## 1. Introduction

An inherent difficulty in measurement of precipitation is that of change of support. The support of a measurement may be a point or an areal region, and at small scales it may be tempting to assume the two are equivalent. In this paper we study the small-scale subgrid variability of the drop size distribution (DSD) in order to answer two questions. First, what error is introduced by assuming that a point measurement of precipitation represents an areal region? Second, if an estimate of the DSD is derived from areal precipitation measurements, how representative is it of the underlying subgrid precipitation process?

The DSD statistically characterizes the microstructure of precipitation. All bulk rainfall variables of interest can be derived from the DSD (e.g., Ulbrich 1983; Testud et al. 2001). It is well known that the DSD, and thus precipitation, is highly variable in time and space (e.g., Jameson and Kostinski 2001; Uijlenhoet et al. 2003; Jaffrain and Berne 2012b). Subgrid variability of the DSD and nonlinearities between rainfall variables imply that any assumption of equivalence between areal and point precipitation measurements introduces error. Take, for example, a weather radar that measures electromagnetic radiation reflected off hydrometeors within a particular volume of air. This radar reflectivity *Z* (mm^{6} m^{−3}) is related to rain intensity *R* (mm h^{−1}) via the DSD (e.g., Marshall and Palmer 1948; Uijlenhoet 2001), and this relationship is known to be scale dependent (Verrier et al. 2013; Sassi et al. 2014). To calculate *R* from *Z*, one of two assumptions regarding scale is usually made. The first assumption is that a point measurement can represent a pixel. For example, a point measurement of the DSD is used to relate *R* to *Z*, or a point measurement of *R* is related to an areal measurement of *Z*. In the second assumption, an areal measurement is assumed to be representative of the subgrid process. For example, the radar measurements are used to infer properties of a DSD model that is assumed to describe the areal DSD from which *R* can be calculated. This approach makes the assumption that the retrieved DSD model is representative at the pixel scale. None of these approaches takes subgrid variability of the DSD into account.

Previous studies of DSD variability have involved the use of single disdrometers and investigation of DSD variability over time series measurements (e.g., Uijlenhoet et al. 2003; Lee and Zawadzki 2005; Chapon et al. 2008). Jameson (2015) provided a technique for upscaling single disdrometer measurements, which produces gridded simulations of nonparametric point DSDs that honor the statistical properties of the observations. The resulting gridded fields are useful for statistical characterization of the rainfall field, but are not appropriate for direct comparison to fields of observations. Lee et al. (2007) derived the spatial and temporal distributions of the DSD, using radar data, a time series of point DSD measurements, and a double-moment normalized DSD model. They found that with two DSD moments (measured *Z* and simulated *R*) they were able to capture the bulk, but not all, of the variability in the DSD.

Other studies have used networks of disdrometers to examine DSD variability in space. Miriovsky et al. (2004) used a network of four disdrometers of different types within a 1-km^{2} region in Iowa, United States, and reported that radar reflectivity was highly variable within this area. However, large instrumental differences meant they were unable to determine quantitative variability, and the study was focused on the *Z*–*R* relationship. Lee et al. (2009) used four disdrometers with interspacings up to about 33 km, combined with radar data, to study four stratiform rain events in Montreal, Canada. Variability in DSD moments and bulk variables was found even over small distance lags, and lower-order DSD moments were found to be slightly more correlated than moments of higher order, indicating that large drops play a larger role than small drops in the variability of the DSD. This study was limited by the network that was not specifically designed for DSD variability research. Tokay and Bashor (2010) used a network of three disdrometers to quantify variability in DSD model parameters and bulk variables along a 1.7-km linear transect on Wallops Island, Virginia (United States), with the goal of studying DSD variability within a typical ground-based radar pixel (2 km × 2 km). This network was limited by the small number of instruments. Tapiador et al. (2010) set up a network of 16 disdrometers at eight locations over a 4 km × 4 km area near Ciudad Real, Spain. They found large spatial variability at kilometer scale, but the analysis focused on bulk variables and not the DSD itself. Jaffrain and Berne (2012b) used a network of 16 disdrometers over a 1-km^{2} region [described in Jaffrain et al. (2011)] to observe the DSD over the area of a typical operational weather radar pixel in Lausanne, Switzerland. Using stochastic simulation of characteristic drop diameter, total drop concentration, and rain rate, they found that the assumption that a point measurement represents an areal estimate introduced close to Gaussian errors with zero mean and significant standard deviations. For example, at 60-s resolution and over the whole network, the error standard deviation was up to 25% for rain rate in convective cases. Noise in the data overtook natural DSD variability for time steps longer than 30 min. Using the same network of disdrometers, Jaffrain and Berne (2012a) studied the subgrid variability of the power laws used for rain-rate estimation from radar. When the power laws were trained using point measurements, then applied at 1-km^{2} scale, an error in rain-rate estimation of between −2% and +15% was observed. More recently, Jameson et al. (2015a,b) studied DSD variability using a network of 21 logarithmically spaced disdrometers over a very small area (distance lags of 1–100 m) in South Carolina, United States. Jameson et al. (2015a) found that spatial and temporal clustering structures of the DSD were significantly different from each other and that they differed across drop size classes.

Capturing the full variability of the DSD is nontrivial. In an instrument network, it is a challenge to place enough instruments to measure the full DSD variability. For example, Tapiador et al. (2010) found that at least six disdrometers would be required to accurately capture the full spatial variability of the DSD at kilometer scale. Efforts to address this issue have included stochastic simulation techniques to estimate areal DSD gamma model parameters (e.g., Schleiss et al. 2012). Stochastic simulation was used by Jaffrain and Berne (2012b) to quantify DSD model parameter variability. Jameson et al. (2015b) highlighted that it is inadequate to evaluate DSD variability using only integral variables, because different DSDs can produce the same integral variable values. Studies using disdrometer networks have drawn some broad conclusions: that DSD variability within an event can be larger than that between events (Tapiador et al. 2010; Jaffrain and Berne 2012b), and that variability increases with greater domain size (Jaffrain and Berne 2012b; Jameson et al. 2015b) and greater drop size (Jameson et al. 2015b) and decreases with temporal integration (Jaffrain and Berne 2012b; Tokay and Bashor 2010).

In this paper, we focus on DSD volumetric drop concentrations in their measured drop size classes, without assuming any functional form of the DSD. This is in contrast to previous studies that have, for the most part, focused on bulk variables or parameters of a DSD model. Jameson (2015) also simulated nonparametric DSDs; their approach differed from ours in that they used Bayesian upsizing and only a single disdrometer. We present results that used a geostatistical approach (Raupach and Berne 2016) to perform stochastic simulation of measured DSDs on a regular, high-resolution grid of points. Simulations were constrained by measured experimental DSDs from a network of instruments. These gridded simulations allowed us to estimate the DSD at both grid and subgrid scales and thus make comparisons between these scales.

We considered the variability of the DSD over a range of scales from 500 m × 500 m to 7.5 km × 7.5 km. Special focus was put on two scales specifically chosen to correspond to real-world DSD applications. The first was 5 km × 5 km, about the size of the ground footprint of the Global Precipitation Measurement (GPM) spaceborne weather radar (Hou et al. 2014). The second was 2.8 km × 2.8 km, the operational pixel size of the Consortium for Small-Scale Modeling (COSMO)^{1} atmospheric model as used in Germany (Baldauf et al. 2011). Simulated grids of DSDs allowed us to quantify the error introduced by assuming that a point measurement of the DSD represents an areal region at various scales. The results were generalized by normalizing the scale by the decorrelation distance of rainfall intensity. The errors found should be taken into account, for example, in ground validation of radar and model outputs using point measurements. Further, using the stochastic simulation gridded output, we simulated the way in which GPM and COSMO would observe pixel-scale DSD processes and tested how representative these areal estimates were of the underlying subgrid processes. The use of fitted model parameters allowed us to determine the primary sources of errors in the areal DSD retrievals.

The rest of this manuscript is structured as follows. In section 2 we introduce the DSD, define the bulk variables we studied, and outline two DSD models and their parameters. In section 3 the data used in this study are described. In section 4 we explain the stochastic simulation approach, the methods used to calculate variabilities and errors, and the GPM- and COSMO-style retrievals of the DSD at pixel scale. Results are presented in section 5, with subsections provided for results concerning raw DSD concentrations (section 5a), bulk variables (section 5b), GPM-style retrieval of rain rates (section 5d), and COSMO-style retrieval of rain rates (section 5e). A brief discussion in section 6 puts the results into perspective. Conclusions are drawn in section 7.

## 2. The raindrop size distribution

The volumetric DSD ^{−1} m^{−3}) is the number of raindrops with equivolume diameter in the range

### a. DSD bulk variables

^{−3}) is the total number of raindrops per unit volume and is writtenThe mass-weighted drop diameter

*W*(g m

^{−3}) is calculated from the DSD aswhere

^{−3}) is the density of liquid water. The rain rate

*R*(mm h

^{−1}), one of the most important bulk variables, represents the flux of liquid water at a surface. It is derived from the DSD aswhere

^{−1}) is the terminal fall velocity of a drop with equivolume diameter

*D*(mm) (vertical air motion is ignored). Terminal fall velocities can be accurately predicted from drop size (e.g., Atlas et al. 1973; Beard 1976; Brandes et al. 2002).

*Z*(dB

*Z*) is derived as the sixth moment of the DSD:In the Mie regime, which applies when the particles are similar in size than the wavelength, the radar reflectivity can be found from the DSD usingwhere

*λ*(cm) is the radar wavelength,

^{2}) is the back-scattering cross section for a drop with equivolume diameter

*D*at wavelength

*λ*.

*k*(dB km

^{−1}), the amount of radar signal attenuated by the precipitation per kilometer of distance traveled, which is calculated from the DSD aswhere

^{2}) is the extinction cross section for a drop with equivolume diameter

*D*(mm) at wavelength

*λ*(cm).

### b. DSD models

*μ*(unitless), the slope parameter

^{−1}), and the intercept

^{−3}mm

^{−1 − μ}). When

*μ*= 0 the model reduces to the Marshall and Palmer exponential form. For the rest of this section, we assume integration over drop sizes from 0 to ∞. While measured DSD data are necessarily truncated at some minimum and maximum observable

*μ*are related via

*μ*, all variables in the normalized DSD model have conventional units. It is formulated [using

*D*

_{m}as in Seto et al. (2013)]where

^{−1}m

^{−3}) is a scaling factor, and

### c. Finding model parameters

When DSD measurements are available, DSD model parameters can be fitted to the observed data. However, it is often the case that the DSD itself is not measured, and the DSD model parameters for an areal region must be estimated from observations of other rainfall variables. In this section we outline how DSD model parameters are retrieved in two separate systems that correspond to the two scales we focus on in this paper. First, we describe algorithms proposed for use by the Dual-Frequency Precipitation Radar (DPR) on board the GPM Core Observatory satellite (Hou et al. 2014). These algorithms infer the DSD from radar reflectivities. Second, we show how the COSMO atmospheric model (e.g., Baldauf et al. 2011) infers DSD parameters from the modeled total mass fraction of liquid water.

#### 1) GPM-style DSD retrieval

The GPM DPR measures radar reflectivity at Ku (13.6 GHz) and Ka band (35.55 GHz) (Hou et al. 2014). For each pixel, a measurement is made in either single- (Ku or Ka band) or dual-frequency (Ku and Ka band) mode. From the returned radar reflectivities, the parameters of a DSD model are estimated. Here we briefly review the single- and dual-frequency methods described in Seto et al. (2013) and Liao et al. (2014), in which the DSD model used is the normalized DSD model shown in Eq. (10). In this paper, we assume an idealized system in which radar reflectivities are known perfectly, which is to say that attenuation is assumed to have been estimated and corrected [for discussion of an attenuation-correction algorithm used with this DSD retrieval method, see Seto et al. (2013)]. Further, we assume a uniform distribution of energy within the radar beam.

^{−1}mm m

^{3}), in which

^{−1}m

^{−3}(Liao et al. 2014). Using this assumption, and substituting the DSD model defined in Eq. (10) into Eqs. (6) and (7), the normalized reflectivity and normalized specific attenuation are obtained (Liao et al. 2014). Normalized reflectivity is given byand normalized specific attenuation is given by

*Z*(dB

*Z*). DFR is independent of

^{6}m

^{−3}) is used to determine DSD model parameters (Seto et al. 2013). Dividing

^{−1}mm

^{−6}m

^{3})By assuming a constant value of

*μ*, lookup tables can be precalculated to determine the values of DFR and

*μ*assumed and

*μ*can then be used with the normalized DSD model [Eq. (10)]. The model can be substituted for

#### 2) COSMO-style DSD retrieval

*μ*and

*q*(unitless; Doms et al. 2011). The total mass fraction of water is defined from the DSD aswhere

^{−3}) is the total density of the air–water mixture and

^{−3}) is the air density. Note that the liquid water content

*μ*= 0 and

^{6}m

^{−4}= 8000 m

^{−3}mm

^{−1}for rain. Since version 4.21, the gamma model has been in use and

*μ*(Seifert et al. 2011). The current operational code uses a modified Ulbrich (1983)

*μ*relationship, such that for

*N*

_{0}(m

^{−4 − μ}),

The value of *μ* in the current operational code is 0.5 and thus ^{−3} mm^{−3/2}. Once parameters *μ* and *W*) of the DSD is used to describe the DSD. Two-moment microphysical schemes, in which, for example, particle concentration is also predicted and used, have been considered (e.g., Baldauf et al. 2011; Van Weverberg et al. 2014); however, Baldauf et al. (2011) found that in the specific case of current COSMO operational use, the benefits of a two-moment scheme did not outweigh the increased computational cost. As a result, the one-moment scheme is currently in operational use, and in this paper we focus on that scheme.

## 3. Data

The DSD data used in this study were measured using a network of disdrometers deployed during two field campaigns. The campaigns lasted about 3 months each and were held in Ardèche, France, in the autumns of 2012 and 2013, as part of the Hydrological Cycle in the Mediterranean Experiment (HyMeX^{2}; Drobinski et al. 2014). This area is subject to heavy precipitation events during the autumn months, when large masses of warm, moist air are produced over the Mediterranean Sea and pushed by southerly flow onto the land, where topography triggers convective activity (Frei and Schär 1998; Miniscloux et al. 2001; Delrieu et al. 2005; Ricard et al. 2012). Such synoptic systems are often quasi-stationary, leading to events that can last from hours to days and produce high rainfall totals (Ricard et al. 2012). These broad meteorological conditions are well understood (Lin et al. 2001; Nuissier et al. 2011), but orographic rainfall on the kilometer scale is a complex process that requires further study (Roe 2005; Houze 2012).

The instrumental network was made up of Particle Size and Velocity (Parsivel) disdrometers. Parsivels are laser optical disdrometers that classify raindrops into classes of equivolume diameter and velocity, by detecting drops as they fall through and interrupt a laser sheet (Löffler-Mang and Joss 2000). The first generation was released by OTT Hydromet in 2005, and the second generation (Parsivel^{2}) was introduced in 2011, offering improvements over the earlier model (Tokay et al. 2014). The network used in this study was composed of 11 (nine in 2012) Parsivels and two Parsivel^{2} disdrometers in a region of about 13 km × 7 km. Figure 1 shows a map of the network, and Table 1 shows basic station information. Stations were deployed over both autumn campaigns, with the exceptions of Montbrun and Pradel-Grainage (2013 only). Collocated stations must be handled with care for geostatistical analysis, because measurement differences should not be interpreted as spatial effects. To avoid these problems, we used the best-performing Parsivel at each collocated station, judged by comparison of rain rates with collocated tipping-bucket rain gauges at 5-min resolution. Thus, three more collocated Parsivel^{2} stations (two at Villeneuve and one at Pradel-Grainage) were excluded from this analysis. The collocated stations Pradel 1 and Pradel 2 were nevertheless used to determine variogram nuggets [see Raupach and Berne (2016), who used the same instrument network], but of these stations, only data from Pradel 1 were used in the generation of stochastic simulations. The maximum difference in disdrometer to rain gauge relative error between two collocated disdrometers was 13%. DSDs recorded by each disdrometer were corrected using the technique of Raupach and Berne (2015). The equivolume diameter classes in which drops appeared in the corrected Parsivel data are shown in Table B1.

Station information, showing the station name, its World Geodetic System 1984 latitude and longitude, the number of hours and total amount the station recorded during event times (calculated using 1-min-resolution data and including all within-event time steps), raw measurement integration time *T*, and altitude. St-Etienne stands for Saint-Etienne-de-Fontbellon, Villeneuve stands for Villeneuve-de-Berg, Grainage stands for Pradel-Grainage, and Vignes stands for Pradel-Vignes. All were Parsivel 1 instruments, except for St-Etienne and Villeneuve, which were Parsivel^{2}. Table modified from Raupach and Berne (2016).

In both campaigns, a transportable dual-polarimetric X-band Doppler weather radar called MXPol [for full instrument details, see Schneebeli et al. (2013)] was located to the northeast of the disdrometer network. Among other scans, the radar performed horizontal plan position indicator (PPI) scans at various elevations above the disdrometer network. We used the lowest reliable scan elevation, which was 4° above horizontal. These scans were made about every 5 min.

Raw Parsivel measurements were made using an integration time of either 30 or 60 s. These measurements were resampled to 1-, 5-, and 10-min resolution by averaging the drop concentrations per equivolume diameter bin for each time step. Time steps at each resolution were only considered if no solid precipitation was observed, the Parsivel indicated no error condition, the rain intensity observed was greater than 0.1 mm h^{−1}, and there was at least one radar scan made within the Parsivel integration time. Events were defined using 1-min-resolution DSD data, as periods of rain in which there was no more than 1 h of completely dry time across the network, and for which at least one station recorded an amount greater than 1 mm. The use of geostatistical analysis restricted the events that could be used to those with a sufficient number of observation pairs [see Raupach and Berne (2016), in which the same events were used]. Definitions of the analyzed events are shown in Table 2.

Analyzed event times, showing the start time (end of first time step), the event length, the peak 1-min rain rate, the number of stations that measured DSDs during the event, and the number of within-event time steps suitable for geostatistical analysis per time resolution. For each time resolution, events were included if it was possible to estimate variograms with at least 30 pairs of observations per distance lag class. Table modified from Raupach and Berne (2016).

The equations in sections 2b and 2c require the assumption of integration over drop sizes from 0 to ∞. Disdrometers necessarily have minimum and maximum drop sizes that they measure. The processed DSD data used in this paper were truncated at minimum and maximum drop sizes of 0.2495 and 7 mm, respectively. Willis (1984) showed that for the gamma DSD model, as long as the maximum drop diameter is greater than 2.5 times the median drop diameter ^{−1}. The effect of truncation on the measured and simulated DSDs was hence assumed to be negligible.

## 4. Methods

In this section we describe the methods that were used. First, the two typical pixel areas on which most analyses were performed are introduced. Second, the method for generation of gridded DSD estimates from measured DSD data is explained. Third, we show the method used to compare point- to pixel-scale DSDs and bulk variables. Fourth, the steps involved in estimating and comparing GPM- and COSMO-style pixel-scale outputs to subgrid values are outlined. Finally, we briefly describe how we calculated accumulated rain amounts per method.

### a. Regions of interest for typical scales

DSD stochastic simulation was used to estimate DSDs on regular grids. There were two main regions of interest (ROIs). The ROI areas were 5 km × 5 km and 2.8 km × 2.8 km, so as to represent the typical size of a spaceborne weather radar footprint and the horizontal pixel size of a high-resolution numerical weather prediction (NWP) model, respectively. The locations of the ROIs were judiciously chosen so as to have a reasonable trade-off between maximizing the number of disdrometer stations within each one and the ROIs being as far as possible from the edge of the radar coverage area. In this way, the DSD fields simulated in the chosen areas were able to be well constrained by ground and radar measurements. The resulting ROIs are shown in Fig. 1. The smaller ROI is contained entirely within the larger ROI. Grids were made at 100-m resolution using coordinates in the Universal Transverse Mercator (UTM) zone 31 projection, including ROI edge points.

### b. Estimation of gridded DSDs

The DSD simulation method of Raupach and Berne (2016) was used to estimate the DSD at a regularly spaced grid of locations covering the area of the disdrometer network. The approach is summarized briefly in appendix A. Each estimate in the grid represents the DSD at the point scale at a given location, and the grid as a whole is conditioned by the measurements from the network of disdrometers.

DSD simulation was performed for integration times of 1, 5, and 10 min. A 1-min resolution was the highest temporal resolution available at all stations and therefore provided the closest possible approximation of DSD variability within an instantaneous radar scan. We produced 100 stochastic realizations for every available time step. Each time step was treated independently, with the simulations conditional on the disdrometer measurements for each. Bulk variables were calculated from the estimated DSDs. Rain-rate calculations require the use of a drop velocity model, which in turn requires the altitude and latitude of the point in question. The altitude was taken as the mean station altitude within the ROI, which was 275 m. Digital elevation data (Jarvis et al. 2008) showed that the large ROI has a mean elevation of 260 m with a standard deviation of 44 m. The mean latitude of grid points in the large ROI, 44.568 26°N, was used as the latitude. Raindrop terminal fall velocities were found using the technique of Beard (1976), assuming a sea level temperature of 15°C and relative humidity of 0.95. This assumed sea level temperature led to a temperature for the large region of about 13°C, based on a standard atmospheric temperature lapse rate of −6.5°C km^{−1} (Wallace and Hobbs 2006). Radar reflectivities were calculated using vertical incidence. Drop shapes were calculated using the model of Andsager et al. (1999), and backscattering coefficients were found using the T-matrix code of Mishchenko and Travis (1998).

Because the stochastic simulation method we use assumes normally distributed variables, unrealistically large drop concentrations were occasionally predicted by the stochastic simulation algorithm. To remove these values, we set a threshold on rain rate: any simulated DSDs for which the corresponding rain rate was greater than 200 mm h^{−1} were removed and not counted. This threshold removed less than 0.05% of simulated DSDs. The largest measured 1-min rain rate from the disdrometer network was 73 mm h^{−1}.

Example grids of simulated fields of radar reflectivity are shown in Fig. 2, where it can be seen that the stochastic simulation results for *Z* are consistent with radar observations. The stochastic simulation results are less smooth than interpolated fields (using kriging). This is because the result of interpolation by kriging is equivalent to the mean of all possible stochastic outputs. An interpolated output would present the single most likely grid of point values. In contrast, the 100 stochastically simulated realizations per time step are different but equally likely sets of values, each of which honors the spatial properties and values of the measured inputs. We used simulation instead of interpolation for two reasons. First, it allows for inclusion of more extreme values that would be “smoothed out” by interpolation using kriging. Second, stochastic simulation generates many realizations of the same process and thus creates a larger sample set to analyze.

### c. Comparison of point-to-areal estimates

Areal DSDs were calculated by taking the mean drop concentration per equivolume drop diameter class for every point in each available grid. Areal bulk variables were then calculated from the mean DSD. These areal DSDs could then be compared to point DSDs. To ensure that any DSD estimation error resulting from the simulation technique was not misclassified as error resulting from assuming a point measurement to be areal, we compared simulated areal estimates to individual simulated point estimates. Comparisons were made for those points that were within 250 m of the center of each region of interest and that had rain intensity greater than or equal to 0.1 mm h^{−1}. Points with rain intensity less than 0.1 mm h^{−1} were considered to contain zero rain.

*k*th drop equivolume diameter class

^{−1}m

^{−3}) and

^{−1}m

^{−3}) are point and areal simulated estimates of drop concentration for the

*k*th drop diameter class, which contains drops with equivolume diameter in the range

### d. Simulation of GPM- and COSMO-style DSD retrieval

For each 5 km × 5 km stochastic realization, the way in which a satellite-based system such as GPM might estimate the DSD for the region was simulated. To do so, radar reflectivities *μ* = 3 [as in Seto et al. (2013)] and calculated a lookup table for

As is well known (e.g., Seto et al. 2013), in the dual-frequency case, low values of DFR correspond to two values of *k* must be estimated (e.g., Seto et al. 2013). In this work, in which the GPM algorithm is simulated, no beam effects were taken into account. To more accurately simulate GPM-like results, only records for which the radar reflectivities were greater than or equal to the DPR sensitivities of 18 dB*Z* for Ku band and 12 dB*Z* for Ka band (Hou et al. 2014) were analyzed. Records for which the DFR value was outside the range of DFR values in the lookup table were not analyzed.

In a similar approach, for each 2.8 km × 2.8 km stochastic realization of gridded DSDs, the way that an NWP model such as COSMO might estimate the DSD was simulated. The liquid water content for the mean DSD for each realization was calculated, and using these values of *W* the value of *μ* = 0, ^{−3} mm^{−1}) and the newer gamma model constants quoted in Baldauf et al. (2011) (*μ* = 0.5, ^{−3} mm^{−3/2}).

For both GPM- and COSMO-style DSD model retrieval, the retrieved parameters were used in the corresponding DSD models to find drop concentrations for each Parsivel drop size class. To make comparisons fair, the resulting modeled DSDs were truncated to the same range of drop sizes (0.2495–7 mm) as the measured and simulated DSDs. Areal bulk variables were then calculated for the retrieved modeled DSDs.

### e. Total rain amount

The accumulated rain amount is of primary interest to hydrologists. To see how accumulated rain amount is affected by the scale on which the measurement is taken, we calculated the sum of all 1-min rain amounts, for time steps on which all methods estimated a DSD. Simulation realizations were grouped together, so each method provided 100 accumulated rain amounts. The point simulations used were chosen randomly, one per realization, from the 2.8 km × 2.8 km region’s previously sampled points. Note that these rain accumulations include only 1-min time steps for which radar data were available, which at their most frequent occurred about every 5 min.

## 5. Results

In this section, the results are shown. The error introduced by the assumption that a point measurement represents an areal measurement is addressed for both DSD concentrations and bulk variables. We then show how precipitation variables simulated using GPM- and COSMO-style retrieval of the DSD relate to the subgrid distributions of corresponding values.

### a. Drop diameter class concentrations

The difference ^{−1}, so intermittency caused point estimates to miss areal rainfall about one-quarter of the time. Again for 1-min resolution, 2 785 674 nonzero point estimations of rainfall were compared to areal estimations. Errors introduced by the assumption that a point measurement represents an areal measurement are shown in absolute and relative terms for a 2.8 km × 2.8 km area in Tables B2 and B3, respectively, and for a 5 km × 5 km area in Tables B4 and B5, respectively. Figure 4 shows absolute error by equivolume drop diameter class, by time resolution, and by estimation type. In these tables and plots, we show only those drop classes for which the 10th and 90th percentile absolute differences were not both zero at 1-min resolution. At 1-min resolution, the median errors show negative bias. Median relative error increased for larger drop classes (greater than 2 mm), due to the fact that large drops occur infrequently and thus their areal concentrations were often small. The greatest range of errors occurred for drops with equivolume diameters between about 0.3 and 2 mm.

Figure 5 shows the interquartile range (IQR) of relative error by drop diameter, temporal resolution, and scale. The spread of relative error was always larger in the larger area than the smaller area. Error spreads in 5- and 10-min-resolution results were similar, but the results for 1-min temporal resolution show a larger spread of relative errors for drops up to about 2.5 mm in equivolume diameter. The spread of relative errors decreased from the minimum drop size to about 0.75 mm, then increased with drop size. The 1-min-resolution error spread decreased for drops larger than about 2.25 mm. We conclude that in general terms, the variability of the drop concentration per diameter class increases with drop size and areal scale and decreases with temporal resolution. This conclusion agrees with that of Jameson et al. (2015b), who gave a physical explanation: as the network size increases, the probability of sampling rarer parts of the DSD spectrum increases, and the variability of the DSD increases. This effect is greater for larger (and rarer) drop sizes.

For larger drops, areal concentrations become smaller with greater integration time, which leads to outlier values of relative error and increased IQRs. To decrease the effect of outlier relative errors, we divided the drops as evenly as possible into three diameter classes: the small third of drops in the range [0.2495, 0.6245 mm) (Parsivel classes 3–5), the medium third of drops in the range [0.6245, 0.8745 mm) (classes 6 and 7), and the large third of drops in the range [0.8745, 7 mm) (classes 8–22). In the measured Parsivel data from all stations, at 1-min resolution, these classes contained 29%, 32%, and 39% of the drops, respectively. Figure 6 shows, for these classes, the distributions of relative errors introduced by the assumption that a point measurement represents an areal region, by temporal resolution and spatial scale. The quantile statistics for these errors are shown in Table B6.

The range of relative error was largest for the class of small drops. Error range decreased and median error became more positive with decreasing temporal resolution. For high-resolution 1-min data, the median error was always negative, meaning that more than half of the time the point measurement underestimated the areal measurement. At its largest (1-min, large area) this median error was −10%. The distributions were positively skewed. This means that the largest errors occurred when the point value overestimated the areal value. These results make intuitive sense. Light rainfall is more common than heavy rainfall, so given an areal region that contains both light and heavy rain, a point measurement is more likely to sample the light rain and thus underestimate the areal amount. In the rarer cases when the point does sample heavy rain, however, the error is likely to be a large overestimation: a point measurement inside a storm will overestimate an areal measurement in which there are many dry or light rain regions. As the integration time increases, severe underestimation becomes less likely. We note that the IQR of relative errors did not increase with increasing drop size when the classes contained similar numbers of drops. This observation supports the physical explanation mentioned above, that increased variability of large drops is due to their appearance in a rarer part of the DSD spectrum.

The resulting error distributions were grouped by rain rate estimated at each simulated point measurement. The classes of rain rate were chosen to each contain roughly one-third of the rain rates at compared points for 1-min resolution and were (0.1, 0.5 mm h^{−1}], (0.5, 2 mm h^{−1}], and (2, 200 mm h^{−1}]. This grouping highlights how common light rainfall was, with almost two-thirds of the estimated rain rates being below 2 mm h^{−1}. The results of the grouped analyses are shown in Fig. 7 and show that there is a strong influence of the rain intensity on the observed difference between point and areal DSDs. For very light rain, the median errors were always negative, while for heavier rain there were often positive median errors, and the distributions were more strongly positively skewed. These results confirm the intuitive reasoning given for the overall differences observed.

The results in this section show that the assumption that a point measurement of the DSD represents an areal measurement can introduce significant error. The distributions of relative errors for the chosen drop size classes had median values that showed bias between −10% and +8%. The distributions had large IQRs and were positively skewed. Very light rainfall produced negatively biased point DSD estimates, while simulated point measurements of heavier rainfall produced generally positive bias and some large overestimations of the areal value.

### b. Bulk variables

Variability in the DSD implies, of course, variability in precipitation bulk variables. We quantified the error introduced by assuming that a point measurement represents an areal region, for total drop concentration *R*; mm h^{−1}), liquid water content (i.e., *W*; g m^{−3}), mass-weighted mean drop diameter

The results show trends that are similar to those of the drop concentration relative errors. In both small and large areas, the spread of relative errors for all variables decreased when the temporal resolution was decreased from 1 to 5 min. IQRs were larger for the larger areal scale. Using the IQR as an indication of how much a variable could be affected by an assumption that a point measurement is areal, the least affected variable was *W*, and *R*. We hypothesize that the fact that the radar reflectivities are expressed in logarithmic units (dB*Z*), and the fact that *W*, *R*, and

The same analysis as for drop concentrations, of splitting the differences into classes of simulated point measurement rain rate, was performed. These results are shown in Fig. 9. Again there was a strong influence of the rain rate at the point compared: median errors were negative for all variables for very light rain and were all positive for rain with intensities greater than 2 mm h^{−1}. For *W*, and *R*, the error distributions were more positively skewed for the heaviest class of precipitation than for the other classes.

### c. Generalized comparisons

This study focuses on areal regions that correspond in size to real-world areal sizes used by the COSMO model and GPM satellite. The results of the point-to-areal comparisons can be generalized somewhat by normalizing the areal size (side length of the square area) by some characteristic distance or scale, here taken to be the decorrelation distance of the rainfall process. We found the decorrelation distance for each event using experimental variograms on rain rate from the disdrometer network. Each time step was taken as a separate realization of the process, a log transform and Cressie’s robust variogram estimator (Cressie 1993) were used, and experimental variograms were fitted with a spherical model. Across events for which the variogram model range was within the observation distance, the mean decorrelation distance was 5.5 km. We used this as a climatological value for the studied region. Note that this also means that even in the GPM-size pixel, the full variability of the rainfall process was not always sampled.

The same areal-to-point value comparisons as described above were performed for areal regions around the center of a region formed by adding 2500 m to the north and west sides of the large ROI. The regions were square, with side lengths from 500 to 7500 m in 500-m increments. The interquartile range of relative errors was used as an indication of the error spread for each areal size. Figures 10 and 11 show the interquartile range of relative errors for drop concentration classes and bulk variables, respectively, by normalized areal size. The spread of error increases almost linearly with the size of the areal region, and again the error spread is largest for the smaller third of drops and decreases with temporal integration. We expect that the rate of increase in error range would slow toward some finite limit as the ratio of areal side length to decorrelation distance becomes larger than one, but our network size did not allow us to simulate such large areal measurements. Linear model intercepts and slopes are given for normalized areal sizes less than one in Table B9. These results provide an estimate of expected error spread that can be applied to other regions with similar climatologies, if the decorrelation distance of rain rate is known.

### d. GPM-style retrieval of bulk variables

The previous sections showed that if a point DSD is assumed to represent an area, an error is introduced. This error, in general, increases with the size of the area and with drop size. We now turn to the inverse case in which a retrieved areal DSD is assumed to be representative of the subgrid process. We first address GPM-style DSD retrieval. The GPM DPR makes areal measurements of Ku- and/or Ka-band radar reflectivity at a resolution of about 5 km × 5 km. In this section we show how simulated GPM-inferred bulk variables compared to subgrid values in the observed area.

Recall from section 4d that using the mean-DSD-derived ^{−3}), rain rate (mm h^{−1}), and sixth-moment radar reflectivity (mm^{6} m^{−3}) for each realization. A total of about 38% of the areal DSDs were excluded from these analyses: 27% for having

Figure 12 shows GPM-derived rain rates compared to DSD-derived rain rates for the dual- and single-frequency cases. The results show that, assuming an accurate estimation of the specific attenuation, both dual- and single-frequency methods for estimation of rain rate produced good matches to DSD-derived rain rates. We note again that these are idealized results that ignore any beam effects or error in the estimation of specific attenuation. To test whether the resulting areal values were representative of the subgrid values, we found the percentile of each GPM-simulated variable within the respective subgrid distribution of values, in order to look at where within the distribution of subgrid values each GPM-simulated value appeared. Densities of percentiles across all realizations are shown for *R*, and *Z* in Fig. 13 for the two GPM-style retrieval methods, as well as for values obtained using the mean areal DSD. These plots show, graphically, where each areal value appeared most often within the subgrid distributions of values.

The mean DSD provided a good approximation of all values for the area. From these results it is clear that extreme values affected the mean DSD, and therefore the mean-derived values were usually larger than the median subgrid values. The small sampling size of the simulated point measurements implies that the subgrid median values are underestimates of the real population values, with this effect increasing with the moment order of the bulk variable (Uijlenhoet et al. 2006). The GPM-style methods were able to produce values of *R* and *Z* that were generally as representative of the subgrid values as the mean DSD for the area. However, they both produced values of lower-order moments

In the case of dual-frequency retrieval, GPM-simulated values of low DSD moments *R* and *Z* were well estimated, although, like the results for the mean DSDs, they produced density peaks above the median subgrid value. For single-frequency retrieval, GPM-simulated variables were always within the subgrid range, but lower-order moments tended to be overestimated, with density peaks greater than the median subgrid value.

Error in GPM-style retrieval of the DSD is a combination of error caused by inexact estimation of *μ*. We fitted *μ* to each observed DSD using the technique of Johnson et al. (2014) [modified such that *μ* used by the GPM algorithms compared to the fitted gamma model parameters for the areal DSDs. The fixed value of *μ* = 3 used by the GPM models was the 4th percentile of the distribution of *μ* fitted to experimental DSDs.

We tested the GPM DSD retrieval using 1) DSD-derived values of *μ*, thus eliminating the retrieval of *μ*; and 3) GPM-derived *μ*. The performance was improved for the first option, with lower-order moments more representative of the subgrid values, but *R*, and *Z* were slightly degraded with respect to their proximity to the median subgrid values, though they were still representative of the subgrid distributions. In the third option, the performance was similar to the performance when *μ* = 3, with *Z* and *R* slightly degraded. We conclude that in GPM-style retrieval,

### e. COSMO-style retrieval of bulk variables

A numerical weather model such as COSMO must also infer the parameters of a DSD model. In the case of COSMO, the parameters are inferred from an areal estimation of total mass fraction of water. Recall from section 4d that the COSMO areal DSD retrieval was simulated for each stochastic realization on the 2.8 km × 2.8 km scale region. COSMO retrievals were simulated using both the exponential DSD model and updated gamma DSD model parameters. COSMO-inferred values of *R*, and sixth-moment *Z* (in linear units) were compared to subgrid simulated values to see whether they were representative of the underlying processes.

A comparison between the COSMO-simulated rain rates and the mean-DSD rain rates for the 2.8 km × 2.8 km region is shown in Fig. 15 for the gamma and exponential models. The gamma DSD model reproduced *R* slightly more accurately than the exponential DSD model. Just as for the GPM-style retrieval analysis presented above, the percentiles of the bulk variables within the corresponding subgrid distributions were determined for each realization. Density plots for these percentiles are shown in Fig. 16. The two DSD models both had a tendency to overestimate *Z* and to produce values of *R* values were always within the range of subgrid values. The exponential model had *R* and *Z*. The gamma model had *Z* above the subgrid range 21% of the time. Although they returned reasonable values of *R*, the DSDs found using the COSMO-derived exponential and gamma models were often unrealistic in terms of *Z*.

The accuracy of COSMO-style retrieval is affected by the choice of DSD model parameters *μ* and *μ* and *μ* were both at the 0th percentile of fitted values. The fixed value of

We reran the COSMO retrievals, using the fitted values for *μ* and *μ* and *R* and *Z* were always within the subgrid range. Some overestimation of *μ* and

### f. Total rain amount

Figure 18 shows the distributions of total rain amounts by method. Rain amounts were calculated per simulation number, so there are 100 values in each distribution shown in this plot. The differences between the accumulations calculated using point estimates and those calculated using an area are clear. Recall that we use the areal mean DSD as the reference amount. Simulated point measurements typically overestimated the total rain amount. COSMO with the gamma model also shows an overestimation, which agrees with earlier results. The accumulation amounts for the larger areal region were often smaller than those for the small area, which we hypothesize is due to the larger region containing more low or zero values. The density of values of *R* inside the 5 km × 5 km region is shown in Fig. 19; it is clear from this plot that the distributions of *R* in the subgrid regions are indeed highly skewed. Also shown are the ranges of accumulated amounts for the idealized GPM case in which *μ* was used, and for the COSMO cases in which fitted DSD model parameters were used. In both cases, the idealized versions are closer to the areal reference values, which supports our previous hypotheses that the sources of error are primarily the estimation of

## 6. Discussion

The results in this paper are primarily symptomatic of the large variability that exists in DSDs, even over relatively small regions such as those studied here. In this section we put the results into perspective by discussing their meaning in a broader context.

While the use of a point measurement to represent an areal region is common practice, our results show that even if the bias is low, the error introduced by such an assumption has wide variability, both for drop concentrations and bulk variables. Especially at high temporal resolution, great caution should be used if a point measurement of the DSD or a related variable is assumed to represent an area. As shown also by other studies (e.g., Jameson et al. 2015b), variability on the DSD increases with areal size and drop size and decreases with temporal integration. The variability of the mass-weighted mean drop diameter

The variability in the DSD poses challenges for the estimation of areal DSDs. Because of the skewed nature of drop concentration distributions, even simply taking the mean DSD over dense measurements within the area produces an areal DSD that is likely to represent the higher-valued portion of the subgrid distribution. Algorithms that attempt to derive an areal DSD from other areal measurements have a harder task. Our results show that, at least in the case of idealized GPM and COSMO algorithms, rainfall rate and radar reflectivity are estimated more reliably than lower-order moments of the DSD. Low-order moments from estimated areal DSDs should therefore be treated with more caution than higher-order moments.

It is normal to use a DSD model to represent an areal (or point) DSD. These models generally do an excellent job of capturing the DSD and its bulk variables. That being said, our results have shown large variability in the DSD parameters that are appropriate for DSDs measured at high temporal resolution. In the case of the COSMO algorithm, errors in the bulk variable outputs were largely due to the fixed DSD model shape and slope parameters, which we showed were often different to values fitted to the areal DSDs. On the other hand, errors in the GPM-style algorithm were largely due to error in the estimation of the characteristic drop diameter

Our results are based on stochastic simulation of the rainfall process over typical areas that correspond to pixel sizes in real rainfall products (e.g., GPM and COSMO). Simple experimental variograms on *R* recorded by the Parsivel network can give an approximation of the decorrelation distances in the rain process in time and space for each event. In our case, the large region was large enough to cover the decorrelation distance for 80% of the events. The small areal region, however, was never large enough to observe the full variability of the rainfall process. In time, the decorrelation distance was larger than any integration time we used for all but one event. It should not be assumed, therefore, that our simulated domains captured the full variability of the rainfall process.

Finally, it should be noted that these results are for a specific area in Ardèche, France. This is a midlatitude region subject to Mediterranean rainfall, and we studied events that occurred during the autumn months, when rainfall is known to be higher in these regions (see section 3). The results shown here are specific to this region. It is our hypothesis, however, that because these results depend primarily on the variability in the DSD, the broad conclusions just outlined are valid wherever there is large DSD variability present.

## 7. Conclusions

In this paper we have shown results of analyses designed to quantify the effects of DSD variability on areal measurements of precipitation. We focused on two typical scales: 5 km × 5 km, corresponding to the footprint of the GPM satelliteborne weather radar, and 2.8 km × 2.8 km, a high-resolution horizontal pixel size of the COSMO NWP model. The analyses were based on high-resolution grids of simulated experimental DSDs that were found using the DSD interpolation and simulation technique of Raupach and Berne (2016), constrained using radar and disdrometer network data from the HyMeX 2012 and 2013 autumn campaigns in Ardèche, France. The mean DSD over the simulated grid was taken as the areal DSD. These areal DSD estimations were then compared to simulated DSDs on the point scale and used to investigate whether GPM- and COSMO-style areal DSD retrievals were representative of subgrid values.

The effect of assuming that a point measurement of the DSD represents an areal measurement was tested first. The results showed that this assumption introduces an error that, in general, is greater for larger drop sizes, increases with the size of the areal measurement, and decreases when measurements are integrated in time. Although these errors are most often from small to moderate, with median errors on equally distributed drop size classes between −10% and 8%, the distributions of errors on both DSD concentrations and resulting bulk variables had wide interquartile ranges. Thus, for any one application of this assumption, it is possible for a large error to occur. While point estimates generally underestimated areal estimates for very light rainfall, they overestimated areal values for heavier rainfall, and the error distributions were mostly positively skewed. Generally, bias was more positive and the error distributions had larger spread for rainfall of greater than 2 mm h^{−1}. Bulk variables also exhibited large interquartile ranges of errors. Of the bulk variables, rain rate, liquid water content, and drop concentration were all more affected than radar reflectivities and mass-weighted median drop diameter. To generalize the results, we provided linear relationships between the expected interquartile range of point-to-areal error and areal side length normalized by a reference distance, which we took to be the climatological decorrelation distance of rain intensity. For a similar region in which this climatological information is known, these relationships could be used to give an estimate of the point-to-areal error range for common bulk rainfall variables.

Second, the case in which retrieved areal DSDs are assumed to represent underlying subgrid processes was investigated. We simulated GPM-style retrieval of the DSD on the pixel scale with specific attenuation assumed perfectly known, and COSMO-style retrieval on a smaller pixel scale. Both single- and dual-frequency GPM-style retrieval methods were able to reproduce rain rate and radar reflectivity in a representative way, but the retrieved lower-order moment values were not always properly representative of the subgrid DSDs. Similarly, COSMO-style retrieval performed well for rain rate and radar reflectivity retrieval but produced values of drop concentration and characteristic drop diameter that were often unrealistic in comparison to subgrid ranges. Assumed and derived DSD model parameters used by GPM- and COSMO-style retrieval were compared to parameters fitted to the simulated areal DSDs. The results showed the large variability in appropriate DSD model parameters. The GPM-style retrieval error was largely due to error in estimation of the characteristic drop size from radar reflectivities. In contrast, error in COSMO-style estimation of lower-order DSD moments was largely due to the use of fixed DSD model parameters. In this paper we focused only on horizontal variability of the DSD. Future work will turn toward investigating vertical variability of the DSD and associated precipitation variables.

## Acknowledgments

The authors thank the LTE team members who set up and helped maintain the LTE disdrometer network for HyMeX SOPs 2012 and 2013: Joël Jaffrain, Marc Schleiss, Jacopo Grazioli, and Daniel Wolfensberger. We thank those who set up, maintained, and provided data from the HPicoNet rain gauge and disdrometer network: Brice Boudevillain, Gilles Molinié, and Simon Gérard from the Laboratoire d’étude des Transferts en Hydrologie et Environnement (LTHE) at Grenoble University. Data from the LTHE network were obtained from the HyMeX program, sponsored by Grants MISTRALS/HyMeX, ANR-2011-BS56-027 FLOODSCALE project, and OHMCV. The authors thank the Swiss National Science Foundation for financial support under Grant 2000021_140669. The authors thank Remko Uijlenhoet and an anonymous reviewer for their helpful manuscript reviews.

## APPENDIX A

### Summary of Stochastic Simulation Method

A summary of the stochastic simulation process of Raupach and Berne (2016) is as follows:

- A binary rainfall occurrence map is calculated for each event time step, using radar data.
- The measured drop concentrations per equivolume diameter class are log transformed to make their distributions as normal as possible.
- The “dry drift” (Schleiss et al. 2014a) is calculated for each measured drop equivolume diameter class, using the occurrence map. The dry drift is the relationship between the measured drop concentration and the distance of the measurement location from a dry region in space. A spherical or Gaussian functional form is fitted to each observed dry drift. Using these fitted models, the dry drift for any class at any distance can be calculated.
- The dry drift is subtracted from the observed DSDs, thus removing much of the nonstationarity introduced to the rainfall process by precipitation intermittency.
- Principal component analysis (PCA) is used to transform the detrended DSDs into a set of orthogonal components that can be treated separately.
- The PCA components are individually analyzed using geostatistics. The variogram of each component is estimated, using nuggets defined by collocated stations. A spherical variogram model is fitted to each experimental variogram. Multiple equally likely realizations of possible values of each PCA component are estimated for each point, using Gaussian conditional simulation (e.g., Pebesma 2004; Schleiss et al. 2014b). In this paper, up to 100 nearest neighbors were used for each point.
- The gridded PCA component estimations are back transformed to detrended DSD concentrations, the dry drifts are added for each component at each point, and the log transformation is back transformed. The result is a set of realizations of the DSD field. Each realization contains simulated DSDs for each grid point, in the same equivolume diameter classes used for the measurements.

## APPENDIX B

### Tables of Results

In this appendix, detailed tables of results are shown (Tables B1–B9).

Parsivel equivolume drop diameter classes for which drops were recorded. Variable *D* is the center of each class to two decimal places, while *δ* is the width of each class. Class definitions are from the OTT Parsivel operating instructions.

Absolute error statistics comparing mean DSDs at 2.8 km × 2.8 km pixel scale to point scale. Variable *D* (mm) is the center of each class to two decimal places, and *q*_{25}, *q*_{50}, and *q*_{75} are the 25th, 50th, and 75th percentiles of the differences, respectively (mm^{−1} m^{−3}).

Relative error statistics comparing mean DSDs at 2.8 km × 2.8 km pixel scale to point scale. Variable *D* (mm) is the center of each class to two decimal places, and *q*_{25}, *q*_{50}, and *q*_{75} are the 25th, 50th, and 75th percentiles of the relative differences, respectively, expressed as percentages of areal concentrations.

Absolute error statistics comparing mean DSDs at 5 km × 5 km pixel scale to point scale. Variable *D* (mm) is the center of each class to two decimal places, and *q*_{25}, *q*_{50}, and *q*_{75} are the 25th, 50th, and 75th percentiles of the differences, respectively (mm^{−1} m^{−3}).

Relative error statistics comparing mean DSDs at 5 km × 5 km pixel scale to point scale. Variable *D* (mm) is the center of each class to two decimal places, and *q*_{25}, *q*_{50}, and *q*_{75} are the 25th, 50th, and 75th percentiles of the relative differences, respectively, expressed as percentages of areal concentrations.

Relative error statistics by drop class small [0.2495, 0.6245 mm), medium [0.6245, 0.8745 mm), and large [0.8745, 7 mm) and scale side length *S* (km), comparing DSDs at pixel scale to point scale. Column headers *q*_{25}, *q*_{50}, and *q*_{75} are the 25th, 50th, and 75th percentiles of the relative differences, respectively, expressed as percentages of areal concentrations.

Absolute error statistics by bulk variable and scale side length (km), comparing DSDs at pixel scale to point scale. Column headers *q*_{25}, *q*_{50}, and *q*_{75} are the 25th, 50th, and 75th percentiles of the relative differences, respectively.

Relative error statistics by bulk variable and scale side length (km), comparing DSDs at pixel scale to point scale. Column headers *q*_{25}, *q*_{50}, and *q*_{75} are the 25th, 50th, and 75th percentiles, respectively, of the relative differences expressed as percentages of areal amounts.

Linear model intercepts and slopes to provide IQR by areal side length proportion of decorrelation distance of *R*. Linear models are fitted to proportions less than one. Results are organized by temporal resolution and bulk variable, and the

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