Las PYME internacionalizadas reflejan una productividad mayor con relación a las 

While this chapter is primarily about interpolation of the measured DSD spectrum, it is worth mentioning that the concepts presented here can equally be applied to perform stochastic simulation of DSD spectra. By replacing the kriging process with a sequential conditional Gaussian simulation approach (see e.g. Pebesma, 2004; Schleiss et al., 2014b), multiple similar realisations of the DSD can be simulated. In contrast to interpolation, in which the most likely value is found for each point, stochastic simulation produces many equally-likely realisations of the field, all of which have identical spatial properties (variograms). The mean of the stochastic fields approaches the kriging result as the number of realisations increases. While kriging has a smoothing effect, individual stochastic realisations are more realistic fields that are not as smooth as kriging outputs. When stochastic simulation is used with our

3.6. Conclusions

DSD estimation technique, the principal components of the detrended DSD processes are simulated at points that are indicated by the occurrence process to contain precipitation. Once values for each realisation are obtained, the same process outlined in Section 3.2.5 is used to back-transform the components into simulated DSDs.

We used conditional sequential Gaussian simulation from the R Gstat package (Pebesma, 2004) to calculate 100 simulated realisations of the DSD for the same grid, region, and example time step used in Section 3.3. For computational efficiency, the simulation algorithm was restricted to using at most 500 nearest neighbours at each iteration. The same occurrence field was used for all simulations. Figure 3.13 shows two rain intensity fields derived from DSDs simulated using this technique, while Figure 3.14 shows measured and simulated radar reflectivity. The radar reflectivities are derived from the same simulated DSDs as the rain rate in Figure 3.13 (a). These simulation results are equally-likely realisations for the same time step. The realisations each have the same spatial properties and the same intermittency, but the values of individual points are different. The per-location mean of the simulated DSDs across all realisations would converge to the same values given by the kriging method, as the number of realisations increased. These individual simulated fields are obviously less smooth than the kriged results and may contain extreme values. Stochastic simulation offers a useful probabilistic approach to generate ensembles of realistic DSD fields, and to examine the variability of the DSD while taking into account possible extreme values that would be “smoothed out” by the interpolation technique.

(a) Simulation one (b) Simulation two

0 2 4 6 8 0 2 4 6 0 2 4 6 x [km] y [km] 2.5 5.0 7.5 10.0 12.5 R[mmh−1]

Figure 3.13 – Two example simulated fields for rain rate R, for an example time step (2013-10-20 03:00 UTC) during event 13. Note that the fields differ and are not smooth.

3.6 Conclusions

We have presented a new approach for the interpolation of experimental raindrop size distri- bution spectra. Using the technique, non-parametric drop size distribution spectra can be estimated at unmeasured locations. We showed that raindrop concentrations per diameter

(a)MeasuredZH (b)SimulatedZH 0 2 4 6 8 0 2 4 6 0 2 4 6 x [km] y [km] 20 30 40 ZH[dBZ]

Figure 3.14 – Comparison to measured values for an example time step (2013-10-20 03:00 UTC) during event 13: (a) the measured radar reflectivity (attenuation corrected horizontal polarisation ZH[dBZ] at 100 m resolution, SNR > 5 dB and ZH> 10), and (b) an example simulated field for ZH. The simulated field is less smooth and more realistic than the interpolated field shown in Figure 3.9.

class are subject to a dry drift. Given a rainfall occurrence field, the dry drift can be subtracted from the DSD concentration fields to obtain detrended fields that are not affected by this source of non-stationarity. The DSD interpolation technique works on these detrended fields, and uses geostatistical methods to characterise and interpolate principal components of the detrended DSD. We applied the technique to disdrometer network data from HyMeX cam- paigns in Ardèche, France. The method is equally applicable to data from other sensors. We used radar data to determine occurrence fields which were used to calculate the dry drifts. If dedicated radar data were not available, the occurrence field could be calculated from other sources such as numerical weather prediction models or operational weather radar products, or indeed estimated from disdrometer time series (Schleiss et al., 2014a). The presented technique will also operate effectively, although not as accurately, without any consideration of the dry drift.

Leave-one-out testing using HyMeX data demonstrated that the new technique is able to estimate the DSD spectrum with low bias at unmeasured locations. Since the method is geo- statistical, an associated measure of estimation uncertainty is provided for every estimation. Stochastic simulation of the DSD is possible through simple modification of the technique, and could be used to estimate the probability densities of DSD values at points in space, or to examine possible extreme values of the DSD. All bulk rainfall variables can be calculated from the DSD, so the DSD interpolation method is effectively able to interpolate or simulate all bulk variables simultaneously. However, its main utility comes from the fact that it interpolates or simulates the non-parametric DSD spectrum assuming no prescribed functional form. The method is useful for studies that use networks of disdrometers to investigate the small-scale variability of the drop size distribution and its associated precipitation variables. Possible extensions to the technique include the use of kriging with external drift to take topographical

3.6. Conclusions

4

Small-scale variability of the raindrop

size distribution and its effect on

areal rainfall retrieval

• T. H. Raupach and A. Berne. Small-scale variability of the raindrop size distribution and its effect on areal rainfall retrieval. Journal of Hydrometeorology, 17:2077–2104, July 2016. doi: 10.1175/JHM-D-15-0214.1. © American Meteorological Society. Used with permission.

This work was completed by T. Raupach under the supervision of A. Berne. Research, analyses, and writing are by T. Raupach. For data acknowledgements, see Appendix A.

4.1 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 chapter we study the small-scale sub-grid variability of the 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 sub-grid precipitation process?

Sub-grid variability of the DSD and non-linearities 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 [mm6m−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 sub-grid 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 sub-grid variability of the DSD into account.

Previous studies of DSD variability have involved the use of single disdrometers and inves- tigation of DSD variability over time-series measurements (e.g. Uijlenhoet et al., 2003b; Lee and Zawadzki, 2005; Chapon et al., 2008). Jameson (2015) provided a technique for up-scaling single disdrometer measurements, which produces gridded simulations of non-parametric point DSDs that honour the statistical properties of the observations. The resulting gridded fields are useful for statistical characterisation 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 the DSD, using radar data, a time series of point DSD measurements, and a double-moment normalised 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. Miri-

ovsky et al. (2004) used a network of four disdrometers of different types within a 1 km2region

in Iowa, United States (US), 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 inter-spacings up to about 33 km, combined with radar data,

4.1. Introduction

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, US, with the goal of

studying DSD variability within a typical ground-based radar pixel (2×2 km2). 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 4×4 km2area near Ciudad Real, Spain. They found large

spatial variability at kilometre 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 km2region

(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-seconds 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 minutes. Using the same network of disdrometers, Jaffrain and Berne (2012a) studied the sub-grid 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 km2scale, an error in rain rate estimation of

between −2% and +15% was observed. More recently, Jameson et al. (2015a) and Jameson et al. (2015b) studied DSD variability using a network of 21 logarithmically-spaced disdrometers over a very small area (distance lags of 1 to 100 m) in South Carolina, US. 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 non-trivial. In an instrument network, it is a chal- lenge 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 kilometre scale. Efforts to address this issue have in- cluded 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 chapter, 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 non-parametric DSDs; their approach differed from ours in that they used a Bayesian up-sizing approach and only a single disdrometer. We present results that used the geostatistical approach shown in Chapter 3 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 sub-grid scales, and thus make comparisons between these scales.

We considered the variability of the DSD over a range of scales from 500×500 m2to 7.5×7.5

km2. Special focus was put on two scales specifically chosen to correspond to real-world

DSD applications. The first was 5×5 km2, about the size of the ground footprint of the Global

Precipitation Measurement (GPM) space-borne weather radar (Hou et al., 2014). The second

was 2.8×2.8 km2, the operational pixel size of the COSMO 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 generalised by normalising 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 sub-grid 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 chapter is structured as follows. In Section 4.2 we outline two DSD models and their parameters. In Section 4.3 the data used in this study are described. In Section 4.4 we explain the stochastic simulation approach, the methods used to calculate variabilities and errors, and the GPM-style and COSMO-style retrievals of the DSD at pixel scale. Results are presented in Section 4.5, with subsections provided for results concerning raw DSD concentrations (4.5.1), bulk variables (4.5.2), GPM-style retrieval of rain rates (4.5.4) and COSMO-style retrieval of rain rates (4.5.5). A brief discussion in Section 4.6 puts the results into perspective. Conclusions are drawn in Section 4.7.