# Structural characteristics of rainfields

Frédéric Mesnard, Henri Sauvageot

► To cite this version:

Frédéric Mesnard, Henri Sauvageot. Structural characteristics of rainfields. Journal of Geophysical Research, 2003, 108 (3), pp.D002808. 10.1029/2002JD002808. hal-00137821

HAL Id: hal-00137821

https://hal.science/hal-00137821v1

Submitted on 25 Jun 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Structural characteristics of rain fields

Frédéric Mesnard

Laboratoire de Physique de l'Atmosphère Tropicale, Université des Antilles et de la Guyane, Pointe-à-Pitre, Guadeloupe

Henri Sauvageot

Observatoire Midi-Pyrénées, Laboratoire d'Aérologie, Université Paul Sabatier, Toulouse, France

Received 29 July 2002; revised 11 December 2002; accepted 20 March 2003; published 8 July 2003.

[1] This paper analyzes the shape of rain area size distributions (RASDs) observed by radar in tropical and midlatitude regions. The rain area is defined, with respect to a rain threshold $\tau$, as the area, inside a contour, where the rain rate is higher than $\tau$. The size considered is the diameter of the equivalent circular area $D$. The reflectivity peaks inside the rain areas are numbered and the rain areas put together by classes having the same peak number. The size distributions of the rain areas containing the same number of peaks $p$ are well fitted by lognormal functions, the parameters of which, $\mu_p$ and $\sigma_p$, are weakly dependent on the conditions of the peak determination. The parameters $\mu_p$ and $\sigma_p$ are found to be linked by a power relation. For $D < 6$ km most of the rain areas are single peaked. The RASD without peak number distinction is a lognormal mixture. Using the relations obtained for $\mu_p$, $\sigma_p$, and the rain area number as functions of $p$ enables one to simulate RASDs with results in good agreement with the observations. Because of the limitation in the radar sampling at both ends of the RASD and the decrease of area number with increasing size, the radar-observed RASDs are severely truncated. The possibility of an approximate fitting of the truncated RASD with diverse functions is discussed. At the intermediary values of the threshold, i.e., for $\tau$ ranging between 3 and $12 \text{ mm h}^{-1}$, where the RASD is easiest to compute, the slope of these truncated distributions is not strongly dependent on the threshold. This slope appears not to be very sensitive to the minimum and maximum distances bounding the domain where the rain field is observed. The RASD is found to be sloping more over sea than over land. The slope of the RASDs shows a small diurnal variation but no significant annual variation. The “perimeter to area” (fractal) dimension of the rain area for the sample without peak number distinction is around 1.35. Single-peaked and multipeaked rain areas have different perimeter to area dimensions, ~1.23 and 1.38, respectively. These values are almost independent from the threshold and from the location of the domain observed.
**INDEX TERMS:** 3314 Meteorology and Atmospheric Dynamics: Convective processes; 3329 Meteorology and Atmospheric Dynamics: Mesoscale meteorology; 3354 Meteorology and Atmospheric Dynamics: Precipitation (1854); 3360 Meteorology and Atmospheric Dynamics: Remote sensing; 3374 Meteorology and Atmospheric Dynamics: Tropical meteorology; **KEYWORDS:** rain field, rain cells, radar meteorology, lognormal distribution, fractal dimension, rainfall distribution

**Citation:** Mesnard, F., and H. Sauvageot, Structural characteristics of rain fields, *J. Geophys. Res.*, 108(D13), 4385, doi:10.1029/2002JD002808, 2003.

## 1. Introduction

[2] A knowledge of the rain area size distribution (RASD) is relevant to many fundamental and applied topics, notably in the atmospheric circulation models used for the parametrization of the water and energy cycles [e.g., Houze, 1993; Georgakakos and Krajewski, 1996; Cheng and Arakawa, 1997], in radio communication for the modeling of the tropospheric and earth-space microwave

propagation [e.g., Crane, 1996; Goldhirsh, 2000], in hydrology for rain-runoff modeling, and also for the assessment of beam filling and target homogeneity problems for the precipitation measurement from a spaceborne radar [Simpson et al., 1988; Durden et al., 1994; Sauvageot et al., 1999; Meneghini et al., 2001].

[3] A brief review of the literature on RASD is presented in section 2. This review leads to the conclusion that there is confusion and a lack of accuracy in the ideas concerning the representation of the RASD shape as well as the natural variability of the distribution parameters. No discussion can be found in the literature on the influence of the rain area
structure on the parameters of the RASD. The natural variations of the RASD are thus far from being sufficiently known and documented to be used in models.

[4] The object of this paper is to revisit the distribution of rain areas by size, in order to look at aspects of the topic not covered before (see section 2), notably the influence of the peak numbers inside the rain areas on the shape and parameters of the rain area size distribution. The goal is also to assess the variability of the parameters of RASDs and its climatological dependence. The data were collected in two climatological regions not covered by previous studies: West Africa, where the main rain-bearing system is the tropical squall line, notably important for rain measurement from space, and France, where the most significant rain-producing system is the midlatitude frontal system, including a wide stratiform component.

## 2. Brief Review of the Literature

[5] The space and time distribution of rain areas is related to the organization of atmospheric motions, since rain is the consequence of convective motions able to ensure the droplet growth up to a size sufficient to be detected. When convective structures are embedded in a cloud continuum, as in the large cloud systems of midlatitude frontal depressions, or in tropical squall lines, rain field and atmospheric structure can only be observed by radar. When the atmosphere is not uniformly saturated, the observation of the structural characteristics of convective fields (and sometimes of rain fields) can be approached by cloud field analysis from photographic or radiometric infrared (IR) data. That is why, in the present study on rain field structural characteristics, it is relevant to take into account the results and conclusions of the studies on the cloud field organization, notably the cumulus cloud fields. Many such studies were performed because of the cloud influence on the climate system through radiative effects.

[6] Cumulus cloud size distribution, in the size range of ~100 m to 15 km, have been reported by, e.g., *Wielicki and Welch* [1986], *Welch and Wielicki* [1986], *Welch et al.* [1988], *Cahalan and Joseph* [1989], *Joseph and Cahalan* [1990], *Sengupta et al.* [1990], *Weger et al.* [1992], *Zhu et al.* [1992], *Kuo et al.* [1993], *Benner and Curry* [1998], and *Gotoh and Fujii* [1998]. The cloud-size parameter considered in most of these works is the diameter of the equivalent circular area $D$, i.e., $D = (4A/\pi)^{1/2}$, where $A$ is the cloud horizontal area. The size distribution can be easily represented by a functional form. For cloud populations including sizes smaller than 1 km, many authors have proposed a double power law [e.g., *Cahalan and Joseph*, 1989; *Kuo et al.*, 1993; *Benner and Curry*, 1998], even if other forms were also discussed, notably for large clouds, cloud clusters, and for rain areas (see below). The main point is that the size distribution displays a clear change in the slope around a break diameter $D_b$ of ~1 km. For $D < D_b$ the slope $\alpha$ of the cloud size distribution $N(D)$, such as $N(D) = N_0D^{-\alpha}$, where $N_0$ is a parameter, is found around $\alpha \approx 1.7$, and for $D > D_b$, $\alpha \approx 3.4$. Most authors observe that clouds can be classified into two populations: unicellular or multicellular, according to the number of cells found inside each cloud. The counting of the cells is obtained by considering the

bump of the cloud top. The break diameter is interpreted as the transition between the two populations, the cumulus clouds evolving toward a multicellular structure as their effective diameter becomes larger than $D_b$. It has been suggested that $D_b$ corresponds to the largest individual cell size in the observed population [*Joseph and Cahalan*, 1990; *Kuo et al.*, 1993]. $D_b$ displays some variability. For example, *Benner and Curry* [1998], from an analysis of 22 distributions of tropical cumulus clouds, found $D_b$ ranging between ~0.5 and 1.5 km. The number of cells which make up a multicellular cloud increases with the effective diameter in such a way that almost all clouds with $D > 5$ km are multicellular with a typical number of five cells for $D \approx 5$ km [*Kuo et al.*, 1993]. For small clouds, that is, unicellular or multicellular clouds with a low cell number, the horizontal aspect ratio ($a/b$ where $a$ and $b$ are the minor and major axes, respectively, of an assumed elliptically shaped cloud contour) is ~2 but diminishes toward 1, that is, becomes circular, when the size increases [*Welch et al.*, 1988; *Kuo et al.*, 1993].

[7] In the present paper dedicated to the study of the structural characteristics of rain fields from radar data, the peak number of the radar reflectivity factor inside the precipitation areas is considered. It is suggested that the cells numbered on the photograph or IR image of cumulus cloud fields could have some qualitative relations with the peak of the reflectivity field discussed in the present paper.

[8] If a power law seems more appropriate than an exponential to fit cloud size distributions [*Benner and Curry*, 1998], the size distribution of the cells (that is, unicellular clouds and individual cells of multicellular clouds) is found rather exponential [*Wielicki and Welch*, 1986; *Kuo et al.*, 1993]. It can be observed, however, that the two kinds of functions are very different from a mathematical point of view since the power law is a fat-tailed distribution, while the exponential is a thin-tailed one. This aspect is not discussed in the above-quoted references, most authors using only one of them and making no comparison between the two.

[9] Many authors have investigated the fractal nature of cloud and rain fields using the perimeter to area (PtA) relationship, probably for the sake of simplicity, since more rigorous approaches such as the functional box counting could be used. The PtA relationship is written $P = cA^{d/2}$, where $c$ is a coefficient and $d$ is the “fractal dimension,” that is, a parameter depending on the complexity of cloud or rain area contours [*Mandelbrot*, 1982]. For simple shapes such as a circle or a rectangle, $d=1$, and $d \to 2$ when the perimeter becomes increasingly ragged and contorted. For clouds and rain areas (as observed from satellite imagery and by radar) between 1 and $10^4$ km$^2$, most authors found a single fractal dimension of $d$ around 1.35, suggesting that the shapes of clouds and rain areas are “self-similar” throughout their whole range of size [e.g., *Lovejoy*, 1982; *Rhys and Waldvogel*, 1986; *Lawford*, 1996; *Féral and Sauvageot*, 2002]. For cumulus clouds, including sizes smaller than 1 km up to ~20 km, most investigators have found that PtA exponent $d$ is different for small and large sizes, with the same breaking diameter as for the size distribution. For small cumulus clouds with $D < D_b$, $d$ is reported to be in the 1.08–1.29 range, and for $D > D_b$, $d$ is between 1.27 and 1.73 [*Benner and Curry*, 1998]. These
differences in *d*, with respect to $D_b$, are attributed to the influence of the multicellular character on the cloud contour. The values observed for $\alpha$ as well as for $D_b$ are not sensitive to the temperature threshold used for the analysis [Kuo et al., 1988, 1993; Mapes and Houze, 1993] and are preserved as the spatial resolution of the data is degraded [Zhu et al., 1992; Kuo et al., 1993]. Cirrus clouds are also observed to have a bi-power size distribution and a bi-fractal PtA relation [Kuo et al., 1988]. *Féral and Sauvageot* [2002] found that supercellular convective storms, because they are unicellular, have a fractal size smaller than the ordinary multicellular storms.

[10] The distribution of the deep convective cloud clusters with sizes larger than 20 km has been discussed from satellite data by, e.g., Machado et al. [1992], Machado and Rossow [1993], Mapes and Houze [1993], and Carvalho and Silva Dias [1998]. For convective clusters up to $D \approx 500$ km, Machado and Rossow [1993] found that the size distribution of the equivalent circular diameter is correctly represented by a power function with a power coefficient of -2. For oceanic warm pool cloud clusters (between 16 and 300 km), Mapes and Houze [1993] observed a somewhat lognormal distribution of size and a constant fractal dimension of ~1.4. Carvalho and Silva Dias [1998] showed that the fractal dimension offers a potential for mesoscale cloud pattern recognition.

[11] Comparatively with cloud size distribution the rain area size distribution has not received very close attention in the past (review by Crane [1996] or by Sauvageot et al. [1999]). From hourly photographs of an analogic PPI radar scope, Dennis and Fernald [1963] found that the rain shower number, as a function of the size, decreases exponentially. In their work the only threshold $\tau$ considered for the RASD computation was presumably the minimum detectable rain rate, which depends on the distance from the radar. Their data set came from several locations in the United States, in the Pacific Ocean, and in western Europe. The range of shower sizes considered for the computation was 1–10 km. The slope of the exponential distribution, $\lambda$, ranges from -0.21 to -0.38 km⁻¹. A similar study (that is, using the noise level as threshold) with similar results was performed by Miller et al. [1975] on data collected in western North Dakota. They stratified their data as a function of seasons for *D* ranging between 2 and 15 km and found no seasonal trend in the slope values. In these two references, no indication is available on the minimum and maximum range of sizes and distances considered for the RASD computation. Thus a possible influence of the counting conditions on the result cannot be discussed. Drufuca [1977] also found from radar data that the area of rain cells within various thresholds can be described by exponential functions. Konrad [1978] analyzed radar data from the mid-Atlantic coast of the United States. Distributions for a radar reflectivity factor (Z) threshold ranging from 10 to 60 dBZ were considered. The minimum and maximum cell diameter (it is assumed that rain areas are approximately circular), $D_{min}$ and $D_{max}$, were 2 and 14 km, respectively. The conclusion was that “exponential representation appears reasonable.” The slope values range from -0.36 to -0.50 km⁻¹. From a radar data base obtained on the same site of the mid-Atlantic coast and spanning over 5 years and all seasons, Goldhirsh and Musiani [1986]

found that the distribution of the rain cell equicircle diameter *D*, for *D* ranging between 1 and 5 km, can be represented by an exponential function whose slope and average and median diameters are almost invariant with respect to the rain rate threshold $\tau$ over an interval expanding between 3 and 88 mm h⁻¹. They found a slope of -0.79; the reason for such a high value will be discussed in section 8. Lognormal distributions were also proposed to represent radar-observed rain areas (without peak number consideration), height and duration distributions [Lopez, 1977; Houze and Cheng, 1977; Lopez et al., 1984], rain rate [Atlas et al., 1990; Kedem et al., 1990; Sauvageot, 1994], and the size distribution of cloud clusters and superclusters observed over the ocean [Mapes and Houze, 1993].

[12] For all the above-quoted studies the correlation coefficient of the fitting to an exponential distribution, when specified, is higher than 0.96. The structure inside the rain areas, notably the cellular structures, is not taken into account in the discussion of the slope of the RASD. The fractal properties are not discussed. The reality of a highly interactive cellular structure inside some convective rain clouds is now well-established evidence. For example, the cellular structure (supercell or multicell) is one of the main characteristics of severe convective rainstorms. Small-scale cellular rain structure is also present in deep stratiform rain clouds in the form of the “generating cells.” These cellular characteristics are observed to maintain their identity over relatively long durations.

[13] Concerning the minimum size of the precipitation area (or cell) to consider, in situ measurements made by instrumented aircraft or with a ground-based disdrometer, as well as radar observations, show that clouds and precipitation have spatial fluctuations at a scale as small as a few tens of meters [e.g., Sauvageot et al., 1984; Seliga et al., 1986; Duroure and Guillemet, 1990; Fabry, 1996]. However, these fluctuations are mainly due to turbulence and mechanical effects on the spatial distribution of the raindrops (such as size sorting by the wind shear). The taking into consideration of the duration required for the growth of precipitating size droplets [e.g., Byers, 1965; Young, 1993] and the reflectivity minima for precipitating cumulus clouds [Sauvageot and Omar, 1987] suggest that a cell with a size below a few hundred meters would not be sufficiently persistent to develop a reflectivity core identifiable as a rain cell, even with a low threshold. Benner and Curry [1998] point out that clouds smaller than ~1 km are not seen to precipitate. In fact, drizzle can be observed in clouds smaller than 1 km such as fair weather cumulus clouds [Lhermitte, 1987]; the radar reflectivity factor, however, is probably too low to be analyzed as a peak in the reflectivity field, owing to the antenna resolution limit. That is why, in the reflectivity field, a rather arbitrary low truncation limit around 1 km has to be accepted for the area size distribution. The difficulties encountered for the radar observation of small areas, as well as the limitation for large areas, are discussed in section 4.

### 3. The Data Set

[14] The data were collected in four regions (Figure 1 and Table 1). Two of them are in the tropical zone of West
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Location of the radar data collection areas (see Table 1). A, Abidjan; B, Bordeaux; N, Niamey; T, Toulouse.</figcaption></figure>

Africa; one at Niamey, in the continental area, and one at Abidjan, on the Gulf of Guinea coast. The other two regions are in southwestern France, one inland, at Toulouse, and the other on the Atlantic coast, at Bordeaux.

[15] Niamey, in southwestern Niger, is situated in the middle of the Sahelian Sudanese strip. The mean annual total rainfall is ~500 mm. The rainy season, which lasts ~3 months, from early July to late September, is associated with the northward move of the Intertropical Convergence Zone (ITCZ). The area is very flat. The radar, located at the Niamey airport (13°29'N, 2°11'E, altitude 220 m), is a C band EEC/WR 100.5, with a peak power of 250 kW, a pulse repetition frequency of 250 Hz, a pulse width of 2 μ, and a half-power beam width θ = 1.5°. As all the radars used in this study, it is a non-Doppler, single-polarization radar. The processing and storage of the data were performed with a numerical system analyzing the reflectivity on 8 bits, called SANAGA [Sauvageot and Despaux, 1990; Ramos-Buarque and Sauvageot, 1997]. The data set contains 163 plan position indicators (PPIs) acquired in July and August 1993.

[16] Abidjan, in southern Ivory Coast, on the Gulf of Guinea coast, is ~1000 km southwest of Niamey. The country, north of the coast, is also rather flat. The rainy season is linked to the meridional motion of the ITCZ, with

a long rainy season from March to mid-July associated with the northward migration of the ITCZ and with a short rainy season from mid-September to late November, associated with the southward motion of the ITCZ. The mean annual rainfall is ~1800 mm. The prevailing character of the rainfall is convective. The rainy systems move mainly eastward, that is, parallel to the Gulf of Guinea coast. The radar is located at the Abidjan airport (5°25'N, 4°W, altitude 40 m). It is a Thomson-CSF, RODIN, C band radar with the same technical characteristics as that of Niamey, except for the half-power beam width, which is 1.3° for the radar at Abidjan. The data acquisition was also performed with a SANAGA processing system. The data set contains 15,462

<table><caption>**Table 1.** Number of Rain Areas for Various Rain Thresholds τ in the Four Data Sets</caption><thead><tr><th>Radar Location</th><th>τ = 1 mm h<sup>-1</sup></th><th>τ = 5 mm h<sup>-1</sup></th><th>τ = 12 mm h<sup>-1</sup></th><th>τ = 16 mm h<sup>-1</sup></th></tr></thead><tbody><tr><td>Bordeaux</td><td>401,720</td><td>70,967</td><td>12,456</td><td>6348</td></tr><tr><td>Toulouse</td><td>2630</td><td>1685</td><td>247</td><td>172</td></tr><tr><td>Niamey</td><td>700</td><td>1333</td><td>1313</td><td>1178</td></tr><tr><td>Abidjan</td><td>148,277</td><td>93,232</td><td>58,268</td><td>45,774</td></tr></tbody></table>
PPIs acquired from March to December 1995 and from March to December 1996.

[17] Toulouse in southwestern France is ~100 km north of the central Pyrénées and 230 km east of the Atlantic coast, in a flat region. The mean annual rainfall is ~650 mm, that is not very different from that of Niamey but distributed throughout the year. Most of the rainy events are zonal frontal cyclonic disturbances [Bourrel et al., 1994]. The climatic conditions are thus rather homogeneous, although, during the summer season, the frontal systems are often reduced to a convective line associated with a cold front in surface (the stratiform component is almost absent). The radar is located on the site of Météo-France at Toulouse-le-Mirail (43°34′N, 1°29′E, altitude 200 m). It is a Thomson-CSF, RODIN, C band radar with the same technical characteristics as the Abidjan radar and a processing system similar to SANAGA. The data set contains 286 PPIs acquired in September 1996.

[18] Bordeaux is also in the southwest of France, 36 km east of the Atlantic coast, on a flat region. The climate of the Bordeaux area is of oceanic type with the same seasonal effect as in the Toulouse area. The mean annual rainfall is ~900 mm. The radar is located at the Bordeaux-Mérignac airport (44°52′N, 0°30′W, altitude 70 m). It is an OMERA, S band radar with a peak power of 700 kW. The pulse width is 2 μ, the pulse repetition frequency 250 Hz, and the half-power beam width 1.8°. The data processing was similar to that of Toulouse. As the radar at Toulouse, the radar at Bordeaux is a part of the French operational radar network managed by Météo-France. The data set contains 35,286 PPIs acquired during the year 1996. The cell number in each sample of the present data set (Table 1) is similar to or larger than in the samples used by the authors quoted in section 2.

# 4. Definitions and Data Processing

## 4.1. Rain Areas, Rain Cells, and Thresholding

[19] When describing the structure of a rain field, difficulty comes from diversity and overlapping of the space and timescales acting inside the rain systems. This also leads to a terminology difficulty.

[20] In the study of cumulus cloud fields from satellite data (section 2), most authors consider individual clouds and refer to “cell” when describing the inner structure of the cloud area from the distribution of the cloud top temperature. The cell is identified as a clear cold area with brightness temperature increasing around a minimum. The cell is proposed as a basic structure entity associated with an individual convective updraft. Then the cloud population can be described in terms of single-cell or multicell clouds, located on each side of a break diameter. For large cloud system studies from satellite data the structural elements are termed “clusters” and “superclusters” [e.g., Machado and Rossow, 1993; Mapes and Houze, 1993].

[21] In the studies of radar-observed rainclouds and rain fields (including convective and stratiform structure) the terms used are “cell” or “area.” Cell seems prefered when there is an explicit or implicit reference to the associate dynamic structure as the cause of rain (e.g., the well-known multicell and supercell thunderstorms or the generating cells of the stratiform clouds). In the present study, the term area

is perfered since the main goal is the discussion of the rain field structure.

[22] In the literature, rain areas (or cells) have been defined mainly in two ways. First, as the area interior to a closed contour, down a local maximum (or peak) of the reflectivity by 3 to 10 dB, and surrounding this local maximum [e.g., Konrad, 1978; Crane, 1996]. Second, when analyzing the area size distribution with respect to a rain rate threshold τ, the rain area is defined as the area inside which the rain rate $R \ge \tau$ [e.g., Drufuca, 1977; Goldhirsh and Musiani, 1986]. The rain area is continuous and, along the contour that bounds it, the rain rate is at threshold value. In this study the second definition is used, since the interest bears on discussing RASDs as a function of the peak number. However, the results obtained when using the first definition are presented in section 8.

[23] Whatever the definition considered, the analysis of rain field structural characteristics in terms of rain area size distribution is again confronted with the space scale diversity of the rain system. Indeed, rain fields are made up of an overlapping structure of different scales, one element of a given scale frequently enclosing one or more elements of a smaller scale in such a way that, when varying the threshold, the region of the rain field that supplies the statistics partly changes. Figure 2 represents an arbitrary rain field with several peaks of reflectivity. For a reflectivity threshold of 0, 10, and 20 dBZ this rain field is counted as a single area, as five areas, or as two areas, respectively. The case of a tropical squall line reflectivity field, made up of a compact line of intense convective rain showers, followed by a large stratiform part [e.g., Houze, 1993; Ramos-Buarque and Sauvageot, 1997], can be taken as an extreme example. For a low threshold, for instance $\tau = 1$ mm h$^{-1}$, most of the compact line forms a huge single area, the size of which is larger than the maximum size taken into account in the computed distribution. Thus for this level the statistics turn on the peripheral region of the convective line where the low rain rate areas are and on the stratiform part. For intermediate thresholds, for instance $\tau \ge 10$ mm h$^{-1}$, it is the opposite: Peripheral and stratiform parts do not

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Drawing of an arbitrary rain field. According to the threshold, the number of rain areas varies between one and five.</figcaption></figure>
<table><caption>**Table 2.** Parameters of the Radar Pulse Resolution and of RASD Analysis<sup>a</sup></caption><thead><tr><th>Radar Location</th><th>θ, deg.</th><th>r<sub>max</sub></th><th>θ r<sub>max</sub>, km</th><th>D<sub>min</sub></th><th>MDRR at r<sub>max</sub>, mm h<sup>-1</sup></th><th>τ<sub>min</sub>, mm h<sup>-1</sup></th></tr></thead><tbody><tr><td>Bordeaux</td><td>1.75</td><td>100</td><td>3</td><td>3</td><td>0.15</td><td>0.2</td></tr><tr><td>Toulouse</td><td>1.3</td><td>100</td><td>2.3</td><td>3</td><td>0.15</td><td>0.2</td></tr><tr><td>Niamey</td><td>1.6</td><td>80</td><td>2.8</td><td>3</td><td>0.15</td><td>0.2</td></tr><tr><td>Abidjan</td><td>1.3</td><td>100</td><td>2.3</td><td>3</td><td>0.15</td><td>0.2</td></tr></tbody></table>

<sup>a</sup>RASD, rain area size distribution, θ, half-power beam width; r<sub>max</sub>, distance from the radar of the farthest part of the observed area; D<sub>min</sub>, minimum diameter considered in the computed RASD; MDRR, minimal detectable rain rate at distance r; τ<sub>min</sub>, lowest rain rate threshold used for a RASD computation.

participate in the statistics, only the line of intense convective rain is taken into account. That is why the peak number inside the RASDs has been considered.

## 4.2. Bias of the Radar Sampling

[24] The minimum detectable rain rate and the minimum rain area size that can be resolved by the radar (owing to the filtering effect of the pulse volume) increase with the distance. Thus, as discussed by Sauvageot et al. [1999], in order to avoid introducing biases in the radar sampling, two conditions have to be fulfilled: (1) The lowest rain rate threshold used for the RASD computation has to be equal to or higher than the minimum detectable rain rate at the farthest part of the studied area, and (2) the minimum size taken into account in the statistics (that is, the left truncation of the RASD) must not be smaller than the (minimum) area size resolved by the radar at maximum distance r<sub>max</sub>. This is the main cause of limitation on r<sub>max</sub> when specifying the size of the radar-observed area used to compute the RASDs. In Table 2 the parameters of the radar resolution volume and the size of the left truncation of the distribution are indicated for each radar. Of course, the parameters of the RASD computation are higher than the corresponding minimum values of the radars.

[25] For a given elevation angle the height of the radar beam above the ground increases with the distance. This raises problems concerning the influence of the beam height on the size of the rain areas due to the variability of the vertical profile of the reflectivity (VPR). Konrad [1978] shows, from a statistical analysis of high-resolution radar data, that in the rain showers, below melting level, the VPR is weakly dependent on the height. Above this level, an average vertical reflectivity gradient from -2.5 to -3.5 dB km<sup>-1</sup> is observed (of course, such a negative gradient is not present on individual convective rain shafts). For the two tropical sites of the present work, during the rainy season, the melting level is above an altitude of 5000 m, so that the radar beam remains in the rain up to a distance of ~170 km, for a PPI scan at the minimum elevation angle (~0.8°). The condition on the distance associated with the minimal rain area size observable is found to be more restrictive than that on the VPR. Any distinction based on the VPR has thus been considered in the present data set for the tropical sites. For the two midlatitude regions the cases when the VPR includes a melting layer have been removed from the data set processed.

[26] Another point which has to do with the influence on the size of the rain areas of the beam height increasing with

the distance is the variation of the shape of the rain areas with the height. Common experience shows that the horizontal extent of a given rain area for a given threshold increases when the height decreases (of course, the above statement would not be true if a very low reflectivity threshold was considered, thus taking into account the generating cloud, notably when this cloud spreads out below a statically stable level; cloud areas are larger than precipitation areas [Austin and Bellon, 1982]). Such an influence has not been considered in the present work.

[27] In relation with the biases linked to the radar sampling, the influence of the vertical wind shear on the vertical structure and shape of precipitation has to be stressed. The vertical shear of the wind spreads the falling precipitation. As a result, the horizontal area of the intersection of a quasi-horizontally azimuthally scanning radar beam with a sheared precipitation shaft depends on the beam altitude and on the vertical profile of the shear vector. For example, the spreading of individual snow trails can be larger than 100 km, with a sedimentation of the contiguous trails at low levels. Moreover, as discussed by Sauvageot and Koffi [2000], in some dynamic configurations of contiguous generating clouds the differential effects of vertical wind shear on rain showers with different rain rates result in a merging of the individual falling showers. Another point is that the time life of generating clouds and precipitation is not exactly coincident (precipitation can be aloft even though the generating convective current is no longer active). Consequently, the rain area distribution does not exactly mirror the generating cloud distribution and dynamic organization, at least on a small scale. In the present study, these distortions have to be accepted. PPIs including anomalous propagation echoes have been removed from the data set.

## 4.3. Shape of the Observed Domains

[28] The shapes of the domains observed with the radars were chosen in order to satisfy the conditions on r<sub>max</sub> given in Table 2, to avoid the ground echoes at short distance around the radar, and also to take into account some peculiarities of the sites. These peculiarities are the following: for Niamey, the presence of a ground rain gauge network east of the radar so that the observed domain is limited to the half plan east of the radar; and, for Toulouse, the presence of the Pyrénées range, which means that the observed domain is limited to a short distance southward.

## 4.4. Calibration of the Radar Data

[29] During the period of data collection the four radars were carefully maintained and electronically calibrated at least once a week. At two of the four sites, Niamey and Toulouse, ground rain gauge networks were available for the calibration of the radar data [Lebel et al., 1992; Bourrel et al., 1994]. To obtain the rain rate estimates from the reflectivity factor measurements, using the Z-R relations (of the form Z = aR<sup>b</sup>, where a and b are coefficients), the cumulative rain height observed with the radar at all the calibration sites is compared to the cumulative rain height observed with the rain gauges. A calibration factor is then obtained for each rainy event and applied to the whole rain field [Brandes, 1975].

[30] At Bordeaux and Abidjan the calibration of the Z-R relation is achieved using the probability matching method
[Calheiros and Zawadzki, 1987], which means that coefficients a and b of Z-R are obtained by fitting P(Z), the probability density function (pdf) of Z, to P(R), the “climatic” pdf of R. The climatic P(R) distributions for the two sites are computed from large annual disdrometer data sets [Sauvageot, 1994; Sauvageot and Lacaux, 1995] not taken simultaneously with the radar observation. The P(Z) distributions are computed from large samples, corresponding to several hours of continuous observations including convective and stratiform rainfall components, in order to obtain a steady P(Z) [Rosenfeld et al., 1994]. The accuracy of the probability-matching method has not yet been clearly evaluated, notably for the midlatitude rain fields. However, it is considered in the present work as a method sufficient for an approximate radar calibration, bearing in mind that the RASD parameters (notably the slope) are not strongly dependent on the rain threshold (i.e., on the calibration).

## 4.5. Computing the Rain Area Size Distribution

[31] The rain area size estimator used in this paper is the equivalent circular diameter. The PPI radar scans, in polar coordinates were converted along a Cartesian grid, with a uniform pixel size of 1 × 1 km², using a technique similar to that described by Mohr and Vaughan [1979]. The radar images were then thresholded by elimination of the pixels where R < τ. Then the area inside the contours for R ≥ τ and their equivalent circular diameter D were computed. To compute the rain areas, the threshold images were segmented using an eight-connected-pixel algorithm similar to that described in Kuo et al. [1993]. When, inside a closed curve defining the limit of a rain area, there remained a small area where R < τ (which was rather unfrequent), this area was not taken into account in the rain area.

[32] For each rain area the peak number was computed. A peak is associated with a finite area which is the area S enclosed by the line of relative minimum of reflectivity surrounding the local maximum. The peak was validated only for S ≥ Smin and when its reflectivity was ΔZ above the surrounding local minimum line. The peak number inside a rain area thus depends on Smin and ΔZ. The sensitivity of the results to these two parameters is discussed in section 5. In the present work the values used were Smin = 9 pixels and ΔZ = 2 dB.

[33] The rain areas astride on the boundary of the radar-observed domain were not considered. The fraction of areas concerned by this problem increases with the size of the areas. In section 5 of this paper the rain areas are grouped in classes with the same area peak number. Clearly, the more a class contains large areas, the more the large-size end of the RASD inside it is biased by the selective rejection of the astride areas. The result is a decrease of the mean diameter and of the number of areas in this class. This problem leads to the limitation of the maximal size that can be correctly sampled with a radar.

[34] Section 4.2 emphasizes the fact that the areas with D ≤ 3 km, the minimum diameter resolved by the radar at distance rmax, are not correctly sampled. This bias affects the area size distribution (the area number around 3 km increases at the expense of the smaller size categories) as well as the reflectivity observations due to a nonuniform beam-filling effect. That is why in the present paper the data points for D ≤ 3 km are not used.

[35] For the sake of clarity most of the curves and tables are presented for only three values of the threshold τ, namely 23, 24, and 39 dBZ, corresponding to a rain rate of ~1, 5, and 12 mm h⁻¹, respectively (using standard Z-R relations). The choice of these thresholds can be justified as follows: The threshold 1 mm h⁻¹ corresponds to the major part of the rain field observed by the radar, including the stratiform regions. For this threshold, as pointed out above, the RASD statistic is supplied by the peripheric rainy areas, notably the convective cells at the beginning of their growth and the convective cells at the end of their decaying phase. The threshold 5 mm h⁻¹ is a threshold enabling the resolution of the rain cells of the stratiform region but poorly adapted to the resolution of the more intense convective areas (notably the tropical systems). The threshold 12 mm h⁻¹ removes the stratiform regions and resolves the heavy convective areas. The higher thresholds are associated with a small rain area number, which means that the distribution is thus more severely truncated on its right side. They are consequently not deeply interesting for the present analysis.

# 5. Rain Area Size Distributions With Peak Number Distinction

## 5.1. Frequency Distributions

[36] This section is a discussion of the occurrence frequency distributions of the rain areas containing a given peak number p (referred to as RASDp hereinafter), as a function of diameter. Figure 3 shows the RASDp data points and the fitting by a lognormal function for five values of p for Bordeaux and Abidjan. The curves in Figure 3 for Toulouse and Niamey are very similar to those for Bordeaux and Abidjan and are not displayed, to avoid overloading the paper. For p > 1 the fitting is very good. The fitting was computed using only the data points with D > 3 km, since the areas with smaller diameters are not correctly sampled. For p = 1 this condition removes a large fraction of the data, notably at Abidjan. The corresponding curve in Figure 3 emphasizes the overestimate of the frequency for D = 3 km and its underestimate for D < 3 km. For p > 1 the modal diameter of the curves in Figure 3 increases with p, while the relative frequency decreases and the span of the associated D variations increases in such a way that the overlapping between the curves increases. For p = 1 the overlapping with the other curves in Figure 3 is very small; for D ≤ 7 km at Bordeaux and D ≤ 6 km at Abidjan most of the rain areas are single peaked. The modal diameters are larger at Bordeaux than at Abidjan. As pointed out in section 4.5, when p increases, the right side of the curve in Figure 3 is more and more concerned by the sampling bias for large areas.

[37] The lognormal fitting function [e.g., Crow and Shimizu, 1988] is

$$P_p(D) = \frac{1}{(2\pi)^{1/2} D \sigma_p} \exp \left[ - \frac{(X - \mu_p)^2}{2\sigma_p^2} \right], \quad (1)$$

where X = ln(D). The subscript p indicates that the concerned data sample is for a constant peak number, and ln(D is the Napierian logarithm of D. The two parameters of Pp(D), μp
<figure><img src="image_4.png" /><figcaption>Figure 3. Size distribution of rain areas having the same peak number p, or rain area size distribution p (RASDp), and fitting by a lognormal function for five values of p at Bordeaux and Abidjan for threshold 23 dBZ. The curves are fitted on the data points for D > 3 km.</figcaption></figure>

and σp, are the mean and standard deviation of X, respectively, given by

$$ \mu_p = \ln \mu_D - \frac{1}{2} \ln \left( 1 + \frac{\sigma_D^2}{\mu_D^2} \right) \quad (2) $$

$$ \sigma_p^2 = \ln \left( 1 + \frac{\sigma_D^2}{\mu_D^2} \right), \quad (3) $$

where μD and σD are the mean and standard deviations of D. Reciprocally, we have

$$ \mu_D = \exp \left( \mu_p + \frac{\sigma_p^2}{2} \right), \quad (4) $$

$$ \sigma_D^2 = \left[ \exp(2\mu_p + \sigma_p^2) \right] \left( \exp \sigma_p^2 - 1 \right). \quad (5) $$

[38] Other forms of distributions have been tested to fit the RASDp, notably the gamma distribution. The best results are obtained with the lognormal distribution.

## 5.2. Sensitivity to Smin and ΔZ

[39] To illustrate the sensitivity of Pp(D) on the choice of Smin and ΔZ, the variation of μp and σp as a function of p, and for various values of Smin and ΔZ, is presented in Figure 4. Figure 4 shows that the rate of increase of μp and decrease of σp diminishes when p increases. The value of ΔZ has a very small influence on μp and σp. The slight increase of μp with Smin and ΔZ, for a same value of p, observed in Figure 4, is associated with the filtering of small areas when Smin and ΔZ increase, which induces a shift toward the right of the distribution parameter (Figure 3). This effect is not clear on σp. Figure 4 is for Bordeaux. The curves for the other sites are similar to those in Figure 4. The values of ΔZ and Smin used for the computation of Figure 3 were 2 dB and 5 pixels, respectively.

## 5.3. Relation σp-μp

[40] The variations μp(p) and σp(p) such as presented in Figure 4, can be fitted very tightly with power relations. For example, for Bordeaux and for the thresholds 23 dBZ the relations are, with D in kilometers,

$$ \mu_p = 1.66 p^{0.28}, \quad (6) $$

with a correlation coefficient of 0.98, and

$$ \sigma_p = 0.36 p^{-0.33}, \quad (7) $$

with a correlation coefficient of 0.99.

[41] Eliminating p between relations (6) and (7) gives a relation between σp and μp. For Bordeaux and the threshold 23 dBZ this relation is

$$ \sigma_p = 0.65 \mu_p^{-1.18}, \quad (8) $$

with D in kilometers.

[42] Figure 5 presents as an example the plot of σp as a function of μp for the observed RASDp (without considering the values for p = 1, since they are biased) for Bordeaux at threshold 23 dBZ. For μp < 3.6 the data points are found to be tightly lined up. Beyond this value the rain areas are not correctly sampled by the radar (this point will be clarified further). The curve representing relation (8) is also drawn in Figure 5 for Bordeaux. It holds well with the data points. For the other regions and thresholds, results similar to Figure 5 and relations (6), (7), and (8) are obtained, with almost the same coefficients. Figure 5 shows, for example, the case of Abidjan, for the threshold 23 dBZ, with the curve of relation (8). The curve for Bordeaux holds for Abidjan.

[43] For μp between 2 and 3.5 the pair (σp and μp) of Figure 5 can also be fitted with a linear relation. The mean relation for the various sites and thresholds is σp = -0.12 μp + 0.51. Replacing μp and σp in equation (1) by their expression as a function of p, i.e., relations (6) and (7), respectively, shows that the shape of P(D) depends only on p.

## 5.4. Mean Diameter

[44] Figure 6 presents the observed mean diameter of rain areas having p peaks inside as a function of p, that is, D̄(p),
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Variation as a function of *p* of the parameters *μ*<sub>*p*</sub> and *σ*<sub>*p*</sub> for the RASD<sub>*p*</sub> for various values of (top) S<sub>min</sub> and (bottom) ΔZ. The upper part was computed for ΔZ = 1 dB and the lower part for S<sub>min</sub> = 3 pixels. The curves are for the Bordeaux data set.</figcaption></figure>

for Bordeaux and Abidjan. An analytical expression of this
function can be obtained using relations (6) and (7) in
equation (4), namely, for Bordeaux

$$
\begin{align}
\mu_D &= \exp\left[1.66 p^{0.28} + \frac{1}{2} (0.36 p^{-0.33})^2\right] \tag{9} \\
\mu_D &= \exp(1.66 p^{0.28} + 0.06 p^{-0.66}). \nonumber
\end{align}
$$

For *p* ≥ 1 the second term in the bracket is negligible with
respect to the first one and then

$$
\mu_D \approx \exp(1.66 p^{0.28}) = \bar{D}_p, \quad (10)
$$

with *D* in kilometers. A relation similar to equation (10) can
be written for σ<sub>D</sub> from equation (5) and relations (6) and (7).
The abscissa of the mode of *P*<sub>*p*</sub>(*D*) is *D* = exp (μ<sub>*p*</sub> − σ<sub>*p*</sub><sup>2</sup>).

[45] The curve representing equation (10) is drawn on
Figure 6 both for Bordeaux and for Abidjan. It agrees very
well up to about D̄ ≅ 35 km and p ≅ 15. Beyond, the points
bend toward smaller D̄ because of the radar sampling bias on
large areas discussed in sections 4 and 5. Figure 6 thus shows
that the upper limit of area size whose distribution can be
correctly observed with the radar, i.e., ~35–40 km, corre-
sponds to ~15 peaks (using relation (6) and equation (10),
p = 15 corresponds to D̄ = 35 km and μD = 3.5). The D̄ values
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5. Variation of $\sigma_p$ versus $\mu_p$ for Bordeaux and Abidjan at threshold 23 dBZ and curve of equation (8).</figcaption></figure>

for $p=1$ were not used because they are biased, as illustrated in Figure 3. The points for Toulouse and Niamey are similar to those of Figure 6 and almost superimposable.

[46] The data points of Figure 6 for $p$ between 1 and 15 can also be fitted with a power relation of the form $\bar{D} \propto p^{0.73}$. It may be observed that, if each peak was associated to the same (elementary) area, $D^2$ would be proportional to $p^{0.5}$. The power 0.73 shows that the higher the peak number is inside an area, the larger the elementary area associated with a peak is.

### 5.5. Area Number

[47] Figure 7 displays the variations of the relative frequency of the rain areas as a function of $p$, that is, $N_A(p)$. The log $N_A$ decreases linearly as $\log p$ increases up to about $p \approx 20$. The result of the fitting of the data points with a curve of the form $N_A = cp^d$ is given in Table 3. The intercept $c$ increases with $\tau$. It is the reverse for slope $d$. No clear tendency appears for the site influence. A mean relation is

$$N_A = 0.71 p^{-2.77}, \qquad (11)$$

where $N_A$ is the relative frequency. Figure 7 shows that relation (11) holds as well for Bordeaux as for Abidjan. It is worth observing that between $p=2$ and $p=15$ the relative frequency of the rain areas reduces by more than 250.

[48] To conclude on this section, the RASD$_p$ are found to be well represented by lognormal distributions. The distributions are weakly influenced by the parameters $\Delta Z$ and $S_{min}$ used for the peak number validation. The parameters of the fitted curves $\mu_p$ and $\sigma_p$ are functionally related to $p$. There is thus a tight power relation between $\mu_p$ and $\sigma_p$, and the distribution $P_p(D)$ depends only on $p$.

## 6. Rain Area Size Distribution (RASD) Without Peak Number Distinction

[49] The results of section 5 show that the size distribution of rain areas without peak number distinction is not lognormal or exponential as proposed by previous authors (section 2) but a lognormal mixture.

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6. Variation of mean area diameter $\bar{D}$ as a function of the peak number $p$ for various values of threshold $\tau$ at Bordeaux and Abidjan and curve of equation (10).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Variation of the relative frequency of rain area as a function of the peak number *p* at Bordeaux and Abidjan for various values of threshold *τ* and curve of equation (11).</figcaption></figure>

[50] Figure 8 shows the RASD observed at Bordeaux for $τ = 23$ dBZ. Using relations (6) and (7), sets of random number simulating, for each *p* value a normalised RASD<sub>p</sub> has been generated. Using relation (11) (with the coefficient of Bordeaux at 23 dBZ given in Table 3) as a weighting function of the simulated RASD<sub>p</sub>, the RASD has been retrieved. The resulting distribution is drawn in Figure 8. It agrees very well with the observed one. The dispersal of the computed points is larger than that of the observed ones,

<figure><img src="image_11.png" /><figcaption>**Figure 8.** RASD observed at Bordeaux for $τ = 23$ dBZ and RASD simulated using equations (6), (7), and (11).</figcaption></figure>

notably for $D > 30$ km, because the number of computed areas is smaller than the observed one.

## 7. Approximation of RASD for Truncated Observations

[51] As seen above, radar-observed rain areas distributions are truncated on the left, at sizes larger than the modal value (Figure 3). This limit of ground radars is also present for space radars; for example, the precipitation radar of the Tropical Rainfall Measuring Mission (TRMM) [Simpson et al., 1988] has a minimum resolution of 4 km. On the right the maximum rain area size that can be correctly observed is restricted by the size of observed domain (sections 4 and 5) but also, and mainly, by the rapid decrease of the frequency observed for the large rain areas (Figure 7), to such a point that the area number of the large-size classes is low and then affected by a prohibitive variance. This large-area size limit decreases as $τ$ increases (Table 1). With the present data set collected over long duration observing campaigns, that is, with a wide sample, this size limit is rather remote (around 15 km for $τ = 5$ mm h<sup>-1</sup> for Niamey and Toulouse and around 30 km for Bordeaux and Abidjan). For application using RASD collected at the spatiotemporal scale of a convective rain system, such as tropical squall lines observed with a ground or space radar, the size of the data sample is such that the RASD description will be limited to diameter $3 < D < 15$ km. This is the interval considered in this section.

<table><caption>**Table 3.** Coefficients of the Relation $N_A = cp^d$ for the Various Sites and Thresholds<sup>a</sup></caption><thead><tr><th rowspan="2"></th><th colspan="4">τ = 23 dBZ</th><th colspan="4">τ = 34 dBZ</th><th colspan="4">τ = 39 dBZ</th></tr><tr><th>P<sub>M</sub></th><th>c</th><th>d</th><th>R</th><th>P<sub>M</sub></th><th>c</th><th>d</th><th>R</th><th>P<sub>M</sub></th><th>c</th><th>d</th><th>R</th></tr></thead><tbody><tr><td>Bordeaux</td><td>15</td><td>0.60</td><td>-2.48</td><td>0.99</td><td>15</td><td>0.60</td><td>-2.88</td><td>0.99</td><td>12</td><td>0.83</td><td>-3.53</td><td>0.99</td></tr><tr><td>Toulouse</td><td>15</td><td>0.40</td><td>-1.78</td><td>0.99</td><td>15</td><td>0.57</td><td>-2.09</td><td>0.99</td><td>8</td><td>0.89</td><td>-2.86</td><td>0.99</td></tr><tr><td>Niamey</td><td>12</td><td>0.92</td><td>-2.62</td><td>0.99</td><td>12</td><td>0.59</td><td>-2.39</td><td>0.99</td><td>8</td><td>0.74</td><td>-2.44</td><td>0.99</td></tr><tr><td>Abidjan</td><td>15</td><td>0.58</td><td>-2.31</td><td>0.99</td><td>12</td><td>0.83</td><td>-3.74</td><td>0.99</td><td>6</td><td>0.94</td><td>-4.12</td><td>0.99</td></tr></tbody></table>

<footnote><sup>a</sup>P<sub>M</sub>, maximum peak number used for the computation; c, linear coefficient; d, exponential coefficient; R, correlation coefficient.</footnote>
<figure><img src="image_12.png" /><figcaption>**Figure 9.** RASD without peak number distinction for Bordeaux and Abidjan for $\tau = 23$ dBZ. The lognormal fitted curve was computed for $D > 3$ km since the data points for $D \le 3$ km are biased.</figcaption></figure>

[52] To fit the truncated RASDs, we have considered the three functions most discussed in the literature to describe the observed cloud and rain area distributions (section 2 for the relevant references), that is, the lognormal, power an exponential distribution. Even if two of them (lognormal and exponential) are not mathematically compatible with the results of section 5, it is useful to know the accuracy reachable with it for a fitting by pieces of the RASDs.

[53] The data point and the fitted lognormal function at Bordeaux and Abidjan for $\tau = 23$ dBZ are shown in Figure 9. The fitting was computed for $D > 3$ km, but the data points for $D \le 3$ km are shown. For $D > 3$ km the fitting is rather good. The value of the parameters of the fitted curves for the various sites and thresholds is given in Table 4. The values $\mu$ and $\sigma$ decrease slightly when $\tau$ increases. They are smaller for Bordeaux and Abidjan than for Toulouse and Niamey. This difference may be linked to the coastal situation of Bordeaux and Abidjan comparatively with Toulouse and Niamey, which are continental, that is, to the differences between the organization of convection over land and over

sea. This hypothesis can be confirmed by the fact that lightning activity has a very different intensity over land and over sea [Orville and Henderson, 1986]. Lightning activity is strongly influenced by the organization of the convective field [Zipser and Lutz, 1994; Seity et al., 2001]. Convective fields are assumed to be more homogeneous over sea than over land, that is, with a lower proportion of intense rain areas and a higher proportion of moderate and small rain areas, without strong differences of the average rain rate as shown by Nzeukou and Sauvageot [2002]. This corresponds to a RASD slope higher (i.e., more sloping) over sea than over land, which is what is observed in Table 4 and with the other fitting function discussed below. So the suggestion offered here is that this hypothesis is the most probable and deserves serious consideration.

[54] Figure 10 shows the RASD for Bordeaux and Abidjan at 23 dBZ with the curves of the power function fitted to the data points. Table 5 gives the parameters $N_0$ and $\alpha$ of the fitted curves for the four sites and the three levels, with $P(D) = N_0 D^{-\alpha}$, where $N_0 = P_0/N_T$ and $N_T$ is the total number of areas. The fitting is very good. The slope increases with $\tau$. RASD is found more sloping for Bordeaux and Abidjan than for Toulouse and Niamey for the reason discussed in the previous paragraph.

[55] Figure 11 shows the RASD for Bordeaux and Abidjan at the threshold 23 dBZ and for the curves fitted with an exponential function. The corresponding parameters, $M_0$ and $\lambda$ where $P(D) = M_0 \exp(-\lambda D)$ with $M_0 = E_0/N_T$, are given in Table 6. The fitting is not bad but poorer than that of the power law. Exponential function was selected by many authors to fit RASD. It has the advantage of having moments with simple writing, which is suitable for some computations such as the area average rain rate estimate from RASD [Sauvageot et al., 1999]. The slope $\lambda$ is not strongly dependent on the rain threshold: For $\tau$ ranging between 5 and 15 mm h$^{-1}$ the maximum deviation from the average does not exceed 16%.

[56] Section 4 pointed out that in order to avoid a cause for bias for the RASD, $r_{max}$, the outer limit of the radar-observed domain, has to be maintained at a value such that

<table><caption>**Table 4.** Mean and Standard Deviation of the Lognormal Function Fitted to the RASD for the Various Sites and Thresholds and for $D > 3$ km<sup>a</sup></caption><thead><tr><th>τ, dBZ</th><th>μ</th><th>σ</th><th>R</th></tr></thead><tbody><tr><td colspan="4" style="text-align: center;">Bordeaux</td></tr><tr><td>23</td><td>1.30</td><td>0.62</td><td>1.00</td></tr><tr><td>34</td><td>1.17</td><td>0.53</td><td>1.00</td></tr><tr><td>39</td><td>1.09</td><td>0.51</td><td>1.00</td></tr><tr><td colspan="4" style="text-align: center;">Niamey</td></tr><tr><td>23</td><td>1.50</td><td>0.78</td><td>0.96</td></tr><tr><td>34</td><td>1.41</td><td>0.68</td><td>0.98</td></tr><tr><td>39</td><td>1.48</td><td>0.68</td><td>0.98</td></tr><tr><td colspan="4" style="text-align: center;">Abidjan</td></tr><tr><td>23</td><td>1.24</td><td>0.62</td><td>1.00</td></tr><tr><td>34</td><td>1.01</td><td>0.47</td><td>1.00</td></tr><tr><td>39</td><td>0.89</td><td>0.53</td><td>1.00</td></tr><tr><td colspan="4" style="text-align: center;">Toulouse</td></tr><tr><td>23</td><td>1.52</td><td>0.91</td><td>0.90</td></tr><tr><td>34</td><td>1.47</td><td>0.77</td><td>0.98</td></tr><tr><td>39</td><td>1.34</td><td>0.63</td><td>0.98</td></tr></tbody></table>

<footnote><sup>a</sup>D, diameter; μ, mean; σ, standard deviation; R, correlation coefficient.</footnote>
<figure><img src="image_13.png" /><figcaption>Figure 10. Same as Figure 8 for power fitting.</figcaption></figure>

<table><caption>Table 5. Parameters of the Function $P(D) = N_0 D^{-\alpha}$ Fitted to the RASD for the Various Sites and Thresholds<sup>a</sup></caption><thead><tr><th>τ, dBZ</th><th>N<sub>0</sub></th><th>α</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td colspan="4">Bordeaux</td></tr><tr><td>23</td><td>2.08</td><td>2.07</td><td>0.98</td></tr><tr><td>34</td><td>3.73</td><td>2.58</td><td>0.99</td></tr><tr><td>39</td><td>5.24</td><td>2.88</td><td>0.99</td></tr><tr><td colspan="4">Niamey</td></tr><tr><td>23</td><td>0.71</td><td>1.35</td><td>0.97</td></tr><tr><td>34</td><td>0.70</td><td>1.28</td><td>0.92</td></tr><tr><td>39</td><td>0.74</td><td>1.24</td><td>0.94</td></tr><tr><td colspan="4">Abidjan</td></tr><tr><td>23</td><td>1.91</td><td>2.03</td><td>0.99</td></tr><tr><td>34</td><td>2.39</td><td>2.20</td><td>0.99</td></tr><tr><td>39</td><td>3.35</td><td>2.47</td><td>0.99</td></tr><tr><td colspan="4">Toulouse</td></tr><tr><td>23</td><td>1.17</td><td>1.68</td><td>0.98</td></tr><tr><td>34</td><td>1.25</td><td>1.70</td><td>0.99</td></tr></tbody></table>

the minimum size taken into account in the statistics is not smaller than the minimum area size resolved by the radar resolution cell at $r_{max}$. To test their sensitivity to $r_{min}$ and $r_{max}$, the truncated RASD were computed for three concentric annular domains, centered on the radar, with $r_{max} - r_{min} = 60$ km, namely 20 to 80 km, 60 to 120 km, and 100 to 160 km for $τ = 3$ mm h$^{-1}$ and for $D < 15$ km. The test was performed on the Bordeaux data set because of (1) the large size of the data set, (2) the wide beam width (1.75°) of the Bordeaux S band radar, and (3) the altitude of the melting layer (0°C isotherm), which is lower than in any other region considered in this work. Of course, all the available data were used, including the winter events. The result appears in Table 7. It shows that the RASD is not very sensitive to the increase of $r_{min}$ and to $r_{max}$. The same result is obtained with the other fitting functions. This conclusion agrees with Zhu *et al.* [1992], who stress that cloud size distributions are not much influenced by the data resolution.

[57] As seen in section 4.1, some authors define rain areas as the areas interior to a closed contour down a local reflectivity peak by, for example, 3 dB. The resulting distributions are, in such conditions, close to single-peaked RASD. Also, we found that 95% of the rain areas having a

<figure><img src="image_14.png" /><figcaption>Figure 11. Same as Figure 8 for exponential fitting.</figcaption></figure>

<footnote><sup>a</sup>R, correlation coefficient; D, diameter.</footnote>
<table><caption>**Table 6.** Parameters of the Function $P(D) = M_0 \exp(-\alpha D)$ Fitted to the RASD for the Various Sites and Thresholds<sup>a</sup></caption><thead><tr><th>τ, dBZ</th><th>M<sub>0</sub></th><th>λ, km<sup>-1</sup></th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td colspan="4" style="text-align: center; font-style: italic;">Bordeaux</td></tr><tr><td>23</td><td>0.43</td><td>0.30</td><td>0.98</td></tr><tr><td>34</td><td>0.51</td><td>0.37</td><td>0.96</td></tr><tr><td>39</td><td>0.57</td><td>0.42</td><td>0.96</td></tr><tr><td colspan="4" style="text-align: center; font-style: italic;">Niamey</td></tr><tr><td>23</td><td>0.45</td><td>0.25</td><td>0.94</td></tr><tr><td>34</td><td>0.46</td><td>0.24</td><td>0.94</td></tr><tr><td>39</td><td>0.48</td><td>0.23</td><td>0.97</td></tr><tr><td colspan="4" style="text-align: center; font-style: italic;">Abidjan</td></tr><tr><td>23</td><td>0.40</td><td>0.29</td><td>0.96</td></tr><tr><td>34</td><td>0.45</td><td>0.32</td><td>0.97</td></tr><tr><td>39</td><td>0.51</td><td>0.36</td><td>0.97</td></tr><tr><td colspan="4" style="text-align: center; font-style: italic;">Toulouse</td></tr><tr><td>23</td><td>0.66</td><td>0.31</td><td>0.99</td></tr><tr><td>34</td><td>0.66</td><td>0.31</td><td>0.96</td></tr></tbody></table>

diameter smaller than 5 km are single-peaked and that single-peaked rain areas larger than 15 km are very unlikely. That is why the case of single-peaked RASDs deserves a special attention.

[58] Figure 12 represents the single-peaked RASDs for Bordeaux and Abidjan for $3 < D < 15$ km and thresholds between 1 and 12 mm h$^{-1}$. Figure 12 shows that these truncated distributions can be very well approximated by an exponential with slope weakly dependent on $τ$. Table 8 gives the parameters of the distribution for the various sites and thresholds. For Bordeaux and Abidjan, the two sites where a large data set is available, the slope $λ$ is very similar, around 0.6–0.7.

## 8. Diurnal and Annual Variation

[59] The diurnal and annual variation of the slope of truncated RASD was investigated. For $τ = 1$ mm h$^{-1}$ the slope displays no diurnal variation. For a higher threshold the slope is minimum for the period 1900 to 0100 UT. The minimum slope value can be interpreted as corresponding to the maximum of the convective field development, including a significant component of large rain areas. This is in agreement with the climatology of thunderstorms which usually shows a maximum of occurrences by the end of the afternoon. Our observations also show that for coastal areas, in the southwest of France, the maximum of convective storms is observed later in the day than for the inland area. No significant annual variation of the RASD slope was observed.

<table><caption>**Table 7.** Slope of an Exponential Function Fitted to the Bordeaux RASDs for Three Concentric Annular Domains Centered on the Radar<sup>a</sup></caption><thead><tr><th rowspan="2">τ, mm h<sup>-1</sup></th><th colspan="2">20 ≤ r ≤ 80</th><th colspan="2">60 ≤ r ≤ 120</th><th colspan="2">100 ≤ r ≤ 160</th></tr><tr><th>λ</th><th>R</th><th>λ</th><th>R</th><th>λ</th><th>R</th></tr></thead><tbody><tr><td>1</td><td>0.31</td><td>0.99</td><td>0.30</td><td>0.99</td><td>0.31</td><td>0.99</td></tr><tr><td>2</td><td>0.36</td><td>0.99</td><td>0.35</td><td>0.99</td><td>0.31</td><td>0.99</td></tr><tr><td>3</td><td>0.41</td><td>0.98</td><td>0.37</td><td>0.98</td><td>0.32</td><td>0.98</td></tr></tbody></table>

## 9. Perimeter-to-Area Relation

[60] As recalled in section 1, the coefficient $d$ of the relation between perimeter $P$ and horizontal area $A$ of clouds and rain areas has proven a pertinent structural parameter of cloudfields and rain fields. The results of the fitting of a $P = aA^{d/2}$ function to the data points for the four sites show that the “fractal dimension” $d$ is significantly different for the single-peaked and multipeaked rain areas. However, for the multipeaked rain areas, there is no influence of the peak number on $d$. The mean values are between 1.13 and 1.28 for single-peaked rain areas and between 1.35 and 1.42 for multipeaked ones. No significant diurnal or annual variation of these parameters was observed for the four data sets. The value of $d$ for the samples without peak number distinction is ~1.35, that is similar to the one found by most authors (section 2) for rain and clouds areas over a wide range of sizes. However, for the samples without peak number distinction, $d$ displays a tendency to decrease when the threshold $τ$ increases, because the increase of $τ$ leads to a decrease of the multipeaked areas with respect to single-peaked ones.

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Single-peaked RASD at Bordeaux and Abidjan for various thresholds.</figcaption></figure>

<footnote><sup>a</sup>R, correlation coefficient; D, diameter.</footnote>

<footnote><sup>a</sup>r, distance from the radar; τ, rain threshold; R, correlation coefficient.</footnote>
<table><caption>**Table 8.** Parameters of the Function $P(D) = M_0 \exp(-\lambda D)$ Fitted to the Single-Peaked RASD for the Various Sites and Thresholds<sup>a</sup></caption><thead><tr><th>τ, dBZ</th><th>M<sub>0</sub></th><th>σ</th><th>R</th></tr></thead><tbody><tr><td colspan="4">Bordeaux</td></tr><tr><td>23</td><td>1.61</td><td>0.60</td><td>1.00</td></tr><tr><td>34</td><td>1.95</td><td>0.68</td><td>1.00</td></tr><tr><td>39</td><td>1.80</td><td>0.69</td><td>0.99</td></tr><tr><td colspan="4">Niamey</td></tr><tr><td>23</td><td>0.63</td><td>0.36</td><td>0.99</td></tr><tr><td>34</td><td>0.65</td><td>0.35</td><td>0.98</td></tr><tr><td>39</td><td>0.69</td><td>0.37</td><td>0.99</td></tr><tr><td colspan="4">Abidjan</td></tr><tr><td>23</td><td>1.57</td><td>0.61</td><td>1.00</td></tr><tr><td>34</td><td>1.91</td><td>0.67</td><td>1.00</td></tr><tr><td>39</td><td>2.24</td><td>0.72</td><td>1.00</td></tr><tr><td colspan="4">Toulouse</td></tr><tr><td>23</td><td>1.00</td><td>0.47</td><td>0.99</td></tr><tr><td>34</td><td>0.98</td><td>0.48</td><td>0.99</td></tr><tr><td>39</td><td>1.89</td><td>0.65</td><td>0.99</td></tr></tbody></table>

[61] It is suggested that these differences of *d* for single-peaked and multipeaked rain areas are due to the greater raggedness of the multipeaked rain area perimeters in comparison with the single-peaked ones. A similar explanation was proposed for the fractal dimension differences for the satellite-observed single-cell and multicell cumulus clouds by Cahalan and Joseph [1989], Sengupta et al. [1990], Kuo et al. [1993], Benner and Curry [1998], and for severe convective storms by Féral and Sauvageot [2002]. The values reported for *d* are recalled in section 2. As for rain areas, they are smaller for single-cell than for multicell structures. The values found in the present study for single-peaked and multipeaked rain areas are enclosed inside the range of values reported for single-cell and multicell cumulus clouds respectively.

## 10. Summary and Conclusions

[62] This paper presents the results of a study bearing on the shape of rain area size distributions observed by radar in tropical and midlatitude areas and on the sensitivity of the parameters of the RASDs to the rain area inner structure, to the rain rate threshold $τ$, and to the geographical location of the observed region. The data were collected in four regions, Niamey, Abidjan, Toulouse, and Bordeaux. The equivalent circular diameter *D* is used as the rain area size estimator.

[63] The rain areas were defined, from a thresholding of the reflectivity fields, as the area inside which the rain rate $R \ge \tau$. The sampling of the areas, small with respect to the radar beam size at the maximum observed distance and large with respect to the radius of the domain monitored by the radar, was shown to be biased. The reflectivity peak having an area $\ge S_{min}$ and an amplitude, with respect to the surrounding field, $\ge \Delta Z$, were numbered. The rain areas were classified as a function of the peak number *p* observed inside. The size distribution of rain areas having the same peak number inside ($RASD_p$) is found to be well fitted by a lognormal function. The two parameters of the

fitted function, $\mu_p$ and $\sigma_p$, the mean and standard deviation of $\ln D$, where *D* is the equivalent circular diameter of the rain areas, are weakly dependent on the value of $S_{min}$ and $\Delta Z$. The parameter $\mu_p$ increases and $\sigma_p$ decreases as *p* increases, and they are functionally linked with *p*. They are also tightly linked by a power relation, in such a way that the $RASD_p$ depends only on *p*. The mean diameter and the area number of the $RASD_p$ are found to be tightly linked with *p* with a distribution very similar at mean and tropical latitude.

[64] The above results lead to the conclusion that RASD without peak distinction is a lognormal mixture. Using the relations obtained for $\mu_p$, $\sigma_p$, and the rain area number as functions of *p* enables one to simulate RASDs with results in good agreement with the observations.

[65] Because of the sampling bias of the radars at both ends of the RASD, the approximate fitting of the RASD, truncated on the left and on the right with lognormal, power, and exponential functions, was discussed. For $3 < D < 15$ km, good results are obtained. The slope of the distribution is weakly dependent on the rain rate threshold $τ$. The slope of the RASD is not very sensitive to the minimum and maximum distances bounding the domain where the rain field is observed either.

[66] For $D < 6$ km most of the rain areas are found to be single peaked. The distribution of the truncated single-peaked rain area sizes very well fits an exponential function, with correlation coefficients close to 1.00. Most single-peaked rain areas have a diameter smaller than 15 km. The slope of the distribution of the single-peaked rain area sizes is not very sensitive to the threshold.

[67] The RASDs are found sloping more over sea than over land. It is suggested that this difference is linked to the organization of the convection which is more homogeneous over sea than over land.

[68] The diurnal variation of the slope of the RASDs shows a slight minimum at the end of the day, in relation with the usual maximum of the convective storm occurrence. No significant annual variation of the RASD slope is observed.

[69] The perimeter-to-area (fractal) dimension for the sample without peak number distinction is around 1.35, that is, similar to the value found by other authors. However, it is found that single-peaked and multipeaked rain areas have different perimeter-to-area dimensions, between 1.13 and 1.28 for the single-peaked rain areas and between 1.35 and 1.42 for the multipeaked ones. These values are almost independent from the threshold and from the location of the observed domain.

[70] Acknowledgments. The authors would like to thank all those who contributed to the radar data set used, notably Méteo-France for the Toulouse and Bordeaux data, the Direction of the Meteorology of Niger, the Agence pour la Sécurité de la Navigation Aérienne (ASECNA), the Institut de Recherche pour le Développement (IRD) for the Niamey data, and the Agence Nationale de la Météorologie (ANAM) for the Abidjan data. Thanks also to the three reviewers for valuable comments and suggestions.

## References

Atlas, D., D. Rosenfeld, and D. A. Short, The estimation of convective rainfall by area integrals: 1. The theoretical and empirical basis, *J. Geophys. Res.*, 95(D3), 2153–2160, 1990.

<footnote><sup>a</sup>R, correlation coefficient.</footnote>
Austin, G. L., and A. Bellon, Very short range forecasting of precipitation by objective extrapolation of radar and satellite data, in *Nowcasting*, edited by K. A. Browning, pp. 177–190, Academic, San Diego, Calif., 1982.

Benner, T. C., and J. A. Curry, Characteristics of small tropical cumulus clouds and their impact on the environment, *J. Geophys. Res.*, **103**(D22), 28,753–28,767, 1998.

Bourrel, L., H. Sauvageot, J. J. Vidal, D. Dartus, and J. P. Dupouyet, Radar measurement of precipitation in the cold mountainous areas: Case of the Garonne basin, *Hydrol. Sci. J.*, **39**, 369–389, 1994.

Brandes, E. A., Optimizing rainfall estimates with the aid of radar, *J. Appl. Meteorol.*, **14**, 1339–1345, 1975.

Byers, H. R., *Elements of Cloud Physics*, 191 pp., Univ. of Chicago Press, Chicago, Ill., 1965.

Cahalan, R. F., and J. H. Joseph, Fractal statistics of cloud fields, *Mon. Weather Rev.*, **117**, 261–272, 1989.

Calheiros, R. V., and I. Zawadzki, Reflectivity-rain rate relationships for radar hydrology in Brazil, *J. Clim. Appl. Meteorol.*, **26**, 118–132, 1987.

Carvalho, L. M. V., and M. A. F. Silva Dias, An application of fractal box dimension to the recognition of mesoscale cloud patterns in infrared satellite images, *J. Appl. Meteorol.*, **37**, 1265–1282, 1998.

Cheng, M. D., and A. Arakawa, Inclusion of rainwater budget and convective downdrafts in the Arakawa-Schubert cumulus parametrization, *J. Atmos. Sci.*, **54**, 1359–1378, 1997.

Crane, R. K., *Electromagnetic Wave Propagation Through Rain*, 273 pp., Wiley-Interscience, New York, 1996.

Crow, E. L., and K. Shimizu (Eds.), *Lognormal Distributions: Theory and Applications*, 387 pp., M. Dekker, New York, 1988.

Dennis, A. S., and F. G. Fernald, Frequency distributions of shower sizes, *J. Appl. Meteorol.*, **2**, 767–769, 1963.

Drufuca, G., Radar-derived statistics on the structure of precipitation patterns, *J. Appl. Meteorol.*, **16**, 1029–1035, 1977.

Durden, S. L., E. Im, F. K. Li, W. Ricketts, A. Tanner, and W. Wilson, ARARM: An airborne rain-mapping radar, *J. Atmos. Oceanic Technol.*, **11**, 727–737, 1994.

Duroure, C., and B. Guillemet, Analyse des hétérogénéités spatiales des stratocumulus et cumulus, *Atmos. Res.*, **25**, 331–350, 1990.

Fabry, F., On the determination of scale ranges for precipitation fields, *J. Geophys. Res.*, **101**(D8), 12,819–12,826, 1996.

Féral, L., and H. Sauvageot, Fractal identification of supercell storms, *Geophys. Res. Lett.*, **29**(14), 1686, doi:10.1029/2002GL015260, 2002.

Georgakakos, K. P., and W. F. Krajewski, Statistical-microphysical causes of rainfall variability in the tropics, *J. Geophys. Res.*, **101**(D21), 26,165–26,180, 1996.

Goldhirsh, J., Two-dimension visualization of rain cell structures, *Radio Sci.*, **35**, 713–729, 2000.

Goldhirsh, J., and B. Musiani, Rain cell size statistics derived from radar observations at Wallops Island, Virginia, *IEEE Trans. Geosci. Remote Sens.*, **24**(6), 947–954, 1986.

Gotoh, K., and Y. Fujii, A fractal dimensional analysis on the cloud shape parameters of cumulus over land, *J. Appl. Meteorol.*, **37**, 1283–1292, 1998.

Houze, R. A., *Cloud Dynamics, Int. Geophys. Ser.*, vol. 53, 570 pp., Academic, San Diego, Calif., 1993.

Houze, R. A., and C. P. Cheng, Radar characteristics of tropical convection observed during GATE: Mean properties and trend over the summer season, *Mon. Weather Rev.*, **105**, 964–980, 1977.

Joseph, J. H., and R. F. Cahalan, Nearest neighbor spacing of fair weather cumulus clouds, *J. Appl. Meteorol.*, **29**, 793–805, 1990.

Kedem, B., L. S. Chiu, and Z. Karni, An analysis of the threshold method for measuring the area-average rainfall, *J. Appl. Meteorol.*, **29**, 3–20, 1990.

Konrad, T. G., Statistical models of summer rainshowers derived from fine-scale radar observations, *J. Appl. Meteorol.*, **17**, 171–188, 1978.

Kuo, K. S., R. M. Welch, and S. K. Sengupta, Structural and textural characteristics of cirrus clouds observed using high spatial resolution LANDSAT imagery, *J. Appl. Meteorol.*, **27**, 1242–1260, 1988.

Kuo, K.-S., R. M. Welch, R. C. Weger, M. A. Engelstad, and S. K. Sengupta, The three-dimensional structure of cumulus clouds over the ocean: 1. Structural analysis, *J. Geophys. Res.*, **98**(D11), 20,685–20,711, 1993.

Lawford, R. G., Temporal variations in the scaling properties of rain echoes during the development of a cold low in Saskatchewan, *J. Appl. Meteorol.* **35**, 796–809, 1996.

Lebel, T., H. Sauvageot, M. Hoepffner, M. Desbois, B. Guillot, and P. Hubert, Rainfall estimation in the sahel: The EPSAT-Niger experiment, *Hydrol. Sci. J.*, **37**, 201–215, 1992.

Lhermitte, R. M., Small cumuli observed with a 3 mm wavelength radar, *Geophys. Res. Lett.*, **14**, 707–710, 1987.

Lopez, R. E., The lognormal distribution and cumulus cloud populations, *Mon. Weather Rev.*, **105**, 865–872, 1977.

Lopez, R. E., D. O. Blanchard, D. Rosenfeld, W. L. Hiscox, and M. J. Casey, Population characteristics, development processes and structure of radar echoes in south Florida, *Mon. Weather Rev.*, **112**, 56–75, 1984.

Lovejoy, S., Area-perimeter relation for rain and cloud areas, *Science*, **216**, 185–187, 1982.

Machado, L. A. T., and W. B. Rossow, Structural characteristics and radiative properties of tropical cloud clusters, *Mon. Weather Rev.*, **121**, 3234–3260, 1993.

Machado, L. A. T., M. Desbois, and J. P. Duvel, Structural characteristics of deep convective systems over tropical Africa and the Atlantic Ocean, *Mon. Weather Rev.*, **120**, 392–406, 1992.

Mandelbrot, B., *The Fractal Geometry of Nature*, 461 pp., W. H. Freeman, New York, 1982.

Mapes, B. E., and R. A. Houze, Cloud cluster and superclusters over the Oceanic warm pool, *Mon. Wea. Rev.*, **121**, 1398–1415, 1993.

Meneghini, R., J. A. Jones, T. Iguchi, K. Okamoto, and J. Kwiatkowski, Statistical methods of estimating average rainfall over large space-time-scales using data from the TRMM precipitation radar, *J. Appl. Meteorol.*, **40**, 568–585, 2001.

Miller, J. R., A. S. Dennis, J. H. Hirsch, and D. E. Cain, Statistics of shower echoes in western North Dakota, in *Preprints 16th Radar Meteorology Conference, Houston*, pp. 391–396, Am. Meteorol. Soc., Boston, Mass., 1975.

Mohr, C. G., and R. L. Vaughan, An economical procedure for Cartesian interpolation and display of reflectivity factor data in three-dimensional space, *J. Appl. Meteorol.*, **18**, 661–670, 1979.

Nzeukou, A., and H. Sauvageot, Distribution of rainfall parameters near the coasts of France and Senegal, *J. Appl. Meteorol.*, **41**, 69–82, 2002.

Orville, R. E., and R. W. Henderson, Global distribution of midnight lightning: September 1977 to August 1978, *Mon. Weather Rev.*, **114**, 2640–2653, 1986.

Ramos-Buarque, S., and H. Sauvageot, The estimation of rainfall in the Sahelian squall line by the area-threshold method, *Atmos. Res.*, **43**, 209–216, 1997.

Rhys, F. S., and A. Waldvogel, Fractal shape of hail clouds, *Phys. Rev. Lett.*, **56**, 784–787, 1986.

Rosenfeld, D., D. Atlas, and E. Amitai, The window probability matching method for rainfall measurements with radar, *J. Appl. Meteorol.*, **33**, 682–693, 1994.

Sauvageot, H., The probability density function of rain rate and the estimation of rainfall by area integrals, *J. Appl. Meteorol.*, **33**, 1255–1262, 1994.

Sauvageot, H., and G. Despaux, SANAGA: Un système d'acquisition numérique et de visualisation des données radar pour la validation des estimations satellitaires de precipitations, *Vei. Clim. Satellitaire*, **31**, 51–55, 1990.

Sauvageot, H., and M. Koffi, Multimodal raindrop size distributions, *J. Atmos. Sci.*, **57**, 2480–2492, 2000.

Sauvageot, H., and J. P. Lacaux, The shape of average drop size distributions, *J. Atmos. Sci.*, **52**, 1070–1083, 1995.

Sauvageot, H., and J. Omar, Radar reflectivity of cumulus clouds, *J. Atmos. Oceanic Technol.*, **4**, 264–272, 1987.

Sauvageot, H., R. Auria, and B. Campistron, Microscale structure of convection in stratiform clouds: An observational and numerical study, *Proc. 9th Int. Conf. On Cloud Phys.*, pp. 363–366, Tallinn, Boston, Mass., 1984.

Sauvageot, H., F. Mesnard, and R. S. Tenorio, The relation between the area-average rain rate and the rain cell distribution parameters, *J. Atmos. Sci.*, **56**, 57–70, 1999.

Seity, Y., S. Soula, and H. Sauvageot, Lightning and precipitation relationship in coastal thunderstorms, *J. Geophys. Res.*, **106**(D19), 22,801–22,816, 2001.

Seliga, T. A., K. Aydin, and H. Direskeneli, Disdrometer measurements during an intense rainfall event in central Illinois: Implications for differential reflectivity radar observations, *J. Clim. Appl. Meteorol.*, **25**, 835–846, 1986.

Sengupta, S. K., R. M. Welch, M. S. Navar, T. A. Berendes, and D. W. Chen, Cumulus cloud field morphology and spatial patterns derived from high spatial resolution Landsat imagery, *J. Appl. Meteorol.*, **29**, 1245–1267, 1990.

Simpson, J., R. F. Adler, and G. R. North, A proposed tropical rainfall measuring mission (TRMM) satellite, *Bull. Am. Meteorol. Soc.*, **69**, 278–295, 1988.

Weger, R. C., J. Lee, T. Zhu, and R. M. Welch, Clustering, randomness, and regularity in cloud fields: 1. Theoretical considerations, *J. Geophys. Res.*, **97**(D18), 20,519–20,536, 1992.
Welch, R. M., and B. A. Wielicki, The stratocumulus nature of fog, *J. Clim. Appl. Meteorol.*, **25**, 101–111, 1986.

Welch, R. M., K. S. Kuo, B. A. Wielicki, S. K. Sengupta, and L. Parker, Marine stratocumulus cloud fields off the coast of southern California observed using Landsat imagery: 1. Structural characteristics, *J. Appl. Meteorol.*, **27**, 341–362, 1988.

Wielicki, B. A., and R. M. Welch, Cumulus cloud properties derived using Landsat satellite radar, *J. Clim. Appl. Meteorol.*, **25**, 261–276, 1986.

Young, K. C., *Microphysical Processes in Clouds*, 427 pp., Oxford Univ. Press, New York, 1993.

Zhu, T., J. Lee, R. C. Weger, and R. M. Welch, Clustering, randomness, and regularity in cloud fields: 2. Cumulus cloud fields, *J. Geophys. Res.*, **97**(D18), 20,537–20,558, 1992.

Zipser, E. J., and K. R. Lutz, The vertical profile of radar reflectivity of convective cells: A strong indicator of storm intensity and lightning probability, *Mon. Weather Rev.*, **122**, 1751–1759, 1994.

F. Mesnard, Laboratoire de Physique de l'Atmosphère Tropicale, Université des Antilles et de la Guyane, 97 159 Pointe-à-Pitre, Guadeloupe. (fmesnard@univ-ag.fr)

H. Sauvageot, Observatoire Midi-Pyrénées, Laboratoire d'Aérologie, Université Paul Sabatier, F-31400 Toulouse, France. (sauh@aeroobs-mip.fr)