#### 5.1. Dimensions (Diameter) of a Cell

[23] In this article, the statistical distributions of rain cell diameters were calculated for 1998 in Barcelona and for 1996 in Brue. Although the database comprised more years of data, those were specifically selected because 1998 is the year in which more rain events were recorded by the Barcelona network (see Table 1) and 1996 belongs to the central period of the Brue network in which the greatest number of gauges was operational. A total of 2983 rain cells were analyzed for Barcelona and 854 for Brue. For the sake of comparison and to avoid the errors caused by the sampling mechanism of the tipping buckets, the rain fields have been generated with an integration time of 5 min.

[24] The experimental probability density functions (PDF) are shown in Figure 4a. They were calculated using the standard Gaussian kernel density estimator [*Bowman and Azzalini*, 1997]. The criterion used for the calculation of the diameters was the minimum distance between the rain cell center or maximum intensity point, and the point half that value. Thus the cell dimensions were calculated for each rain snapshot obtained from the interpolation of all the rain events presenting spatial structure. For both databases, Figure 4a shows that the most probable diameter of a rain cell ranges between 2 km and 10 km. That is, a radius ranging from about 5 km down to 1 km, assuming circular contours for simplicity. Experience shows, however, that a slight oblateness (ellipticity) is experienced, often due to the effect of a “squashing” wind [*Changnon*, 1981; *Callaghan and Vilar*, 2002].

[25] The fact that some of the diameters are comparable to the network diagonal (16.2 km for Brue and 16.9 km for Barcelona) is explained by the procedure used to calculate the diameters as twice the size of the half-intensity radius. However, the radius is never larger than the diagonal of the network, which is the largest dimension the radius can reach within the catchment studied. As shown by the statistics, big rain cells are not likely to occur. This topic is clarified in more detail in section 5.3.

[26] The cumulative distribution functions (CDF) of the rain cell diameters are shown in Figure 4b together with a second-order polynomial fit applied to the logarithm of the rain cell diameter and obtained in the least squares sense. For practical applications, these polynomials and their range of validity are given in equations (5) and (6) below. The polynomials were preferred among other fitting methods because they are easily applicable to the generation of synthetic rain cells whose diameters can be obtained using the inversion method.

[27] The average diameter is of the order of 6 km and the most probable diameter in statistical terms is about 3.5 km. The 10% and 90% percentiles of the rain cell diameter distributions were found to be 2 km and 17 km respectively. These values were found to be similar for both Brue and Barcelona suggesting similar mechanisms for the generation of convective rainfall, which is the predominant type of rainfall causing structures in the form of rain cells. Similar values of diameters can be found in literature obtained from either radar scans or rain gauge measurements. For instance, the method proposed by *Harden et al.* [1974] gives a value of the diameter at the half intensity point that ranges between 0.6 and 2.7 km. These values have been obtained in the United Kingdom and after applying the synthetic storm (SS) technique [*Drufuca*, 1974] to rapid response rain gauge measurements. In the work of *Pawlina* [2002], equivalent rain cell diameters were obtained from applying the SS technique to point rain gauge measurements in Italy. The results are compared with radar scans showing that the rain cell diameter ranges between 2 and 20 km depending on the intensity threshold used to identify the rain cells.

[28] In the work of *Changnon* [1981], the convective cells in Illinois (USA) were investigated in terms of their length, width and orientation. Several days of data were used from three regular rain gauge networks (one gauge per 23 km^{2}) showing that the most frequent rain cell, defined there as the rain–no rain isohyet, is 16 km long and 6 km wide.

[29] Other research studies carried out recently using weather radars are reviewed by *Pawlina* [1997] for Italy and by *Crane* [1996] for the USA. Assuming circular rain cells, in the work of Pawlina the average diameter was found to be dependent on the type of rain and ranging between 7.5 and 60 km using an intensity threshold of 0.5 mm h^{−1} for the cell identification. On the other hand, in the work of *Crane* [1982] the cell area (−3 dB contour) was found to range between 3.4 and 10 km^{2} depending on the peak reflectivity (i.e., a diameter ranging between 2.1 and 3.6 km).

[30] The dissimilarities found in the different experiments could be due to the different criteria used for the selection of rain events, to the interpolation scheme, or to the Z-R hypothesis employed to convert radar reflectivities into rainfall rates. The comparison between radar cell dimensions and heavy rainy areas determined at the ground by rain gauges is not direct [*Steiner et al.*, 1999]. In any case, more studies in other climatic zones are needed to complete the description of the rain cell dimensions and structure.

[31] In this paper, the fact of obtaining similar results in two locations, Barcelona and Brue, with different climatic characteristics and during two different years seems to indicate that the threshold is a commanding factor influencing the spatial structure of rain intensities. This appears to be a common feature either in the Mediterranean or in the Atlantic areas leading to the thinking that the phenomenological causes of these structured storms (formation, discharge and end) are similar.

[32] The spatial characterization of rain cells for Brue and Barcelona has been carried out with rain fields generated every five minutes. It has been verified by the authors that the integration period does not have an effect on the statistics of the rain cell dimensions. For this purpose, the dimensions of the rain cells occurred in Barcelona in 1998 have been calculated for the two integration periods of 1 and 5 min, showing good agreement. Obviously, a similar study cannot be performed with the Brue database because of the sampling properties of the tipping buckets employed. Nevertheless, similar conclusions would be expected.

[33] As regards the spatial density of rain gauges, it can be concluded that extra resolution does not necessarily lead to extra information about the statistical distribution of cell sizes. This is the case for the Brue experimental network, whose density is higher but it provides similar results as the Barcelona database. These results are probably related to the “irregularity nature” (fractal dimension) of the network itself, as pointed out in section 2. Nonetheless, a high-resolution rain gauge network may provide greater detail about the effects that the local topography and the measurement network itself have on the rain cell structure and trajectories.

#### 5.2. Distribution of the Maxima

[34] The peak intensity reached within the rain cells was also analyzed. This was calculated as the maximum intensity within a given rainfall rate snapshot using an integration time of 5 min. The maximum of the precipitation rate reached inside the rain cell is an important parameter when combined with the cell size. Together, they have implications in drainage planning (flooding) and radio communications networking subjected to fading.

[35] Figure 5a shows the PDF of the maximum intensities for Barcelona and Brue. As expected, the probability decreases rapidly when the maximum intensity increases although this tendency is more evident for Brue. It has been found that once a cell structure appears, the maximum rainfall rate within the cell can take almost any value up to about 100 mm h^{−1}.

[36] Figure 5b shows the cumulative distribution functions and polynomial fittings applied to the logarithm of the maximum intensity. The cumulative distribution of the maxima is different for Barcelona and Brue, with percentiles of 6 mm h^{−1} and 4 mm h^{−1} (10%) and 50 mm h^{−1} and 31 mm h^{−1} (90%) respectively. For the sake of completeness, both CDFs have been given polynomial fittings (in the least squares sense) in (7) and (8) respectively.

[37] The differences observed in the maximum intensities for BCN and Brue are unlikely to be due to the different thresholds used in identifying the rain events or the number of rain cells analyzed for each location. Instead, it seems plausible that the reason is the dominant rainfall generating mechanism associated with the local climatologic features of either site (Mediterranean and Atlantic coasts).

#### 5.3. Dependence Between the Rain Cell Size and Its Peak of Intensity

[38] Once the distributions of the rain cell diameters and the maximum intensities reached are known, one is lead to study their joint distribution. The joint histogram for the Barcelona database has been obtained from the computation of the probabilities given by

which is the probability that a rain cell has a diameter and peak intensity that belongs to the ranges [d_{i}, d_{i} + Δd) and [R_{Mj}, R_{Mj} + ΔR_{M}) respectively. Where d_{min} and d_{max} are the minimum and maximum values of rain cell diameters and R_{M min} and R_{M max} are the minimum and maximum values of peak intensities obtained from the analysis of the interpolated fields. These joint probabilities are shown in Figure 6. The number of bins used was 50 (N = K = 50). A detailed study reveals that the course of a storm follows the well-known three-stage general trend.

[39] 1. An initial period where the rainfall rate is still weak and the rain cell dimensions can take almost any value. This stage corresponds to the rain cell formation.

[40] 2. In the middle and more important part of the storm, the rain rate reaches its maximum or peak. The cell size stabilizes and does not change much. It is in this period that the most probable diameters appear (Figure 4a) with a value ranging between 2 and 10 km.

[41] 3. In the last stages, the rainfall rate in the storm decreases and the value of the rain cell diameter increases.

[42] The results obtained for Brue reveal the same conclusions. Strictly speaking, since stages 1 and 3 correspond to the birth and decay of the rain cell, the rain intensities are almost uniform along any direction in space and it is meaningless to talk about rain contours. This is why the bigger diameters (up to 30 km) in Figure 6, take place during episodes of “relatively” weak rainfall rates (<28 mm h^{−1}). Bigger diameters usually appear during widespread rain events or stages 1and 3 of a storm. On the other hand, once the rain event is spatially structured (see Figure 3) the extent of the rain cell does not vary too much during the central part (stage 2) of the rain event. The extent (microscale) of the rain gauge networks in Barcelona and Brue limits the period of observation of the rain cells. In this way, most of the rain cells appear when they are on stage 2 and would traverse a radio communications network. Other cells are partially measured when they dissipate or develop inside the network.

[43] Regarding the relationship between cell diameter *d* and rainfall rate R_{M}, some models available in the literature affirm that *d* decreases as *R*_{M} increases. To clarify this point, the family of conditional distributions has been calculated for Barcelona and Brue according to the conditional probability

in which P is the probability that a rain cell has a diameter in the range [d_{i}, d_{i} + Δd) given that its peak intensity belongs to the range [R_{Mj}, R_{Mj} + ΔR_{M}). The average diameter as a function of R_{M}, or statistical expectation E{} (km), has been obtained following the standard expression for the conditional expectation given by (11) below and has been plotted in Figure 7 for Brue and Barcelona:

[44] When the average diameter is zero in Figure 7, it means that there are no rain cells in the database with maximum rainfall rates comprising the range of values represented in the abscissa. For instance, this happens in Brue and for heavy storms (>85 mm h^{−1}). However, these are more frequent in Barcelona.

[45] Figure 7 also shows the results obtained with the exponential cell (EXCELL) model given by *Capsoni et al.* [1987] and *Awaka*'s [1989] model. The EXCELL model was derived from the rain cell analysis of 1964 maps of radar data recorded in Italy between April and October of 1980 whereas the values given by Awaka result from the modification of the EXCELL model for small intensities. For the sake of comparison all diameters plotted in Figure 7 have been obtained using the EXCELL criterion, that is, they have been calculated as twice the distance from the rain cell peak where the rainfall rate falls to 1/e of the maximum value R_{M}. The rain cell size given by the EXCELL and the Awaka models is dependent on the intensity peak and decreases as the peak intensity increases. This tendency was also observed in other studies as in the work of *Pan and Bryant* [1994] and *Crane* [1982]. The rain cell sizes obtained by *Pan and Bryant* [1994] were derived from the comparison between slant path attenuation and rainfall rate, both recorded at the earth station. On the other hand, in the work of *Crane* [1982], the diameter of the volume cells (−3 dB contour) was found to range between 2 and 4 km (assuming circular contours) and dependent on the peak rainfall rate. This analysis was based on a radar sample of 25 storm days in Kansas (USA).

[46] One can observe significant differences between the average diameters obtained for Brue and Barcelona and those given by the EXCELL and Awaka models. These differences are to be expected since the half intensity diameters are already larger than the values given by the EXCELL model (see Figures 4a and 4b). The disagreement can be explained by several facts. The EXCELL and Awaka expressions were derived from the modeling of rain cells and not from a direct classification using radar scans. In fact, in the work of *Capsoni et al.* [1987], the rain cell sizes included in the radar database are considerably higher than the ones obtained from the modeling. Moreover, the radar maps used to extract the EXCELL model were recorded during the warm season (April–October) of 1980, at a height of 1.5 km above ground level, and using the Marshall-Palmer raindrop size distribution. In contrast, the results presented here, in this paper, were derived from the analysis of a whole year of spatial rainfall rate data recorded at the ground level and in two different locations. Moreover, only events exhibiting a cell structure have been considered in this paper whereas the variation of the rain cell dimensions given by the EXCELL and Awaka models indicates the presence of two different mechanisms (light rain and heavy rain) probably related to a mixture of stratiform/convective types.

[47] In any case, Figure 7 represents only the expectation of the rain cell dimension and this does not seem to vary too much with the maximum rainfall rate for Brue and Barcelona. *Goldhirsh and Musiani* [1992] also observed this fact showing that that the size distribution of the core isopleths obtained from radar in the Atlantic Coast of USA are independent of the core intensity. This would be in agreement with our results.

[48] In order to clarify the disagreements shown in Figure 7, the PDFs associated to the average cell diameters were obtained for a range of peak intensities and are shown for Brue and Barcelona in Figures 8a and 8b, respectively. The spreading of the curves as well as the location of the maximum depend only slightly on the value of the maximum rainfall rate reached. However, if the average value of the diameters is what is taken, then that value can safely be assumed to be independent of the peak intensity.