999: Fluctuation properties of precipitation. Part IV: Finescale clustering of drops in variable rain. J. Atmos. Sci., 56, 8291, doi:10.1175/1520-0469(1999)056<0082:FPOPPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Joss, J., and Waldvogel A. , 1967: Ein Spektrograph für Niederschlagstropfen mit automatischer Auswertung (A spectrograph for raindrops with automatic interpretation). Pure Appl. Geophys., 68, 240246, doi:10.1007/BF00874898.

    • Search Google Scholar
    • Export Citation
  • Kostinski, A. B., and Jameson A. R. , 1997: Fluctuation properties of precipitation. Part I: On deviations of single-size drop counts from the Poisson distribution. J. Atmos. Sci., 54, 21742186, doi:10.1175/1520-0469(1997)054<2174:FPOPPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Krajewski, W. F., and Coauthors, 2006: DEVEX-disdrometer evaluation experiment: Basic results and implications for hydrologic studies. Adv. Water Resour., 29, 311325, doi:10.1016/j.advwatres.2005.03.018.

    • Search Google Scholar
    • Export Citation
  • Kruger, A., and Krajewski W. F. , 2002: Two-dimensional video disdrometer: A description. J. Atmos. Oceanic Technol., 19, 602617, doi:10.1175/1520-0426(2002)019<0602:TDVDAD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ladoy, P., Schmitt F. , Schertzer D. , and Lovejoy S. , 1993: The multifractal temporal variability of Nimes rainfall data. C. R. Acad. Sci. Ser. II, 317 (6), 775782.

    • Search Google Scholar
    • Export Citation
  • Lavergnat, J., and Golé P. , 1998: A stochastic raindrop time distribution model. J. Appl. Meteor., 37, 805818, doi:10.1175/1520-0450(1998)037<0805:ASRTDM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lazarev, A., Schertzer D. , Lovejoy S. , and Chigirinskaya Y. , 1994: Unified multifractal atmospheric dynamics tested in the tropics: Part II, vertical scaling and generalized scale invariance. Nonlinear Processes Geophys., 1, 115123, doi:10.5194/npg-1-115-1994.

    • Search Google Scholar
    • Export Citation
  • Lilley, M., Lovejoy S. , Desaulniers-Soucy N. , and Schertzer D. , 2006: Multifractal large number of drops limit in rain. J. Hydrol., 328, 2037, doi:10.1016/j.jhydrol.2005.11.063.

    • Search Google Scholar
    • Export Citation
  • Lovejoy, S., and Schertzer D. , 1990: Fractals, raindrops and resolution dependence of rain measurements. J. Appl. Meteor., 29, 11671170, doi:10.1175/1520-0450(1990)029<1167:FRARDO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lovejoy, S., and Schertzer D. , 1995: Multifractals and rain. New Uncertainty Concepts in Hydrology and Hydrological Modelling, A.W. Kundzewicz, Ed., Cambridge Press, 62–103.

  • Lovejoy, S., and Schertzer D. , 2008: Turbulence, rain drops and the l1/2 number density law. New J. Phys.,10, 075017, doi:10.1088/1367-2630/10/7/075017.

  • Lovejoy, S., Duncan M. R. , and Schertzer D. , 1996: Scalar multifractal radar observer’s problem. J. Geophys. Res., 101, 26 47926 492, doi:10.1029/96JD02208.

    • Search Google Scholar
    • Export Citation
  • Lovejoy, S., Schertzer D. , and Allaire V. , 2008: The remarkable wide range spatial scaling of TRMM precipitation. Atmos. Res., 90, 1032, doi:10.1016/j.atmosres.2008.02.016.

    • Search Google Scholar
    • Export Citation
  • Mandapaka, P. V., Lewandowski P. , Eichinger W. E. , and Krajewski W. F. , 2009: Multiscaling analysis of high resolution space–time lidar-rainfall. Nonlinear Processes Geophys., 16, 579586, doi:10.5194/npg-16-579-2009.

    • Search Google Scholar
    • Export Citation
  • Nykanen, D. K., and Harris D. , 2003: Orographic influences on the multiscale statistical properties of precipitation. J. Geophys. Res., 108, 8381, doi:10.1029/2001JD001518.

    • Search Google Scholar
    • Export Citation
  • Olsson, J., 1995: Limits and characteristics of the multifractal behavior of a high-resolution rainfall time series. Nonlinear Processes Geophys., 2, 2329, doi:10.5194/npg-2-23-1995.

    • Search Google Scholar
    • Export Citation
  • Royer, J.-F., Biaou A. , Chauvin F. , Schertzer D. , and Lovejoy S. , 2008: Multifractal analysis of the evolution of simulated precipitation over France in a climate scenario. C. R. Geosci., 340, 431440, doi:10.1016/j.crte.2008.05.002.

    • Search Google Scholar
    • Export Citation
  • Salles, C., Creutin J. D. , and Sempere-Torres D. , 1998: The optical spectropluviometer revisited. J. Atmos. Oceanic Technol., 15, 12151222, doi:10.1175/1520-0426(1998)015<1215:TOSR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., and Lovejoy S. , 1987: Physical modelling and analysis of rain and clouds by anisotropic scaling and multiplicative processes. J. Geophys. Res., 92, 96939714, doi:10.1029/JD092iD08p09693.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., and Lovejoy S. , 1997: Universal multifractals do exist!: Comments on “A Statistical analysis of mesoscale rainfall as a random cascade.” J. Appl. Meteor., 36, 12961303, doi:10.1175/1520-0450(1997)036<1296:UMDECO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., and Lovejoy S. , 2011: Multifractals, generalized scale invariance and complexity in geophysics. Int. J. Bifurcation Chaos, 21, 34173456, doi:10.1142/S0218127411030647.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., Tchiguirinskaia I. , Lovejoy S. , and Hubert P. , 2010: No monsters, no miracles: In nonlinear sciences hydrology is not an outlier! Hydrol. Sci. J., 55, 965979, doi:10.1080/02626667.2010.505173.

    • Search Google Scholar
    • Export Citation
  • Schertzer, D., Tchiguirinskaia I. , and Lovejoy S. , 2012: Getting higher resolution rainfall estimates: X-band radar technology and multifractal drop distribution. IAHS Publ.,351, 105–110.

  • Tchiguirinskaia, I., Salles C. , Hubert P. , Schertzer D. , Lovejoy S. , Creutin J. D. , and Bendjoudi H. , 2003: Multifractal analysis of the OSP measured rain rates over time scales from millisecond to day. IUGG 2003, Sapporo, Japan, IUGG, JSM18/01A/B20-003.

  • Tessier, Y., Lovejoy S. , and Schertzer D. , 1993: Universal multifractals: Theory and observations for rain and clouds. J. Appl. Meteor., 32, 223250, doi:10.1175/1520-0450(1993)032<0223:UMTAOF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tessier, Y., Lovejoy S. , Hubert P. , Schertzer D. , and Pecknold S. , 1996: Multifractal analysis and modeling of rainfall and river flows and scaling, causal transfer functions. J. Geophys. Res., 101, 26 42726 440, doi:10.1029/96JD01799.

    • Search Google Scholar
    • Export Citation
  • Thurai, M., and Bringi V. N. , 2005: Drop axis ratios from a 2D video disdrometer. J. Atmos. Oceanic Technol., 22, 966978, doi:10.1175/JTECH1767.1.

    • Search Google Scholar
    • Export Citation
  • Thurai, M., Peterson W. A. , Tokay A. , Schutz C. , and Gatlin P. , 2011: Drop size distribution comparisons between Parsivel and 2-D video disdrometers. Adv. Geosci., 30, 39, doi:10.5194/adgeo-30-3-2011.

    • Search Google Scholar
    • Export Citation
  • Tokay, A., Petersen W. A. , Gatlin P. , and Wingo M. , 2013: Comparison of raindrop size distribution measurements by collocated disdrometers. J. Atmos. Oceanic Technol., 30, 16721690, doi:10.1175/JTECH-D-12-00163.1.

    • Search Google Scholar
    • Export Citation
  • Uijlenhoet, R., Stricker J. N. M. , Torfs P. J. J. F. , and Creutin J. D. , 1999: Towards a stochastic model of rainfall for radar hydrology: Testing the Poisson homogeneity hypothesis. Phys. Chem. Earth, Part B, 24, 747755, doi:10.1016/S1464-1909(99)00076-3.

    • Search Google Scholar
    • Export Citation
  • Verrier, S., de Montera L. , Barthes L. , and Mallet C. , 2010: Multifractal analysis of African monsoon rain fields, taking into account the zero rain-rate problem. J. Hydrol., 389, 111120, doi:10.1016/j.jhydrol.2010.05.035.

    • Search Google Scholar
    • Export Citation
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