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  • View in gallery

    (top to bottom) Visual illustration of the DSD simulation algorithm.

  • View in gallery

    Radar rain-rate maps for (top) event 1 and (bottom) event 2. The size of the domain is 50 × 50 km2. The location of the 16 disdrometers is represented by a white cross in the lower-left corner.

  • View in gallery

    Time series of (top) μ, (middle) Nt (m−3), and (bottom) R (mm h−1) at 30-s time resolution for event 1 (frontal). The three lines represent the min, avg, and max values measured by the 16 disdrometers at each time step.

  • View in gallery

    Same as Fig. 3 but for event 2 (convective).

  • View in gallery

    Fitted temporal variograms of and for (top) event 1 and (bottom) event 2. For details about the variogram parameters, see Table 3.

  • View in gallery

    Snapshot of a simulated field of μ, Nt, D0, and R for the frontal event. The simulation size is 50 × 50 km2 and corresponds to the domain shown in Fig. 2, and the pixel resolution is 500 × 500 m2.

  • View in gallery

    Same as Fig. 6 but for the convective event.

  • View in gallery

    Fields of Zh, Zdr, Ah, and Kdp corresponding to Fig. 6.

  • View in gallery

    Fields of Zh, Zdr, Ah, and Kdp corresponding to Fig. 7.

  • View in gallery

    Simulated distributions of (left) and (right) for (top) event 1 and (bottom) event 2. Each box plot represents the 10%, 25%, 50%, 75%, and 90% quantiles of all simulated values. The arrow next to each box plot represents the range of and for the eight control disdrometers.

  • View in gallery

    Simulated and measured temporal variograms of (left) and (right) for (top) event 1 and (bottom) event 2. The black dotted lines represent the 10% and 90% quantiles of the simulated variograms for each pixel in the simulation domain. The continuous black line represents the median of all simulated variograms. The dashed red lines represent the min and the max of the eight variograms computed using the control disdrometers.

  • View in gallery

    Estimated drift functions for log10(μ + 1) and log10(Nt) for the convective event.

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Stochastic Simulation of Intermittent DSD Fields in Time

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  • 1 Laboratoire de Télédétection Environnementale, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
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Abstract

A method for the stochastic simulation of (rain)drop size distributions (DSDs) in space and time using geostatistics is presented. At each pixel, the raindrop size distribution is described by a Gamma distribution with two or three stochastic parameters. The presence or absence of rainfall is modeled using an indicator field. Separable space–time variograms are used to estimate and reproduce the spatial and temporal structures of all these parameters. A simple and user-oriented procedure for the parameterization of the simulator is proposed. The only data required are DSD time series and radar rain-rate (or reflectivity) measurements. The proposed simulation method is illustrated for both frontal and convective precipitation using real data collected in the vicinity of Lausanne, Switzerland. The spatial and temporal structures of the simulated fields are evaluated and validated using DSD measurements from eight independent disdrometers.

Corresponding author address: Alexis Berne, GR C2 564, Station 2, 1015 Lausanne, Switzerland. E-mail: alexis.berne@epfl.ch

Abstract

A method for the stochastic simulation of (rain)drop size distributions (DSDs) in space and time using geostatistics is presented. At each pixel, the raindrop size distribution is described by a Gamma distribution with two or three stochastic parameters. The presence or absence of rainfall is modeled using an indicator field. Separable space–time variograms are used to estimate and reproduce the spatial and temporal structures of all these parameters. A simple and user-oriented procedure for the parameterization of the simulator is proposed. The only data required are DSD time series and radar rain-rate (or reflectivity) measurements. The proposed simulation method is illustrated for both frontal and convective precipitation using real data collected in the vicinity of Lausanne, Switzerland. The spatial and temporal structures of the simulated fields are evaluated and validated using DSD measurements from eight independent disdrometers.

Corresponding author address: Alexis Berne, GR C2 564, Station 2, 1015 Lausanne, Switzerland. E-mail: alexis.berne@epfl.ch

1. Introduction

Precipitation is an important and complex part of the earth’s water cycle and climate system. Although it is very beneficial in everyday life (it produces most of the freshwater on the earth’s surface and is therefore vital for plants and animals), it can also have dramatic effects on humans and infrastructures (e.g., floods, droughts, landslides, and avalanches). Consequently, many efforts have been devoted over the last decades to measure, quantify, model, and predict precipitation with increasing accuracy. Nowadays, most of the individual physical mechanisms (e.g., nucleation, diffusional growth, collisional growth, evaporation, and breakup) involved in the formation of clouds and precipitation are fairly well understood (Pruppacher and Klett 1997). The study and modeling of their complex interactions is, however, still an active field of research. At larger scales, additional complexity is introduced through the influence of land and sea surface processes (e.g., changes in temperature and evaporation) that significantly contribute to the spatial and temporal variability of precipitation (Koster and Suarez 1995; Roe 2005). As a result, rainfall is generally nonstationary, inhomogeneous, intermittent, and largely variable in space and time.

Modern meteorological and hydrological models require accurate rainfall estimates with high spatial and temporal resolutions. Weather radars provide relatively good inputs to these models but their estimates can be affected by large uncertainties (Wilson and Brandes 1979; Krajewski and Smith 2002; Villarini and Krajewski 2010). More robust and better data can be expected if measurements coming from several independent rainfall sensors (e.g., satellites, ground-based radars, rain gauges, disdrometers, and microwave links) are combined. Finding robust and reliable merging techniques that take into account the relative uncertainties and error contributions of each instrument is known to be a very difficult problem. In particular, there is a strong lack of reference data against which the different instruments and retrieval algorithms could be compared and evaluated objectively. A common approach therefore consists of using stochastic rainfall simulators. The latter are very interesting and useful because they allow for generation of large numbers of statistically homogeneous fields at high spatial and temporal resolutions (far beyond the capabilities of traditional rainfall sensors) that can be used as reference data in further analysis.

In this article, a stochastic rainfall simulator at the mesoscale (domain size between 10 and 100 km) is presented that adequately reproduces the main statistical features of a given rain event—that is, the spatial and temporal structures, the rainfall intermittency, and, most importantly, the (rain)drop size distribution (DSD). From the DSD it is then possible to derive or approximate most physical quantities of interest—that is, the shape, fall speed, and energy of the individual drops. In addition, quantities like the rain rate, the median drop diameter, the radar reflectivity, the differential reflectivity, specific attenuation, and differential phase shift (which are very important for remote sensing of precipitation and explicitly depend on the DSD) can also be derived. As a result, simulated DSD fields are very valuable for many practical applications, including soil erosion problems (Kinnell 2005), runoff and infiltration (Smith et al. 2009), atmospheric deposition processes (Beier et al. 1993), the ZR relationship for weather radars (Battan 1973; Ulbrich 1985), attenuation correction algorithms at C and X bands (e.g., Bringi et al. 1990; Delrieu et al. 1999; Testud et al. 2000), rainfall measurement using satellites [e.g., Tropical Rainfall Measuring Mission (TRMM) and Global Precipitation Measurement (GPM)], and microwave links from telecommunication networks (Messer et al. 2006; Leijnse et al. 2007).

So far, most of the stochastic rainfall simulation techniques have been focusing on the simulation of daily, monthly, or annual precipitation amounts using chain-dependent stochastic processes (e.g., Katz 1977; Richardson 1981; Wilks 1999). Also, significant efforts have been devoted to the simulation of spatially correlated rain-rate fields using, for example, principal components (Bouvier et al. 2003), clustering processes (Waymire et al. 1984; Rodríguez-Iturbe and Eagleson 1987; Sivapalan and Wood 1987), self-similarity (Gupta and Waymire 1993; Menabde et al. 1997), and geostatistics (Guillot 1999). Only a few studies have investigated the possibility of simulating rainfall fields (or rain profiles) with higher level of detail—for example by including the (rain)drop size distribution (e.g., Krajewski et al. 1993; Berne and Uijlenhoet 2005; Lavergnat and Golé 2006; Lee et al. 2007). More recently, Schleiss et al. (2009) proposed a DSD simulator based on geostatistics. Their method, however, was limited to small, nonintermittent, and instantaneous rainfall fields. In this article, the method proposed by Schleiss et al. (2009) is extended and a new DSD simulator is presented that can be used to generate both intermittent and temporally correlated rainfall fields.

This article is structured as follows: section 2 is devoted to the description of the models and mathematical tools needed for the simulation. Section 3 explains how to parameterize the simulator. Section 4 provides a detailed step-by-step description of the simulation algorithm and explains how the main variables of interest can be computed from the simulated DSD fields. In section 5, the capabilities of the simulator are illustrated and evaluated using real data collected in the vicinity of Lausanne, Switzerland. The advantages and limitations of the proposed method are discussed in section 6, and the conclusions and perspectives are given in section 7.

2. Modeling

A substantial amount of material presented in this section is similar to sections 2 and 3 of Schleiss et al. (2009) and will not be repeated here. Instead, the major differences between the two approaches are highlighted.

a. Drop size distribution

The drop size distribution is described using a Gamma distribution (Ulbrich 1983; Willis 1984; Ulbrich and Atlas 2007) with three parameters (μ, Λ, and Nt), exactly as in section 2.1 of Schleiss et al. (2009):
e1
Because of the natural variability of rainfall, the DSD parameters (μ, Λ, and Nt) can be seen as realizations (in time and space) of an underlying multivariate random function. Furthermore, some studies (Zhang et al. 2001; Seifert 2005) suggest that there is a deterministic relation between the shape and the rate parameter:
e2
where f depends on the type of precipitation and on the local climatology. Such relations are interesting because they allow for reducing the number of stochastic parameters to be simulated. However, Moisseev and Chandrasekar (2007) argued that μ–Λ relations must be handled with extreme care because they might be the result of statistical errors and data filtering of disdrometer measurements. In any case, this is not a critical issue because the proposed simulation method can be easily adapted to include an additional third stochastic parameter, as in Schleiss et al. (2009). But for simplicity, only the first case (two stochastic parameters μ and Nt and a deterministic relationship between Λ and μ) is presented in this article. Finally, it must be noted that the proposed simulation technique can easily be adapted to any other parametric DSD model (e.g., exponential or lognormal).

b. Intermittency

Intermittency (i.e., the presence or absence of rainfall) is modeled using an indicator field
e3
where R(x) (mm h−1) represents the instantaneous rain rate at location x. Using this notation, R(x) can be seen as the product of two random functions:
e4
where R+(x) > 0 represents the nonzero rain rate at location x (for consistency, R+ is only defined for rainy locations). Using the same decomposition, Barancourt et al. (1992) derived an interesting scheme for the interpolation and simulation of intermittent rain-rate fields. Following the same idea, one can extend this result to the simulation of intermittent DSD fields by applying the same decomposition to the two DSD parameters:
e5
Using this notation, an intermittent DSD field (at a given time t and location x) is given by a triplet (I, μ+, and ) where I represents the rainfall indicator field and (μ+, ) the DSD parameters. But for conciseness (and because the DSD is not defined for dry locations anyway), we will drop the “+” sign and simply denote the DSD field by (I, μ, Nt).

c. Anamorphosis

For now, suppose that a given rainfall indicator field I has been simulated and that the rainy and dry locations are known. The only parameters that remain to be simulated are μ and Nt. Simulating realistic DSD fields means generating independent realizations of (μ, Nt) according to some unknown bivariate distribution, which is a problem known to be very difficult in general. A possible solution, known as Gaussian anamorphosis (Leuangthong and Deutsch 2003), is to transform the original parameters (μ, Nt) into independent Gaussian variables for which a variety of efficient simulation algorithms are known—for example, turning bands (Journel and Huijbregts 1978; Guillot 1999) and sequential simulation (Ripley 1987). For details about the anamorphosis procedure, readers are referred to section 2.2 of Schleiss et al. (2009).

d. Space–time structure

The space–time structure of I, , and is modeled using a space–time variogram (Matheron 1965; Chilès and Delfiner 1999):
e6
where is a separation vector, is a given time lag, and E is the expectation operator. Unfortunately, fitting a valid space–time variogram can be very difficult in general, especially when few data are available. Therefore, a common approach consists in separating the spatial and the temporal variations:
e7
where γS(h) and γT(τ) represent the spatial respectively temporal variogram of Z(x, t). For the rainfall indicator field I, both γS(h) and γT(τ) can be derived and fitted using radar rain rate or reflectivity data. For the DSD parameters, this is usually not possible because disdrometer networks are usually not dense and large enough for a direct estimation of γS(h). Hence, most of the time, the only variogram that can be computed is the one in the time domain. In this case, the spatial variogram must be approximated using external information (e.g., the speed and direction of advection and the anisotropy of the rainfall field). If available, polarimetric radar quantities like the differential reflectivity Zdr and the reflectivity at horizontal polarization Zh can also be used to estimate the spatial structure of and .
In the following, the authors propose a technique based on Taylor’s hypothesis of frozen turbulence (Taylor 1938) that allows for approximating the spatial structure of and without using polarimetric radar data. Taylor’s hypothesis states that the rainfall field moves, during short periods of time, with constant velocity v, and that its evolution for time lags up to 20–30 min is small compared to the advection process (Li et al. 2009). As a consequence, the temporal covariance of a variable at time lag τ is equal to the spatial covariance at space lag h = τv, or equivalently, γS(τv) = γT(τ). Hence (short) variations in time can be converted into small variations in space, but only in the direction of advection. For all other directions, the relation between temporal and spatial variations is not known a priori. However, it is reasonable to assume that similar relations exist and can be described using a geometric anisotropy parameter derived from radar rain-rate or reflectivity measurements:
e8
e9
where 0 < α ≤ 1 is called the anisotropy ratio, the vector giving the direction of minimum variability (not necessarily identical to the direction of advection), 〈h, a〉 denotes the standard scalar product between h and a, and the double vertical bars denote the norm in . The anisotropy factor α(h) is supposed to reflect a geometrical property of the rainfall field and is therefore assumed to be identical for the rain rate, the reflectivity, , and . Under these assumptions, the complete space–time variograms of and are given by
e10
Note that it is possible to generalize Eq. (10) by taking into account a time-dependent speed and direction of advection. In this case, each time period (e.g., 20–30 min) is represented by an average advection vector vi, an average anisotropy direction ai, and an average anisotropy ratio αi(h). The space–time variogram of each time period is then given by
e11

3. Parameterization

a. Required data

Optimal parameterization of the simulator is achieved if both spatial and temporal DSD data are available. Unfortunately, and as far as the authors know, such data are not yet available. In the meantime, alternative parameterizations using fewer data must be considered. In the following, the authors provide a method that allows for parameterizing the simulator using solely DSD time series (collected using one or several disdrometers) and radar rain-rate or reflectivity data for the considered domain and event of interest. The temporal sampling resolution of the disdrometers must be high enough to capture most of the natural variations occurring in the DSD but not too high to avoid strong sampling effects. Typically, temporal resolutions between 30 s and 1 min are adequate. The size of the simulation domain must be large enough to catch the spatial structure of the rainfall field but not too large for the variogram to be representative over the entire domain. Also, the size of the simulation domain must be consistent with the length of the collected DSD time series and the average speed of advection. A 30-min time series with an average speed of advection of 5 m s−1 only corresponds to 9 km along the direction of advection and is unlikely to be representative of a 50 × 50 km2 domain. Depending on these parameters, domain sizes between 10 and 100 km can be considered.

b. Parameters derived from radar data

Radar data are used to estimate 1) the rainfall intermittency, 2) the spatial and temporal variograms of the rainfall indicator field, 3) the speed and direction of advection, and 4) the anisotropy parameters of the rainfall field. All these parameters are assumed constant during the considered event or, alternatively, time dependent. For simplicity, the procedure is only illustrated for constant parameters.

The rainfall intermittency is estimated using the radar data by computing the proportion pW of wet and pD = 1 − pW of dry locations within the simulation domain. Note that if the intermittency changes significantly with time, it is better to consider separate time periods with varying intermittency. The spatial structure of the rainfall indicator field is then estimated (for each time period) by computing its average spatial sample variogram (aggregating the fields in time) and by fitting a valid variogram model on it. The temporal structure of the rainfall indicator field is estimated by computing the average sample time variogram (aggregating the time series in space) and by fitting a valid variogram model on it. This defines the complete space–time variogram of the rainfall indicator field.

The average speed and direction of advection υ is determined using a cell or echo-tracking algorithm on each pair of successive radar images (e.g., Rinehart 1979; Crane 1989; Tuttle and Foote 1990). The anisotropy parameters a and α given in Eqs. (8) and (9) are estimated by computing the spatial sample (2D) variogram of the radar rain rate or reflectivity data. It is then possible to identify the (average) direction of minimum variability a and the corresponding (average) anisotropy ratio α.

c. Parameters derived from DSD data

First, the three-parameter DSD model (μ, Λ, and Nt) given in Eq. (1) is fitted on each collected DSD spectrum. The scatterplot between μ and Λ is used to investigate if there is a relation (possibly nonlinear) between μ and Λ. For simplicity, and because of the high correlation that is usually observed, only linear relations between μ and Λ are considered in this article. The generalization of this technique to nonlinear relations (e.g., power laws or exponential models) is straightforward. For linear relations, the standard Pearson correlation coefficient is used to decide whether two or three stochastic parameters should be included in the model. As a rule of thumb, a correlation of 0.9 or higher is considered sufficient to drop one of the parameters. Otherwise, all three DSD parameters are kept and the simulation scheme is adapted to include an additional stochastic parameter. Note that this also includes the case of (complex) nonlinear relations, which are then captured automatically by the Gaussian anamorphosis and do not need to be parameterized. However, it must be emphasized that three-parameter simulations require far more DSD data (at least 10 times more than with two parameters) and significantly increase the simulation time. It is therefore highly recommended to work with two stochastic parameters whenever it is possible.

In the next step, the DSD parameters (μ, Nt) are detrended following a procedure described in the appendix. This is particularly important for highly intermittent rain fields where the constant alternating between dry and rainy periods significantly affects the variograms of μ and Nt. The detrended time series of (μ, Nt) are then transformed into independent Gaussian variables and using a Gaussian anamorphosis (see section 2c). The temporal variograms of and are estimated and fitted using two valid temporal variogram models and . Finally, the complete space–time variograms and needed for the simulations are estimated using Eq. (10).

4. Simulation

Generating a simulation means choosing a simulation algorithm. The choice of this algorithm depends on the type of variables to be simulated. An intermittent DSD field consists of two parts: 1) a rainfall indicator field I(x) and 2) a bivariate (μ, Nt) DSD field (for rainy locations only), which can be transformed into two independent Gaussian fields and with zero mean and unit variance. Although different simulation methods can be considered, we focus on a very general technique called sequential simulation (Ripley 1987).

a. Sequential simulation

Consider the case of a random function Z(x) with M known realizations z(x1), …p, z(xM). Suppose that Z(x) needs to be simulated at N new data points xM+1, … , xM+N, conditionally to the previous values z(x1), … , z(xM). The sequential simulation paradigm states that Z(xi), i = M + 1, … , M + N can be simulated sequentially by randomly sampling from the conditional distribution P{Z(xi) < z(xi) | z(x1), … , z(xi−1)} and including the outcome z(xi) in the conditioning dataset for the next step. Hence, the same scheme can be used in theory to generate both conditional (M > 0) and unconditional simulations (M = 0). The practical difficulty is that, in general, the conditional probabilities are not known, except for the Gaussian case (with known mean) for which the conditional distribution of Z(xi) knowing {z(xj)}j<i is Gaussian with mean Z*(xi) (the simple kriging estimator at xi) and variance (the associated kriging variance) (e.g., Chilès and Delfiner 1999, p. 164). Note also that for indicator fields (i.e., fields that take only 0 and 1 values) the conditional distribution is equal to the conditional expectation E[Z(xi) | z(x1), … , z(xi−1)], which is not known in general but can be estimated like in the Gaussian case by the simple kriging estimate Z*(xi) (Alabert 1987). The only theoretical problem with this method is that Z*(xi) can be less than 0 or greater than 1, even for very simple variogram models. If this is the case, the corresponding probabilities are set to 0 or 1 accordingly. At the end of the simulation, the variogram of the simulated indicator field is checked against the model to discard possible artifacts due to these order-relation violations.

b. DSD simulation algorithm

This paragraph provides a detailed step-by-step description of the algorithm used to generate intermittent DSD fields in space and time. For a visual diagram of the simulation algorithm, see Fig. 1. First, sequential indicator simulation (SIS) is used to generate an indicator field with mean pW and space–time structure given by γI(h, τ). Locations and periods for which the outcome of the simulation is 1 are considered wet. The others are considered dry. Only the wet periods and locations are used to simulate the DSD parameters. In the next step, sequential Gaussian simulation (SGS) is used to generate a Gaussian field for with mean 0 and space–time structure given by . A second Gaussian field with mean 0 and space–time structure given by is generated for . At the end of the simulations, an inverse anamorphosis is applied to back transform (, ) into the original parameter space (μ, Nt). If the original time series of μ and Nt were detrended prior to analysis (see appendix), the external drifts (mainly caused by the transitions between dry and rainy periods or locations) are added back to the simulated fields at this stage. Finally, Λ is derived through its deterministic relation with μ. If desired, the simulation procedure can be repeated several times to obtain different realizations in space and time.

Fig. 1.
Fig. 1.

(top to bottom) Visual illustration of the DSD simulation algorithm.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

c. Postprocessing

One of the main advantages of simulating DSD fields is the fact that it is possible to use them to derive (through direct numerical integration) many quantities of interest, including the rain rate R (mm h−1), the median drop diameter D0 (mm), the radar reflectivity Zh|υ (dBZ) at horizontal and vertical polarization, the differential reflectivity Zdr (dB), the specific attenuation Ah|υ (dB km−1) at horizontal and vertical polarization, and the differential phase shift on propagation Kdp (° km−1).

For example, the rain rate is given by
e12
where υ(D) (m s−1) represents the terminal fall speed of a drop of diameter D (e.g., Beard 1977). The median volume diameter D0 is given by
e13
Parameters related to (polarimetric) weather radars like Zh|υ, Ah|υ, and Kdp are given by
e14
e15
e16
where w (cm) represents the radar wavelength; m the complex refractive index of water (at a given temperature); and (cm2) the backscattering extinction cross sections at horizontal and vertical polarization, respectively, for a drop of diameter D; and Re(Shh|υυ) (m) the real part of the forward-scattering amplitudes at horizontal and vertical polarization. All the scattering computations can be done using the T-matrix code by Mishchenko et al. (2002) for a given raindrop axis ratio relationship (e.g., Andsager et al. 1999).

5. Illustration using real data

This section illustrates the capabilities of the previously described DSD simulator using real data collected in the vicinity of Lausanne, Switzerland. Two rain events (frontal and convective) that are very different in terms of magnitude and temporal dynamics are selected to better illustrate the simulator’s capability of reproducing different rainfall types and structures. Note that the goal is neither to predict nor to reproduce the observed events but to generate synthetic rainfall fields sharing the same statistical properties—that is, spatial and temporal structures, intermittency, and raindrop size distributions.

a. Data

The DSD data are collected using a network of 16 optical disdrometers of type Parsivel (Loffler-Mang and Joss 2000) deployed over the École Polytechnique Fédérale de Lausanne (EPFL) campus, Lausanne, Switzerland. The entire network covers an area of approximately 0.5 km2 (Jaffrain et al. 2011). The distances between the stations are between 80 and 800 m. The sampling temporal resolution is 30 s. The uncertainty on the collected measurements has been quantified and extensively documented in Jaffrain and Berne (2011). For validation purposes, the disdrometer data are divided into two groups: eight disdrometers are used for the parameterization and the remaining eight for the validation.

The radar data are provided by MeteoSwiss. The complete rain-rate map is of size 610 × 538 km2 with 1 × 1 km2 resolution. The temporal resolution is 5 min. The rain rates are estimated by combining the measurements of three C-band weather radars at different elevations, correcting for the main sources of errors (ground clutter, beam shielding, and vertical variability) according to the procedure described in Germann et al. (2006). The estimated rain rates are then coded using 16 (irregularly spaced) intensity classes from 0 to 100 mm h−1.

b. Simulation domain

A simulation domain of size 50 × 50 km2 covering the city of Lausanne and the disdrometer network described in section 5a is considered. The domain mostly lies in the so-called “Swiss Plateau,” between the Jura Mountains and the Swiss Alps. The minimum and maximum altitudes are 370 and 2010 m MSL but most of the domain (75%) is lower than 800 m MSL and only 10% is higher than 1100 m MSL. The simulation domain mostly avoids regions like the mountains of Savoie in the south, the Swiss Alps in the east, and the Jura Mountains in the northwest where radar data are known to be noisy and affected by larger uncertainties.

c. Considered events

Two events that occurred on 17 June and 5 August 2010 are selected. The first event is classified (based on visual inspection of disdrometer and operational radar data) as frontal, and the second is convective. The radar rain-rate maps corresponding to these events are shown in Fig. 2 at 0050 and 1355 UTC, respectively. The main characteristics of the selected events are given in Table 1. For simulation purposes, only the time periods during which the events passed over the disdrometer network are considered—that is, from 0050 to 0345 UTC for the first event and from 1355 to 1455 UTC for the second. The time series of μ, Nt, and R corresponding to these time periods are shown in Figs. 3 and 4. Note that Λ is not shown because it is strongly correlated with μ (correlation coefficient of 0.96) and is therefore expressed as a linear model of μ in the simulations. It is also important to point out the differences between the two events (represented on different scales for better readability). The frontal event is characterized by small to moderate values of Nt, moderate to large values of μ (small drops), and limited temporal variability. The convective event, on the other hand, exhibits a lot of temporal variability, large values of Nt (lots of drops), and small values of μ (large drops).

Fig. 2.
Fig. 2.

Radar rain-rate maps for (top) event 1 and (bottom) event 2. The size of the domain is 50 × 50 km2. The location of the 16 disdrometers is represented by a white cross in the lower-left corner.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

Table 1.

Rainfall type, intermittency, advection, and anisotropy for each selected event.

Table 1.
Fig. 3.
Fig. 3.

Time series of (top) μ, (middle) Nt (m−3), and (bottom) R (mm h−1) at 30-s time resolution for event 1 (frontal). The three lines represent the min, avg, and max values measured by the 16 disdrometers at each time step.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

Fig. 4.
Fig. 4.

Same as Fig. 3 but for event 2 (convective).

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

d. Parameterization

For each selected event, the spatial and temporal sample variograms of the rainfall indicator field are computed using the radar rain-rate data (see section 2b). Each sample variogram is fitted using a spherical variogram model:
e17
where n is called the nugget, s the (partial) sill, and r the range. The fitted values of nugget, sill, and range for each variogram are given in Table 2.
Table 2.

Fitted values of nugget, partial sill, and range for the rainfall indicator variograms.

Table 2.

Next, the time series of and are used to compute the temporal variograms of the DSD parameters. A sum of two spherical variograms—one for the small-scale variability and one for the large-scale variability—is fitted to each sample variogram (see Fig. 5). The fitted values of nugget, sills, and ranges are given in Table 3. The temporal variograms are then used to define the complete space–time variograms of and as described in section 2d.

Fig. 5.
Fig. 5.

Fitted temporal variograms of and for (top) event 1 and (bottom) event 2. For details about the variogram parameters, see Table 3.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

Table 3.

Fitted values of nugget, partial sills, and ranges for the temporal variograms of and .

Table 3.

e. Simulated DSD fields

For illustration purposes, two 1-h simulations with a temporal resolution of 1 min and a spatial resolution of 500 × 500 m2 have been generated. This turned out to be a good trade-off between the required simulation time (a few minutes on a standard desktop computer) and the ability to reproduce most of the space–time dynamics occurring in rainfall and DSD at these scales. All the simulations were performed on a standard desktop computer using the statistical computing software “R” (R Development Core Team 2011) and the “gstat” package by Pebesma (2004). The complete animations of the simulated DSD fields with associated rain rate and median drop diameter are given in the supplemental online material. Snapshots illustrating the simulated fields (at some given time) are provided in Figs. 6 and 7. The associated radar bulk variables Zh, Zdr, Ah, and Kdp at X band (9.41 GHz) and 20°C are shown in Figs. 8 and 9.

Fig. 6.
Fig. 6.

Snapshot of a simulated field of μ, Nt, D0, and R for the frontal event. The simulation size is 50 × 50 km2 and corresponds to the domain shown in Fig. 2, and the pixel resolution is 500 × 500 m2.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

Fig. 7.
Fig. 7.

Same as Fig. 6 but for the convective event.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

Fig. 8.
Fig. 8.

Fields of Zh, Zdr, Ah, and Kdp corresponding to Fig. 6.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

Fig. 9.
Fig. 9.

Fields of Zh, Zdr, Ah, and Kdp corresponding to Fig. 7.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

f. Evaluation

The quality and realism of the simulated DSD fields are difficult to evaluate quantitatively. It is, for example, not possible to compare the simulated fields with observed radar data because the latter were used to parameterize the simulator. An “indirect” evaluation of the simulated fields based on the measurements of the eight remaining disdrometers discarded during the parameterization is nevertheless proposed. The idea is to compare the statistical properties and the temporal structure of the simulated and observed time series of μ and Nt. For each location x in the simulation domain, the simulated values of μ and Nt (at this particular location) are extracted. This gives, for each location, a 1-h time series of simulated DSD parameters. For each location, the mean and of the extracted time series is computed. To validate the simulations, the distributions of and are compared to their equivalents obtained using the time series from the eight “control” disdrometers (see Fig. 10). One can see that the simulated and observed values of and are in good agreement but that the simulated mean values have a much larger dispersion than the control values. This can, however, be explained by the fact that there are far more simulated time series (one for each pixel in the simulation domain) than measured time series (only eight control disdrometers).

Fig. 10.
Fig. 10.

Simulated distributions of (left) and (right) for (top) event 1 and (bottom) event 2. Each box plot represents the 10%, 25%, 50%, 75%, and 90% quantiles of all simulated values. The arrow next to each box plot represents the range of and for the eight control disdrometers.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

To evaluate the simulator’s capabilities to reproduce correct temporal structures, the variograms of the simulated time series of and are compared to the variograms obtained from the eight control disdrometers (see Fig. 11). Again, the simulated variograms exhibit a larger dispersion than the control variograms but the overall agreement between the variograms is good. This suggests that the simulated fields exhibit realistic temporal structures that are consistent with the control disdrometers. Finally, it is also important to point out that both events (frontal and convective) were statistically well reproduced by the simulator despite their different nature.

Fig. 11.
Fig. 11.

Simulated and measured temporal variograms of (left) and (right) for (top) event 1 and (bottom) event 2. The black dotted lines represent the 10% and 90% quantiles of the simulated variograms for each pixel in the simulation domain. The continuous black line represents the median of all simulated variograms. The dashed red lines represent the min and the max of the eight variograms computed using the control disdrometers.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

6. Discussion

One of the main advantages of the proposed simulation technique is its simplicity: it uses less than 10 parameters to characterize the complete space–time structure of the DSD fields, including intermittency and advection. Furthermore, each of these parameters can be easily estimated using DSD time series and radar rain-rate or reflectivity data. As a result, hundreds of simulations can be performed (in a reasonable time) on a single standard desktop computer. There are, however, some limitations that have to be mentioned.

The size of the simulation domain is limited by the fact that the variograms of and must be representative of the variability of the DSD over the entire domain. Areas with different variabilities (due, for example, to orographic effects) should be separated and treated individually. As a rule of thumb, a reasonable simulation domain for convective events should not exceed 50 × 50 km2 and 100 × 100 km2 for stratiform events. Another factor that strongly limits the size of the simulation domain is the particular form of the space–time structure assumed in Eq. (10), which essentially states that the advection field must be constant over the entire domain. Local changes in advection within the simulation domain (for example, due to mountains or a coastal line) cannot be reproduced correctly. The only possible generalization is to consider an average advection field that changes over periods of 20–30 min and to simulate each time block separately.

The presented simulation technique only considers two-dimensional fields of DSD at the ground level (plus an additional temporal dimension). No assumptions with respect to the vertical variability and structure of the DSD are made. In theory, three-dimensional fields of DSD could also be simulated using the technique presented in this article. In practice, however, this turns out to be far more difficult because the horizontal and vertical dimensions exhibit completely different dynamics and structures. Nevertheless, if adequate parameterizations of the vertical variability (i.e., variogram) of μ and Nt can be provided, an extension to three-dimensional fields of DSD (using the generated 2D plus time fields as a starting point) is possible. This is particularly interesting for applications in which the vertical properties of the DSD in the atmosphere are more important than on the surface (e.g., for space-borne weather radars and telecommunication applications).

Finally, it must be noted that there is currently no direct way of controlling complex physical processes like the growth, the decay, the merging, and the splitting of individual rain cells in the simulations. Individual rain cells may emerge or split up during the simulations by “chance” (i.e., without any particular forcing or additional parameterization) but this is essentially due to “side effects” (e.g., a rain cell that is advected outside the simulation domain causes another cell to appear elsewhere) and cannot be related to any microphysical processes or atmospheric dynamics. Introducing such a forcing into the simulations is difficult and far beyond the scope of this paper. In the simplified case with just one rain cell, additional external drifts on I, , and (similar to the one presented in the appendix) could be used to introduce such dynamics into the simulations. For example, a temporal drift on I can be used to control the percentage of dry and wet locations in the simulation domain and therefore the size of the considered cell with respect to its lifetime. Similarly, a temporal drift on can be used to control the increase or decrease of the average drop diameter within the cell. Finally, a temporal drift on allows for controlling the average drop concentration with respect to the lifetime of the cell. It is, however, not clear how this technique could be extended to the case of multiple interacting rain cells with different lifetimes and dynamics.

7. Conclusions

Precipitation is an essential part of the hydrological, atmospheric, and climatic system and must therefore be monitored and measured with great accuracy. Yet the large spatial and temporal variability of precipitation appears to be an inevitable source of uncertainty in many practical applications. Modern hydrological and atmospheric models require accurate rainfall estimates with high spatial and temporal resolutions, which is sometimes far beyond the capabilities of current rainfall sensors. In these cases, a stochastic rainfall simulator can be a very valuable tool because it provides large amounts of reference data at high spatial and temporal resolutions. These reference data can be used to investigate various aspects and algorithms in the models. For example, they can be used to evaluate the performance of new algorithms for the merging of data collected using different sensors and techniques (e.g., weather radars, rain gauges, disdrometers, and microwave links). Other possible applications include the study of the ZR relationship for weather radars, attenuation correction algorithms at C and X bands, downscaling of precipitation and reflectivity fields, soil erosion problems, atmospheric deposition, and signal attenuation of telecommunication microwave links.

In this article, a new method for the stochastic simulation of intermittent DSD fields in time has been presented. The method generalizes previous work done by Barancourt et al. (1992) and Schleiss et al. (2009) and uses a combination of sequential indicator simulation and sequential Gaussian simulation. A multivariate Gaussian anamorphosis is used prior to the simulation in order to transform the original DSD parameters into independent normalized Gaussian variables. External drifts are used to ensure the continuity between dry and rainy locations. In section 5, the simulator’s ability to reproduce complex and different rainfall patterns using real data collected in Lausanne, Switzerland has been illustrated. An indirect evaluation of the simulations suggests that the simulated DSD fields exhibit realistic space–time structures that are in good agreement with independent DSD measurements.

The main advantage of the presented simulator is its simplicity and its short computation time. The main limitation is the inability to reproduce the interactions (growth, decay, merging, and splitting) of multiple rain cells. Future work will mainly focus on practical applications of the DSD simulator to problems related to the downscaling of DSD fields and to the statistical representativity of radar and satellite measurements.

Acknowledgments

The authors acknowledge the financial support from the Swiss National Science Foundation (Grants 200021-118057 and 200020-132002).

APPENDIX

Detrending the DSD Time Series

Recall that the variogram is only defined for intrinsic random functions (see Chilès and Delfiner 1999). Time series of μ and Nt usually do not satisfy this condition because E[μ(t)] and E[Nt(t)] depend (nonlinearly) on t. Typically, the beginning and the end of each event is characterized, on average, by lower rain rates. Therefore, the expected rain-rate drifts toward zero as one approaches the next dry period. Similar continuity conditions also affect each of the DSD parameters. The characteristics of this drift are event dependent and must be estimated and removed before computing the sample variograms. Finding appropriate methods to estimate these drifts can be difficult and should be done with great care. In the following, a possible method for detrending DSD time series that takes into account the proximity (in time) to the nearest dry period is proposed. First, a logarithmic transform is applied to μ > −1 and Nt > 0:
ea1
ea2
The distribution of the log-transformed parameters is usually more symmetric and hence better suited for statistical analysis. For a measured DSD time series (μ, Nt, I)(ti), the sets D = {i | I(ti) = 0} and W = {i | I(ti) = 1} of all dry and wet time periods are identified. For all iW, the distance τi = minjD{|titj|} to the nearest dry period is computed. Finally, the drifts mμ(τ) and are estimated for each (transformed) DSD parameter:
ea3
ea4
where N(τ) represents the number of observations at an (approximate) time lag τ from the nearest dry period. A (simple) theoretical drift model is fitted to the sample estimates and used to detrend the time series of μ and Nt (in the logarithmic space):
ea5
ea6
An example of this technique is shown in Fig. A1, where the estimated drift functions mμ(τ) and mμ(τ) are shown for the convective rainfall event presented in section 5. Note that usually increases with τ, whereas for mμ(τ), it is the opposite.
Fig. A1.
Fig. A1.

Estimated drift functions for log10(μ + 1) and log10(Nt) for the convective event.

Citation: Journal of Hydrometeorology 13, 2; 10.1175/JHM-D-11-018.1

After removing the drift, a Gaussian anamorphosis is applied to the DSD time series. All the simulations are performed in the Gaussian space. At the end, the simulated variables are back transformed using an inverse anamorphosis and the drift (depending on the distance to the nearest simulated dry location or period) is added. Finally, the inverse transform corresponding to (A1) and (A2) is applied to retrieve the parameters in the original parameter space. The logarithmic transform ensures that only valid values of μ > −1 and Nt > 0 are produced after adding the drifts to the simulations.

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