## Abstract

The area–time integral (ATI) method has previously been successfully used to estimate the area-averaged rain-rate distribution and the rainfall volume over an area from radar or from satellite infrared (IR) data. In most cases, the method was implemented over regions or test areas with an assumed homogeneous climatic character, that is, without a strong spatial variation of the rain regime throughout the test area. In the present paper, the behavior of the ATI method is discussed for a test area displaying two strong gradients of the cumulative annual rainfall: one meridional, at the transition between regions having, respectively, a desertic and an equatorial climate and the other zonal, at the transition between land and sea. The studied area is divided into four subtest areas (north, south, land, and sea) over which the ATI computation is applied separately. The linear coefficient relating the radar-observed area-averaged rain rate and the fractional area where the rain is higher than a threshold calculated over the four subtest areas is found to be almost constant, in agreement with the ergodic character of the rain-rate distribution observed in this region. Similarly, the linear coefficient relating the rain volume over the subtest areas to the IR satellite-derived ATI, a parameter analogous to the Geostationary Operational Environmental Satellite (GOES) Precipitation Index (GPI), is found to be very steady, with a mean value of 3.02 mm h^{−1} and a coefficient of variation of only 8%. These coefficients, as well as the underlying dynamic and microphysical processes, do not seem significantly influenced by the climatic character, even at a short space scale, in the studied area. The ratio of radar rain areas to cloud areas is, notably, almost constant. For a brightness temperature of 235 K, the ratio of the cloud area to rain area is around 1.68.

## 1. Introduction

Quantitative observation of tropical rain fields is necessary to the study of global climatic mechanisms. The lack or insufficiency of ground-based networks (both rain gauge and radar) in tropical areas, leads to consider the paramount importance of achieving a correct interpretation of satellite data in term of precipitation at the ground (e.g., Jenkins et al. 2002). Among the means proposed to reach this objective, the threshold methods, which are the object of the present paper, are attractive because of their simplicity and the quality of the results they yield. The threshold methods consider the relation between the area occupied by the clouds having a top with an equivalent blackbody brightness temperature (TB) colder than a threshold and the rain volume (*V*), or the area-averaged rain rate 〈*R*〉, inside an observed area of finite size. This approach has notably led to the Geostationary Operational Environmental Satellite (GOES) Precipitation Index (GPI; see section 2 for definition) method (Barrett 1970; Arkin 1979; Richards and Arkin 1981; Jobard and Desbois 1992; Atlas and Bell 1992; Meneghini and Jones 1993; Arkin and Xie 1994; Johnson et al. 1994; Joyce and Arkin 1997; Wang et al. 1998; Laurent et al. 1998; Meneghini et al. 2001; among others). A similar methodological development has occurred for the hydrological processing of radar data with the area–time integral (ATI) method (Doneaud et al. 1984; Chiu 1988; Lopez et al. 1989; Atlas et al. 1990a, b; Rosenfeld et al. 1990; Short et al. 1993; Morrissey et al. 1994; Ramos-Buarque and Sauvageot 1997; Sauvageot et al. 1999; Meneghini et al. 2001; among others).

The rainfall volume, or the area-averaged rain rate, are the quantities of interest for many applications of rainfall measurement for terrestrial hydrology and for general circulation models. The threshold methods show that the rainfall volume (or the mean rain rate, or the rainfall accumulation) integrated over an area for some time (e.g., the duration of a rainy event such as the crossing of a squall line over the observed area) is linearly related to the corresponding integral of the area occupied by the cloud whose TB at the top is lower than a threshold *τ*_{TB} or by the radar-observed rain rate higher than a threshold *τ _{R}*. This implies that the ratio between the precipitation and the cloud-top areas, with respect to the thresholds, is equal to the ratio of the linear coefficients of the ATI and GPI methods (see section 2), as emphasized by Atlas and Bell (1992). Are these coefficients stable for various areas and seasons? Most of the quoted authors acknowledge that the climatic variability of these coefficients has to be documented in order to better know the limits of the threshold method applicability to small spatiotemporal scales. For the GPI coefficients more particularly, Richards and Arkin (1981) and Joyce and Arkin (1997) demonstrate the stability of the linear GPI coefficient over tropical oceans, but, for example, Arkin and Meisner (1987), Morrissey and Greene (1993), Atlas and Bell (1992), Arkin and Xie (1994), and Ba et al. (1995) emphasize the fact that a calibration is necessary for various continental areas and seasons. It is notably the case for the Sahelian region (Ba et al. 1995). Most of the integrating or observed areas over which the threshold methods were demonstrated in the above-quoted works however were assumed to be climatologically homogeneous (i.e., areas in which the “climatic” rainfall regime does not display strong variations of the annual mean rainfall accumulation).

The question addressed in the present paper is the behavior of the ATI and GPI coefficients (or the conditions of implementation of these methods) over an integration area inside which there is a strong climatological gradient of the seasonal rainfall accumulation. In other words, the question is whether the linear coefficients linking *V* or 〈*R*〉 to the integrated cloud or rain areas are dependent on the presence of this gradient. The area considered to discuss this point is located on the Atlantic coast, at the western end of Sahelian Africa, around Cape Verde, a region where two strong gradients, land–sea and north–south, of the seasonal cumulative rain depth are observed. Much of the rainfall in this region is provided by mesoscale westward-moving convective systems. The choice of this test area is also justified by the availability of a reliable radar dataset. Some aspects of the climatology of the rain in this area are given in Nzeukou and Sauvageot (2002, hereafter NS02) and Nzeukou et al. (2004).

## 2. Basic concepts of the threshold method

The present section begins with the radar threshold (ATI) method for the sake of clarity, because it seems conceptually clearer than the satellite threshold method. The basic concepts of the precipitation estimate from the radar ATI method were chronologically reviewed in several of the papers quoted in section 1 (e.g., Doneaud et al. 1984; Atlas and Bell 1992; Short et al. 1993; Johnson et al. 1994; Morrissey 1994; Meneghini et al. 2001). That is why this section is restricted to a short, nonchronological presentation of the aspects that are useful to define the terms and quantities used in the present study.

At the basis of the method is the experimental obviousness that the conditional (i.e., where it rains) probability density function (pdf) of the rain rate *R*, that is *P*(*R*), for a given climatic area is defined and reproducible (Atlas et al. 1990a). A data sample of *R* with a size large enough for the various rain-rate classes be represented with a convenient weighting is thus described by *P*(*R*). This sample can be obtained from data storage in space and/or in time. The data sample is sufficient when *P*(*R*) is stable (e.g., Atlas et al. 1990a; Rosenfeld et al. 1994). Observation shows that *P*(*R*) is correctly represented by a lognormal distribution (e.g., Atlas et al. 1990a; Kedem et al. 1990; Sauvageot 1994), that is, Λ(*μ _{R}*,

*σ*

^{2}

_{R}), where

*μ*and

_{R}*σ*are the mean and standard deviation of

_{R}*R*with, between these two parameters, the relation

*σ*

_{R}= 5

*μ*(Sauvageot 1994; NS02). Now it is known that the radar reflectivity factor of rain is linked to

_{R}*R*by the power relation

*Z*=

*aR*, where

^{b}*a*and

*b*are coefficients (e.g., for the Dakar area see Nzeukou et al. 2004). In such conditions, if

*P*(

*R*) is lognormally distributed,

*P*(

*Z*), the pdf of

*Z*, is also lognormal, that is, Λ(

*μ*,

_{Z}*σ*

^{2}

_{Z}) (e.g., Crow and Shimizu 1988).

If, in an area of observation *A*_{0}, we have a field of *R* represented by *P*(*R*), the area-averaged conditional (climatic) rain rate can be written as

and the area occupied by the rain having a rate higher than a threshold *τ*_{R}, that is, the fractional area, is

with

showing that 〈*R*〉 is linearly linked to the fractional area *F*(*τ _{R}*) by a factor

*S*(

*τ*) depending only on

_{R}*P*(

*R*) and on the threshold

*τ*. Equation (4) shows that for

_{R}*τ*= 0 the denominator of the right-hand side of (4) is 1 and then we have

_{R}*S*(

*τ*) = 〈

_{R}*R*〉, the conditional climatic rain rate. Here

*S*(

*τ*) has the physical meaning of a mean rain rate to be associated with a rainy area (

_{R}*R*>

*τ*) reduced with respect to the effective rainy area (

_{R}*R*> 0). This factor enables to retrieve the correct climatic rain rate 〈

*R*〉 and thus to calculate the real rain volume inside an observed area

*A*

_{0}even when the effective rain area is not entirely observed.

Equation (3) can be applied to a snapshot over the area *A*_{0}. Considering the condition on the size of the data sample required to ensure the validity of the method, the area with *R* > *τ _{R}* inside the snapshot has to be large enough so that the associated

*P*(

*R*) be stable. In the presence of a rain field of small extent, a time series of snapshots observed at times t

_{i}separated by intervals Δ

*t*large enough to ensure the statistical independence from each other, can be used. If, at time

_{i}*t*,

_{i}*A*(

_{i}*τ*) is the area with

_{R}*R*>

*τ*, with

_{R}*A*(

_{i}*τ*= 0) =

_{R}*A*and 〈

_{ri}*R*〉 is the area-averaged rain rate over

_{i}*A*, the summation over the time series gives

_{ri}The left member of (5) represents the rain volume *V* fallen during the observation time *T _{i}* = Σ

_{i}Δ

*t*inside the observed area and the summation in the right member represents (ATI)

_{i}*, the radar-observed ATI for the threshold*

_{R}*τ*

_{R}in the same area. Thus, (5) can be written as

With a time series, an observed area moving with the rainy system can be considered (e.g., Doneaud et al. 1984).

Clearly, for a fixed P(R), the conditional 〈*R*〉 in (1) is fixed and, for a given threshold *τ _{R}*, quantities

*F*(

*τ*) in (2) and

_{R}*S*(

*τ*) in (4) are also fixed. However equations similar to (1)–(6) can be written if the rainy area (

_{R}*R*> 0) is replaced by another wider area including no-rain region, notably a radar-observed area

*A*

_{0}. Area

*A*

_{0}can be made up of a time series of radar Plan Position Indicators (PPIs) or a particular fraction of PPI (e.g., the north part), in order to reach the data sample large enough to reduce the statistical variance on 〈

*R*〉 and

*F*(

*τ*). In this case, 〈

_{R}*R*〉 and

*F*(

*τ*) in (3) are not fixed parameters since unconditional and depending on

_{R}*A*

_{0}, but

*S*(

*τ*) is not changed. In fact the discrete form of (3) can be written for the conditional case:

_{R}where Δ*A _{i}* is the area of pixels I, where

*R*> 0 and Δ

_{i}*A*the area of pixels

_{j}*j*where

*R*≥

*τ*. Equation for the unconditional case is obtained by replacing the denominator on both sides of (3′) by

_{R}*A*

_{0}.

That is why *S*(*τ _{R}*) can be obtained by observing pairs of unconditional [〈

*R*〉,

*F*(

*τ*)]

_{R}_{A0}inside

*A*

_{0}. Least squares fitting of the data pairs produces an estimate of the climatic

*S*(

*τ*) of (4) (see section 4).

_{R}The concept of ATI, whose implementation to radar data is rather straightforward, can also be used with satellite data, notably the cloud-top IR equivalent blackbody TB, since there is an empirical relation between the rainfall generated by a convective cloud and the area occupied by its top for TB lower than a threshold *τ*_{TB}. In fact, the concept of rain measurement from a thresholded area was initially proposed for implementation on the IR satellite cloud observation by Barrett (1970), Arkin (1979), Richards and Arkin (1981), and Arkin and Meisner (1987). These last authors developed the GPI, with which the rainfall accumulation averaged over a mesh of 2.5° × 2.5° and cumulated over a time t can be written as

where *F _{c}* is the fraction of pixels colder than a threshold TB = 235 K averaged over the 2.5° × 2.5° mesh during

*t*, with

*t*in hours and GPI in millimeters. The factor of 3 in (7) is equivalent to a rain rate of 3 mm h

^{−1}. This value was determined by Arkin (1979) and Richards and Arkin (1981) for 235 K, the threshold corresponding to the best correlation (about 0.86), from the Global Atmospheric Research Program (GATE) Atlantic Tropical Experiment (GARP) data. These authors show that the correlation between the rainfall accumulation and the cold cloud-top area improves when the area and time of integration increase (up to 0.90 for 2.5° × 2.5° and 24 h). Arkin and Meisner (1987) confirm that the best correlations for tropical ocean areas are obtained around TB = 235 K. The GPI method has been used operationally from 1982 in the Global Precipitation Climatology Project (GPCP; Huffman et al. 1997). However, Arkin and Xie (1994) found, for the Japan area, that although the relation (7) from GATE is valid, the optimum value of the TB threshold can vary substantially according to the area considered. Several variations of the GPI method have been proposed (e.g., Morrissey and Greene 1993; Joyce and Arkin 1997) using different values of the TB threshold and numerical coefficient of (7).

In the present paper, the relation between rainfall and cold cloud-top area, that is (7), is written in a form identical to (6), in which *τ _{R}* is replaced by

*τ*

_{TB},

*S*(

*τ*) by a linear coefficient

_{R}*G*(

*τ*

_{TB}), and (ATI)

*by (ATI)*

_{R}_{TB}:

Relation (8) can also be obtained from (7) by replacing in (7) the constant factor of 3 (mm h^{−1}) by a variable, 3*α*, and by multiplying the two sides of (7) by the observed area *A*_{0}. Then with *G*(*τ*_{TB}) = 3*α*, Eq. (8), results, since (GPI)*A*_{0} = *V* and *F _{c}t A*

_{0}= (ATI)

_{TB}.

Combining (6) and (8), the ratio of cloud area to rain area can be written, as suggested by Atlas and Bell (1992), as

For *τ _{R}* = 0, (ATI)

*represents the total area occupied by the rain.*

_{R}Nevertheless the relation between 〈*R*〉 or *V* and the cold top (ATI)_{TB} is not so tight as with the radar (ATI)* _{R}* because of the spreading of the convective cloud top under a statically stable layer such as the tropopause, not directly related to the rain inside the cloud. The relation between 〈

*R*〉 or

*V*and the cold top (ATI)

_{TB}has been discussed by many authors as pointed out in section 1. The IR data of geostationary satellites are used rather than the microwave radiometric data of low-orbiting satellites because the sampling time of microwave data is not adapted to the fast evolving and moving mesoscale convective systems.

## 3. Data

Rain rates estimated from ground-based radar observations were collected in the region of Dakar (Cape Verde), on the Senegalese Atlantic coast (Fig. 1). The site and the radar data are described in detail in NS02 and Nzeukou et al. (2004). In brief, the region of observation is flat, without relief higher than 200 m of altitude. It is located at the west end of the Sahelian strip, which is the zone of transition between the Saharan desertic latitude, north of about 17°N, and the rainy equatorial latitude, south of 12°N. Typical of the Sahelian climate, there is a single rainy season of about 3 months, from early July to late September, when the intertropical convergence zone is at its most northern latitude. Almost all the rainy events are squall lines moving quickly westward (e.g., Desbois et al. 1988; Mathon and Laurent 2001). Most of them decay over the nearby Atlantic while a few systems grow stronger when moving over the sea (Gray and Landsea 1992). Figure 1 shows the 39-yr-averaged isohyets of the annual rainfall accumulation for the western Sahel. It emphasizes the strong climatic meridional rainfall gradient of the Sahelian strip (about 300 mm at 16°N and 1500 mm at 12°N, that is a factor 5 over 400 km). Figure 2 displays with more details the isohyets of the observed area over land and sea averaged for 7 yr of observation (1993–99) with the radar at Dakar-Yoff Airport. It shows that the combination of the meridional and the zonal land–sea rainfall gradient results in coastal area isohyets approximately directed southwest–northeast.

NS02 show that, in spite of these gradients, the rain field in the observed area is approximately ergodic (i.e., the spatial average is equal to the temporal average) and that the mean conditional climatological rain rate is around 5.1 mm h^{−1}, that is about 6.4 times higher than the value observed at midlatitude (about 0.8 mm h^{−1}; Sauvageot 1994). Consequently the rain duration in the observed area is very short (80% of the total rain falls in less than 5 h). Besides, the areal distribution of the annual rainfall accumulation displays a strong heterogeneity. Figure 3 shows, as an example, a smoothed distribution of the rainfall accumulation for 1995; the interannual deviations from the long-term average can reach a factor of 4 in excess or in default.

To analyze the behavior of the ATI method in the observed area, four subareas have been considered, as in NS02. The four subareas are shown in Fig. 4 and their superficies are given in Table 1. The land and sea subareas occupy the hatched part of the annular area between 60 and 180 km of radial distance *r* from the radar (*r* < 60 km is rejected because of the presence of ground echoes on the land side). The north and south subareas are the annular domains bounded by distances 60 and 180 km, lying north and south of the Dakar latitude. In the observed area, during the rainy season, the altitude of the precipitation melting level is higher than 5000 m, so that the radar beam is in the rain up to 200 km of distance, considering the elevation angle used for the PPI scan of 0.8° and the half-power beam width of 1.3°. The values of the averaged rainfall parameters for the four subareas are recalled, from NS02, in Table 1. Table 1 shows that the annual average of the rainfall accumulation 〈*H*〉 is very different in the four subareas but that 〈*H*〉 is proportional to 〈*T*〉, the rainfall duration, while the conditional average rain rate *μ _{R}* is almost constant, which is consistent with the ergodic character of the rain field in the observed area. In Table 1,

*μ*and

_{R}*σ*are corrected for the truncation of the raw data for reflectivity lower than the minimum detectable rain rate. This behavior of the rain field is emphasized by Fig. 5 where the cumulative distribution function (CDF) of rain rate for the four subareas deduced from the radar data is presented. The four CDFs are very similar. These subareas are used for the computation of the area integrals both for radar and for satellite data.

_{R}The radar reflectivity data were acquired with the meteorological radar of Dakar-Yoff (see Table 2 for technical characteristics) in the form of PPI at a constant elevation of 0.8°. Acquisitions were performed in manual mode by an operator with time sampling intervals between 10 and 20 min (irregular because they are manual; this airport radar is not devoted to a single use). The parameters of the radar acquisition are given in Table 3. The reflectivity measured along the radials was corrected for attenuation by gas, clouds, and rain and for the global effects of distance and reflectivity differences between the precipitation observed aloft by the radar and rain at surface level with the iterative algorithm of Hildebrand (1978) and the probability matching method (PMM) proposed by Calheiros and Zawadzki (1987) as specified in NS02. The correctness of this calibration was then verified by comparison of the rainfall accumulation obtained by radar and by ground rain gauges in five sites located east of the radar for *r* < 100 km. The observed differences were smaller than 20% for individual rainy events. The radar dataset is summarized in Table 3. In the present study, only 4 yr of data (1996–99) have been used. The difference in the annual number of PPI scans (Table 3) is only due to particular circumstances and choice of the radar operator concerning the time-sampling intervals.

The space data used to compute the cloud-top temperature ATI are the IR images of the geostationary satellite Meteosat at the time-sampling interval of 30 min.

## 4. Results and discussion

### a. Variability of the area-averaged rain rate to fractional area relation from radar data

To emphasize the behavior of the linear coefficient *S*(*τ _{R}*) of (3), the pairs [〈

*R*〉,

*F*(

*τ*)] were computed for each PPI in each of the four subareas for seven thresholds

_{R}*τ*

_{R}between 1 and 20 mm h

^{−1}. Then

*S*(

*τ*) was calculated by least squares fitting from the pairs [〈

_{R}*R*〉,

*F*(

*τ*)] for the four subareas and for each of the 4 yr of the dataset. Figure 6 shows, as an example, the plots of the data points and the fitted curves for 1998 and for

_{R}*τ*= 5 and 10 mm h

_{R}^{−1}. This figure looks like those obtained by other authors in other places (e.g., Atlas et al. 1990a; Rosenfeld et al. 1990; Johnson et al. 1994; Ramos-Buarque and Sauvageot 1997). Notably the correlation coefficient is higher than 0.97 and

*S*(

*τ*) increases with

_{R}*τ*. The interest and originality of Fig. 6 is to show the behavior of

_{R}*S*(

*τ*) for the four subareas. The values of

_{R}*S*(

*τ*) for the other thresholds are given with the standard deviation in Table 4 (top) and in Fig. 7, where 1999 has been added for comparison. Table 4 (top) and Fig. 7 show that, for a given value of

_{R}*τ*,

_{R}*S*(

*τ*) is almost constant for the four subareas. It is slightly smaller for the north and sea than for the land and south, with the largest difference of 23% between the north and south at

_{R}*τ*= 5 mm h

_{R}^{−1}for 1998, but this difference is much smaller for 1999 (Fig. 7).

In Table 4 (top), the mean and standard deviation of *S*(*τ _{R}*), that is, 〈

*S*(

*τ*)〉 and

_{R}*σ*

_{S(τR)}, respectively, are given as well as

*S*(

*τ*) for the total area, that is, for the entire annular area, for distances between 60 and 180 km from the radar. The coefficient of variation [

_{R}*σ*

_{S(τR)}/

*S*(

*τ*)] is smaller than 0.082 and

_{R}*S*(

*τ*) for the total area is found almost equal to 〈

_{R}*S*(

*τ*)〉, which is an average giving the same weight to the four subareas.

_{R}All these parameters have also been computed separately for the convective line and for the stratiform region of the squall lines. As recalled in section 3, most of the rainfall systems in the observed area are Sahelian squall lines, which are known to have a very typical organization made up of a narrow convective line releasing heavy rainfall with a high rain rate at the front of the system, followed by a wide stratiform area giving widespread rain with low intensity. An example of such a squall line observed by radar in the Dakar area is shown in Fig. 4 of Nzeukou et al. (2004). To delimit the convective and stratiform areas in tropical squall lines, a frequently used automatic criterion is obtained by observing that most rain rates in the convective line are higher than 10 mm h^{−1}, while rain rates in the stratiform region are usually lower than 10 mm h^{−1} (Gamache and Houze 1982; Tokay et al. 1999; Sauvageot and Koffi 2000; Nzeukou et al. 2004; among others). There is an interest in considering separately the convective and stratiform rainfall component in relation with problems of surface hydrology, erosion, and agriculture. The threshold *R* = 10 mm h^{−1} has been used as a distinguishing criterion to separate convective lines and stratiform regions in order to apply (3′) to calculate the values given in Table 4. What Table 4 shows is that the above conclusion concerning the total system can be applied to the convective line and stratiform region considered separately.

These results are coherent with the finding of NS02 on the ergodicity of the rain fields in the Cape Verde region and of Nzeukou et al. (2004) on the rather low variability of *S*(*τ _{R}*) in West Africa. These are, however, not redundant with the previous results since they concern the behavior of

*S*(

*τ*) in smaller areas and for shorter durations than previously considered, and especially in a climatologically transitional area, with a strong rainfall gradient. The results for the other years are not given because they are similar to those of 1998.

_{R}### b. Variability of the volumetric rainfall to ATI relation from satellite IR data

From the radar data, the volumetric areal rainfall *V* for each rainy event, that is for each squall line crossing the radar-observed area, has been calculated for the four subareas of Fig. 4 and for various thresholds *τ _{R}*. The (ATI)

_{TB}corresponding to each event and each subarea of Fig. 4 have also been calculated for various TB thresholds from the IR Meteosat satellite data. Then the pairs of corresponding values [

*V*, (ATI)

_{TB}] have been regressed in order to determine the factor

*G*(

*τ*

_{TB}) of the relation (8). Because each rainy event crossing the observed area provides only one pair [

*V*, (ATI)

_{TB}], the number of data points corresponding to a single rainy season (i.e., 1 yr) is not sufficient to perform a fitting, notably for the north and sea subareas where the rain duration is low (i.e., where the number of events is small). That is why the data points of 4 yr (1996–99) have been gathered in a single set.

Figure 8 presents, as an example, the plot of the [*V*, (ATI)_{TB}] pairs for the four subareas of Fig. 4 and for the total area at two thresholds of *τ*_{TB}, with the fitted curves and the corresponding value of the linear coefficient *G*(*τ*_{TB}) of (8). Figure 8 shows that, though less tight than *S*(*τ _{R}*), the relation between

*V*and (ATI)

_{TB}is well defined. Most of the correlation coefficients for the

*G*(

*τ*

_{TB}) calculation are around 0.8 and 0.9. The choice of

*τ*used for the computation of

_{R}*V*almost has no influence on the resulting

*G*(

*τ*

_{TB}) values [for

*τ*increasing from 3 to 15 mm h

_{R}^{−1},

*G*(

*τ*

_{TB}) decreases by about 5%]. This low sensitivity of

*G*(

*τ*

_{TB}) to

*τ*mirrors the low sensitivity of (6) to the rain-rate threshold used for the calculation of

_{R}*V*from radar data.

Table 5 gathers the *G*(*τ*_{TB}) values for five *τ*_{TB} thresholds and Fig. 9 displays the corresponding variations. The rain-rate threshold used to compute the radar-derived rain volume *V* necessary for the *G*(*τ*_{TB}) calculation is *τ _{R}* = 5 mm h

^{−1}. The correlation coefficients of the

*G*(

*τ*

_{TB}) given in Table 5 are everywhere higher than 0.80. Table 5 shows that

*G*(

*τ*

_{TB}) increases by about 47% for

*τ*

_{TB}decreasing from 243 to 203 K. This variation compensates for the reduction of the cloud-top area associated to the

*τ*

_{TB}decrease. For

*τ*

_{TB}decreasing below 220 K,

*G*(

*τ*

_{TB}) increases more for the sea and south than for the north and land. At 203 K,

*G*(

*τ*

_{TB}) is almost the same for north and land but 17% and 34% higher for south and sea respectively. This suggests for the deepest convection a lesser “rain efficiency” for the storms occurring in the north and land areas than in the sea and south areas. Such differences could be linked to specificities of the observed region such as cyclogenesis processes boosting the convection for some rain systems over the near ocean, in the common part of the sea and south areas (Gray and Landsea 1992; Sall and Sauvageot 2005) or inversely to a damping or inhibiting effect of the Saharan air layer over the north and land areas. The main interest of Table 5 and Fig. 9 is to emphasize the steadiness of

*G*(

*τ*

_{TB}) for the four subareas around 243 to 235 K, including the threshold of 235 K for which the best correlation was obtained for tropical ocean in previous studies (quoted in section 2). At 243–235 K, the variability for the four subareas is the smallest and the coefficient of variation of

*G*(

*τ*

_{TB}) is only 8%. For 235 K, the average value of

*G*(

*τ*

_{TB}) is 3.02 mm h

^{−1}almost exactly the value of 3 mm h

^{−1}proposed by Arkin (1979) for the GATE area. The smallest value of

*G*(

*τ*

_{TB}) is for the north and the largest is for south. For the land and sea they are not significantly different. These results suggest that the dynamic and microphysical processes involved in the rain generation in the Sahelian squall lines are moderately dependent on the climate character such as the dryness of the climate. In addition squall lines are made up of a compact cluster of intense but short lifetime convective elements in such a way that the life cycle effect of individual cells is smoothed and does not perturb the area-averaged parameters.

The surface of the subtest areas is given in Table 1. The two smallest land and sea are about 1.5° × 1.5°, half the two largest north and south , which are 2.1° × 2.1°. It is found that for areas smaller than that of Fig. 4, *G*(*τ*_{TB}) decreases as observed by Richards and Arkin (1981).

Using (9), the ratio of the cloud area at a given IR threshold to the rain area can be obtained. This ratio can be calculated for the total rainy area if *S*(*τ _{R}*) for

*τ*= 0 is used in (9), as pointed out in section 2. Now

_{R}*S*(

*τ*= 0) can be obtained by extrapolating the computed values of

_{R}*S*(

*τ*) of Table 4 to

_{R}*τ*

*= 0, or from*

_{R}*S*(

*τ*) values computed from disdrometer data, given in Table 7 of Nzeukou et al. (2004), or again directly from Table 1 since

_{R}*S*(

*τ*= 0) is equal to the climatic conditional rain rate [i.e., 〈

_{R}*μ*〉 (= 〈

_{R}*R*〉)]. These three ways give a same value of 〈

*R*〉 that is around 5.1 mm h

^{−1}. With this value, the ratio of the cloud area at 235 K to the rainy area is 5.1/3.03 = 1.68. This value is only 15% larger than that computed by Atlas and Bell (1992) for GATE at 235 K, which is 1.47.

## 5. Summary and conclusions

The relation between the cloud area and the rain area at the transition from an arid to an equatorial climate has been analyzed. The studied area, located at the west end of the Sahelian strip, around Cape Verde, displays strong north–south and land–sea gradients of the rainfall accumulation. The studied area has been segmented into four subtest areas: north, south, land, and sea. The basic concepts of the threshold method applied to rain radar and to cloud satellite data processing have been presented and some relations between the two approaches were discussed.

Using a radar dataset of 4 yr collected with the radar of Dakar-Yoff, the linear parameter *S*(*τ _{R}*) relating the rain volume

*V*(or the area-averaged rain rate) to the rain ATI (ATI)

*(or the fractional area) with respect to a rain threshold*

_{R}*τ*

_{R}has been shown to be almost the same for the four subtest areas, which is locally independent of the climatic character. This result is of interest for rainfall measurement by ground-based and space radar or microwave radiometer.

Using the IR data of the Meteosat satellite and the rain volume estimated by radar, the relation between the rain volume *V* and the cloud ATI (ATI)_{TB}, with respect to a brightness temperature threshold *τ*_{TB}, has been calculated. The scattering of *G*(*τ*_{TB}), the parameter of the linear relation between *V* and (ATI)_{TB}, which is similar to the GPI coefficient, is found minimal for *τ*_{TB} around 243 to 235 K, with a coefficient of variation (CV) of only 8%, that is with a low variability for the four subtest-areas, which suggests it is moderately dependent on the climatic character, semiarid or equatorial, continental or maritime. For *τ*_{TB} = 235 K, the mean value of *G*(*τ*_{TB}) is 3.02 mm h^{−1}, almost exactly that found for the GPI coefficient for the GATE area.

The ratio of *S*(*τ _{R}*), for

*τ*= 0, to

_{R}*G*(

*τ*

_{TB}) enables the estimation of the ratio of the cloud area (at

*τ*

_{TB}) to the rain area. This ratio is found almost constant around 1.68 for

*τ*

_{TB}= 235 K.

These results stress that the cause of rainfall accumulation gradients and the main difference between the rain regime in the four subareas of the studied region is rain duration.

The stability of the parameters *S*(*τ _{R}*) and

*G*(

*τ*

_{TB}) suggests that the dynamic and microphysical processes underlying the relations between the rain and the generating clouds in the studied area are not strongly influenced by the climatic character considered in term of rainfall accumulation. However, scatter of

*G*(

*τ*

_{TB}) for

*τ*

_{TB}lower than about 220 K suggests some physical differences between the cloud and rain processes in the four subareas for the deepest convection. In other words, similarities of the averaged rainfall parameters do not imply that generating rain systems are similar.

These results strengthen the conclusions of previous authors that the cloud ATI is a very efficient method for the rainfall estimate. At least in the studied part of the Sahelian strip, and on the condition that the considered ATI be large enough, the cloud ATI deserves to be considered for an accurate estimate of the rainfall accumulation, even in the area where the annual rainfall accumulation height is low.

## Acknowledgments

The authors are grateful to all those who contributed to the collection of the radar dataset used in this study, notably the National Meteorological Office of Senegal, the ASECNA, and the team of the Laboratoire de Physique de l’Atmosphère Simeon Fongang of the Ecole Supérieure Polytechnique, University Cheikh Anta Diop at Dakar. Many thanks also to Dr. Dominique Dagorne from IRD, for helping with satellite data processing and for providing his Triskel software. The authors would also like to thank the Association Universitaire de la Francophonie (AUF) for granting support to C.M.F. Kebe.

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## Footnotes

* Current affiliation*:* Ecole Supérieure Polytechnique, Université Cheikh Anta Diop de Dakar, Dakar, Senegal

*Corresponding author address:* Dr. Henri Sauvageot, Université Paul Sabatier, Observatoire Midi-Pyrénées, Laboratoire d’Aérologie, CRA, 65300 Lannemezan, France. Email: sauh@aero.obs-mip.fr