## 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 *Z*–*R* 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

*μ*, Λ, and

*N*), exactly as in section 2.1 of Schleiss et al. (2009):Because of the natural variability of rainfall, the DSD parameters (

_{t}*μ*, Λ, and

*N*) 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:where

_{t}*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

*N*and a deterministic relationship between Λ and

_{t}*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

*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: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:Using this notation, an intermittent DSD field (at a given time

*t*and location

*x*) is given by a triplet (

*I*,

*μ*

^{+}, and

*I*represents the rainfall indicator field and (

*μ*

^{+},

*I*,

*μ*,

*N*).

_{t}### 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 *N _{t}*. Simulating realistic DSD fields means generating independent realizations of (

*μ*,

*N*) 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 (

_{t}*μ*,

*N*) into independent Gaussian variables

_{t}### d. Space–time structure

*I*,

*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:where

*γ*(

_{S}**h**) and

*γ*(

_{T}*τ*) represent the spatial respectively temporal variogram of

*Z*(

*x*,

*t*). For the rainfall indicator field

*I*, both

*γ*(

_{S}**h**) and

*γ*(

_{T}*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

*γ*(

_{S}**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

*Z*

_{dr}and the reflectivity at horizontal polarization

*Z*can also be used to estimate the spatial structure of

_{h}**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**=

*τ*

**v**, or equivalently,

*γ*(

_{S}*τ*

**v**) =

*γ*(

_{T}*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:where 0 <

*α*≤ 1 is called the anisotropy ratio,

**h**,

**a**〉 denotes the standard scalar product between

**h**and

**a**, and the double vertical bars denote the norm in

*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,

**v**

_{i}, an average anisotropy direction

**a**

_{i}, and an average anisotropy ratio

*α*(

_{i}**h**). The space–time variogram of each time period is then given by

## 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 km^{2} 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 *p _{W}* of wet and

*p*= 1 −

_{D}*p*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

_{W}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 **a** and the corresponding (average) anisotropy ratio *α*.

### c. Parameters derived from DSD data

First, the three-parameter DSD model (*μ*, Λ, and *N _{t}*) 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*, *N _{t}*) 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

*N*. The detrended time series of (

_{t}*μ*,

*N*) are then transformed into independent Gaussian variables

_{t}## 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*, *N _{t}*) DSD field (for rainy locations only), which can be transformed into two independent Gaussian fields

### a. Sequential simulation

Consider the case of a random function *Z*(*x*) with *M* known realizations *z*(*x*_{1}), …p, *z*(*x _{M}*). Suppose that

*Z*(

*x*) needs to be simulated at

*N*new data points

*x*

_{M+1}, … ,

*x*

_{M+N}, conditionally to the previous values

*z*(

*x*

_{1}), … ,

*z*(

*x*). The sequential simulation paradigm states that

_{M}*Z*(

*x*),

_{i}*i*=

*M*+ 1, … ,

*M*+

*N*can be simulated sequentially by randomly sampling from the conditional distribution

*P*{

*Z*(

*x*) <

_{i}*z*(

*x*) |

_{i}*z*(

*x*

_{1}), … ,

*z*(

*x*

_{i−1})} and including the outcome

*z*(

*x*) in the conditioning dataset for the next step. Hence, the same scheme can be used in theory to generate both conditional (

_{i}*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*(

*x*) knowing {

_{i}*z*(

*x*)}

_{j}_{j<i}is Gaussian with mean

*Z**(

*x*) (the simple kriging estimator at

_{i}*x*) and variance

_{i}*E*[

*Z*(

*x*) |

_{i}*z*(

*x*

_{1}), … ,

*z*(

*x*

_{i−1})], which is not known in general but can be estimated like in the Gaussian case by the simple kriging estimate

*Z**(

*x*) (Alabert 1987). The only theoretical problem with this method is that

_{i}*Z**(

*x*) 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.

_{i}### 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 *p _{W}* and space–time structure given by

*γ*(

_{I}*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

*μ*,

*N*). If the original time series of

_{t}*μ*and

*N*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

_{t}*μ*. If desired, the simulation procedure can be repeated several times to obtain different realizations in space and time.

### 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 *D*_{0} (mm), the radar reflectivity *Z*_{h|υ} (dB*Z*) at horizontal and vertical polarization, the differential reflectivity *Z*_{dr} (dB), the specific attenuation *A*_{h|υ} (dB km^{−1}) at horizontal and vertical polarization, and the differential phase shift on propagation *K*_{dp} (° km^{−1}).

*υ*(

*D*) (m s

^{−1}) represents the terminal fall speed of a drop of diameter

*D*(e.g., Beard 1977). The median volume diameter

*D*

_{0}is given byParameters related to (polarimetric) weather radars like

*Z*

_{h|υ},

*A*

_{h|υ}, and

*K*

_{dp}are given bywhere

*w*(cm) represents the radar wavelength;

*m*the complex refractive index of water (at a given temperature);

^{2}) the backscattering extinction cross sections at horizontal and vertical polarization, respectively, for a drop of diameter

*D*; and Re(

*S*

_{hh|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 km^{2} (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 km^{2} with 1 × 1 km^{2} 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 km^{2} 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*, *N _{t}*, 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

*N*, moderate to large values of

_{t}*μ*(small drops), and limited temporal variability. The convective event, on the other hand, exhibits a lot of temporal variability, large values of

*N*(lots of drops), and small values of

_{t}*μ*(large drops).

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

### d. Parameterization

*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.

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

Next, the time series of

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

### 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 m^{2} 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 *Z _{h}*,

*Z*

_{dr},

*A*, and

_{h}*K*

_{dp}at X band (9.41 GHz) and 20°C are shown in Figs. 8 and 9.

### 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 *N _{t}*. For each location

*x*in the simulation domain, the simulated values of

*μ*and

*N*(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

_{t}To evaluate the simulator’s capabilities to reproduce correct temporal structures, the variograms of the simulated time series of

## 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 ^{2} and 100 × 100 km^{2} 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 *N _{t}* 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*, *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

## 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 *Z*–*R* 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

*μ*and

*N*usually do not satisfy this condition because

_{t}*E*[

*μ*(

*t*)] and

*E*[

*N*(

_{t}*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

*N*> 0:The distribution of the log-transformed parameters is usually more symmetric and hence better suited for statistical analysis. For a measured DSD time series (

_{t}*μ*,

*N*,

_{t}*I*)(

*t*), the sets

_{i}*D*= {

*i*|

*I*(

*t*) = 0} and

_{i}*W*= {

*i*|

*I*(

*t*) = 1} of all dry and wet time periods are identified. For all

_{i}*i*∈

*W*, the distance

*τ*= min

_{i}_{j∈D}{|

*t*−

_{i}*t*|} to the nearest dry period is computed. Finally, the drifts

_{j}*m*(

_{μ}*τ*) and

*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

*N*(in the logarithmic space):An example of this technique is shown in Fig. A1, where the estimated drift functions

_{t}*m*(

_{μ}*τ*) and

*m*(

_{μ}*τ*) are shown for the convective rainfall event presented in section 5. Note that

*τ*, whereas for

*m*(

_{μ}*τ*), it is the opposite.

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 *N _{t}* > 0 are produced after adding the drifts to the simulations.

## REFERENCES

Alabert, F., 1987: Stochastic imaging of spatial distributions using hard and soft information. M.S. thesis, Dept. of Applied Earth Sciences, Stanford University, 198 pp.

Andsager, K., , Beard K. V. , , and Laird N. F. , 1999: Laboratory measurements of axis ratios for large rain drops.

,*J. Atmos. Sci.***56**, 2673–2683.Barancourt, C., , Creutin J.-D. , , and Rivoirard J. , 1992: A method for delineating and estimating rainfall fields.

,*Water Resour. Res.***28**, 1133–1144.Battan, L. J., 1973:

*Radar Observation of the Atmosphere*. University of Chicago Press, 324 pp.Beard, K. V., 1977: Terminal velocity adjustment for cloud and precipitation drops aloft.

,*J. Atmos. Sci.***34**, 1293–1298.Beier, C., , Hansen K. , , and Gundersen P. , 1993: Spatial variability of throughfall fluxes in a spruce forest.

,*Environ. Pollut.***81**, 257–267.Berne, A., , and Uijlenhoet R. , 2005: A stochastic model of range profiles of raindrop size distributions: Application to radar attenuation correction.

,*Geophys. Res. Lett.***32**, L10803, doi:10.1029/2004GL021899.Bouvier, C., , Cisneros L. , , Dominguez R. , , Laborde J.-P. , , and Lebel T. , 2003: Generating rainfall fields using principal components (PC) decomposition of the covariance matrix: A case study in Mexico City.

,*J. Hydrol.***278**(1–4), 107–120.Bringi, V. N., , Chandrasekar V. , , Balakrishnan N. , , and Zrnić D. S. , 1990: An examination of propagation effects in rainfall on radar measurements at microwave frequencies.

,*J. Atmos. Oceanic Technol.***7**, 829–840.Chilès, J.-P., , and Delfiner P. , 1999:

*Geostatistics: Modeling Spatial Uncertainty*. Wiley Series in Probability and Statistics, Applied Probability and Statistics, Wiley, 695 pp.Crane, R. K., 1989: Automatic cell detection and tracking.

,*IEEE Trans. Geosci. Electron.***17**, 250–262.Delrieu, G., , Hucke L. , , and Creutin J.-D. , 1999: Attenuation in rain for X- and C-band weather radar systems: Sensitivity with respect to the drop size distribution.

,*J. Appl. Meteor.***38**, 57–68.Germann, U., , Galli G. , , Boscacci M. , , and Bolliger M. , 2006: Radar precipitation measurement in a mountainous region.

,*Quart. J. Roy. Meteor. Soc.***132**, 1669–1692, doi:10.1256/qj.05.190.Guillot, G., 1999: Approximation of Sahelian rainfall fields with meta-Gaussian random functions—Part 1: Model definition and methodology.

,*Stochastic Environ. Res. Risk Assess.***13**(1–2), 100–112.Gupta, V. K., , and Waymire E. , 1993: A statistical analysis of mesoscale rainfall as a random cascade.

,*J. Appl. Meteor.***32**, 251–267.Jaffrain, J., , and Berne A. , 2011: Experimental quantification of the sampling uncertainty associated with measurements from PARSIVEL disdrometers.

,*J. Hydrometeor.***12**, 352–370.Jaffrain, J., , Studzinski A. , , and Berne A. , 2011: A network of disdrometers to quantify the small-scale variability of the raindrop size distribution.

,*Water Resour. Res.***47**, W00H06, doi:10.1029/2010WR009872.Journel, A. G., , and Huijbregts C. J. , 1978:

*Mining Geostatistics*. Academic Press, 600 pp.Katz, R. W., 1977: An application of chain-dependent processes to meteorology.

,*J. Appl. Probab.***14**, 598–603.Kinnell, P. I. A., 2005: Raindrop-impact-induced erosion processes and prediction: A review.

,*Hydrol. Processes***19**, 2815–2844.Koster, R. D., , and Suarez M. J. , 1995: Relative contributions of land and ocean processes to precipitation variability.

,*J. Geophys. Res.***100**(D7), 775–790.Krajewski, W. F., , and Smith J. A. , 2002: Radar hydrology: Rainfall estimation.

,*Adv. Water Resour.***25**, 1387–1394.Krajewski, W. F., , Raghavan R. , , and Chandrasekar V. , 1993: Physically based simulation of radar rainfall data using a space–time rainfall model.

,*J. Appl. Meteor.***32**, 268–283.Lavergnat, J., , and Golé P. , 2006: A stochastic model of raindrop release: Application to the simulation of point rain observations.

,*J. Hydrol.***328**(1–2), 8–19.Lee, G., , Seed A. W. , , and Zawadzki I. , 2007: Modeling the variability of drop size distributions in space and time.

,*J. Appl. Meteor. Climatol.***46**, 742–756.Leijnse, H., , Uijlenhoet R. , , and Stricker J. N. M. , 2007: Rainfall measurement using radio links from cellular communication networks.

,*Water Resour. Res.***43**, W03201, doi:10.1029/2006WR005631.Leuangthong, O., , and Deutsch C. V. , 2003: Stepwise conditional transformation for simulation of multiple variables.

,*Math. Geol.***35**, 155–173.Li, B., , Murthi A. , , Bowman K. , , North G. , , Genton M. , , and Sherman M. , 2009: Statistical tests of Taylor’s hypothesis: An application to precipitation fields.

,*J. Hydrometeor.***10**, 254–265.Loffler-Mang, M., , and Joss J. , 2000: An optical disdrometer for measuring size and velocity of hydrometeors.

,*J. Atmos. Oceanic Technol.***17**, 130–139.Matheron, G., 1965:

*Les Variables Régionalisées et Leur Estimation*. Masson et Cie, 305 pp.Menabde, M., , Seed A. , , Harris D. , , and Austin G. , 1997: Self-similar random fields and rainfall simulation.

,*J. Geophys. Res.***102**(D12), 13 509–13 515.Messer, H., , Zinevich A. , , and Alpert P. , 2006: Environmental monitoring by wireless communication networks.

,*Science***312**, 713, doi:10.1126/science.1120034.Mishchenko, M., , Travis L. , , and Lacis A. , 2002:

*Scattering, Absorption, and Emission of Light by Small Particles*. Cambridge University Press, 445 pp.Moisseev, D. N., , and Chandrasekar V. , 2007: Examination of the

*μ*-*δ*relation suggested for drop size distribution parameters.,*J. Atmos. Oceanic Technol.***24**, 847–855.Pebesma, E. J., 2004: Multivariate geostatistics in S: The gstat package.

,*Comput. Geosci.***30**, 683–691, doi:10.1016/j.cageo.2004.03.012.Pruppacher, H. R., , and Klett R. L. , 1997:

*Microphysics of Clouds and Precipitation*. 2nd ed. Atmospheric and Oceanographic Sciences Library, Vol. 18, Kluwer Academic Press, 954 pp.R Development Core Team, 2011:

*R: A Language and Environment for Statistical Computing*. R Foundation for Statistical Computing, 3460 pp. [Available online at http://www.R-project.org/.]Richardson, C., 1981: Stochastic simulation of daily precipitation, temperature and solar radiation.

,*Water Resour. Res.***17**, 182–190.Rinehart, R. E., 1979: Internal storm motions from a single non-doppler weather radar. Ph.D. dissertation, Colorado State University, 280 pp.

Ripley, B. D., 1987:

*Stochastic Simulation*. Wiley, 237 pp.Rodríguez-Iturbe, I., , and Eagleson P. S. , 1987: Mathematical models of rainstorm events in space and time.

,*Water Resour. Res.***23**, 181–190.Roe, G. H., 2005: Orographic precipitation.

,*Annu. Rev. Earth Planet. Sci.***33**, 645–671.Schleiss, M. A., , Berne A. , , and Uijlenhoet R. , 2009: Geostatistical simulation of two-dimensional fields of raindrop size distributions at the meso-

*γ*scale.,*Water Resour. Res.***45**, W07415, doi:10.1029/2008WR007545.Seifert, A., 2005: On the shape-slope relation of drop size distributions in convective rain.

,*J. Appl. Meteor.***44**, 1146–1151.Sivapalan, M., , and Wood E. F. , 1987: A multidimensional model of nonstationary space-time rainfall at the catchment scale.

,*Water Resour. Res.***23**, 1289–1299.Smith, J. A., , Hui E. , , Steiner M. , , Baeck M. L. , , Krajewski W. , , and Ntelekos A. A. , 2009: Variability of rainfall rate and raindrop size distributions in heavy rain.

,*Water Resour. Res.***45**, W04430, doi:10.1029/2008WR006840.Taylor, G. I., 1938: The spectrum of turbulence.

,*Proc. Roy. Soc. London***164**, 476–490.Testud, J., , Le Bouar E. , , Obligis E. , , and Ali-Mehenni M. , 2000: The rain profiling algorithm applied to polarimetric weather radar.

,*J. Atmos. Oceanic Technol.***17**, 332–356.Tuttle, J. D., , and Foote G. B. , 1990: Determination of the boundary layer airflow from a single Doppler radar.

,*J. Atmos. Oceanic Technol.***7**, 218–232.Ulbrich, C. W., 1983: Natural variations in the analytical form of the raindrop size distribution.

,*J. Climate Appl. Meteor.***22**, 1764–1775.Ulbrich, C. W., 1985: The effects of drop size distribution truncation on rainfall integral parameters and empirical relations.

,*J. Climate Appl. Meteor.***24**, 580–590.Ulbrich, C. W., , and Atlas D. , 2007: Microphysics of raindrop size spectra: Tropical continental and maritime storms.

,*J. Appl. Meteor. Climatol.***46**, 1777–1791.Villarini, G., , and Krajewski W. F. , 2010: Review of the different sources of uncertainty in single polarization radar-based estimates of rainfall.

,*Surv. Geophys.***31**, 107–129.Waymire, E., , Gupta V. K. , , and Rodriguez-Iturbe I. , 1984: A spectral theory of rainfall intensity at the meso-

*β*scale.,*Water Resour. Res.***20**, 1453–1465.Wilks, D. S., 1999: Simultaneous stochastic simulation of daily precipitation, temperature and solar radiation at multiple sites in complex terrain.

,*Agric. For. Meteor.***96**(1–3), 85–101.Willis, P. T., 1984: Functional fits to some observed drop size distributions and parameterization of rain.

,*J. Atmos. Sci.***41**, 1648–1661.Wilson, J. M., , and Brandes E. A. , 1979: Radar measurement of rainfall—A summary.

,*Bull. Amer. Meteor. Soc.***60**, 1048–1058.Zhang, G., , Vivekanandan J. , , and Brandes E. , 2001: A method for estimating rain rate and drop size distribution from polarimetric radar measurements.

,*IEEE Trans. Geosci. Remote Sens.***39**, 830–841.