Rainfall monitoring during HAPEX-Sahel. 2. Point and areal estimation at the event and seasonal scales

Journal of

Hydrology ELSEVIER

Journal of Hydrology 188-189 ( i 997) 97-122

Rainfall monitoring during HAPEX-Sahel. 2. Point and areal estimation at the event and seasonal scales T. Lebel a'*, L. Le Barb6 b aORSTOM/LTHE, Groupe PRAO, BP 53, 38041 Grenoble Cedex 9, France bORSTOM/LTHE, Groupe PRAO, BP 5045, 34032 Montpellier Cedex 1, France

Abstract The water and energy balance models that will be used in HAPEX-Sahel to analyze the data collected during the experiment are strongly conditioned by the rainfall estimation accuracy over the areas of interest. Following the description of the rainfall conditions that prevailed during HAPEXSahel and the computation of statistics characterising the point rainfall process for the Sahel, presented in a companion paper, it is examined here how accurate are the areal rainfall estimates provided by the EPSAT-Niger network at the event and seasonal scales. Using a geostatistical framework, it is shown that it is possible to infer a climatological variogram that represents the average variability of the event rainfields, within the limits imposed by the resolution and sampling window of the network. Average event rainfall estimation errors are derived. The density of the EPSAT-Niger mesoscale network (one station for 150 km 2) allows the estimation of the event areal rainfall over the 1° x 1° square with an average uncertainty around 5%. On the supersites (100750 krn 2) the average uncertainty can be as large as 20%, depending on the site and rainfall considered. This uncertainty is reduced by two-thirds, or more, when using the denser network designed specifically to cover the supersites. It is then demonstrated how the variability within the event rainfields and the space-distribution of the rainy events combine to determine the spatial structure of a rainfield resulting from the accumulation of several event rainfields. Over the HAPEX-Sahel study area, the within-the-event-rainfield variability is dominant over the variability resulting from the distribution of the events in space. Consequently the correlation length does not vary much when shifting from the event to the seasonal scale. At an unsampled location the seasonal rainfall error magnitude is on the order of the average error at the event scale, divided by the square root of the number of events recorded during the season.

* Corresponding author. 0022-1694/97/$17.00 © 1997- Elsevier Science B.V. All rights reserved PII S0022-1694(96)03325-2

98

T. Lebel, L. Le Barb3/Journal of Hydrology 188-189 (1997) 97-122

1. Introduction A recent paper by Peters-Lidard and Wood (1994) has accentuated the importance of assessing storm areal average rainfall in field experiments. Their study shows that large estimation errors may be expected when estimating the event rainfall from raingauge networks having a density below a 'critical' level. In such cases, the reliability of the 'ground truth' used for satellite and radar rainfall measurement systems and of the computed inputs to water balance models is questionable. Finding an appropriate rainfall ground truth for remote sensing validation and for water balance studies in the Sahel was one of the main reasons that led to initiate the EPSATNiger (E-N) experiment (Lebel et al., 1992). Latter, the implementation of HAPEX-Sahel (H-S) increased the need for obtaining as accurate as possible rainfall estimates in this region, especially at the field (a few hectares to 1 km2), supersite (15-20 km 2 x 1520km 2) and 1° x 1° scales (12 000 km2). These estimates must be accompanied by a measure of uncertainty that only statistically based algorithms can provide. The rainfall conditions that prevailed during the H - S years (1990-1993) have been analyzed in a companion paper (Lebel et al., 1997). Some basic statistics characterising the point rainfall process have been given (rain rates, event rainfall) and a certain stability of these statistics from year to year has been shown. One main conclusion was that, when there is a seasonal rainfall deficit, it is more due to a smaller number of rainy events than to a variation of the efficiency of these events. Based on these first results, the goal of the present paper is twofold: (i) to provide the H - S investigators with an assessment of the event rainfall estimation accuracy over an array of space scales, when using the E - N raingauge network only; (ii) to develop a conceptual framework allowing the analysis of the structure of the rainfields obtained when several event rainfields are accumulated (10day, monthly or seasonal rainfields). The analysis is carried out in a geostatistical framework. Other approaches could have been chosen (see, e.g. Rodriguez-Iturbe and Mejia, 1974; Jones et al., 1979). However the key point when statistically analysing rainfall estimation errors is to quantify the spatial organisation of the rainfields through a spatial correlation function. In that respect, variograms are an efficient tool since they allow the filtering of the field mean which is not always accurately known. In addition, the more sophisticated formalism of generalised covariance functions allow for dealing with fields that are not first order stationary. This is an interesting possibility when the rainfall process is known to present some trend, as the south-north gradient of the interannual rainfall in the Sahel (Lebel et al. (1992) have found an average gradient of 1 mm km -I in the region of Niamey). A good review of the use of geostatistics in rainfall analysis may be found either in Obled (1986) or in Bacchi and Kottegoda (1995). The reader is referred to those papers (other relevant references are also provided in the course of the text) and we will not enter into theoretical considerations when using here the most classical techniques of kriging (that is, variogram inference and interpolation). This is the case for Sections 2 and 3, devoted to the analysis of the event rainfield variograms and the assessment of event rainfall estimation errors. In Section 4, we deal with a question which, at least to our knowledge, has been little considered in the rainfall-geostatistics literature and which is of great importance for both water balance studies and remote sensing validation in the

T. Lebel, L Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

99

Sahel: how to relate the spatial organisation of the accumulation of N event rainfields to that of a single event rainfield. For instance it is generally considered (e.g. Dugdale et al., 1991) that some satellites can provide meaningful estimates at best over 10 days, and more securely over 1 month, which is equivalent to the accumulation of a few to a few tens of rainfall events. Also, a pending interrogation regarding the water balance and vegetation dynamics of the Sahel is how the soil moisture content and the vegetation respond to the rainfall distribution over time scales ranging from the event scales to the number of months comprising the rainy season. In all these applications, what is needed is a proper modelling of the rainfall distribution at both the event scale (because it conditions the rapid dynamics following the rain) and at the N-event scale (because it tells how the strong local gradients observed at the event scale may possibly be smoothed out when aggregating several events). The theoretical link between these two scales is investigated in Section 4, and its implications for the structure of the Sahelian seasonal rainfields are presented in Section 5, before giving some conclusions and perspectives in Section 6.

2. Spatial structure of the event raintieids 2.1. Notation The point cumulative event rainfall is denoted H. The ith realisation of the corresponding random process (rainy event) is denoted Hi. This realisation is sampled at a number K of raingauge stations, giving the measurements Hik. The locations of station k in the 2D space is denoted Xk. The spatial mean and standard deviation of the sampling of the ith rainy event are denoted H i and si. The spatial structure of each rainy event is characterised by its variogram Yi. The scaled variogram is defined as:

• 7 =~i/s~

~1)

The scaled climatological variogram, -y~, is then defined as an average over N rainy events: 1 N • =E* "tn = -~ i Yi

(2)

2.2. The variograms of the E P S A T - N i g e r event rainfields The EPSAT-Niger setup was designed with the idea that a robust inference of the variogram was a critical issue for a proper assessment of rainfall estimation errors, as pointed out by, among others, Delhomme (1978), Creutin and Obled (1982), Bastin et al. (1984), Lebel et al. (1987). Therefore, in addition to the roughly regular mesoscale network covering the entire 1° x 1° square (DS, 12 000 km 2) a target area was selected in the centre of the study area, characterised by an increasing ralngauge density from its borders towards its centre. This target area coincides with the Central Supersite (CSS, 750 km 2) which is the combination of the West Central Supersite (WCSS, 225 km 2) and the East

lO0

T. Lebel, L Le Barb6/Journal of Hydrology 188-189 (1997) 97-122

1992 100

!

EPSAT-Niger

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I

80

,

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100

,

I

120

i

140

(km)

Fig. 1. Distribution of the number of gauge couples in each distance class of equal length (! km), when considering the total network (mesoscaleplus supersites). Central Supersite (ECSS, 400 kin2). At the centre, the gauge spacing is 1 km. This configuration allows for a more even distribution of the pair of gauges into the distance classes used to infer the variogram (Fig. 1). The network is shown in Fig. 1 of Lebel et al. (1997). The mesoscale network numbers 72 stations, while the denser network covering the CSS numbers 28 stations. The EPSAT-Niger study area is in fact larger than the DS, an additional ten gauges being used to cover 4000 k m 2 located to the west of the H - S square. This means that the entire mesoscale network is made of 82 raingauges and covers 16 000 km 2. Following the climatological approach proposed by Bastin et al. (1984) and Lebel and Bastin (1985) which allows the computation of an average structure function for several realisations of the same random process, the climatological variograms of the E - N event rainfields have been computed for each of the 3 years 1990-1992 (Fig. 2). The mesoscale network has been reduced to 31 stations in 1993 and the pattern of the CSS network has been totally changed, so that the 1993 variogram can not be compared with those of the previous years. The three climatological variograms are characterised by two main features: (i) a nugget effect on the order of 10-15% of the field variance; (ii) a correlation length of about 30 kin. Obviously, these values reflect both some intrinsic properties of these rainfields and the sampling characteristics of the measurement network. Since the HAPEX-Sahel study area is rather flat and the E - N network regularly covers space scales ranging from a few kilometres to slightly more than 100 km, neither the orographic nor the

101

1". Lebel, L Le Barb6/Journal of Hydrology 188-189 (1997) 97-122

Climatological 1,4

~

il "i

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~

' l,

o

li.

:

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0,2 ~

........

1990

0,0

................. 1992

(o >

~ ~

0

20

40

60 Distance

80

100

120

140

(kin)

Fig. 2. Climatological variograms of the event rainfields for the 3 years 1990-1992 (events for which more than 70% of the operating raingauges reported rainfall). Note the similarity between the correlation lengths of the three variograms which are all around 30 km.

irregular sampling effects are a concern. Rather, the two main points likely to influence the parameters of the E - N variograms are the scale of observation on the one hand, and the statistical homogeneity of the fields on the other hand. After 3 years of observation it was realised that the small structures associated with convective cells had very little chance of being observed in the CSS. Their size is on the order of 1-2 km, a value which is matched by the station spacing of the CSS network only at its very centre, where four stations are installed on the four corners of a 1 km 2 square. The high probability of having convective cells undetected by the E - N network likely explains much of the nugget effect observed on the climatological variograms of Fig. 2. The other end of the E - N observation scale spectrum is characterised by a 'window' effect: structures beyond 100 km are not detectable either, which entails the impossibility of observing an entire rainfall system. These limitations cannot be addressed further unless resorting to other sources of data (radar for small scales; satellite for large scales). The ongoing work on the E - N radar data, especially in order to account for the attenuation and sampling effects (Benichou et al., 1995), shows how difficult it is to retrieve the statistical properties of the 'true' rainfields from the reflectivity fields (a similar diagnostic is delivered by Krajewski et al. (1995) using a simulation approach). Some time will thus be needed before the radar data can provide the required complementary information for studying the small scale structures revealed by the nugget on the climatological variograms of Fig. 2.

T. Lebel, L. Le Barb(/Journal of Itydrology 188-189 (1997) 97-122

102

2.3. How far may classification help in the identification of the climatological variogram ? The second point to consider in the variogram analysis is the statistical homogeneity of the events chosen to compute the climatological variograms. The Sahelian rainfall is produced by similar meteorological patterns all through the rainy season. It is mostly convective, even though a stratiform anvil develops at the rear of the squall lines. Three main types of convective systems have been identified by Desbois et al. (1988) on satellite images: local convective systems, moving organised convective systems, and squall lines. The latter two are grouped here under the term 'Sahelian Mesoscale Convective Systems' (SCMS). To date, no meteorological criteria have been found to objectively differentiate between those three types of rainfall systems. However, based on the study of Amani et al. (1996), Lebel et al. (1997) make a link between the type of rainfall system and the associated rainfield intermittency, as measured by the percentage of rainy stations. Using the Minimum Percentage of Rainy Stations (MPRS) as a criterion to build subsets of rain events, the influence of the intermittency on the climatological variogram has been studied. Originally a value of MPRS equal to 70% (that is, all events with at least 70% of operating stations reporting rainfall) was chosen to minimise the window effect mentioned above as well as to eliminate local convective events, characterised by a high degree of intermittency. Such events represent more than 80% of the total seasonal rainfall and 60% of the total number of rainy events (Table 1). They constitute the subset that was used to compute the variograms of Fig. 2. Then, alternate values of MPRS have been considered, ranging from 30% to 90%. Increasing MPRS leads to a decrease of the correlation length of the climatological variogram (Table 1 and Fig. 3). In fact, for highly intermittent events, the cross covariance between rainy and non-rainy areas adds a linear component to the variogram of the rainy areas and the identification of a meaningful correlation length becomes difficult since the sill tends to disappear. An attempt was also made at using the classification proposed by Amani et al. (1996). The events of each group produced by this classification have significantly different point statistics (Table 2), which may influence their spatial structure as well. One point especially worth noting in Table 2 is that the events of group 3 are characterised by an average probability of zero rainfall larger than 30% (that is a MPRS lower than 70%). When the Table l Event rainfall statistics for the groups of events characterised respectively by over 70% and over 90% of operating stations recording rainfall

1990 1991 1992

More than 70% of raingauges recording rainfall

More than 90% of raingauges recording rainfall

Number of events (% of the seasonal total)

Rainfall (% of the seasonal rainfall)

Average correlation length

Number of events (% of the seasonal total)

Rainfall (% of the seasonal rainfall)

Average correlation length

60 72 60

79 85 84

25 28 28

32 39 33

57 61 60

22 20 24

103

7". Lebel, L. Le BarbO/Journal of Hydrology 188-189 (1997) 97-122

Climatological

Variogram

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40

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>= 5 0 % >= 30%

i

S0

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100

I

120

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Distance (km) Fig. 3. Climatologicalvariograms (1992) for the events where at least 70% of the operating stations reported rainfall and those where at least 90% of the operating stations reported rainfall. events of groups 1 and 2 are pooled together the average probability of zero rainfall is smaller than 15% (Table 3) which may be considered as acceptable for a geostatistical approach. The climatological variograms computed separately for each group (Fig. 4) exhibit a similar behaviour for distances below 50 kin. The linear trend on the variogram of group 2 at distances larger than 50 krn is attributable to the average intermittency of this group being higher (25%) than that of group 1 (5%). Altogether the statistics of the MPRS = 70%-events (85) and that of the events (77) of the combined groups 1 - 2 of Amani et al. (1996) are almost the same (Table 3). Therefore, in the following, the variogram analysis will be limited to the MPRS = 70%-events (hereafter referred to as the 'mesoscale events'), the MPRS criterion being easier to compute than A m a n i ' s criterion and less sensitive to the sampling procedure. The climatological variogram of the mesoscale events for the 3 years is characterised by Table 2 Event rainfallstatistics(conditionalmean and standarddeviation;probabilityof zero rainfall:F0) for each group of the classificationproposed by Amani et al. (1996) Group 1

1990 1991 1992 1990-1992

Group 2

Group 3

Conditional (H > 0) mean; SD (mm)

F0(%)

Conditional (H > 0) mean; SD (ram)

F0(%)

Conditional (H > 0) mean; SD (mm)

F0(%)

20.2; 11.3 20.3; 11.6 23.5; 10.8 21.5; 11.3

8 3 5 5

13.2; 12.1 12.8; 10.5 13.6; 9.6 13.2; 10.6

28 21 27 25

10.3; 10.6 8.3; 9.6 8.6; 9.5 9.0; 9.8

38 45 35 41

104

T. Lebel, L. Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

Climatological !

variogram

I

1992 I

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40

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I

80

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Fig. 4. Climatological variograms (1992) for the events of groups 1 and 2 of the classification proposed by Amani et al. (1996).

the following features: (i) a nugget on the order of 10-15% of the field variance; (ii) a correlation length on the order of 30-40 km; (iii) a sill slightly above its expected value of 1.0 (this fact was already noticed by Lebel and Bastin (1985) working on hourly rainfields in southeast France). A search for possible anisotropy led to the conclusion that the Sahelian event rainfields can be considered as isotropic. Again this is a diagnostic based on the E - N sampling only. Were the latitude range of the study area wider, both anisotropy and another correlation length could have appeared. The important point to recall is that the variogram identified is valid over the range of distances that will be used Table 3 Rainfall statistics (1990-1992) of the mesoscale events, as defined by Amani et al. (1996) (combination of groups 1 and 2), or by the MPRS = 70% criterion Number of events

Amani 77 MPRS = 70% 85

F0

0.13 0.15

Non-conditional

Conditional (H > 0)

Mean

SD

Mean

SD

13.8 12.9

10.7 11.6

15.9 15.2

10.8 11.9

F0 is the probability of zero rainfall. The total number of events over the period is 133. Mean and standard deviation (SD) are in miilimetres.

T. Lebel, L. Le Barb~/Journal of H.vdrology 188-189 (1997) 97-122

105

in the kriging estimation procedure. The identified correlation length characterises the clustering scale. The nugget may be seen as the price to pay for the uncertainty linked to the inability of the E - N network to detect convective cells. The effect of the nugget on the estimation error is visible in Fig. 7, further presented in Section 3. A last important question regarding the use of this climatological variogram for assessing the event rainfall estimation errors is related to the dispersion of the individual event variograms around their average. This dispersion is illustrated in Fig. 5 for 1992 (30 events having recorded more than 70% of non-zero rainfall). This dispersion is quite large and for a given event one would be best advised to use the variogram of that particular event rather than the climatological variogram. Nevertheless, the climatological variogram remains one of the simplest and most synthetic tools in our possession to assess the average errors involved when estimating the event rainfall in HAPEX- Sahel. One should simply keep in mind that in the worst cases, the event rainfields display a white noise pattern. In such cases, the standard deviation of the point estimation error is that of the rainfield itself.

3. Event rainfall estimation errors

Rainfall estimation errors are a function of the intensity and degree of organisation of the rainfields, of the density and geometry of the network used for the estimation and of the size of the area over which the estimation is needed. Since the work of Huff (1970), several papers have been devoted to assessing areal rainfall estimation errors for a variety of aggregation scales and climates. Lebel et al. (1987) compute scaled standard deviations of estimation errors for time aggregation scales ranging from 1 h to 1 day. Seed and Austin (1990) work on daily and monthly rainfall. More recently, Peters-Lidard and Wood (1994) assess the uncertainty in estimating areal precipitation for single storm events recorded during FIFE and MAC-HYDRO '90. For that purpose, they use a conceptual rainfall simulation model. Correlation lengths are computed on a case by case basis, by running the rainfall simulator for each observed storm. Parameters of the model account for both the convective cells and the clusters. However, because of the small size of the experimental sites, the authors acknowledge that the computed correlation lengths are representative of the rain cell radius, rather than of the clustering scale. Among the 19 storms studied for FIFE, three have a correlation length lower than 1 km and only one larger than 1.5 km. The study by Peters-Lidard and Wood (1994) confirms that convective cells are responsible for important rainfall gradients at the 1 km scale. Such gradients could be detected by the E - N network only by chance, when one cell develops over the very centre of the CSS. On the other hand, the supersites and the mesoscale areas (DS, E - N study area) are large enough to include several convective cells so that the averaging effect diminishes the influence of the local gradients on the estimation accuracy over such areas. A priori, the E - N network was appropriately designed to sample the variability (essentially linked to the clustering of the convective cells) at such scales. An example of how the rainfall estimate over the E - N study area (16 000 km 2) converges towards its 'true' value, when the number of stations is increased, is given in Fig. 6 for the squall line of 21/08/1992,

T. Lebel, L Le Barbd/Journal of Hydrology 188-189 (1997) 97-122

106

sI ~

//

..

el

s

°'1

t i

i

i

r

i

i

i

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i

10

20

30

40

50

60

70

80

!

90

dl stance (km) Fig. 5. The individual scaled event variograms for 1990 (events with more than 70% of stations recording rainfall).

21108/92 100

80

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4)

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4o

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20

Ul

0

0

10

20

30

40

Number

50

60

70

80

90

100

of stations

Fig. 6. Rainfall estimate for the event of 25/08/1992 as a function of the number of stations used for the estimation. The 'true' average for the event is 45.2 mm and the standard deviation is 15.8 mm (based on the measurements of the mesoscale network only).

T. Lebel, L Le Barbd/Journal of Hydrology 188-189 (1997) 97-122

107

during which all the operating stations recorded rainfall. The mean raindepth over the 82 stations of the mesoscale network is 45.2 mm and the standard deviation is 15.8 mm. The average over the E - N study area, estimated by kriging of the 82 stations, is 44.6 mm. Adding the values recorded at the CSS stations not belonging to the mesoscale network change this average by less than 0.5 ram. In order to study how the estimation errors evolve with the number K' (1 --< K' <-- 82) of stations used in the estimation procedure, random subsarnples of the mesoscale network are created. Since the total number of possible subsamples, Nr,, becomes rapidly untractable (88 560 subsamples for K' = 3 and more than 150 x 1012 subsamples for K' = 10), a selection of 'intelligent' networks is performed by imposing a minimum distance, d(K'), between the stations of the simulated networks: d(K')= 2V -~-7,

(3)

where A EN is the area of the EPSAT-Niger study zone. When a maximum of 5000 networks is obtained, the selection process is stopped. The standard deviation of the estimates produced by all these networks, is then computed for each value of K'. In that particular instance it appears that the estimate standard deviation is approximately a linear function of (I/K')I~, at least up to K' = 25-30 (station spacing of 20-25 kin). The resulting curves 'Mean + St.Dev.' and 'Mean - St.Dev.' are plotted in Fig. 6, along with the absolute minimum and maximum. A minimum of ten stations is required in order to get a standard deviation on the order of 5 ram, or 10% of the average. However, with such a density, the range between the extremes remains high (maximum estimate: 71 mm; minimum estimate: 19 mm). This range decreases slowly, compared with what is observed for the standard deviation, when the number of stations increases. The scaled kriging variance, derived from a scaled climatological variogram which is event-independent, provides a more general assessment of the average estimation uncertainty. This scaled climatological variogram 0¢*represents the average spatial variability of the mesoscale events, and the scaled estimation error variance, valid for a given network of K stations over a given area A is computed as: K

K

K

°e2(A) =2 ~ )~k'Y~- y, ~ hkhk"Y~'--)'~tA, k=l

(4)

k=lk'=l

where:

L

"/ka =

"LIA

'y*(X~,X)dX, and 'YAA =

"?*(X,X')dXdX'

(5)

are computed numerically. The weighting factors {~,k, )~k'} are the kriging coefficients. The appropriate error variance for a given event i, conditioned by the requirement of having reported more than 70% of rainy stations, is then computed as: K

K

°2e(A,i) =2 ~, ~k~.~ -- ~, k=l

K

~, Xk)W)'ia, --~/i,.,,

k=lk'=l

(6)

108

T. Lebel,

L Le BarbtVJoumal

of Hydrology

HAPEX-Sahel:

oscale network

1.

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0

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188-189

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(1997)

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km2

1

T. Lebel, L Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

109

where: ~=s~'3~*,

(7)

is used as an estimate of the event variogram %. Combining Eq. (5), Eq. (6) and Eq. (7) yields:

oe(A, i)=si'o*e(A ),

(8)

The standard deviation of estimation error, ae(A, i), is thus computed as the product of the scaled standard deviation of estimation error, oe(A), (which is computed once and for all for a given measurement configuration) and of the experimental standard deviation of the event rainfall considered, s~. However, sampling limitations have a great effect on the estimation of si. Ideally s~ should be computed from a sample of Kindependent realisations of Hi, K being at least equal to 30. With a decorrelation length on the order of 20-30 kin, A must be larger than 10 000 knl 2 to meet such a requirement. Accordingly, even when one is interested in rainfall estimation over small to medium areas (10-1000 kin2), a larger view of the rainfield is needed to assess its overall variability and compute s~. Provided the ability of estimating si,, a~(A) gives access to the computation of ae(A, i) and subsequently to the relative estimation uncertainty, e(A,i) as:

si.cre(a )

e(A, i)= - -

(9)

Hi(a) '

The curves of Fig. 7 show the variation of a~ (A) as a function of the network density for the supersites and the mesoscale areas. Starting from a gauge spacing of 30 krn, the value of o*~(A) over the supersites decreases rapidly when the network density increases until a critical density is reached, beyond which a*~(A) converges slowly towards an asymptote. This critical density varies from about 50 gauges per 1000 km 2 for the Southern supersite (SSS, 100 km 2) to 25 for the ECSS (400 krn 2) and 15 for the CSS (750 kin2). With an average density of about seven gauges per 1000 krn 2 the E - N mesoscale network is below the critical density for all the supersites. This justifies a posteriori the densifying of the E - N network that was carried out over the CSS (37 gauges per 1000 km 2) and over the SSS (60 gauges per 1000 kin2). Note, however, that even above this critical density o*~(A) remains relatively high. This is especially true over the SSS: oe(SSS) = 15% for a density of 70 gauges per 100 km 2. At the DS scale, the E - N mesoscale network appears to be dense enough, o~(A) being less than 5%. This motivated the suppression of one station out of two for the long term rainfall monitoring period that started in 1993. With 31 stations over 12000 kna 2 (or 25 stations per 10 000 km 1) this network gives a value of ae(DS) equal to 10%. The influence of the nugget on a*e(A) is also shown in Fig. 7 for the supersites. The 15% nugget produces an increase of o'_(A) ranging from 2 to 6%. The curves of Fig. 7 provide a global and synthetic view of the average uncertainty Fig. 7. Scaled standard deviation of estimation error as a function of the networkdensity for the supersites(top) and the mesoscale areas (bottom). The climatological variogramused for the computation is an exponential model with a shapeparameterof 10 km and a nuggetof 15% of the rainfieldvariance.Curvesare alsodrawn for a zero nugget (supersites only).

110

T. Lebel, L. Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

EPSAT-Niger

mesoscale

events

1990-1992

40

A

v

E E

C 0 m

30

II > Q "0 im

"0 !._

20

"13

=Je..

B []

G II

C 0

Im 4~ im

~[]~r~

10

[]

~B

m"

[]

[] [] []

C 0

0

•

0

0

10 Conditional

20 mean

Observed Y: 1,02"X0,8

30 event

40 rainfall

50

(mm)

Fig. 8. Standard deviation of the event rainfall versus the mean for events with more than 70% of raingauges reporting rainfall (85 events for the years 1990-1992).

attached to the event rainfall estimation for a given combination of estimation area and network density. The main advantage of such a representation is that it is event-independent, and that the relative influences of the density and of the area size may be assessed from a single graph. As such it provides a guidance for network design in the Sahel. However, most people using E - N estimates will be looking for an assessment of e(A,i) rather than o*~(A).This requires the additional determination of si and Hi(A) on an event by event basis. From expression Eq. (9) and the curves of Fig. 7 an estimate of e(A,i) may then be computed. Naturally it would also be convenient to be able to obtain standard values of e(A,i). Obviously this is possible only for rainy events whose structure is not too different from the average structure represented by the climatological variogram. In such cases there is a close relationship between si and H,(A), as shown in Fig. 8. At the DS scale, s~ is

T. Lebel, L Le Barb(/Joumal of Hydrology 188-189 (1997) 97-122

llllreldralnlrldle e ~

1I 1

Arid m i n l l ~ U m ~ i o n em~. ECSS 1400 kml)

error:SSS 1100 kml)

I

I

1

,I Gauge _~,~__'.ng(ig~)

Amal r l i n t t~

I

I

lm~on I

i"airlfiill lillllnlilion ~

en~r: C ~ O'SOkn'l) I

/

I

i

I

I

i

i ~auge ~,~__'_ng _ (kin)

Ii

Gau~ spacing (ion)

Fig. 9. Average rainfall estimation uncertainty (in percent of the areal rainfall) over the supersites and the 1° x 1° square as a function of the gauge spacing of a regular network and of the event rainfall recorded over the I ° x 1° square. The gauge spacing of the E - N mesoscale network is 12.5 km and the average gauge spacing over the ECSS (irregular network) is 4.5 km. Isovalues are every 5%.

proportional to Hi raised at the power 0.8:

si = 1.05H°8 (n=85; r z =0.84)

(10)

Combining Eq. (9) and Eq. (10) yields: e(A, i)= 1.05o*_(A).Hi(A) -°'2,

(11)

This expression gives the expected value e(A, i) of e(A,i) when H,(A) is close to the field average. Isovalues of e(A, i) in the coordinates (density, HD are given in Fig. 9 for the DS and the supersites. Reducing the estimation error to less than 10% on the supersites requires a gauge spacing of less than 2 km for the SSS, whatever the rainfall over the site, and of between 2.5 km (low rainfall) and 3.5 km (heavy rainfall) for the ECSS. Note that even with the dense CSS network (average gauge spacing of 4.5 km over the ECSS),

T. Lebel, L. Le Barbt/Journal of Hydrology 188-189 (1997) 97-122

112

the average estimation uncertainty over the ECSS is 15% for medium rainfall (20-40 mm) and still more than 10% for heavy rainfall (100 mm). The two main factors influencing the estimation errors at the supersite scale are probably the nugget, on the one hand, and, on the other, the possibility of having a supersite rainfall markedly different from the DS rainfall. Significant departures from the average e(A, i) may be observed over small areas since the smaller A is, the greater chances are that Hi(A) be significantly smaller or larger than the field average. Therefore, the uncertainties computed for large event rainfall over small areas are more likely to be underestimated. The influence of the nugget effect in the estimation error diminishes significantly only when a sufficient number of stations (on the order of ten) are present on the estimation area. By contrast to the potentially significant errors on the supersites, the uncertainty over the DS is small (less than 5% whatever the rainfall) with the mesoscale network and remains so (below 5%) for gauge spacings up to 30 km.

4. Spatial structure for an accumulation of event rainfields

4.1. The high spatial variability of the Sahelian seasonal rainfall The high spatial variability of the seasonal rainfall over the H - S square has already been noted in the companion paper of Lebel et al. (1997). In 1992, the maximum seasonal point rainfall (782 mm) recorded on the Southern Super~ite (SSS) was 54% larger than the minimum (507 mm), the distance between the corresponding two stations being inferior to 10 kin. Having in mind that the seasonal rainfall is the accumulation of several dozen of rainy events, a large smoothing out of the variability observed at the event scale could have been expected. Another sign of the high variability of the seasonal rainfall is the low correlation length identified by Taupin et al. (1993a). It ranges from 20 to 40 km for the three years 19891991. These values are close to those found at the event scale, which contradicts the current belief of a progressive increase of the correlation length with the cumulation time. Since rainfall estimates are needed not only at the event scale, or less, but also over time scales ranging from a few days to the whole rainy season, it is important to understand the relationship between the correlation lengths observed at the event scale and those observed for an accumulation of events.

4.2. Derivation of the variogram of a cumulation of event rainfields The cumulative raindepth Zk at station k over a given period (10 days, 1 month, the rainy season) is the sum of N event raindepths:

Zk=

~, Nki,

i=l,N

(12)

where N is a random variable which can be assumed to be independent of H. The period and area of integration are chosen so that H may be considered as a stationary variable,

T. Lebel, L. Le Barb~lJournal of Hydrology 188-189 (1997) 97-122

113

both in time and space. The variograrn is then expressed as:

~(Z2,ZI)= ~E[(Z-~2-ZI)2] : ~E

j

~ H2j- ,~=I Hli "=

(13)

Assume that there are a random number NI2 of events (NI2 ~ N1 and N~2 --< N2) that hit both points XI and X2. Denoting the remaining events that hit Xl only as N'~ (N' ~= N, N~z) and the remaining events that hit X2 only as N'2 (N'z = N2 - N12) we have: F( N,2

N' 2

N'1

Nt2

~21

i~HliNI2

~21

(14) N' z

Each of the three terms of the right hand side will now be treated separately by writing: E[(Zz - Z l )2] =Ji +Jz + 2J3

(16)

Jl = E

(IV)

H2j - ,~ H,i

Jl-

[ [,j=I)'=I

1-12jH2j'+ i=l• i'=l ~" l-llil-lli'-2iZjZl'12jl'lli~l=l:l

]]

(18)

Assuming the rain events to be independent, we have:

E[HkiHk,j] =6ijE[Hki.Hk,j], 6ij= 1 if i=j, 6ij =0 otherwise

(19)

Since the random variable Nl2 is independent of both HI and HE, it becomes:

JI = N,2E[H~ + H2] - 2NtEE[H2"H,]

(20)

JI =Nj2"2~(HI, H2)

(21)

or:

{ J2 =

N" ~

j' = 1

Jz =E

H2j'-

i'=1

}2 Hw

(22)

]~ HzjH2j, + ~, ~, HjiHli,-2 ~ Z H2jHli

L [,j=Ij'=I

i=li'=l

i=lj=l

(23)

T. Lebel,I.. Le Barb~/Journalof Hydrology 188-189 (1997)97-122

114

J2 = (N'l + N'2)E[H 2] + (N' I ).(N' j - 1). {E[H] }2 + (N'2)-(N' 2 - 1). {E[H] }2 - 2(N'2)'(N' i ). {E[H] }2

(24)

J2 = (N'l + N'2)Var[H] + { N'~ + N'~ - 2N'2N'I } {EtH] }2

(25)

J2 = (Nl + N2 - 2Nl2)Var[n] + {'y(N1, N2) } {E[H] }2

(26)

J3 = E L~j~=In2j- i~=leli

j,~=l n2j- i,~=l eli

/]

(27)

rfNi2 N'2

NI2 N'l

Ni2 N'l

Nt2 N'2

Ll, jflj'=l

i=li'=l

j = l i'=1

i

"~]

j'

(28) Since all the cross-products HkcHkj and Hki'Hk7 of the above expression are rain depths corresponding to different and independent events, we have:

e [nkinkj ] = e [nkink,j]

=

(E[H]) 2

(29)

and Eq. (28) becomes: •I3 = {E[H]}2"{NI2(N2-NI2)+NI2(NI -NI2)-NI2(NI -NI2)-N12(N2 -Nl2)} =0 (30) Finally, by combining expressions Eq. (13), Eq. (16), Eq. (21), Eq. (26) and Eq. (30), we obtain the expression of the variogram of the cumulative rainfall (or N-event variogram) as:

'Y(Zl,Z2)=Nl2"7(Hl, HE) + {(Nl + N2)/2 - Nl2 } Var[H] + {qt(Nl,N2) } {E[H] }2 (31) Under the hypothesis of an exponential distribution of the event rainfall (see Lebel et al., 1997), Var(H) = {E(H)} 2, and the N-event variogram, ~m ='t(ZI, Z2), may be written as: ~'Ne =NIE"Ye + m2{(N1 +NE)/2-NI2 +'YN}

(32)

where: mE = {E(H)} 2,

(33)

-ye=3,(HI,HD, and

(34)

"/N =~(NI,N2),

(35)

The N-event variogram is thus the sum of two terms: ~/'e = Nl2"Ye,

(36)

"Y'N =mE {(NI +N2)/2-NI2 +3'u }

(37)

and:

T. Lebel, L Le Barb6/Journal of Hydrology 188-189 (1997) 97-122

115

Since the Nl2 events are independent, the correlation range of 7' e is equal to that of 7e. The term 7'U is thUS the one that can change the shape of the global N-event variogram. The events considered in the analysis are such as MPRS = 70%, which mean that they hit a large portion, if not all, of the DS. Consequently, the differences between N1, N2 and Ni2 are generally small when the distance h is smaller than the correlation range of the event rainfall, r,. Being rn, on the order of 30 kin, that is one-third of the DS side, the term {(N~ + N2)/2 - N12} remains close to 0 for such distances. The expression of 7m may thus be written as: 7Ue(h) ~ 7'e(h) ~ NI2-'Ye(h), for h <- rH

(38)

For h > rH, "y~ asymptotically tends to its sill, the value of which is Var(H)( = m2). The expression of 3'Ne(h) then becomes: 7 m ( h ) = m 2 { ( N i +N2)/2+TN}, for h > rH

(39)

While expression Eq. (38) is only an approximation that requires 3"N to be small compared with 3/¢, expression Eq. (39) is exact and shows that, for h > rn, the behaviour of the N-event variogram is only dependent on the spatial distribution of the number of rainy events.

5. Rainfall estimation at the monthly and seasonal scales 5.1. The MCS seasonal variogram

One interesting consequence of expressions Eq. (38) and Eq. (39) is that one needs only to infer the event climatological variogram and the distribution of the MCS's in space to obtain the N-event variogram. While inferring 3'e requires a dense ralngauge network, such as EPSAT-Niger's, it need not cover a very large area (a 1° x 1° square seems appropriate in that respect). On the other hand, counting and delimiting the MCS's may be carried out with satellite data. Thus, a combined use of those two sources of data, based on their complementary sampling characteristics, could be of great help in modelling the spatial structure of the Sahelian rainfields for accumulation periods ranging from the event to the season. An experimental verification of expressions Eq. (38) and Eq. (39) has been carried out by building what may be termed as a 'MCS seasonal rainfield', defined as the accumulation of all the mesoscale event rainfields observed during one rainy season. As may be seen from Table 1, the MCS seasonal rainfall accounts for 80-85% of the total seasonal rainfall. The variograms of the MCS seasonal rainfields display a nested structure with a zone of slow increase between 30 and 50 km. In 1991, for instance, the value of the experimental variogram rises from 4500 InlTl 2 to 5500 mm 2 between 25 and 50 kin, then linearly increases to above 10000 mm 2 for distances larger than 70 km (Fig. 10). This sill corresponds to the transition zone between a domain where the approximation Eq. (38) is valid (h < 30 kin) and a domain where the variogram of the number of events can no longer be neglected in expression Eq. (37). In 1991, 35 mesoscale events were counted over the E - N study area. According to expression Eq. (38), the value of the seasonal variogram at the

116

T. Lebel, L. Le Barbg/Journal o f Hydrology 188-189 (I997) 97-122 i

|

w

i~o "0

9

es

;I A

,~O

E E

/

•

:

: ,

:

b

v

E

p

I

m L_

l

0

I

I

fi

m L_

/ : :

:~, a O

P : :'.

94 C

I~

o m m 0 w

0

~

2000

0

Model (7oo; 4,s40; to)

.d oOJ] . . . . •o. ....

i

0

I

I

20

40 Distance

i

Observed

I

60

I

I

80

oo

(km)

Fig. 10. Experimental and theoretical (computed by combining expressions Eq. (38) and Eq. (42)) variogram of the 'mesoscale seasonal rainfall' in 1991. The theoretical variogram is of an exponential type (nugget, 700 ram2; sill, 4340 mm2; shape parameter, l0 km).

beginning of the sill should be equal to the sill of "Y'e: 0r2('y'e) = 35Var(H)

(40)

Taking Var(H) ~ 144 m m 2 from Table 3 yields an estimated sill slightly above 5000 mm 2, which corresponds well to the sill observed in Fig. I0. The event unscaled climatological variogram may be defined as:

"Ye= Var(H ).'y*

(41)

or, taking the scaled climatological variogram of 1991, shown in Fig. 2: "ye(h)= 19+ 13511 - exp(1 - h / 1 0 ) ]

(42)

The theoretical mesoscale seasonal variogram resulting from the combination of expressions Eq. (38) and Eq. (42) is then: "ge(h) = 700 + 4350[ 1 - exp(1 - h~ 10)], for h < 30 km

(43)

This variogram has been superposed on the experimental variogram in Fig. 10. The comparison of the two variograms shows that the scale parameters (nugget and sill) are

T. Lebel, L Le Barbd/Journal of Hydrology 188-189 (1997) 97-122

117

similar, but that the shapes are markedly different, with much lower values of the experimental variogram than those anticipated from the theoretical one. It remains that, as far as estimating the overall variability of the seasonal rainfields is concerned, two main conclusions may be derived from this comparison: (i) for distances below 30-40 km, the variogram of the number of events has little influence on the seasonal rainfield variogram (the approximation of expression Eq. (38) gives a good fit to the experimental variogram); (ii) the scale parameters of the seasonal rainfield variogram may be deduced from those of the event climatological variogram by counting the number of MCS' s hitting the region of interest during the rainy season. The variogram of the MCS seasonal rainfield appears to synthesise various statistics that were obtained independently of the variogram inference itself, which validates the decomposition proposed for the seasonal variogram at distances below the event rainfall decorrelation range. To complete the validation of the nesting representation proposed above in expression Eq. (31), the estimation of-tN is needed. This cannot be done from the EPSAT-Niger data only and will require some further work to incorporate some information on the distribution of the MCS' s in space. Such an information may be reached by analysing regional data sets, such as the one used by Le Barb6 and Lebel (1997).

5.2. Monthly and seasonal rainfall estimation accuracy Assessing the accuracy of the monthly or seasonal rainfall ground truth is of prime importance in validation studies, especially when validating satellite estimates or GCM outputs. It is generally taken for granted that, while satellite and GCM estimates are not so good at the event scale, their quality improves with the time step. Discrepancies between these estimates and the so called 'ground truth' could be partly attributed to the errors linked to the computation of this ground truth. Of course these errors are best assessed on a case by case basis. However, it is important in many situations to get an estimate of the average ground truth uncertainty for a given type of rainfall system, since it is rare that all the data required for a fine assessment of the areal rainfall estimation errors are available. Such average errors may also be used as guidance for network design or in retrospective climate studies. The average uncertainty attached to the estimation of the areal rainfall over a 1° × 1° square in the Sahel have thus been computed for regular networks with a gauge spacing ranging from 10 to 100 km and three time scales: the event, the month (June and August were chosen as typical of the fringe and core of the rainy season, respectively) and the season. For these computations the average event rainfall given in Table 3 (15.5 ram) has been considered. Of course, at the event scale there is a strong probability for the event rainfall to be significantly different from this average. However, Fig. 9 provides the relevant curves to account for this effect. The larger the time scale, the greater the number of mesoscale events (N) will be. Also greater will be the probability that the average of the events recorded during that period approaches the population average. In order to determine an average value of N for a 'normal' month (Njun, Naug) or 'normal' season (Ns), a reduction factor of two-thirds, deduced from the statistics of Table 1 showing that about two-thirds of the rainy events are such as MPRS = 70%, has been applied to the average number of events determined by Le Barb6 and Lebel (1997). With an average number of

118

T. Lebel, L, Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

Rainfall

estimation

uncertainty

over

the

DS

30 = 25

t" m

,/I

Event

mesoscale network

- ~

June

---.4---

Augu~

.... -g.---

Season

20

gO

I:

0 U

oC la

15

10

E

m

m

ILl

5

":-"-g---::::-::'-! 0 I.

0

20 Number

40 of

60

80

stations/10,000

100 km2

Fig. 11. Average estimation uncertainty of the areal rainfall estimation over the I ° x 1° square (in percent of the areal rainfall). Values are computed for an average event producing 15.5 mm of rain.

mesoscale events equal to 15 in August and 32 during the whole rainy season, the effect of the variability of the event rainfall onto the average uncertainty computed for these two durations is greatly reduced. In fact Le Barb6 and Lebel (1997) have shown that the main cause of the rainfall fluctuations in the Sahel at the yearly scale is the fluctuation in the number of rainfall events rather than in the mean event rainfall. Thus, departures from the values computed for 'normal' months or season in Fig. 11 are a function of the square root of the ratio between the observed number of events in a given month or season and the average number of events used in our computation. Fig. 11 shows that with the E - N mesoscale network the average uncertainty ranges from slightly above 5% at the event scale to less than 1% at the seasonal scale. With the reduced 1993 mesoscale network (31 gauges over the DS), the average uncertainty at the seasonal scale remains less than 2%. Of course this low uncertainty is largely due to the averaging effect over the 12000 km 2 area, since validation tests carried out on point rainfall estimates have shown an average uncertainty of 15% on the point seasonal rainfall with this same 1993 mesoscale network. The average uncertainty increases with the gauge spacing. It reaches 5 - 7 % at the monthly scale (3% at the seasonal scale) for a density of ten gauges per 10000 km 2 and 7 - 1 0 % at the monthly scale (4% at the seasonal scale) for a density of five gauges per 10000 km 2.

T. Lebel, L Le Barb(/Journal of Hydrology 188-189 (1997) 97-122

119

6. Conclusion and perspectives The main goal of this second paper of a series of two was to examine how accurate were the rainfall estimates computed over the HAPEX-Sahel supersites and mesoscale areas, using the EPSAT-Niger raingauge network. The convective cells making up the basic structure of the Sahelian Convective Mesoscale Systems were not resolved by the E - N network which was designed for the rainfall survey at the various mesoscales (or to 7). Consequently a global approach, based on a variogram analysis, was chosen rather than one based on a conceptual model of cell dynamics. The estimation uncertainty is kept at a reasonably low level when using the E - N mesoscale network (regular gauge spacing: 12.5 kin) for estimating the raindepth over the 1° x 1° square (DS). The event rainfall is estimated with an uncertainty of about 7% for a medium rainfall of 15 mm and of less than 5% for rainfall larger than 30 mm. With a gauge spacing of 30 Ion the estimation uncertainty is still below 10% for a wide range of event raindepths. Larger errors are encountered when averaging over the smaller areas of the supersites. Using the supersite networks (average gauge spacing: 4.5 kin) the event rainfall is estimated with an uncertainty of about 12% for a medium rainfall of 15 mm over the Central Supersite (CSS, 750 km 2) and of more than 10% for rainfall larger than 50 mm. The average uncertainty is still larger over the smaller East Central (ECSS, 400 km 2) and Southern (SSS, 100 km 2) supersites. Nevertheless, since the gauge spacing was smaller than the average supersite gauge spacing over the ECSS the estimation uncertainty there is not beyond 10% for the rainy events observed during the three HAPEX-Sahel years. Over the SSS, errors of 15% are likely to affect the rainfall estimates for several events. While such errors, especially over the DS and the CSS, are probably acceptable for most water and energy budget models, such figures show how delicate it is to obtain accurate rainfall estimates in the Sahel. The E - N network was a research network with a density far above the average density of the African operational networks. Very often only one, or at best, a few raingauges are available over a 1° x 1° square. For such configurations the event rainfall estimation uncertainty may reach 40%. The correlation length computed from the E - N data set is about 30 kin. The fact that good estimates are obtained over the DS derives logically from that point. The station spacing of a network should be on the order of one-half to one-third of the correlation length, which corresponds to the gauge spacing of the mesoscale network. On the other hand a minimum number of gauges (between five and ten) is required to provide a good estimate over any given area, even with an appropriate gauge spacing, so that an optimal gauge spacing of 15 km is appropriate only for estimation areas ranging from 1000 to 10000 km 2. Over smaller areas the density has to be increased: a gauge spacing of 2 km is required over a 100 km 2 area. This is due to the strong local gradients generated by the convective cells, the presence of which is detected on the event rainfall variogram through a nugget amounting to about 15% of the field variance. This effect will be further studied, using radar data collected between 1991 and 1993 and small scale gauge data collected in 1993 and 1994 (strong attenuation effects and calibration problems have prevented from using the radar data for quantitative purposes up to now). In a second step, longer durations (one month, the rainy season) have been considered.

120

T. Lebel, L Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

Rainfall over such durations is the accumulation of several (N) event rainfalls. Given the meteorological homogeneity of the Sahelian rainfall, there should be a link between the spatial structure of the event rainfall and that of the N-event rainfall, a fact rarely, if ever, considered in previous works. The theoretical link between the event variogram and the Nevent variogram has been established. The N-event variogram is shown to be the superposition of the event variogram multiplied by the square root of N and of the variogram of the number of events. The analysis of the seasonal variograms recorded during HAPEXSahel indicate that for distances below 50 km, the variogram of the number of events is negligible compared with the N-event variogram. Consequently the shape of the seasonal (or monthly) variogram is similar to that of the event variogram for such distances. In particular, a sill, or zone of low increase of the variogram, is observed between 30 and 50 km, which means that a correlation length of 30 km has to be considered in network design studies. Basically the estimation uncertainty for a N-event rainfield is that of the event rainfield divided by the square root of N. Thus the Sahelian seasonal (or monthly) rainfields are characterised by nested structures: (i) strong local gradients over distances smaller than 5 km; (ii) a first correlation plateau starting around or slightly above 30 km; (iii) larger structures that were not sampled properly by the EPSAT-Niger network. Those characteristic scales of variability provide the basis for designing a raingauge network according to the pursued goal of the study. They also should be taken into account in satellite rainfall estimation algorithms. Over a 1° x 1° square a network of ten gauges allows for an estimation uncertainty of less than 10% for the monthly rainfall and of less than 3% for the seasonal rainfall. Such precision is largely enough when validating satellite rainfall algorithms or GCM outputs. For areas down to 1000 km 2, or so, the same number of gauges (ten) is needed, meaning that as long as the gauge spacing is larger than half the decorrelation length the prime criterion to consider is the absolute number of gauges present in the surface of estimation rather than the network density. For still smaller areas the number of gauges in the surface of estimation can be assessed by imposing a gauge spacing of half the decorrelation length, that is one gauge every 15 kin. However, below 10 km 2 the spatial variability linked to the convective cells is no longer smoothed out by the spatial averaging and other rules would have to apply. At the opposite scale (areas of several tens of thousands of square kilometres), the spatial variability of the number of events becomes significant and will have to be considered in the future. The above results apply to the Sahelian rainfall in a fairly general sense. However there are a number of points that remain to be investigated. First the classification proposed to build what has been called a 'MCS seasonal rainfield' from homogeneous rainy events has probably an influence on the inferred spatial structure. Then, even when selecting rainy events with a limited probability of zero rainfall, there is still some residual intermittence in the rainfields which is not well accounted for by the traditional geostatistical approach. Following on the path opened by Brand et al. (1993), may help in this respect. Finally, the statistics used to compare the theoretical and observed seasonal variogram are spatial statistics computed from an array of autocorrelated values. The effect of this autocorrelation could be assessed by replacing these spatial statistics by ensemble statistics deduced from the study of Le Barb6 and Lebel (1997) in the calculation. However, there is an order of magnitude difference between the scales of the two studies. Consequently, the effect of

T. Lebel, L. Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

121

the down scaling from 1000 x 1000 krn 2 to 100 x 100 km 2 onto the computation of statistics for the number of events remain to be studied before merging the results obtained from the regional data set of the operational networks, on the one hand, and, on the other, from intensive but more local setups such as EPSAT-Niger.

Acknowledgements This work is the result of a great deal of effort put in by the EPSAT-Niger team during 4 years spent in Niger. Nothing would have been possible without the collaboration of the Niger Direction de la Mrtrorologie Nationale (DMN). This research was funded by the French Ministry for Cooperation and ORSTOM.

References Amani, A., Lebel, T., Rousselle, J. and Taupin, J.D., 1996. Typology of rainfall fields to improve rainfall estimation in the Sahel by the ATI method. Water Resour. Res., 32(8): 2473-2487. Bacchi, B. and Kottegoda, N.T., 1995. Identification and calibration of spatial correlation patterns of rainfall. J. Hydrol., 165:311-348. Bastin, G., Lorent, B., Duque, C. and Gevers, M., 1984. Optimal estimation of the average areal rainfall and optimal selection of raingauge locations. Water Resour. Res., 20(4): 463-470. Benichou, H., Delrieux, G. and Lecocq, J., 1995. Rainfall climatology in tropical Africa using a C-band weather radar: (1) the attenuation problem. Third International Symposium on Hydrological Applications of Weather Radar, Sao-Paulo, August 1995. Braud, I., Creutin, J.D. and Barancourt, C., 1993. The relation between the mean areal rainfall and the fractional area where it rains above a given threshold. J. Appl. Meteorol., 32: 193-202. Creutin, J.D. and Obled, C., 1982. Objective analyses and mapping techniques for rainfall fields: An objective comparison. Water Resour. Res., 18(2): 413-431. Delhomme, J.P., 1978. Kriging in the hydrosciences. Adv. Water Resour., 1(5): 251-256. Desbois, M., Kayiranga, T., Gnamien, B., Guesous, S. and Picon, L., 1988. Characterization of some elements of the Sahelian climate and their interannual variations for July 1983, 1984 and 1985 from the analysis of METEOSAT ISCCP data. J. Climate, 1(9): 867-904. Dugdale, G., Mc Dougall, V.D, and Milford, J.R., 1991. Rainfall estimates in the Sahel from cold cloud statistics: accuracy and limitations of operational systems. In: Soil Water Balance in the Sudano-Sahelian Zone (Proceedings of the Niamey Workshop, February 1991). IAHS Publ. N ° 199, pp. 65-74. Huff, F.A., 1970. Sampling errors in measurement of mean precipitation. J. Appl. Meteorol., 9(I): 35-44. Jones, D.A., Gourney, R.Y. and O'Connell, P.E., 1979. Network design using optimal estimation procedures. Water Resour. Res., 15(6): 1801-1812. Krajewski, W.F., Anagnostou, E.N. and Ciach, G.J., 1995. Effects of radar measurement process on the inferred rainfall statistics. 5th Int. Conf. on Precipitation, Elounda, Greece, 14-16 June 1995. Le Barb~, L. and Lebel, T., 1997. Rainfall climatology of the HAPEX-Sahel region during the years 1950-1990. J. Hydrol., this issue. Lebel, T. and Bastin, G., 1985. Variogram identification by the mean-square interpolation error method with application to hydrologic fields. J. Hydrol., 77:3 i-56. Lebel, T., Bastin, G., Obled, C. and Creutin, J.D., 1987. On the accuracy of areal rainfall estimation: a case study. Water Resour. Res., 23(11): 2123-2138. Lebel, T., Sauvageot, H., Hoepffner, M., Desbois, M., Guillot, B. and Hubert, P., 1992. Rainfall estimation in the Sahel: the EPSAT-NIGER experiment. Hydrol. Sci. J., 37(3): 201-215. Lebel, T., Taupin, J.D. and D'Amato, N., 1997. Rainfall monitoring during HAPEX-SaheI: I. General rainfall conditions and climatology. J. Hydrol., this issue.

122

T. Lebel, L Le Barb~/Journal of Hydrology 188-189 (1997) 97-122

Obled, C., 1986. Introduction au krigeage h l'usage des hydrologues. Proceedings of the 2 "a Joum6es Hydrologiques ORSTOM, Editions de I'ORSTOM (Colloques et S6minaires) Bondy, pp. 263-288. Peters-Lidard, C.P. and Wood, E.F., 1994. Estimating storm areal average rainfall intensity in field experiments. Water Resour. Res., 30(7): 2119-2131. Rodriguez-lturbe, I. and Mejia, J.M., 1974. The design of rainfall networks in time and space. Water Resour. Res., 10(4): 713-728. Seed, A.W. and Austin, G.L., 1990. Sampling errors for raingage-derived mean areal daily and monthly rainfall. J. Hydrol., 118: 163-173. Taupin, J.D, Amani, A. and Lebel, T., 1993a. Small scale spatial variability of the annual rainfall in the Sahel. In: H.-J. Bolle, R.A. Feddes and J. Kalma (Editors), Exchange Processes at the Land Surface for a Range of Space and Time Scales. Proceedings of the Yokohama Symposium, July 1993, IAHS Publ. N ° 212, pp. 593-602.