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Keywords:

  • rain cells;
  • microscale;
  • interpolation;
  • millimeter waves;
  • fade dynamics;
  • scatter interference

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[1] This paper presents a detailed space-time analysis of rainfall rate using two dense networks of rain gauges covering two microscale areas of some 100 km2 each and located in two distinct climatic areas in the United Kingdom and Spain. The study has been carried out with the main objective of addressing dynamic fade mitigation techniques and scatter interference problems in terrestrial and satellite communication systems. The two databases, using a total of 49 and 23 rain gauges, respectively, are described, and the continuous interpolated field used for the subsequent analyses is explained in detail. The suitability of the networks in terms of density, correlation distance, and fractal dimension are briefly addressed. A to-scale comparison grid of the networks is also given. It has been found that when the maximum intensity reached during a rain event surpasses a certain threshold, different for each of the two areas, rainfall rates exhibit a spatial structure in the form of closed contours of thresholds referred to as cells. In statistical terms, the most probable diameter found is about 3.5 km, and distributions are presented. They are similar for the two sites but not so for the maxima reached inside the cells. Using cross-correlation techniques, displacement and velocities are analyzed in detail. Cells “zigzag” around a dominant trend because of the global cloud movements (driven by winds at heights of about 700 hPa). Local topography strongly affects local behavior. Analytical approximations to the experimental statistical distributions and selected histograms of the results are presented.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[2] The space-time variability of precipitation rate is of general interest both at macroscale and at microscale dimensions. Moreover, the timescales required have become progressively shorter and they are of interest to terrestrial and satellite communications systems amongst other areas (e.g., meteorological predictions, drainage planning, disaster monitoring, etc.).

[3] Because of the move to millimeter wave frequencies to support the large bit rates required for multimedia services, the influence of the atmosphere, and rain fade in particular, need to be quantified. Large bandwidths need the higher part of the available spectrum which has now moved to millimeter wavelengths. In Europe in particular, preliminary studies in the telecommunications sector include the CRABS project or the COST actions 210 [Commission of the European Communities (CEC), 1991], 255 [CEC, 2002] and the ongoing COST 280. These studies have been supported and complemented by some Earth-space propagation experiments such as the OPEX campaign (with ESA's OLYMPUS satellite) and the Italian CEPIT campaign using the ITALSAT 1 satellite.

[4] In the field of meteorology and hydrology for urban areas, the requirements are for rainfall data with very fine time and space resolution [Niemczynowicz, 1990]. Several experiments have been developed in the USA by Huff [1967, 1970a, 1970b] and Changnon [1981] and in Europe by Holland [1967], Niemczynowicz [1988], Lorente and Redaño [1990] or Moore et al. [2000], in order to gain a better knowledge about the local distribution of precipitation.

[5] This article uses two experimental databases and analyzes the spatiotemporal variability of rainfall rate. First, the basic tools employed are explained; these include (1) the databases themselves, (2) their 3-D modeling as rain fields and (3) the interpolation scheme. The spatial arrangement of the rain intensities as rain cells is analyzed in various aspects, namely: conditions necessary for their formation, their size, peak values attained inside, displacement and speed, and correlation with cloud movements.

[6] The spatial properties and movement of rain cells can thus be incorporated into microscale propagation models so as to evaluate the behavior of a dense communication system in the presence of rain. In addition, the spatiotemporal modeling of rain fields is essential for the design and simulation of fade and interference mitigation strategies. The conclusions summarize the results and outline their use in telecommunications applications.

2. Characteristics of the Data Fields

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[7] The analysis carried out in this paper has used two dense rain gauge networks located in two different climatic zones: the city of Barcelona (BCN) in the Mediterranean coast (northeast of Spain) and the river Brue catchment in Somerset (southwest of England). Moreover, the International Telecommunication Union in its Recommendation P.837-4 provides a characteristic rainfall intensity of 30 mm h−1 exceeded for 0.01% of the average year for both locations [International Telecommunication Union, 2003]. Figure 1 shows a scale grid of the two networks as it is indicative of their density. Moreover, in order to set the scene for the paper, Figure 2a is a footprint of an event and Figure 2b shows two orthogonal precipitation profiles of that event.

image

Figure 1. Comparative grid of the distribution of rain gauges for Barcelona (BCN) and Brue. Areas A and B represent dense subnetworks (see text).

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image

Figure 2a. Example of a typical rain cell.

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image

Figure 2b. (top) Horizontal and (bottom) vertical cross sections of the example given in Figure 2a.

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2.1. Barcelona Data Field

[8] The rain gauge network in Barcelona has its origin in the control and protection of the city from flooding. This network, covering a rectangular area of 14.7 km × 8.4 km (123 km2) consists of 23 tipping bucket rain gauges with a bucket size of 0.1 mm. The recorded period analyzed spans from 1994 to 1999. The upper part of the experimental network is a few hundred meters above the mean sea level and on the foot of the hills that surround the city. The lower part is parallel to the coastal line. This network, which was initially conceived in 1983 for drainage planning purposes, was previously used in a lesser dense version of 12 gauges for the characterization and modeling of the local convective cells [Lorente and Redaño, 1990].

[9] In Barcelona, the average annual rainfall is about 600 mm, 40% of which is of convective origin. Convective rain affects relatively small areas, takes place mainly during late summer and autumn and the rain events cannot be easily predicted because their origin is in mesoscale and heavily influenced by the coastal line of Catalonia (sea presence). The statistics of the precipitation in Barcelona and their application to communication systems have been widely discussed in the past by Burgueño et al. [1987].

2.2. Brue Data Field

[10] The Brue network has its origin in the Hydrological Radar Experiment (HYREX) project. HYREX aim was to gain a better understanding of rainfall variability in space and time, as sensed by a weather radar. The vital part of the experiment was the provision of a dense network of rain gauges covering a 135-km2 catchment area from 1994 to 1997. The gauges were originally arranged in such a way that each one was approximately located at the center of a 2 × 2 km square, which is equivalent to a pixel of a UK Meteorological Office radar. In addition, there are two squares having dense subnetworks (Figure 1, areas A and B) of eight gauges each in areas of low and high relief (topography) [Moore et al., 2000]. The final operational network comprised 49 Casella 0.2-mm tipping-bucket rain gauges. The precipitation regime in the Brue region is characterized by the presence of Atlantic frontal systems and the average annual rainfall from 1961 to 1990 was 867 mm.

2.3. Density Distribution of the Rain Gauges

[11] The first question arising when using data from a rain gauge network is the spatial resolution and whether it is good enough to characterize the rainfall rate without losing information about its spatial properties. The average density of rain gauges in Barcelona (BCN) is 1 rain gauge/5.5 km2 whereas for Brue is 2 rain gauges/5.5 km2 (Figure 1). The average distance between each gauge and its closest neighbor is 1.7 km for Barcelona and 1.0 km for Brue. The average distance to the rest of the gauges of the network is 5.7 km for Barcelona and 5.4 km for Brue.

[12] In studies concerning the spatial variability of rainfall rate, it is usually convenient to work with regularly spaced data, like in radar scans. Nevertheless, only rain gauges provide a quantitative measurement of rain intensity. The fractal dimension of the acquisition network is a measure of its spatial regularity or its ability to fill a plane. The fractal dimension is also a practical tool to compare the density of the networks in view of the fractal methodology. If the rain gauges were truly spaced filling, as in the case of radar images, one would expect to compute a network fractal dimension of 2.0. However, if the rain gauges are not uniformly distributed along the acquisition network, the fractal dimension can be significantly less than 2.

[13] To study the fractal properties of a discontinuous rain gauge network, it is useful to calculate a parameter called the correlation dimension [Olsson and Niemczynowicz, 1996]. The correlation dimensions for Barcelona and Brue were found to be 1.72 and 1.46 respectively. This result indicates that both rain gauge networks have a scaling structure (in terms of areas) within the range from 5 to 100 km2 and that the Barcelona network is more regularly arranged.

3. Interpolated Rainfall Rate Fields

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[14] In the field of dynamic and dense millimeter wave radio communications, and particularly in the analysis of the spatial properties of microscale rainfall rates, it is often more convenient to work with regular grids than with irregular data. For this purpose, a spatial interpolation algorithm based on the biharmonic splines was applied to the original sampled data in order to generate continuous rainfall rate fields with a spatial resolution of 100 × 100 m. Each one of the available N gauges was assumed to be located within its associated 100 × 100 m square. The biharmonic splines are commonly used for the interpolation and parameterization of sparsely distributed data sets [Sandwell, 1987]. For the 2-D case, the biharmonic spline function (1) returns the interpolated value R′ using N data points.

  • equation image

where R′ is the rainfall rate evaluated at the point x′ which corresponds to the center of the imaginary 100 × 100-m square, Ri is the rainfall rate measured at the point located in xi, x′ − xi is the vector between the location of the data point and the location of the evaluated point, and ϕ is the 2-D Green's function that has been obtained from the solution of the biharmonic equation and represents the weight applied to each data point Ri:

  • equation image

[15] This interpolation method provides an optimal estimate in the least squares sense, smoothes the rainfall distributions and does not reproduce rain peaks off gauge locations. To measure the accuracy of the interpolation scheme and compare its performance, the “fictitious point” method [Dirks et al., 1998] was employed. The interpolation error σ can be defined by the root mean square error (RMSE):

  • equation image

where Ri is the observed rainfall with rain gauge i, and R′i is the interpolated rainfall estimate (using N − 1 values, excluding i) at the i-th gauge location (xi, yi). The coefficient of variation is defined by

  • equation image

[16] Using the above performance measure ρ(t), four representative Barcelona rain events with different maxima have been analyzed, showing that for the interpolation scheme suggested, ρ ranges between 0.4 and 0.7. Dirks et al. [1998] found similar values for longer integration periods and classical interpolation schemes such as Kriging, Thiessen, areal mean or inverse distance. In the case of Brue, similar or even better conclusions would be expected because of the proximity between rain gauges (see Figure 1). From the error analysis, it can be concluded that the interpolation scheme based on the biharmonic splines reproduces and follows with accuracy the real variations of rainfall rate.

4. Conditions for the Existence of a Spatial Structure

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[17] The term “spatial structure” employed throughout this article refers to the microscale organization of rainfall rates into “rain cells.” Earlier studies have employed different definitions for the identification of rain cells regarding either the value of the isohyet relative to the peak [Goldhirsh and Musiani, 1992], the rain–no rain isohyet [Changnon, 1981], the minimum rain rate contour separating two cores or a certain intensity threshold. (An isohyet is a line drawn through geographical points recording equal amounts of precipitation during a specific period.) Despite the different definitions of a rain cell, they all refer to a region (footprint) where the precipitation exceeds a certain value relative, or not relative, to the peak value reached at the cell center.

[18] To identify rain cells, it is essential to select adequately those rain events that will be used to calculate the interpolated rainfall rate fields. This selection was based on the features of the rain gauges and intends to avoid the limitations in precision, particularly for tipping bucket gauges when working with small timescales and light rain. These limitations have been analyzed in the past by Ciach [2002] and Habib et al. [2001].

[19] In Barcelona, the events chosen were those during which the rainfall rate exceeded 6 mm h−1 at least in one of the gauges (1 tip per minute) and during one minute. In the Brue case, the rainfall rate threshold was 12 mm h−1 (1 tip per minute) because of the lower resolution of the gauges (double volume of the tip). The integration period has also been selected accordingly to the tipping bucket resolution. Following the recommendations given by Habib et al. [2001] and Ciach [2002] for rain gauge measurements, an integration time of 5 min have been used for the generation of the rain fields recorded in Brue whereas in Barcelona, the rain fields can be generated with shorter integration times (down to 1 min).

[20] The observation of continuous rainfall rate fields makes it possible the use of an algorithm for the automatic classification of rain cell structures. Rainfall rate fields were calculated and plotted for all the rain events recorded for the years 1995, 1998 and 1999 in Barcelona and for the years 1995, 1996 and 1997 in Brue. Table 1 shows the number of events analyzed each year. These events were classified according to their maximum intensity.

Table 1. Number of Events Analyzed for Each Year and Classification According to Their Maximum Intensity for the Barcelona and Brue Databases (Integration Time 5 min)
Year/Threshold>12 mm h−1>24 mm h−1>36 mm h−1>48 mm h−1>60 mm h−1
Brue
1995103402093
1996924420115
199790291155
 
Barcelona
19954937311813
19987044301813
1999442516128

[21] From the analysis of the rainfall rate fields in Brue and Barcelona, it was found that when the maximum rate exceeds a certain threshold, the rainfall rates tend to be spatially arranged in closed contours or rain cells. It is well documented that spatial uniformity of rain depends on the intensity [Huff, 1967, 1970b]. The number of spatially structured events increases significantly as the maximum intensity recorded increases. Figure 3 shows, for the two areas, the percentage of rain events exhibiting rain cells when the maximum intensity exceeds the indicated threshold. Since for the weaker events (<24 mm h−1) the proportion of rain events presenting rain cells is around the 60%, for the stronger events (>48 mm h−1) the proportion of structured events is more than the 75%. The integration time used to obtain these percentages was 5 min for both databases.

image

Figure 3. Proportion (%) of rain cells against maximum rainfall rate in Brue and BCN.

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[22] Apart from the maximum intensity, the existence of rain cells depends also on the period of the year. This is because shower-type precipitations tend to occur from Spring to Fall in the latitudes and longitudes of the rain fields analyzed [Huff, 1967]. Winter spatial structure can also appear but this is more sporadic. For instance, for Brue, the totality of storms which occurred during July and August of 1997 were accompanied by spatial structure whereas during the months of January and February, no rain cells were detected. Once it is known that there is a high probability for the events to be organized in rain cells because their maximum intensity exceeds a certain threshold, it is possible to investigate the characteristics of these rain cells in terms of their extent or size, and their trajectories.

5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

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].

image

Figure 4. (a) Probability density functions (PDF) of the rain cell diameters for BCN and Brue. (b) Experimental cumulative distribution functions (CDF) of the rain cell diameters and applied polynomial fit for BCN and Brue.

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[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.

  • equation image
  • equation image

[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 km2) 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 km2 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.

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Figure 5. (a) PDFs of the maximum intensities reached inside a rain cell for BCN and Brue. (b) Polynomial fit and CDFs of the maximum intensities for BCN and Brue.

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[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.

  • equation image
  • equation image

[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

  • equation image
  • equation image
  • equation image

which is the probability that a rain cell has a diameter and peak intensity that belongs to the ranges [di, di + Δd) and [RMj, RMj + ΔRM) respectively. Where dmin and dmax are the minimum and maximum values of rain cell diameters and RM min and RM 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.

image

Figure 6. Joint distribution of rain cell diameter and maximum intensity.

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[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 RM, some models available in the literature affirm that d decreases as RM increases. To clarify this point, the family of conditional distributions has been calculated for Barcelona and Brue according to the conditional probability

  • equation image

in which P is the probability that a rain cell has a diameter in the range [di, di + Δd) given that its peak intensity belongs to the range [RMj, RMj + ΔRM). The average diameter as a function of RM, or statistical expectation E{equation image} (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:

  • equation image
image

Figure 7. Theoretical and calculated average rain cell diameters against maximum rainfall rate (1/e criterion).

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[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 RM. 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.

image

Figure 8. PDF of the rain cell diameters (1/e criterion) found for different intervals of maximum rainfall rate in (a) Brue and (b) BCN.

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6. Movement of the Rain Cells

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[49] The study of the direction and speed by which rain cells move can be carried out simply using the cross correlation between the successive hyetograms. This analysis was also done earlier for the city of Lund [Niemczynowicz, 1988]. The limitation of this approach, however, is that the optimal number of rain gauges to be selected for the correlation analysis of a given rain event varies with the direction of the storm. This limitation can be overcome if a continuous two dimensional rainfall rate field is available as, for instance, through radar observations. Alternatively, a continuous field can be obtained through the interpolation of a discrete 2-D rain gauge field, as done in this article.

6.1. Application to Correlation

[50] Once a rain cell has been identified within a rainfall rate field, the vector movement can be extracted using two-dimensional correlation techniques. These techniques are common in fluid mechanics (particle image velocimetry) and remote sensing [Huff, 1970a; Leese et al., 1971]. The two-dimensional cross correlation between successive snapshots is given by

  • equation image

where equation image (x, y) is the 2-D covariance function evaluated at the (x, y) lags and obtained between two rainfall rate snapshots R1 and R2 separated by a time lag of τ (equivalent to the integration time). A cross-correlation analysis measures the size and direction of the relationship between variables. The displacement of the maximum value from the origin or central point of the cross-correlation matrix combined with the snapshot period gives an indication of the speed and direction of the rain cell. In addition, the shape of the correlation matrix indicates the strength of the linear relationship between observations obtained at different points of the grid [Huff, 1970a].

[51] The two-dimensional cross correlation was used to extract the speed and directions of the rain cells during fifteen events measured in 1996, 1997 and 1999 (example of the Barcelona case). This representative set of rain events include episodes exhibiting maxima ranging from 50 to 150 mm h−1. They have been selected in order to study the behavior of the method on spatial structured events in which the rain cells do not change too much during the event and totally traverse a hypothetical urban communications network. The displacement vectors were obtained for the set whose main characteristics are summarized in Table 2.

Table 2. Main Characteristics of the Set of Events Selected and Analyzed Using the Cross-Correlation Method
EventDateDuration, minMaximum, mm h−1
A30 Aug 19963284
B30 Aug 199617120
C3 Sept 19961270.8
D17 Sept 199618130.44
E22 Sept 199624102
F22 Sept 19962299
G6 Dec 19961148
H7 Dec 199624120
I7 Dec 19961096
J4 July 19973078
K23 Oct 199722150
L23 Oct 19973099
M16 Dec 19972275
N1 Jan 19991248
O14 Sept 199970186

[52] The cross-correlation analyses were carried out using consecutive rainfall rate snapshots integrated over one minute and then over two minutes. As explained in preceding sections, the characteristics of the Barcelona database make it appropriate for the use of finer time resolutions. The option from one minute to two minutes was taken when it was required to smooth the trajectory of the cell, thus avoiding local effects; otherwise, one minute was considered sufficient. The superposition of the displacement vectors between rainfall rate snapshots gives the magnitude and direction of the movement of the rain cell. In most cases, the observation of displacement vectors indicates rapid changes in the structure, direction and speed.

[53] An example of a rain cell trajectory is illustrated in Figure 9a. Figure 9a shows the case in which the evolving cell exhibits a footprint trajectory which is far from being a straight line; nevertheless, one can still see a general trend.

image

Figure 9. (a) Example of a rain cell trajectory recorded in BCN on 7 December 1996. (b) Average direction and speed of the rain cells analyzed in BCN.

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[54] The calculated trajectories of the rain cells were found to be more or less chaotic, depending on the synoptic conditions. Despite this randomness, when the successive vectors are averaged over the whole rain event, they show a clear trend along which the cell zigzags. This zigzag is very likely related to the influence of the city and the presence of the coastal area and the sea, which exhibit a (terrain) characteristic roughness height and a temperature evolution (M. R. Soler, University of Barcelona, personal communication, 2002). The overall effect is reflected in the heat budget or the airflow. Moreover, the rain gauges are located at the top of multistory buildings and have influence by the local effects. Research in the field of pollutant dispersion tries to measure and model these effects [Rotach, 2001; Allwine et al. 2002; Roulet, 2002] together with the influence of the turbulence induced by the buildings, radiation trapping and channeling by the street canyons. One should also not discard the random influences contributed by the tipping buckets and the spatial arrangement of the rain gauge network. However, the errors due to the tipping bucket mechanism have been considerably reduced in this study because of the availability of 0.1 mm buckets [Ciach, 2002].

6.2. General Trends

[55] As indicated, the average speed and angle were obtained for each one of the fifteen events analyzed. This was carried out by averaging the displacement vectors over the whole rain event. The results are presented in Figure 9b. Each vector of Figure 9b has two components: the modulus which is the speed in m s−1 and the phase angle which gives the direction of the rain cell movement. It can be noticed that the most probable directions are SW-NE and NW-SE (parallel and perpendicular to the coastline, respectively). Figure 9b also shows the speed movements which range from 1 to 7 m s−1.

[56] Since the extraction of the global movements does not require the use of rain fields generated with fine time resolution, this analysis has been carried out for both sites. Statistically, the global results for three years and the 2 sites are shown in Figures 10a and 10b. The predominant components for Brue are S-N and W-E (frontal episodes from the Atlantic). S-N and W-E are also the predominant components in Barcelona [Lorente and Redaño, 1990].

image

Figure 10. General trend of the “cell-structured” rain events in (a) BCN for the years 1995, 1998, and 1999 and (b) Brue for the years 1995, 1996, and 1997.

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6.3. Comparison With the Global Movement of the Wind

[57] The results obtained from the cross-correlation analysis have been compared with the wind information provided by the European Meteorological Bulletin maps (European Meteorological Bulletin, Deutscher Wetterdienst). These data comprise the speed and direction of the wind at different pressure levels: Surface (SFC), 700 hPa and 850 hPa. Only two measured values (1200 UTC and 0000 UTC) a day are available from the maps, this is the reason why it is not possible to obtain values that are more accurate. If two rain episodes take place the same day, the identical synoptic wind data were used for both cases.

[58] Table 3 shows the comparison between the values obtained from applying the two-dimensional cross-correlation method and the ones obtained from the wind maps. When data from the radiosonde were available, these were also included in the Table 3. After examining the synoptic situations for the fifteen cases, it was possible to classify them into three different groups.

Table 3. Comparison Between the Directions and Speeds Obtained From the Cross-Correlation and the Values in the European Meteorological Bulletin
EventCross-Correlation ValuesMeasured Values
Surface850 hPa700 hPa
  • a

    Radiosonde data: 20° − 10 m s−1.

  • b

    Radiosonde data: 118° − 14 m s−1.

  • c

    Radiosonde data: 194° − 18.7 m s−1.

  • d

    Radiosonde data: 48° − 5 m s−1.

  • e

    Radiosonde data: 152° − 7 m s−1.

  • f

    Radiosonde data: 195° − 10.1 m s−1.

AW − 1 m s−1CalmN − 5 m s−1N − 5 m s−1
BSW − 3.66 m s−1CalmN − 5 m s−1N − 5 m s−1
CNW − 4.6 m s−1WSW − 2 m s−1CalmN − 2 m s−1
DSW − 3.14 m s−1CalmSW − 10 m s−1SW − 15 m s−1
EN − 2.9 m s−1SW − 2 m s−1N − 10 m s−1NW − 14 m s−1
FNW − 3.04 m s−1SW − 2 m s−1N − 10 m s−1NW − 14 m s−1
GS − 1.95 m s−1SSW − 3 m s−1S − 10 m s−1S − 14 m s−1
HS − 3.5 m s−1N − 2 m s−1SE − 10 m s−1SE − 12 m s−1
IE − 20.4 m s−1N − 2 m s−1SE − 10 m s−1SE − 12 m s−1
JSW − 2.52 m s−1NE − 2 m s−1CalmSW − 8 m s−1
KNW − 1.38 m s−1CalmCalmCalm
LW − 0.76 m s−1CalmCalmCalm
MSW − 2.5 m s−1SSE − 2 m s−1aS − 10 m s−1bSW − 15 m s−1c
NSE − 7.27 m s−1CalmSE − 12 m s−1ESE − 10 m s−1
OSW − 0.43 m s−1CalmdSSE − 5 m s−1eSSW − 15 m s−1f

[59] 1. Events A, B, C, K and L fit rain situations with an obvious local character and without a well-defined synoptic condition. In these cases, the rain cell trajectory is undefined.

[60] 2. Events D, G, H, I, J, M, N and O match synoptic situations, which have, as a common feature, the existence of a depression over the Iberian Peninsula. This situation is the cause for the presence of winds with SW component over Catalonia. Under this particular situation, the direction and speed of rainfall rate is well defined, with a SW or NW trajectory in the most of the cases.

[61] 3. Events E and F belong to a north situation and the rain cells movement is from north to south.

[62] The comparison indicates that the rainfall rate evolution correlates well with the wind direction at heights which correspond to 700 hPa (3000 m approx.) and 850 hPa (1500 m approx.) rather than to the surface wind measurements. Nevertheless, one has to be careful when assuming that the velocity of rain cells is the same as the wind velocity at 700 hPa. This cannot be true when instantaneous wind data are not available since significant differences were found from the comparison with the Meteorological Bulletin maps (see Table 3). It has long been accepted that thunderstorms move, on average, with the wind at approximately the 700 hPa level, [Brooks, 1946; Holland, 1967]. As evidenced in the past by Humphreys [1940], it would be at these heights where the strata containing most of the cumulonimbus extend.

7. Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[63] The extensive analyses carried out in this paper have employed the outputs of two dense rain gauge networks located in two different climatic regions. Their density has been characterized in terms of their fractal dimension. The interpolation algorithm used to map the original sparse data into a regularly spaced rainfall rate fields was shown to perform well.

[64] The statistical distributions of rain cell dimensions and maximum intensities have been obtained from the interpolated rain fields. The analysis shows that rain cell diameters do not follow an exponential distribution as shown in some earlier studies. The differences are probably due to the fact that most of the analyses carried out in the past have been performed on radar scans and very few on rain gauge data. The rain gauge networks used here have allowed the calculation of rainfall rate fields with high spatial and temporal resolution. In contrast, radar images provide reflectivities averaged over the pixel and usually with lower spatial and temporal resolution but cover macroscale areas which often show the presence of several cells.

[65] Additionally, the dependence between diameter and peak intensity of a rain cell has been investigated, showing that the average rain cell dimension can be assumed to be quasi-independent of the maximum intensity. On the other hand, differences were found when the size of the rain cells was compared to other studies. Several reasons for these differences have been given in the paper. They include: the differences between the sources, acquisition and processing of the data, and the no differentiation between the two main precipitation mechanisms, stratiform and convective rain. It is evident that more analysis on radar and rain gauge spatial data is needed to conveniently explain the discrepancies because the comparison between the results available in literature, especially between measurements obtained from radar and those obtained from rain gauge measurements is not straightforward. Rain gauges are the most reliable way to determine the rain intensity that actually reaches the ground surface but there are practical limitations in the maximum area that can be studied, that is, microscale against macroscale.

[66] It would appear that in order to predict the dominant trend in the movement of structured rain cells, it is necessary to analyze meteorological data at a height between 1500 m and 3000 m. Wind velocity measurements at 700 hPa have been applied in the past to convert point rainfall rate into spatial rainfall rate. This method, known as the synthetic storm (SS), is based on the hypothesis that convective rain cells move along a line with constant velocity and that advection is the dominant mechanism. As indicated in this article, this hypothesis holds only when there is convective structuration under a well-defined synoptic condition. The type of cloud would then be indicative of the likely occurrence of storms. Thereafter, everything else is dependent on the intensity thresholds.

[67] Finally, the research and information presented in this article are absolutely essential to evaluate the effects of a moving rain cell on relatively short-scale (microcells of a few tens of kilometers) terrestrial and satellite communications systems. Indeed, as a rain cell of a known size zigzags along a dominant direction with its characteristics as described in this paper, it is possible to realistically undertake engineering studies for the positioning of ground stations in a diversity configuration, to satisfy dynamic quality criteria like, keeping the bit error rate below a given threshold. The same applies to the layout of a terrestrial mesh configuration in order to, for instance, minimize the interference due to side scattering; research carried out by the authors outside the scope of this paper shows that side scatter interference can be substantial for millimeter wave systems.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Characteristics of the Data Fields
  5. 3. Interpolated Rainfall Rate Fields
  6. 4. Conditions for the Existence of a Spatial Structure
  7. 5. Statistical Analysis of the Cell Dimensions and Their Peak Rainfall Rate
  8. 6. Movement of the Rain Cells
  9. 7. Discussion and Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information
FilenameFormatSizeDescription
rds5129-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5129-sup-0002-t02.txtplain text document0KTab-delimited Table 2.
rds5129-sup-0003-t03.txtplain text document1KTab-delimited Table 3.

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