• Bellon, A., , and Zawadzki I. , 1994: Forecasting of hourly accumulations of precipitation by optimal extrapolation of radar maps. J. Hydrol., 157, 211233, doi:10.1016/0022-1694(94)90106-6.

    • Search Google Scholar
    • Export Citation
  • Berenguer, M., , Sempere-Torres D. , , and Pegram G. G. S. , 2011: SBMcast: An ensemble nowcasting technique to assess the uncertainty in rainfall forecasts by Lagrangian extrapolation. J. Hydrol., 404, 226240, doi:10.1016/j.jhydrol.2011.04.033.

    • Search Google Scholar
    • Export Citation
  • Berne, A., , Delrieu G. , , and Boudevillain B. , 2009: Variability of the spatial structure of intense Mediterranean precipitation. Adv. Water Resour., 32, 10311042, doi:10.1016/j.advwatres.2008.11.008.

    • Search Google Scholar
    • Export Citation
  • Bowler, N. E., , Pierce C. E. , , and Seed A. W. , 2006: STEPS: A probabilistic precipitation forecasting scheme which merges an extrapolation nowcast with downscaled NWP. Quart. J. Roy. Meteor. Soc., 132, 21272155, doi:10.1256/qj.04.100.

    • Search Google Scholar
    • Export Citation
  • Cloke, H. L., , and Pappenberger F. , 2009: Ensemble flood forecasting: A review. J. Hydrol., 375, 613626, doi:10.1016/j.jhydrol.2009.06.005.

    • Search Google Scholar
    • Export Citation
  • Collier, C. G., 2007: Flash flood forecasting: What are the limits of predictability? Quart. J. Roy. Meteor. Soc., 133, 323, doi:10.1002/qj.29.

    • Search Google Scholar
    • Export Citation
  • Cuo, L., , Pagano T. C. , , and Wang Q. J. , 2011: A review of quantitative precipitation forecasts and their use in short- to medium-range streamflow forecasting. J. Hydrometeor., 12, 713728, doi:10.1175/2011JHM1347.1.

    • Search Google Scholar
    • Export Citation
  • Emmanuel, I., , Andrieu H. , , Leblois E. , , and Flahaut B. , 2012: Temporal and spatial variability of rainfall at the urban hydrological scale. J. Hydrol., 430–431, 162172, doi:10.1016/j.jhydrol.2012.02.013.

    • Search Google Scholar
    • Export Citation
  • Foresti, L., , Panziera L. , , Mandapaka P. V. , , Germann U. , , and Seed A. , 2015: Retrieval of analogue radar images for ensemble nowcasting of orographic rainfall. Meteor. Appl., 22, 141155, doi:10.1002/met.1416.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , and Zawadzki I. , 2002: Scale-dependence of the predictability of precipitation from continental radar images. Part I: Description of the methodology. Mon. Wea. Rev., 130, 28592873, doi:10.1175/1520-0493(2002)130<2859:SDOTPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , and Zawadzki I. , 2004: Scale dependence of the predictability of precipitation from continental radar images. Part II: Probability forecasts. J. Appl. Meteor., 43, 7489, doi:10.1175/1520-0450(2004)043<0074:SDOTPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , Galli G. , , Boscacci M. , , and Bolliger M. , 2006a: Radar precipitation measurement in a mountainous region. Quart. J. Roy. Meteor. Soc., 132, 16691692, doi:10.1256/qj.05.190.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , Zawadzki I. , , and Turner B. , 2006b: Predictability of precipitation from continental radar images. Part IV: Limits to prediction. J. Atmos. Sci., 63, 20922108, doi:10.1175/JAS3735.1.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , Berenguer M. , , Sempere-Torres D. , , and Zappa M. , 2009: REAL—Ensemble radar precipitation estimation for hydrology in a mountainous region. Quart. J. Roy. Meteor. Soc., 135, 445456, doi:10.1002/qj.375.

    • Search Google Scholar
    • Export Citation
  • Grecu, M., , and Krajewski W. F. , 2000: A large-sample investigation of statistical procedures for radar-based short-term quantitative precipitation forecasting. J. Hydrol., 239, 6984, doi:10.1016/S0022-1694(00)00360-7.

    • Search Google Scholar
    • Export Citation
  • Hou, A. Y., and et al. , 2014: The Global Precipitation Measurement Mission. Bull. Amer. Meteor. Soc., 95, 701722, doi:10.1175/BAMS-D-13-00164.1.

    • Search Google Scholar
    • Export Citation
  • Journel, A. G., 1980: The lognormal approach to predicting local distributions of selective mining unit grades. Math. Geol., 12, 285303, doi:10.1007/BF01029417.

    • Search Google Scholar
    • Export Citation
  • Krajewski, W. F., , and Smith J. A. , 2002: Radar hydrology: Rainfall estimation. Adv. Water Resour., 25, 13871394, doi:10.1016/S0309-1708(02)00062-3.

    • Search Google Scholar
    • Export Citation
  • Lepioufle, J.-M., , Leblois E. , , and Creutin J.-D. , 2012: Variography of rainfall accumulation in presence of advection. J. Hydrol., 464–465, 494504, doi:10.1016/j.jhydrol.2012.07.041.

    • Search Google Scholar
    • Export Citation
  • Lewis, H., and et al. , 2015: From months to minutes—Exploring the value of high-resolution rainfall observation and prediction during the UK winter storms of 2013/2014. Meteor. Appl., 22, 90104, doi:10.1002/met.1493.

    • Search Google Scholar
    • Export Citation
  • Marsan, D., , Schertzer D. , , and Lovejoy S. , 1996: Causal space–time multifractal processes: Predictability and forecasting of rain fields. J. Geophys. Res., 101, 26 33326 346, doi:10.1029/96JD01840.

    • Search Google Scholar
    • Export Citation
  • Mascaro, G., , Vivoni E. R. , , and Deidda R. , 2010: Implications of ensemble quantitative precipitation forecast errors on distributed streamflow forecasting. J. Hydrometeor., 11, 6986, doi:10.1175/2009JHM1144.1.

    • Search Google Scholar
    • Export Citation
  • Onof, C., , Northrop P. , , Wheater H. S. , , and Isham V. , 1996: Spatiotemporal storm structure and scaling property analysis for modeling. J. Geophys. Res., 101, 26 41526 425, doi:10.1029/96JD01378.

    • Search Google Scholar
    • Export Citation
  • Panziera, L., , Germann U. , , Gabella M. , , and Mandapaka P. V. , 2011: NORA–Nowcasting of Orographic Rainfall by means of Analogues. Quart. J. Roy. Meteor. Soc., 137, 21062123, doi:10.1002/qj.878.

    • Search Google Scholar
    • Export Citation
  • Panziera, L., , James C. N. , , and Germann U. , 2015: Mesoscale organization and structure of orographic precipitation producing flash floods in the Lago Maggiore region. Quart. J. Roy. Meteor. Soc., 141, 224248, doi:10.1002/qj.2351.

    • Search Google Scholar
    • Export Citation
  • Paschalis, A., , Molnar P. , , Fatichi S. , , and Burlando P. , 2013: A stochastic model for high-resolution space–time precipitation simulation. Water Resour. Res., 49, 84008417, doi:10.1002/2013WR014437.

    • Search Google Scholar
    • Export Citation
  • Paschalis, A., , Fatichi S. , , Molnar P. , , Rimkus S. , , and Burlando P. , 2014: On the effects of small scale space–time variability of rainfall on basin flood response. J. Hydrol., 514, 313327, doi:10.1016/j.jhydrol.2014.04.014.

    • Search Google Scholar
    • Export Citation
  • Potvin, G., 1993: Space–time correlations and Taylor’s hypothesis for rainfall. M.S. thesis, Dept. of Physics, McGill University, 68 pp.

  • Radhakrishna, B., , Zawadzki I. , , and Fabry F. , 2012: Predictability of precipitation from continental radar images. Part V: Growth and decay. J. Atmos. Sci., 69, 33363349, doi:10.1175/JAS-D-12-029.1.

    • Search Google Scholar
    • Export Citation
  • Schleiss, M., , and Smith J. A. , 2015: A method to estimate the 3D–time structure of the raindrop size distribution using radar and disdrometer data. J. Hydrometeor., 16, 12221242, doi:10.1175/JHM-D-14-0182.1.

    • Search Google Scholar
    • Export Citation
  • Schleiss, M., , Chamoun S. , , and Berne A. , 2014: Stochastic simulation of intermittent rainfall using the concept of dry drift. Water Resour. Res., 50, 23292349, doi:10.1002/2013WR014641.

    • Search Google Scholar
    • Export Citation
  • Seed, A. W., 2003: A dynamic and spatial scaling approach to advection forecasting. J. Appl. Meteor., 42, 381388, doi:10.1175/1520-0450(2003)042<0381:ADASSA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Taylor, G. I., 1938: The spectrum of turbulence. Proc. Roy. Soc. London, A164, 476490, doi:10.1098/rspa.1938.0032.

  • Thorndahl, S., , Smith J. A. , , Baeck M. L. , , and Krajewski W. F. , 2014: Analyses of the temporal and spatial structures of heavy rainfall from a catalog of high-resolution radar rainfall fields. Atmos. Res., 144, 111125, doi:10.1016/j.atmosres.2014.03.013.

    • Search Google Scholar
    • Export Citation
  • Turner, B. J., , Zawadzki I. , , and Germann U. , 2004: Predictability of precipitation from continental radar images. Part III: Operational nowcasting implementation (MAPLE). J. Appl. Meteor., 43, 231248, doi:10.1175/1520-0450(2004)043<0231:POPFCR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tuttle, J. D., , and Foote G. B. , 1990: Determination of the boundary layer airflow from a single Doppler radar. J. Atmos. Oceanic Technol., 7, 218232, doi:10.1175/1520-0426(1990)007<0218:DOTBLA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Zawadzki, I., , Morneau J. , , and Laprise R. , 1994: Predictability of precipitation patterns: An operational approach. J. Appl. Meteor., 33, 15621571, doi:10.1175/1520-0450(1994)033<1562:POPPAO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Snapshots of instantaneous radar reflectivity (dBZ) at the ground level for dataset 1 (strongly convective events).

  • View in gallery

    Snapshots of instantaneous radar reflectivity (dBZ) at the ground level for dataset 2 (mostly stratiform events).

  • View in gallery

    Snapshots of instantaneous radar reflectivity (dBZ) for event 1 at 0100, 0105, 0115, and 0130 UTC 9 Aug 2014. The pixel resolution is 0.005°.

  • View in gallery

    Space–time variogram maps (dBZ2) corresponding to the reflectivity fields in Fig. 3. The time lags are 0, +5, +15, and +30 min with respect to 0100 UTC. The plus signs indicate the position of the lowest semivariance value at each time lag. The minimum semivariance values are 0, 7.52, 18.44, and 37.26 dBZ2. The class width used to compute the variogram maps is 0.01°.

  • View in gallery

    Temporal innovation for event 1 (Fig. 3) and all time lags between 0 and +180 min with respect to 0100 UTC.

  • View in gallery

    Empirical semivariance values (dBZ2) for pure spatial displacements at t = 0100 UTC and event 1 (Fig. 3). The black circles represent the predictable scales at τ = 5, 15, 45, and 180 min. The conversion from latitude–longitude to Euclidean space yields predictable scales of approximately 2.23 km at 5 min, 6.00 km at 15 min, 14.26 km at 45 min, and 24.03 km at 180 min.

  • View in gallery

    Temporal innovation vs time lag τ (min) for all events in dataset 1 and dataset 2. The represented quantiles are 10%, 25%, 50%, 75%, and 90%. The red lines represent the fitted exponential models given in Eq. (8).

  • View in gallery

    Lag-1 autocorrelation of for all 25 events in dataset 1 and dataset 2. The represented quantiles are 10%, 25%, 50%, 75%, and 90%.

  • View in gallery

    Predictable scales (km) vs time lag τ for all events in dataset 1 and dataset 2. The represented quantiles are 10%, 25%, 50%, 75%, and 90%. The red line represents the fitted exponential model in Eq. (9).

  • View in gallery

    Time series of temporal innovation at time lag τ = 5 min, mean reflectivity (dBZ), storm size (× 103 km2), and spatial correlation range (km) for event 1.

  • View in gallery

    Average rank correlation of temporal innovation with mean reflectivity (black dots), storm size (red triangles), and spatial correlation (blue crosses) for all events in dataset 1 and dataset 2.

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A Geostatistical Framework for Quantifying the Temporal Evolution and Predictability of Rainfall Fields

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  • 1 Civil and Environmental Engineering, Princeton University, Princeton, New Jersey
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Abstract

A geostatistical framework for quantifying the temporal evolution and predictability of rainfall fields for time lags between 5 min and 3 h is proposed. The method is based on the computation of experimental space–time variogram maps of radar reflectivity fields. Two new metrics for quantifying temporal innovation and predictability based on minimum semivariance values at different time lags are proposed. The method is applied to high-resolution composite radar reflectivity maps over the United States to study the evolution of 25 convective and 25 stratiform events during the warm season of 2014. Results show that the temporal innovation can be modeled as the sum of two exponential functions of time lag, with approximately 50% of the total innovation occurring over the first 60 min. The median predictable scales for convective events are on the order of 1.6 km at 5 min, 5 km at 15 min, and 12.7 km at 1 h. Furthermore, the optimal time lag for predicting future innovation, taking into account measurement uncertainty and natural variability, appears to be between 30 and 60 min.

Corresponding author address: Marc Schleiss, Civil and Environmental Engineering, Princeton University, E-Quad E316, Olden Street, Princeton, NJ 08540. E-mail: schleiss.marc@gmail.com

Abstract

A geostatistical framework for quantifying the temporal evolution and predictability of rainfall fields for time lags between 5 min and 3 h is proposed. The method is based on the computation of experimental space–time variogram maps of radar reflectivity fields. Two new metrics for quantifying temporal innovation and predictability based on minimum semivariance values at different time lags are proposed. The method is applied to high-resolution composite radar reflectivity maps over the United States to study the evolution of 25 convective and 25 stratiform events during the warm season of 2014. Results show that the temporal innovation can be modeled as the sum of two exponential functions of time lag, with approximately 50% of the total innovation occurring over the first 60 min. The median predictable scales for convective events are on the order of 1.6 km at 5 min, 5 km at 15 min, and 12.7 km at 1 h. Furthermore, the optimal time lag for predicting future innovation, taking into account measurement uncertainty and natural variability, appears to be between 30 and 60 min.

Corresponding author address: Marc Schleiss, Civil and Environmental Engineering, Princeton University, E-Quad E316, Olden Street, Princeton, NJ 08540. E-mail: schleiss.marc@gmail.com

1. Introduction

Rainfall is a highly complex and variable process. Understanding its spatial and temporal variability is crucial for many applications like flood forecasting and mitigation. Increasingly, flood risks are estimated based on large ensemble precipitation estimates rather than deterministic forecasts (Bowler et al. 2006; Cloke and Pappenberger 2009; Germann et al. 2009; Mascaro et al. 2010; Cuo et al. 2011; Paschalis et al. 2014). To choose or generate the most representative ensembles, it is essential to know how fast convective systems evolve in time. Thanks to the increasing resolution, coverage, and accuracy of weather radars, the rate at which precipitation changes in space and time can now be studied in great detail (e.g., Krajewski and Smith 2002; Germann et al. 2006a; Collier 2007; Berne et al. 2009; Emmanuel et al. 2012; Thorndahl et al. 2014; Panziera et al. 2015). Considerable effort has been devoted to understanding and reducing the uncertainties affecting radar rainfall estimates. Nevertheless, our modeling capabilities of rainfall at hydrologically relevant scales (e.g., 1 km and 5 min) remain limited.

Since the pioneering work by Bellon and Zawadzki (1994) and Zawadzki et al. (1994), many studies have been devoted to understanding the short-term predictability of precipitation using weather radar. Working with WSR-88D data near Tulsa, Oklahoma, Grecu and Krajewski (2000) performed an extensive evaluation of three radar-based quantitative precipitation forecast techniques: Eulerian persistence, Lagrangian persistence, and a neural network. They defined predictability as the lead time for which the cross correlation between observations and forecasts falls below a certain threshold. By successively smoothing the radar images, they were able to show that large-scale features are characterized by longer Lagrangian persistence, that is, that predictability increases with decreasing resolution.

Using a large number of radar reflectivity composites at 4-km and 15-min resolution over the continental United States, Germann and Zawadzki (2002) presented the first results of a series of papers devoted to radar-based precipitation predictability at the synoptic scale (Germann and Zawadzki 2004; Turner et al. 2004; Germann et al. 2006b; Radhakrishna et al. 2012). A key issue in these papers was to understand how Lagrangian predictability varies with location, time, and scale, leading to important developments in radar-based precipitation nowcasting systems like Spectral Prognosis (S-PROG; Seed 2003), McGill Algorithm for Precipitation Nowcasting by Lagrangian Extrapolation (MAPLE; Turner et al. 2004), and Short-Term Ensemble Prediction System (STEPS; Bowler et al. 2006).

One of the most important points when trying to forecast precipitation is to know where the useful information is coming from. For short lead times, temporal innovation only plays a minor role. Most of the predictability can be attributed to the persistence of the initial spatial correlation structure (e.g., Taylor 1938; Potvin 1993; Onof et al. 1996). As the time lag increases and the system starts to evolve, initial correlation at small spatial scales quickly becomes irrelevant. Larger scales, however, continue to carry useful information (e.g., Marsan et al. 1996). The rate at which spatial scales lose their predictability as a function of time lag is still poorly documented. Zawadzki et al. (1994) found that rainfall fields that have been downscaled to a 16-km resolution remain predictable up to 120 min. Additional studies by Bellon and Zawadzki (1994), Grecu and Krajewski (2000), and Seed (2003) report minimum predictable scales on the order of 10–20 km after 60 min with large differences depending on storm type, spatial organization, and temporal variability. A limitation of currently available methods for measuring predictability as a function of scale is the necessity to apply successive filters to the radar data. This increases the computation time and has the negative effect of making the predictability metrics dependent on the chosen decomposition scheme.

The goal of this paper is to introduce a new and simple geostatistical method for quantifying the temporal evolution and predictable scales of precipitation fields. The approach complements existing structural analysis techniques and Lagrangian predictability metrics based on cross correlation, multifractals, Fourier filtering, and wavelet decompositions. Its main advantage is that the predictable scales can be determined directly from the space–time variograms, without the need to filter or decompose the original radar image. It offers a large number of potential applications, including the possibility to design new stochastic rainfall simulators and ensemble-based nowcasting schemes (e.g., Berenguer et al. 2011; Panziera et al. 2011; Paschalis et al. 2013; Schleiss et al. 2014; Foresti et al. 2015).

This paper is structured as follows. Section 2 introduces the different datasets and events used for the analysis. The metrics used to quantify the temporal innovation and predictable scales are described in section 3. In section 4, the new tools are applied to a large number of events over the central United States. The results of the analyses are separated into four parts: section 4a shows how temporal innovation varies as a function of time lag, section 4b takes a closer look at the accuracy with which future innovation can be predicted, section 4c provides new insight into the range of predictable scales as a function of time lag, and section 4d focuses on the link between temporal innovation and spatial storm properties. The summary and conclusions are given in section 5.

2. Data

Theoretically, the proposed method can be applied to any rainfall-related quantity (e.g., rain rate, liquid water content, and drop size distribution). For data quality and availability reasons, the variable considered in this paper will be the radar reflectivity Z (dBZ). This choice is motivated by the fact that reflectivity is a more “direct” measurement (from a weather radar perspective) of rainfall than rain rate that has to be retrieved by assuming a drop size distribution. The logarithmic scale is chosen to have more symmetric distributions, thus making it easier to analyze the spatial structure (Journel 1980). In case the variable of interest is the rain rate, a log transform should be considered prior to analysis. For simplicity, the analyses are restricted to two-dimensional composite maps of reflectivity, as close as possible to the ground level. Following the footsteps of Germann and Zawadzki (2002), large radar reflectivity composite images are used instead of a single radar.

a. Radar reflectivity maps

The radar reflectivity maps used in this study are provided by the Iowa Environmental Mesonet (IEM). They are based on a feed of NEXRAD Level III products from the UCAR/Unidata Internet data distribution system. This feed includes the base reflectivity (N0Q) and net echo top (NET). Every 5 min, a script runs a General Meteorology Package (GEMPAK) program called nex2img, which composites the data into a large image of size 12 200 × 5400 pixels with 0.005° spatial resolution (e.g., ~0.5 km) and 0.5 dBZ discretization. A clutter suppression algorithm based on Enhanced Echo Tops (EET) and Rapid Refresh, version 2 (RAP2), model temperatures is applied to remove the most evident anomalous propagation. The complete composite extends from 23° to 50°N and from 126° to 65°W and covers most of the continental United States. However, to avoid issues related to beam blockage, only the part east of the Rocky Mountains will be considered in this study. A threshold of 0 dBZ is used to distinguish between “dry” and “wet” pixels. Only the wet pixels are considered for the analyses.

b. Datasets

Two different datasets are selected for the analyses. The first dataset consists of 25 strongly convective events of various types, sizes, and structures that occurred at different times and places over the central and eastern United States during the month of August 2014. Figure 1 shows a snapshot of each event. Some basic information about the location, time, intensity, size, and spatial correlation of these events can be found in Table 1.

Fig. 1.
Fig. 1.

Snapshots of instantaneous radar reflectivity (dBZ) at the ground level for dataset 1 (strongly convective events).

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

Table 1.

Starting time, latitude, longitude, minimum–maximum average radar reflectivity, minimum–maximum storm size, and minimum–maximum spatial correlation range for all 25 events in dataset 1. For more details about how the spatial correlation range is defined, the reader is referred to section 4d and Eq. (10).

Table 1.

The second dataset is composed of 25 stratiform and weakly convective events of various sizes, intensities, and structures in August and September 2014 (see Fig. 2 and Table 2 for more details). For simplicity, the events in this dataset will be referred to as “mostly stratiform,” meaning that the amount and extent of convection is significantly lower than in dataset 1. The main purpose of having two different datasets is to be able to investigate how small-scale convective processes affect the short-term predictability of rainfall.

Fig. 2.
Fig. 2.

Snapshots of instantaneous radar reflectivity (dBZ) at the ground level for dataset 2 (mostly stratiform events).

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

Table 2.

As in Table 1, but for dataset 2.

Table 2.

To facilitate comparison, each of the 50 events has an equal duration of 4 h. In other words, each event consists of a sequence of 49 consecutive frames separated by regular 5-min time intervals. The starting and ending points were selected based on data availability and quality and do not necessarily correspond to the beginning and the end of the rainfall events. To ensure best possible quality, each frame was visually inspected for signs of clutter, blocking, or other major artifacts that may affect the structural analysis. The 4-h duration was chosen to capture a sufficiently large range of time lags while limiting the computation time associated with estimating a large number of space–time variograms (see section 3d for more details). The smallest considered time lag is 5 min and the maximum time lag is 3 h.

3. Methodology

This section describes the statistics used to quantify temporal innovation and predictable scales. The approach is illustrated in Figs. 36.

Fig. 3.
Fig. 3.

Snapshots of instantaneous radar reflectivity (dBZ) for event 1 at 0100, 0105, 0115, and 0130 UTC 9 Aug 2014. The pixel resolution is 0.005°.

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

Fig. 4.
Fig. 4.

Space–time variogram maps (dBZ2) corresponding to the reflectivity fields in Fig. 3. The time lags are 0, +5, +15, and +30 min with respect to 0100 UTC. The plus signs indicate the position of the lowest semivariance value at each time lag. The minimum semivariance values are 0, 7.52, 18.44, and 37.26 dBZ2. The class width used to compute the variogram maps is 0.01°.

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

Fig. 5.
Fig. 5.

Temporal innovation for event 1 (Fig. 3) and all time lags between 0 and +180 min with respect to 0100 UTC.

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

Fig. 6.
Fig. 6.

Empirical semivariance values (dBZ2) for pure spatial displacements at t = 0100 UTC and event 1 (Fig. 3). The black circles represent the predictable scales at τ = 5, 15, 45, and 180 min. The conversion from latitude–longitude to Euclidean space yields predictable scales of approximately 2.23 km at 5 min, 6.00 km at 15 min, 14.26 km at 45 min, and 24.03 km at 180 min.

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

a. Space–time variograms and time nugget

Let denote a sequence of radar reflectivity fields (where curly brackets indicate an ensemble) at times and spatial locations . The empirical semivariance between and is defined as follows:
e1
and
e2
where curly brackets indicate an ensemble, z is the radar reflectivity, is a spatial displacement, is a time lag, is the set of observations at times t and separated by (up to a tolerance level), and denotes the number of elements in the set . The set of all empirical semivariance values is called the space–time variogram of . It is a structure function characterizing the average dissimilarity in the mean square sense between two fields as a function of time t, spatial displacement , and time lag τ.
An important quantity for measuring the rate at which a field evolves in time is the minimum empirical semivariance value at a given time lag τ, hereinafter called the time nugget:
e3
Similarly to a nugget in a variogram model, characterizes the intrinsic variability between and , independently of the considered spatial displacement. The idea is similar to global correlation-based tracking algorithms (e.g., Tuttle and Foote 1990) where rain cells are tracked based on the spatial displacement that maximizes the cross correlation (i.e., minimizes the semivariance) between two consecutive fields. In this case, however, no tracking is performed and the only quantity of interest is the minimum semivariance value as a function of time lag τ.

To illustrate the notion of time nugget, a series of four radar reflectivity composites recorded at 0100, 0105, 0115, and 0130 UTC 9 August 2014 over the states of Nebraska and Iowa are considered (see Fig. 3). For more details about these radar composites and how they were obtained, the reader is referred to section 2. The empirical semivariance values corresponding to these four fields are shown in Fig. 4. To better highlight the details, the range of spatial displacements was restricted to ±0.3° in longitude and latitude, that is, about 33 km along each direction. The locations of the minimum empirical semivariance value at each time lag are indicated by black and white crosses. The colors of the crosses have no particular meaning and are only used to enhance the visibility. One can see that the time nugget is zero at τ = 0, 7.52 dBZ2 at 5 min, 18.44 dBZ2 at 15 min, and 37.26 dBZ2 at 30 min. Moreover, the spatial displacement that minimizes the semivariance increasingly differs from zero. This simultaneous shift and increase in semivariance with time lag τ is caused by the combined effect of motion, temporal variability, and measurement errors.

b. Temporal innovation factor

The main problem with the time nugget is the fact that it is a scale-dependent measure of variability. In particular, it depends on the mean and variance of , which are known to change with time. Comparing values for different events and across time therefore requires some prior normalization.

To facilitate comparisons, we introduce a normalized measure of variability called the temporal innovation factor :
e4
where denotes the sample variance of at time t:
e5
and N is the number of sample locations . The temporal innovation factor is a dimensionless measure of variability, relative to the initial spatial variance of the process. It ranges between 0 (no innovation) and +∞ (maximum innovation). A value of 0.5 means the variability due to temporal evolution along the direction of apparent motion is equal to 50% of the initial spatial variance.

Figure 5 shows the temporal innovation factors for the first time step of event 1 (see Fig. 3). We can see a strong increase in temporal innovation during the first 50 min, followed by a plateau between 50 and 100 min and another but slower increase after 100 min. The plateau at 50 min is not a general feature of temporal innovation but specific to the considered event. It can be explained by the combined effect of 1) birth, growth, decay, and death of convective cells; 2) increase/decrease in average intensity; and 3) sampling effects and measurement errors. For a more complete and detailed analysis of the temporal innovation factor for various events, time lags, and types of precipitation, the reader is referred to the main results in section 4.

c. Minimum predictable scale

The minimum predictable scale is a special metric introduced to quantify the link between spatial and temporal variabilities in Lagrangian coordinates. Its main purpose is to measure how fast the initial correlation structure of a rainfall field is lost. It also makes it possible to determine the range of spatial scales that remain predictable at time lag τ. Traditionally, predictability is defined as the lifetime after which the cross correlation between observations and forecast falls below a certain threshold (e.g., Zawadzki et al. 1994; Grecu and Krajewski 2000; Germann and Zawadzki 2002). The definition of predictable scales adopted in this paper is slightly different. It is based on the concepts of equivalent semivariance values and the idea that time can be “normalized” to make it similar, in terms of variability, to the other spatial dimensions (Potvin 1993; Lepioufle et al. 2012; Schleiss and Smith 2015).

To fully understand the notion of predictable scale, one needs to keep in mind that spatial variability can be anisotropic, that is, that semivariance values usually depend both on the length and the direction of spatial displacement. The minimum predictable scale is always defined with respect to the direction of strongest variability. More specifically, we define the minimum predictable scale at time lag τ (where curly brackets indicate an ensemble) as the length of the longest spatial displacement for which all empirical semivariance values are smaller or equal to :
e6
and
e7

The reason why is called the minimum predictable scale is that, by definition, the semivariance of any shorter displacement will be below that of the minimum measurable semivariance at time lag τ. As a reminder, the curly brackets indicate an ensemble and the lowest measurable semivariance at time lag τ is equal to the time nugget defined in Eq. (3). In other words, all the variability at scales smaller than will be “masked” by the temporal innovation of the field.

An illustration of the concept of minimum predictable scale is provided in Fig. 6. We can see that the predictable scales for event 1 and time lags of 5, 15, 45, and 60 min are approximately 2.23, 6.00, 14.26, and 24.03 km. For a more detailed and complete analysis of predictable scales as a function of time lag, the reader is referred to section 4c.

d. Some practical aspects

Before we move on to the main results in section 4, there are some practical aspects that need to be discussed. The first issue concerns the computational burden associated with estimating a large number of empirical semivariance values in Eq. (1). Fortunately, one does not need to compute the semivariance values for every possible spatial and temporal displacement to obtain reliable estimates of innovation. A good way to speed up the analyses is to compute the semivariance values in increasing order or time lag τ. This type of approach takes advantage of the fact that rainfall fields cannot propagate at arbitrarily large speeds. The location of the minimum empirical semivariance at time lag τ must therefore be relatively close to the minimum at time lag . Thus, it is sufficient to compute the semivariance values in a relatively small neighborhood around the location of the previous minimum. The size of this neighborhood can be fixed (e.g., 20 km for 5-min time lags) or estimated based on previous propagation speeds. As a result, the temporal innovation factors can be retrieved relatively quickly.

The special case τ = 0 also merits some further attention. By definition of the variogram, we have that . In other words, we know that the temporal innovation at time lag τ = 0 is always zero. However, we still need to compute the spatial variogram maps to determine the range of predictable scales . Unfortunately, computing the complete spatial variogram maps at each time step for each possible spatial displacement can quickly become prohibitive. To save time and determine the predictable scales as fast as possible, it is best to compute the variogram for τ = 0 at the end, after all the other time lags τ > 0. Moreover, the spatial semivariance values can be estimated in order of increasing spatial displacements, that is, starting with short displacements and progressively enlarging the search radius until all predictable scales can be determined unambiguously.

The last issue related to space–time variogram computations that needs to be discussed here is sample size. For small time lags, sample sizes are usually large enough. However, as the time lag increases, determining the magnitude and location of the minimum semivariance value becomes increasingly difficult. In other words, there is an upper limit on the time lags for which reliable innovation estimates can be obtained. This limit can be as large as 24 h or as low as 1 h, depending on the shape, size, motion, and intermittency of the event. To avoid sampling effects, the maximum time lag considered in this paper will be 3 h. In addition, a lower limit of 1000 is imposed on the sample size when computing empirical semivariance values. But even so, it is important to keep in mind that the uncertainty affecting the innovation estimates increases with τ. Innovation factors at large time lags must therefore be interpreted with caution.

4. Results

a. Temporal innovation versus time lag

The goal of this section is to investigate how temporal innovation varies with respect to the time lag τ. To do this, we computed the temporal innovation factors for all available events, time steps t, and time lags τ between 5 min and 3 h, which equals 48 innovation values per event for a 5-min time lag and 13 values per event for a 3-h time lag. The results were then pooled together for each dataset and summarized in the form of a series of box plots in Fig. 7. The results for the strongly convective events are shown in Fig. 7a while the weakly convective events are shown in Fig. 7b. Note that the innovation estimates are limited to 3 h to avoid any problems related to sampling effects (see section 3d for more details). Looking at Fig. 7, we can see that, as expected, both the temporal innovation and its spread increase with time lag τ. However, most of the innovation takes place at time lags below 60 min. For the strongly convective events (Fig. 7a), the innovation keeps increasing long after 60 min, but at a rate that is 3–4 times smaller than during the first hour. Comparing the median innovation values in Figs. 7a and 7b, we see that stratiform events evolve faster (relatively to the total spatial variance) than strongly convective events. The absolute magnitude of the variability, given by the time nugget is, however, much larger during convective events.

Fig. 7.
Fig. 7.

Temporal innovation vs time lag τ (min) for all events in dataset 1 and dataset 2. The represented quantiles are 10%, 25%, 50%, 75%, and 90%. The red lines represent the fitted exponential models given in Eq. (8).

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

Based on Fig. 7, a good model for the median innovation is given by the sum of two exponential models with different sills and ranges:
e8
where denotes the median innovation at time lag τ, s1 > 0 and s2 > 0 are the partial sills, and λ1 > 0 and λ2 > 0 are the exponential slopes. The use of two superposed exponential models is motivated by the fact that the total innovation can be attributed to at least two different processes: small-scale variability and large-scale forcing. Note that the large-scale forcing is more difficult to see during weakly convective events (Fig. 7b) because of the larger interevent variability. A nonlinear adjustment with weights inversely proportional to the interquartile range leads to the following fitted model parameters (for τ expressed in minutes). For the first dataset, s1 = 0.229, s2 = 0.479, λ1 = 0.0607 min−1, and λ2 = 0.0145 min−1. For the second dataset, s1 = 0.699 and λ1 = 0.0466 min−1 (s2 and λ2 are dropped). The fitted pseudo ranges, that is, the distances after which 95% of the sills are reached, are 49.4 min and 3.4 h for dataset 1 and 64.3 min for dataset 2. The time lags after which 50% of the initial variance is reached are 59.7 min for the strongly convective events and 27.1 min for the stratiform and weakly convective cases.

b. Predictability of temporal innovation

In the following, we investigate how accurately future innovation can be predicted based on past estimates of . Interestingly, the answer to this question depends on whether we consider the absolute or relative error. Simple regression analysis (not shown here) suggests that the absolute prediction error, assuming the future innovation is identical to the one at the previous time step, increases approximately linearly with time lag τ. The relative prediction error, on the other hand, appears to be a nonmonotonic function of the time lag τ. To study the relative prediction error, we computed the lag-1, that is, 5-min autocorrelation of for each event and time lag τ. The lag-1 autocorrelation offers an easy way of quantifying the temporal persistence of the innovation, that is, where t − 1 denotes the time step immediately before t. A large autocorrelation means future innovation can be easily predicted from past innovation. Figure 8 shows the autocorrelation values as a function of time lag τ for highly convective events (Fig. 8a) and stratiform cases (Fig. 8b). It shows that the autocorrelation of at very short time lags is relatively low (i.e., median values of 0.51 and 0.20, respectively). Contrarily to what is expected, the autocorrelation increases at short time lags, reaches a maximum for τ between 30 and 60 min, and decreases thereafter.

Fig. 8.
Fig. 8.

Lag-1 autocorrelation of for all 25 events in dataset 1 and dataset 2. The represented quantiles are 10%, 25%, 50%, 75%, and 90%.

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

The fact that the autocorrelation at time lags below 30 min is smaller than between 30 and 60 min is interesting. It means it is more difficult, in terms of relative error, to predict the future innovation over very small time scales. A possible explanation for this loss of predictability at small scales could be the presence of turbulence and short-lived convective processes such as the birth, growth, decay, splitting, and merging of convective cells (Lewis et al. 2015). But then, why would stratiform and weakly convective events exhibit a similar decrease? A more plausible explanation for the loss of predictability at small scales is given by the measurement errors and uncertainties in the radar data. Weather radars are known to be affected by several sources of errors and uncertainties (e.g., Krajewski and Smith 2002). The latter might not play a very important role at large time scales where they are dominated by the natural variability of rainfall. But as we move toward smaller scales, measurement errors and uncertainties become increasingly important. Eventually, we reach a point where it becomes impossible to separate the natural variability from the measurement noise. Figure 8 suggests that the optimal range of time scales for predicting temporal innovation lies somewhere between 30 and 60 min. Larger scales are more difficult to predict because of the natural variability of precipitation. Smaller scales, on the other hand, are harder to predict because of measurement errors and uncertainties. This trade-off between natural variability and measurement uncertainty is something that has to be better taken into account when designing short-term radar-based precipitation forecasting systems and assessing the performance of ensemble rainfall forecasts.

c. Predictable scales versus time lag

The focus of this section is on the link between temporal variability and spatial structure of rainfall fields. The objective is to determine which spatial scales remain relevant for prediction at time lag τ. To address this issue, we use the concept of predictable scale defined in section 3c. More specifically, we computed the predictable scale for each event, time step t, and time lag τ. The results for the strongly convective cases are shown in Fig. 9a while the stratiform and weakly convective events are shown in Fig. 9b. Looking at Fig. 9, we see that the predictable scales increase with time lag τ according to what can be approximated by an exponential model:
e9
where s > 0 is the sill, λ > 0 the exponential slope, and denotes the median predictable scale at time lag τ. A nonlinear adjustment with weights inversely proportional to the interquartile range yields the following parameter estimates (for τ expressed in minutes): s = 18.53 km and λ = 0.0198 min−1 for dataset 1 and s = 8.40 km and λ = 0.0277 min−1 for dataset 2. The nonfitted median predictable scales at 5, 15, 30, and 60 min are 1.57, 4.98, 8.10, and 12.69 km for the strongly convective events and 1.11, 3.15, 4.72, and 6.77 km for the stratiform and weakly convective events. The time needed to reach 95% of the sill is 151.3 min for dataset 1 and 108.1 min for dataset 2. This is consistent with previous results presented in Fig. 7 suggesting that weak stratiform events evolve faster (relatively to the total variance) than strong, convective events. The magnitude of the predictable scales is, however, larger during convective events.
Fig. 9.
Fig. 9.

Predictable scales (km) vs time lag τ for all events in dataset 1 and dataset 2. The represented quantiles are 10%, 25%, 50%, 75%, and 90%. The red line represents the fitted exponential model in Eq. (9).

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

It is worth pointing out that these results are in good agreement with earlier findings. For example, Zawadzki et al. (1994) pointed out that radar data downscaled to a 16-km resolution remain predictable up to 120 min. Our results suggest that the 25% and 75% quantiles of predictable scales after 120 min are between 11.0 and 23.4 km for dataset 1 and between 5.4 and 13.5 km for dataset 2. Additional studies by Bellon and Zawadzki (1994), Grecu and Krajewski (2000), and Seed (2003) reported predictable scales on the order of 10–20 km after 60 min. We find a slightly smaller range of 9.7–16 km for convective events and 5.6–10 km for stratiform ones. Most importantly, our results highlight the fact that temporal innovation and predictable scales substantially vary from one storm to another, depending on the type, size, structure, and temporal variability. It is therefore very valuable to have simple and reliable quantitative methods for estimating innovation rates and predictable scales based on past observations before trying to issue a probabilistic forecast over a given region.

d. Temporal innovation versus storm properties

In the last part of this section, we investigate how temporal innovation depends on spatial storm properties. To keep things simple, we restricted the analyses to three basic quantities: 1) the average radar reflectivity (dBZ) at each time step; 2) the maximum storm size, defined as the area of largest connected rainy component at each time step; and 3) the spatial correlation range , defined as the maximum distance for which all empirical semivariance values are smaller or equal to 50% of the total variance:
e10
and
e11

Figure 10 shows the time series of the temporal innovation for time lag τ = 5 min versus the average reflectivity, storm size, and spatial correlation range (where curly brackets indicate an ensemble) for event 1 (see Fig. 3 and Table 1). For this particular event, we can see that the average radar reflectivity and the spatial correlation range are negatively correlated with (rank correlations of −0.60 and −0.42). The storm size, on the other hand, is slightly positively correlated with temporal innovation (rank correlation of 0.29).

Fig. 10.
Fig. 10.

Time series of temporal innovation at time lag τ = 5 min, mean reflectivity (dBZ), storm size (× 103 km2), and spatial correlation range (km) for event 1.

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

To get more representative results, we repeated the same correlation analysis for all events, datasets, and time lags between 5 min and 3 h. The results are shown in Fig. 11. We can see that none of the considered quantities are very good predictors of temporal innovation on their own. Among the three quantities, the storm size was the least correlated variable in absolute value, with correlations at 5 min between −0.24 (dataset 2) and −0.28 (dataset 1), followed by the mean reflectivity (−0.16 for dataset 1 and −0.56 for dataset 2) and the spatial correlation range (−0.46 for dataset 2 and −0.54 for dataset 1). The sign of the correlation coefficients is consistent with earlier results by Grecu and Krajewski (2000) and Germann and Zawadzki (2002), who showed that large, intense, and spatially organized systems tend to be more predictable. It appears, however, to be extremely difficult to predict temporal innovation based solely on spatial storm properties. Surely, better predictions could be obtained by combining two or more spatial predictors (e.g., the size and the spatial correlation range) or by considering other spatial statistics, but finding good combinations of predictors requires more detailed analyses and more events and is beyond the scope of this paper. A simpler and more effective way of estimating future innovation, as shown in section 4b, consists in exploiting the relatively large autocorrelation of over time scales between 30 and 60 min.

Fig. 11.
Fig. 11.

Average rank correlation of temporal innovation with mean reflectivity (black dots), storm size (red triangles), and spatial correlation (blue crosses) for all events in dataset 1 and dataset 2.

Citation: Journal of Hydrometeorology 17, 3; 10.1175/JHM-D-15-0137.1

Finally, it is also worth mentioning that no significant differences in magnitude were found between the temporal innovation in the morning, afternoon, evening, and night. The most probable explanation for this is that the effect the diurnal cycle has on temporal innovation, once the storms are formed, is too small to be distinguished from the variability between and within events.

5. Summary and conclusions

After decades of research, Lagrangian persistence, that is, the simple propagation of the rainfall field along the direction of storm motion, still remains one of the best and most widely used techniques for forecasting precipitation over short lead times. To be truly useful for hydrological applications, Lagrangian-based forecasting techniques must incorporate some measure of uncertainty (e.g., a combination of measurement errors and temporal evolution). So far, few good methods for estimating and predicting this forecast uncertainty have been proposed. Existing techniques almost exclusively involve some sort of tracking algorithm, forecasting method, or heuristic filtering. This may be the most direct and intuitive solution but comes with its own set of challenges.

In this paper, a new geostatistical framework for quantifying temporal innovation and predictable scales at time lags between 5 min and 3 h was introduced. The main advantage of the proposed method is that it requires no specific forecast model, tracking algorithm, or data filtering. It is also slightly more general and versatile than similar techniques based on cross covariance and cross correlations. A limitation of the proposed method is that it only quantifies the average innovation and predictability over the entire domain of study. As large heterogeneous domains are considered (e.g., several hundreds of kilometers), one should be aware of local changes in spatial and temporal structures caused by convection, orographic effects, or land–ocean transitions. In this case, the innovation and predictability of individual cells or groups of cells in the domain might deviate substantially from the average variability over the entire domain.

In the first part of section 4, we showed that temporal innovation can be modeled as a sum of two exponential functions of time lag representing the small-scale variability and large-scale forcing of the system. On average, more than half of the total innovation occurs at time scales below 60 min. In the second part of section 4 we showed that, although the absolute prediction error on increases with time lag, relative errors were minimized for time scales between approximately 30 and 60 min. This range of time scales appears to be the best trade-off between radar sampling noise/uncertainty and natural variability of precipitation.

In the third part of section 4, the concept of minimum predictable scale was used to analyze the rate at which the initial correlation structure of a rainfall field is lost because of temporal innovation. We found median predictable scales for strongly convective events on the order of 1.6 km at 5 min, 5 km at 15 min, and 12.7 km at 1 h. Stratiform and weakly convective events tend to have smaller spatial correlation and minimum predictable scales on the order of 1.11, 3.15, 4.72, and 6.77 km, respectively. In both cases, the median predictable scale approximately followed an exponential model of the time lag, with large differences from one event to another.

The final analyses in section 4d confirmed earlier findings that small and weakly organized events tend to evolve faster (relative to their total variance) than large, intense, and strongly correlated convective systems. However, the size, intensity, and spatial correlation did not turn out to be good predictors of temporal innovation on their own. There was also no measurable difference in magnitude between temporal innovation in the morning, afternoon, evening, and night.

Of course, all the results in this paper depend on the quality and consistency of the radar reflectivity maps. The data were carefully selected to guarantee the best possible quality, but small errors and artifacts in the radar products cannot be ruled out. Additional analyses based on different radar products and longer time periods will have to be performed to obtain a more representative climatology of temporal evolution and predictability. The same techniques could, for example, be applied to satellite rainfall products such as the Integrated Multisatellite Retrievals for GPM (IMERG; Hou et al. 2014). Unfortunately, most of the variability in rainfall seems to take place at scales below 1 h and 10 km, which are not well captured by the GPM satellites. The GPM mission will, however, be invaluable for studying rainfall predictability at intermediate and large scales. Preliminary comparisons between ground-based radar and satellite rainfall estimates suggest a relatively good agreement in terms of spatial and temporal structures, but this is ongoing work and will have to be detailed in another paper.

Acknowledgments

This work is a contribution to the Spatial and Temporal Analysis of Rainfall in Mesoscale Convective Systems (STORMS) project, funded by Grant P300P2_158499 of the Swiss National Science Foundation. The author thanks the Iowa Environmental Mesonet (IEM) and the National Center for Atmospheric Research (NCAR) for collecting, storing, and providing the radar data used in this study. The radar composites are freely available through the website of the IEM (http://mesonet.agron.iastate.edu).

REFERENCES

  • Bellon, A., , and Zawadzki I. , 1994: Forecasting of hourly accumulations of precipitation by optimal extrapolation of radar maps. J. Hydrol., 157, 211233, doi:10.1016/0022-1694(94)90106-6.

    • Search Google Scholar
    • Export Citation
  • Berenguer, M., , Sempere-Torres D. , , and Pegram G. G. S. , 2011: SBMcast: An ensemble nowcasting technique to assess the uncertainty in rainfall forecasts by Lagrangian extrapolation. J. Hydrol., 404, 226240, doi:10.1016/j.jhydrol.2011.04.033.

    • Search Google Scholar
    • Export Citation
  • Berne, A., , Delrieu G. , , and Boudevillain B. , 2009: Variability of the spatial structure of intense Mediterranean precipitation. Adv. Water Resour., 32, 10311042, doi:10.1016/j.advwatres.2008.11.008.

    • Search Google Scholar
    • Export Citation
  • Bowler, N. E., , Pierce C. E. , , and Seed A. W. , 2006: STEPS: A probabilistic precipitation forecasting scheme which merges an extrapolation nowcast with downscaled NWP. Quart. J. Roy. Meteor. Soc., 132, 21272155, doi:10.1256/qj.04.100.

    • Search Google Scholar
    • Export Citation
  • Cloke, H. L., , and Pappenberger F. , 2009: Ensemble flood forecasting: A review. J. Hydrol., 375, 613626, doi:10.1016/j.jhydrol.2009.06.005.

    • Search Google Scholar
    • Export Citation
  • Collier, C. G., 2007: Flash flood forecasting: What are the limits of predictability? Quart. J. Roy. Meteor. Soc., 133, 323, doi:10.1002/qj.29.

    • Search Google Scholar
    • Export Citation
  • Cuo, L., , Pagano T. C. , , and Wang Q. J. , 2011: A review of quantitative precipitation forecasts and their use in short- to medium-range streamflow forecasting. J. Hydrometeor., 12, 713728, doi:10.1175/2011JHM1347.1.

    • Search Google Scholar
    • Export Citation
  • Emmanuel, I., , Andrieu H. , , Leblois E. , , and Flahaut B. , 2012: Temporal and spatial variability of rainfall at the urban hydrological scale. J. Hydrol., 430–431, 162172, doi:10.1016/j.jhydrol.2012.02.013.

    • Search Google Scholar
    • Export Citation
  • Foresti, L., , Panziera L. , , Mandapaka P. V. , , Germann U. , , and Seed A. , 2015: Retrieval of analogue radar images for ensemble nowcasting of orographic rainfall. Meteor. Appl., 22, 141155, doi:10.1002/met.1416.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , and Zawadzki I. , 2002: Scale-dependence of the predictability of precipitation from continental radar images. Part I: Description of the methodology. Mon. Wea. Rev., 130, 28592873, doi:10.1175/1520-0493(2002)130<2859:SDOTPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , and Zawadzki I. , 2004: Scale dependence of the predictability of precipitation from continental radar images. Part II: Probability forecasts. J. Appl. Meteor., 43, 7489, doi:10.1175/1520-0450(2004)043<0074:SDOTPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , Galli G. , , Boscacci M. , , and Bolliger M. , 2006a: Radar precipitation measurement in a mountainous region. Quart. J. Roy. Meteor. Soc., 132, 16691692, doi:10.1256/qj.05.190.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , Zawadzki I. , , and Turner B. , 2006b: Predictability of precipitation from continental radar images. Part IV: Limits to prediction. J. Atmos. Sci., 63, 20922108, doi:10.1175/JAS3735.1.

    • Search Google Scholar
    • Export Citation
  • Germann, U., , Berenguer M. , , Sempere-Torres D. , , and Zappa M. , 2009: REAL—Ensemble radar precipitation estimation for hydrology in a mountainous region. Quart. J. Roy. Meteor. Soc., 135, 445456, doi:10.1002/qj.375.

    • Search Google Scholar
    • Export Citation
  • Grecu, M., , and Krajewski W. F. , 2000: A large-sample investigation of statistical procedures for radar-based short-term quantitative precipitation forecasting. J. Hydrol., 239, 6984, doi:10.1016/S0022-1694(00)00360-7.

    • Search Google Scholar
    • Export Citation
  • Hou, A. Y., and et al. , 2014: The Global Precipitation Measurement Mission. Bull. Amer. Meteor. Soc., 95, 701722, doi:10.1175/BAMS-D-13-00164.1.

    • Search Google Scholar
    • Export Citation
  • Journel, A. G., 1980: The lognormal approach to predicting local distributions of selective mining unit grades. Math. Geol., 12, 285303, doi:10.1007/BF01029417.

    • Search Google Scholar
    • Export Citation
  • Krajewski, W. F., , and Smith J. A. , 2002: Radar hydrology: Rainfall estimation. Adv. Water Resour., 25, 13871394, doi:10.1016/S0309-1708(02)00062-3.

    • Search Google Scholar
    • Export Citation
  • Lepioufle, J.-M., , Leblois E. , , and Creutin J.-D. , 2012: Variography of rainfall accumulation in presence of advection. J. Hydrol., 464–465, 494504, doi:10.1016/j.jhydrol.2012.07.041.

    • Search Google Scholar
    • Export Citation
  • Lewis, H., and et al. , 2015: From months to minutes—Exploring the value of high-resolution rainfall observation and prediction during the UK winter storms of 2013/2014. Meteor. Appl., 22, 90104, doi:10.1002/met.1493.

    • Search Google Scholar
    • Export Citation
  • Marsan, D., , Schertzer D. , , and Lovejoy S. , 1996: Causal space–time multifractal processes: Predictability and forecasting of rain fields. J. Geophys. Res., 101, 26 33326 346, doi:10.1029/96JD01840.

    • Search Google Scholar
    • Export Citation
  • Mascaro, G., , Vivoni E. R. , , and Deidda R. , 2010: Implications of ensemble quantitative precipitation forecast errors on distributed streamflow forecasting. J. Hydrometeor., 11, 6986, doi:10.1175/2009JHM1144.1.

    • Search Google Scholar
    • Export Citation
  • Onof, C., , Northrop P. , , Wheater H. S. , , and Isham V. , 1996: Spatiotemporal storm structure and scaling property analysis for modeling. J. Geophys. Res., 101, 26 41526 425, doi:10.1029/96JD01378.

    • Search Google Scholar
    • Export Citation
  • Panziera, L., , Germann U. , , Gabella M. , , and Mandapaka P. V. , 2011: NORA–Nowcasting of Orographic Rainfall by means of Analogues. Quart. J. Roy. Meteor. Soc., 137, 21062123, doi:10.1002/qj.878.

    • Search Google Scholar
    • Export Citation
  • Panziera, L., , James C. N. , , and Germann U. , 2015: Mesoscale organization and structure of orographic precipitation producing flash floods in the Lago Maggiore region. Quart. J. Roy. Meteor. Soc., 141, 224248, doi:10.1002/qj.2351.

    • Search Google Scholar
    • Export Citation
  • Paschalis, A., , Molnar P. , , Fatichi S. , , and Burlando P. , 2013: A stochastic model for high-resolution space–time precipitation simulation. Water Resour. Res., 49, 84008417, doi:10.1002/2013WR014437.

    • Search Google Scholar
    • Export Citation
  • Paschalis, A., , Fatichi S. , , Molnar P. , , Rimkus S. , , and Burlando P. , 2014: On the effects of small scale space–time variability of rainfall on basin flood response. J. Hydrol., 514, 313327, doi:10.1016/j.jhydrol.2014.04.014.

    • Search Google Scholar
    • Export Citation
  • Potvin, G., 1993: Space–time correlations and Taylor’s hypothesis for rainfall. M.S. thesis, Dept. of Physics, McGill University, 68 pp.

  • Radhakrishna, B., , Zawadzki I. , , and Fabry F. , 2012: Predictability of precipitation from continental radar images. Part V: Growth and decay. J. Atmos. Sci., 69, 33363349, doi:10.1175/JAS-D-12-029.1.

    • Search Google Scholar
    • Export Citation
  • Schleiss, M., , and Smith J. A. , 2015: A method to estimate the 3D–time structure of the raindrop size distribution using radar and disdrometer data. J. Hydrometeor., 16, 12221242, doi:10.1175/JHM-D-14-0182.1.

    • Search Google Scholar
    • Export Citation
  • Schleiss, M., , Chamoun S. , , and Berne A. , 2014: Stochastic simulation of intermittent rainfall using the concept of dry drift. Water Resour. Res., 50, 23292349, doi:10.1002/2013WR014641.

    • Search Google Scholar
    • Export Citation
  • Seed, A. W., 2003: A dynamic and spatial scaling approach to advection forecasting. J. Appl. Meteor., 42, 381388, doi:10.1175/1520-0450(2003)042<0381:ADASSA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Taylor, G. I., 1938: The spectrum of turbulence. Proc. Roy. Soc. London, A164, 476490, doi:10.1098/rspa.1938.0032.

  • Thorndahl, S., , Smith J. A. , , Baeck M. L. , , and Krajewski W. F. , 2014: Analyses of the temporal and spatial structures of heavy rainfall from a catalog of high-resolution radar rainfall fields. Atmos. Res., 144, 111125, doi:10.1016/j.atmosres.2014.03.013.

    • Search Google Scholar
    • Export Citation
  • Turner, B. J., , Zawadzki I. , , and Germann U. , 2004: Predictability of precipitation from continental radar images. Part III: Operational nowcasting implementation (MAPLE). J. Appl. Meteor., 43, 231248, doi:10.1175/1520-0450(2004)043<0231:POPFCR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tuttle, J. D., , and Foote G. B. , 1990: Determination of the boundary layer airflow from a single Doppler radar. J. Atmos. Oceanic Technol., 7, 218232, doi:10.1175/1520-0426(1990)007<0218:DOTBLA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Zawadzki, I., , Morneau J. , , and Laprise R. , 1994: Predictability of precipitation patterns: An operational approach. J. Appl. Meteor., 33, 15621571, doi:10.1175/1520-0450(1994)033<1562:POPPAO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
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