## Introduction

Flood hydrology is based mainly on knowledge of rainfall. The estimation of low-probability discharges requires knowledge of rare-frequency rainfall laws [e.g., for the gradex method developed by Duband and Guillot (Bouvard and Garros-Berthet 1994)]. This knowledge is therefore a prerequisite for the design of engineering structures (e.g., bridges, dams, or roads) and for determining risk areas.

Mountainous areas in a Mediterranean climate are particularly vulnerable, since the small steep catchments found there can generate flash floods in just a few hours. The risk of flooding and mudflows in such areas is high, and there is a risk of drift formation and avalanches in winter. Therefore, a region including the French Alps and the Mediterranean area was felt to be especially appropriate for performing a case study of heavy rainfall. Data were collected to determine the statistical characteristics of heavy (10- and 100-yr) rainfall in the area. The aim of the study was to obtain reliable rainfall-risk maps for short time steps (e.g., from 1 to 24 h).

There is a wide choice of interpolation techniques for mapping variables such as rainfall; these techniques range from the inverse distance technique (Tabios and Salas 1985) to more complex methods such as kriging (Journel and Huijbregts 1978; Delhomme 1978). However, the spatial distribution of the network used for the study is not sufficiently dense to obtain very reliable rainfall-risk maps. In a region with such a steep relief, topography is thought to play a key role in generating heavy rainfall. In many cases, a (usually linear) relationship is sought between the altitude of a location and the rainfall recorded there (Bleasdale and Chan 1972; Zwahlen 1981; Llasat and Puiggerver 1992; Humbert et al. 1997). This approach is based on the fact that rainfall is largely connected with orographic thermal currents: rainfall increases with elevation. For example, Bleasdale and Chan (1972) reviewed research in Great Britain and Ireland from the late 19th century. In Catalonia, Spain, 64% of the variance in daily rainfall can be explained by altitude (Llasat and Puiggerver 1992). However, this linear method is only strictly appropriate in regions of massive relief with scarcely broken topographic profiles. In other conditions, rainfall in a given place may not be linked to altitude alone; additional topographic features may be useful (Spreen 1947; Burns 1953; Basist et al. 1994; Konrad 1996; Thielen and Gadian 1997; Prudhomme and Reed 1998). Spreen (1947) shows that mean winter rainfall in Colorado is linked to altitude, slope, exposure, and orientation. Eighty-eight percent of the variance is explained by these four parameters, with altitude explaining only 30%. Basist et al. (1994) explain mean annual precipitation in 10 distinct mountainous regions with predictors such as elevation, slope, orientation, and exposure, with exposure and the interaction of elevation and exposure being the best predictors. For heavy rainfall, the method is the same as for precipitation, and previous studies have already been carried out on this subject in Lorraine (Laborde 1984), the Cévennes region (Slimani 1985), the French Alps (Saidi Bououdina 1996; Leblois 1997), and in the United Kingdom (Faulkner and Prudhomme 1998).

The aim of this paper is to show how to introduce topographic information for mapping purposes in the French Alps and to evaluate the accuracy of the method thus defined. The following sections describe the methodology used and the results obtained in the French Alps.

## Methodology

### Study area and dataset: Statistical characteristics of heavy rainfall

Rainfall data for the French Alps (see Fig. 1) were obtained from 90 recording rain gauges (apparatus for measurements at short time steps of 1, 2, 3, 6, 12, and 24 h) and 463 daily rain gauges (apparatus for daily measurements) (see Fig. 2). Rain gauges are generally located in or close to villages, and therefore their density at high altitude is low. Most of the stations are located between 0 and 2000 m (see Fig. 3), and the network does not reflect the actual distribution of altitudes in the Alps above 1500 m.

Heavy rainfall can be characterized by several parameters. The most commonly used values are for events with recurrence frequencies of 10 (10-yr rainfall) and 100 yr (100-yr rainfall). Local values of these statistical parameters in the French Alps were estimated for time steps ranging from 1 h to 1 day by fitting a Gumbel distribution using the moments method for samples of seasonal monthly maxima (Kieffer Weisse and Bois 2001). Table 1 gives the minimum, maximum, mean, and standard deviation values of 10- and 100-yr rainfall, calculated at 90 points for time steps ranging from 1 to 12 h, and at 553 points for 24 h.

The records from the various stations used cover periods ranging from 8 to 87 yr (see Fig. 4a). Time sampling errors are represented by the 80% confidence interval for 100-yr hourly rainfall (see Fig. 4b). Generally speaking, the longer the record period, the smaller the corresponding confidence interval. The uncertainty surrounding the values calculated for heavy rainfall is due partly to time sampling errors and partly to the measurements themselves. In this study, the heavy rainfall values are considered to be correct, although this uncertainty may affect the accuracy of the relationships found with topography.

### Relationships between heavy rainfall and topography

The first step is to find possible links between heavy rainfall and topography. Several kinds of topographic parameters are defined for this purpose [using a digital terrain map (DTM)], and the best set explaining heavy rainfall is used to define a practical method for mapping heavy rainfall fields.

#### Topographic variables based on the AURELHY method

The first set of parameters describing the relief (Kieffer 1995) is based on those defined by the Analyse Utilisant le Relief pour l’Hydrométéorologie (AURELHY) method developed by Benichou and Le Breton (1987) at Météo France. The aim of this method is to map any rainfall parameter automatically. The local topography is used to explain rainfall variables by multivariate linear correlation, and regression residuals are interpolated by kriging (Journel and Huijbregts 1978; Delhomme 1978). This method was used in the Cévennes by Canellas and Merlier (1994) to map daily rainfall for return periods ranging from 5 to 75 yr. There, the minimum and maximum percentages of variance explained by relief vary from 46% to 75%. Similarly, Humbert et al. (1993, 1994) compare this method and the “PLUVIA” method for mapping monthly and annual rainfall in the east of the Rhine–Meuse region.

The variables describing the site are based on the altitudes surrounding a point. Therefore, they describe only the close environment and not the geographical position of the site. Each point G is characterized by the 25 altitudes of a regular grid surrounding it (see Fig. 5). The altitude of the central point G is subtracted from each of the 25 altitudes. Thus, G is characterized by 24 differences in altitude and by its real altitude (i.e., 25 variables). These 25 variables are determined at 6431 points in the French Alps. To reduce the number of variables and to condense the information, a principal component analysis (PCA) (Jackson 1991) is applied to each of the 24 variables representing the differences in altitude. The PCA leads to the adoption of 10 principal components, which are found to be significant (they explain 88% of the total variance). Last, the local topography of point G is represented by 11 variables, namely, its real altitude and the 10 first principal components at this point.

#### Topographic parameters describing the local environment and geographical position of a site within the domain

In this section, parameters characterizing the topographic environment of a point are made more explicit (see section 1). Two main groups may be distinguished, namely, local variables and regional variables. Local variables are those that are defined from the topography in the area around the computational point (parameters describing height, exposure, tangents of a site, slope, and azimuths and radii of principal curvature; see Table 2). In contrast, evaluating a regional variable requires knowledge of the entire domain (distance to the Mediterranean, *X* (E–W) and *Y* (S–N) coordinates, distance to the Mediterranean and to the Rhône, characterization of the general shape of the Alps, and the barrier effect;see Table 3) [for further details on these computations, see Kieffer Weisse (1998)]. To assess the influence of investigation distance, all the local variables are calculated for different DTM spatial steps: 525, 1050, 1575, 2100, and 2625 m. The corresponding values are denoted by index *i* (*i* = 1 to 5) for a spatial step of *i* × 525 m. Hence, a set of 134 values could be computed for each grid point in the Alps. These values are, of course, not all independent (e.g., all the height parameters are linked), but the aim of the study is to find which one best explains heavy rainfall.

#### Methodology used to link heavy rainfall and topographic parameters

*P*

*a*

_{0}

*a*

_{1}

*X*

_{1}

*a*

_{n}

*X*

_{n}

*P*is the heavy rainfall variable (10- or 100-yr rainfall at various time steps) and

*X*

_{1},

*X*

_{2}, . . . ,

*X*

_{n}represent the set of topographic variables selected in the regression equation for

*P.*For multivariate linear regression (Baillargeon 1989; Vigier 1981), the stepwise upward regression method is used to link a rainfall variable

*P*and a set of topographic variables

*X*

_{1},

*X*

_{2}, . . . ,

*X*

_{n}. First, the variable that is the most closely correlated to

*P*is identified. The variables causing the greatest increase in

*R*

^{2}, the determination coefficient related to the variance, are then added successively, and, at each stage, Fisher significance tests are performed so as not to introduce a nonsignificant variable or to discard any variable that has already been introduced and is no longer informative considering the previous variables selected.

In the case of the set of topographic parameters consisting of local and regional variables (section 2), the first step involves studying linear relationships *P* = *aX* + *b* (univariate linear regression) between rainfall parameters and topographic parameters to determine which topographic variable best explains heavy rainfall and, more particularly, whether it is a local or a regional variable. In this case, performances are assessed using determination coefficients *r*^{2} between heavy rainfall (10- and 100-yr rainfall; *P*) and the various topographic variables (*X*). Here, uppercase *R*^{2} is used for multivariate determination coefficients, and lowercase *r*^{2} is used for simple determination coefficients.

### Mapping method based on multivariate linear regression with relief

*P*may be considered as the sum of two variables

*f*(

*R*

_{i}) and

*e*:

*P*

*f*

*R*

_{i}

*e,*

*f*(

*R*

_{i}) is the regression equation associated with the variable

*P,*

*R*

_{i}represents the set of topographic variables

*X*

_{i}selected in the regression equation for

*P,*and

*e*is the regression residual (that part of

*P*that is not explained by the variables

*X*

_{i}).

To map heavy rainfall, it is necessary to evaluate the value of *P* at each point in the domain by using Eq. (2). The regression equation *f*(*R*_{i}) may be calculated at each node of a DTM grid, thus producing an initial map of the field obtained by regression. This result is the map of *f*(*R*_{i}), or estimation of *P* by multivariate regression with the relief. The residual *e* is known at each rain gauging station (i.e., at 551 and 88 stations for 24-h and shorter-than-24-h time steps, respectively). This residual must then be interpolated throughout the domain to estimate a residual value at any point, as described below.

Kravchenko et al. (1996) used kriging to interpolate regression residuals between annual mean rainfall in Wyoming and topographic variables (altitude, slope, exposure, and geographical coordinates). Phillips et al. (1992) applied a similar method in order to map annual mean rainfall in Oregon. Laborde (1984) used the autoregressive filter method in a configuration similar to the current one; the residuals to be interpolated were those of regression between heavy rains and topographic variables in Lorraine.

It is therefore necessary to study the spatial variability of the regression residuals to determine whether it is possible to map the residuals in the current case and, if so, by what method. This study is performed with a structural function: the variogram (Journel and Huijbregts 1978; Delhomme 1978). If the variograms show that the phenomena are well structured in space, the fields can be mapped by ordinary kriging. Therefore, the intrinsic hypothesis must be satisfied. This hypothesis assumes that, for a random variable *z*(*x*_{k}), the expected value of *z*(*x*_{k}) does not depend on the position *x*_{k} and that the variance of [*z*(*x*_{k}) − *z*(*x*_{k} + *h*)] does not depend on the position *x*_{k} for any separation distance *h.* A common form of the theoretical variogram model is then defined by the nugget variance (the variance at zero distance), the sill (the variance to which the semivariogram asymptotically rises), and the range (the distance at which the sill or some predetermined fraction of the sill is reached).

### Accuracy of the mapping method: Cross validation

The last step attempts to evaluate the performance of the “regression with relief and kriging of regression residuals” method. A cross-validation technique is used, and the results are compared with those obtained by kriging (Journel and Huijbregts 1978; Delhomme 1978). This mapping method is very simple, because this interpolator uses only the isolated values of the field to be mapped. No additional information is taken into account.

To carry out cross validation, instead of eliminating a single point at a time from the sample, the rainfall measuring network is divided into two subnetworks, each with a random spatial distribution (see Fig. 6). The values taken by the variables at the stations in the first subnetwork are used as a database for interpolation. This calculation leads to an evaluation *z**(*x*_{i}) of the field at any point *x*_{i} of the domain and, in particular, at the stations in the second subnetwork. The latter is then used for validation. The known real values *z*(*x*_{i}) are compared with the estimated values *z**(*x*_{i}). The calculations are then run by reversing the roles of the two subnetworks. The criteria used for comparing real and estimated values are 1) comparison of the mean *m* and standard deviation *s* of the estimated and real values and 2) the determination coefficient between the estimated and real values: *R*^{2}[*z**(*x*_{i}), *z*(*x*_{i})]. The cross-validation technique used here is thus extremely pessimistic, but the main aim is to compare the efficiency of a method with a reference method under the same conditions.

## Results and discussion

### Link between heavy rainfall and topographic variables based on the AURELHY method

Figure 7 summarizes the results of multivariate linear regression for 10- and 100-yr rainfall, with local topographic variables based on the AURELHY method. It shows that short time steps are better explained than longer ones, with, at best, only 47% of the variance explained (this result was obtained for 10- and 100-yr hourly rainfall). For example, the multivariate determination coefficient *R*^{2} has a value of 0.26 for 10-yr daily rainfall. These results are not very satisfactory; relationships explaining less than 50% of the variance are not relevant for mapping purposes, and this set of topographic variables was subsequently abandoned.

### Statistically significant topographic variables

Generally speaking, the 10- and 100-yr rainfall are more closely linked to regional variables (see Table 3) than to local variables (see Table 2), with short time steps (up to 3 h) explained more satisfactorily than longer ones (see Figs. 8 and 9).

#### Local variables (see Table 2)

The determination coefficients decrease significantly for time steps longer than 3 h; after 3 h, *R*^{2} rapidly decreases toward zero (see Fig. 8). For shorter time steps, the determination coefficients decrease more slowly. Therefore, rainfalls shorter and longer than 3 h do not seem to be influenced by the same phenomena. With short time steps, the greater the smoothing, the better the correlation, at least for DTM spatial steps up to 2625 m (see Fig. 10). Thus, altitudes evaluated with a 2625-m step are the most closely correlated to rainfall. In addition, correlation coefficients between ZFS (smoothed altitude eliminating valleys whose width is less than 2 times the DTM spatial step) and rainfall tend to demonstrate that the wider the valley the greater its influence on short-duration rainfall (see Fig. 10). The correlation of 10- and 100-yr rainfall with the various tangents is significant (except with TMINi). For a given tangent variable (e.g., northeast tangent TNEi), the value calculated with a 2100-m spatial step is the most closely correlated with rainfall. Moreover, the best *r*^{2} values are obtained with the total tangent TT4, which indicates whether or not the horizon is “closed.” Rains are well correlated with the maximum tangents TMAXi, whereas they are hardly correlated with the minimum tangents TMINi, which shows that the highest surrounding relief first influences rainfall by preventing cloud circulation. The determination coefficients show that 10- and 100-yr rainfall values are not connected to exposure parameters (*r*^{2} less than 0.07), to principal curvature radii (*r*^{2} less than 0.06), or to azimuths (*r*^{2} less than 0.03).

#### Regional variables (see Table 3)

The 10- and 100-yr rainfall values are more closely linked to regional variables. For example, the percentage of variance explained by the distances to the sea or to the Rhône River (d1merrhone, d2merrhone) varies from 50% for 1 h to 32% for 24 h for 100-yr rainfall (see Fig. 9). The d1point explains 57% of the 100-yr hourly rainfall variance. The distance to the sea DM and latitude *Y* are the only variables that give a slightly better explanation for long time steps than for short time steps. Indeed, the correlation coefficient *r*^{2} with DM is 0.35 for 100-yr rainfall over 24 h and is only 0.21 for rains lasting 1 h.

### Multivariate linear regressions between rainfall parameters and local and regional parameters

#### Study of the explanatory ability of local and regional variables

The first step is to determine which combination of topographic parameters gives the best explanation of the rainfall parameters. The following sets of variables are used for the multivariate regression: 1) local and regional variables, 2) only the local variables, 3) only the regional variables, and 4) one regional variable and the local variables. The results of the univariate regression study (section 3b) show that regional variables always give a better explanation of rainfall parameters than do local variables. Therefore, the fourth and final multivariate regression test was performed using a set of variables consisting of the single best explanatory regional variable and all the local variables for each rainfall parameter.

Figure 11 shows the variations in determination coefficient *R*^{2} as a function of time step for the four cases of regression (1, 2, 3, and 4) and for 100-yr rainfall. To make a strict comparison between the different values of *R*^{2} from one regression approach to another, only four topographic variables were used in the regression equations in each case (the regression equations are given in section 3c2).

This study shows that regional variables explain the various rainfall parameters better than do local variables. Indeed, the *R*^{2} values obtained using only local variables (case 2) are always lower than those obtained with the other sets of topographic variables. The results are inferior even when the topographic variables include only a single regional variable (case 4). Last, the best determination coefficients *R*^{2} are obtained when all the morphometric variables are considered (case 1). In this case, the regression equations often contain several regional parameters. This situation might appear to be redundant, because the regional variables are well correlated with each other. However, if there is only one regional variable in the topographic variables (case 4), the determination coefficients are not so good. It was therefore decided to use all the defined topographic variables in order to establish the regression equations using the stepwise upward method, which selects only the variables best correlated with the rainfall variable *P.*

#### Selection of linear models and comments

The linear models chosen for the various rainfall parameters depend on four selected variables. This choice is based on the following points. Generally speaking, introducing a fifth variable does not significantly improve the percentage of variance explained (see Fig. 12). In many cases, the fifth variable is not significant in Fisher or Student’s *t* tests. Furthermore, it was decided to standardize the number of topographic variables in the different equations, and a model with a large number of parameters is difficult to use in mapping, which is the ultimate objective of this study.

The final equations are shown in table 4 for 100-yr rainfall. For 10- and 100-yr rainfall, the percentages of variance explained by the topography decrease when the time step increases (see Fig. 13): at 12 and 24 h, the equations are not satisfactory. This is confirmed by the 24-h results, which are based on much larger samples than those for the other time steps.

The regression equations for time steps shorter than 3 h contain the same topographic variables for 10- and 100-yr rainfall: d1point, *Y,* and barrier. It is, therefore, the general shape of the Alps (in the form of a crescent) that gives the best explanation of heavy rainfall at these time steps. The variable *Y* represents the influence of the sea, and barrier is the ability of fluxes to reach the observation point. It is a combination of these effects that explains the spatial distribution of short-duration heavy-rainfall events.

In the case of time steps longer than 6 h, the sets of topographic variables selected in the various equations are different. However, the dmerrhone variables are always present. Therefore, in the case of long time steps, the heavy rains are due either to westerly influences or to Mediterranean influences, depending on the position within the domain. Furthermore, the local variables introduced into the equation are always tangents, and it is therefore the slope of the site that is important for these rainfall parameters.

#### Reliability of the equations: Study of regression coefficients using standardized centered variables

For time steps of 3 h or less, the regression equations contain the same topographic variables for 10- and 100-yr rainfall. Moreover, these variables are introduced into the equation in the same order. In such a case, it is interesting to study the regression coefficients *b*_{i} using standardized centered variables *X*_{i} (see Table 5). For a given time step, the coefficient *b*_{i} corresponding to a given topographic variable is of the same order of magnitude for the variables of 10- and 100-yr rain. For example, the coefficient *b*_{i} corresponding to d1point has a value of 0.276 for 10-yr hourly rainfall and 0.275 for 100-yr hourly rainfall. Moreover, considering the standard deviations of the estimates *σ*_{bi}*b*_{i} are of the same order of magnitude from one time step to another. This result shows that the chosen models only depend on the standard deviation of the rainfall variable *P.* Indeed, the regression coefficient corresponding to the variable *X*_{i} is *b*_{i}*σ*_{Y}/*σ*_{Xi}

For time steps longer than 3 h, two variables out of the four are the same in the regression equations, but they do not appear at the same stage of the regression. In this case, the coefficients *b*_{i} no longer have the same order of magnitude.

To conclude, linear relationships between rainfall parameters and topographic parameters are pertinent and reliable for short time steps. Longer time steps (12 and 24 h) are less satisfactorily explained by topographic variables.

### Mapping method based on multivariate linear regression with relief

#### Spatial analysis of regression residuals

The experimental variograms of the different rainfall regression residuals show that the phenomena are well structured in space. The intrinsic hypothesis (Journel and Huijbregts 1978; Delhomme 1978) is satisfied (see Fig. 14, an example of residuals of 100-yr rainfall). In the case of longer time steps (12 and 24 h), the experimental variograms no longer have a drift as was the case for the other raw rainfall variables (Kieffer Weisse 1998).

In all cases, nugget exponential models were fitted to the experimental variograms (see Table 6 and Fig. 14). At 24 h, the nugget effect is clear. Indeed, the first intermediate-distance class (0–5-km class) includes 130 pairs and the corresponding root mean square deviation *γ*(*h*) is always different from zero, whatever the variable (10- or 100-yr rainfall). Thus, with shorter time steps, nugget models were also chosen, because the first point represented on the graphs is not very reliable in this case (it corresponds to only three pairs). This choice ensures consistency with the 24-h time step. This nugget models the error in estimating the rainfall quantiles. The sills were chosen in such a way that the sum of the nugget and sill would be equal to the variance of the field. The ranges, which are much smaller than those of the initial variables, are larger for short time steps than for long ones. The behaviors are different for time steps shorter than and longer than 3 h.

The good spatial structure of the regression residuals means that they can be mapped by ordinary kriging (Journel and Huijbregts 1978; Delhomme 1978). Examples of maps are shown in section 3d2.

#### Examples of maps obtained by applying the mapping method—comparison with maps obtained by a simple interpolation method: Kriging

Figures 15 and 16 show the different maps obtained by the “regression with relief and kriging of regression residuals” method for the examples of 100-yr rainfall lasting 1 and 24 h: These maps include the maps obtained by multivariate regression, that is, by applying the multivariate linear regression equation between the rainfall variables and morphometric variables at each point of the domain; the maps of regression residuals obtained by kriging; and the final maps of the rainfall field, which are the sum of the multivariate regression and residuals maps. The maps obtained by multivariate regression and the final maps are clearly influenced by the relief: the Alpine chain is clearly distinguished at both 1 and 24 h. The regression residual maps are smoothed, owing to the fact that the fields are modeled with a nugget. In the case of the short-duration events, the regression residuals are small (see Table 7). Adding these residuals to the maps obtained by multivariate linear regression therefore does not significantly modify them. However, this makes it possible to obtain the known heavy-rainfall value at each measuring station.

It was decided to make a visual comparison of the maps obtained by the method defined here and those obtained by a relatively simple method: kriging (Journel and Huijbregts 1978; Delhomme 1978). Figure 17 shows the maps of 100-yr rainfall lasting 1 and 24 h obtained by kriging. These maps can be compared with the final maps of Figs. 15 and 16. Figure 18 shows examples of maps of 100-yr rainfall (3 h and 12 h) obtained by the regression with relief and kriging of regression residuals method and maps of these same fields obtained by kriging. Generally speaking, the maps obtained by kriging are less ragged than those obtained by the method defined here. The latter reflect much more closely the relief of the Alps. But the main characteristics of the fields are represented in both cases. In the case of short time steps, between 1 and 3 h, an area of low heavy-rainfall values is situated in the center of the Alps in a region including the Tarentaise valley and the northern part of the Briançonnais area. Heavier rainfall occurs in the north, west, and south. In particular, the heaviest rainfall is recorded in the south for all time steps, which can be explained by the influence of the Mediterranean. These maps therefore reveal an increasingly pronounced shelter effect as one penetrates into the Alpine massif. As the duration increases, there are noticeable areas of heaviest rainfall, in particular at Coursegoules (southeast of the Alps, not far away from the coast). At 24 h, the areas of greatest rainfall intensity are situated in the southeast of the domain (Var and Alpes Maritimes). Then, several mountain ranges may be distinguished as areas of high intensity: Jura, Chartreuse, Dévoluy, and Baronnies. Last, the areas of lowest intensity are always those in the center of the Alps (Maurienne and Tarentaise).

### Efficiency of the mapping method including relief: Cross validation and comparison with kriging

Table 8 and Fig. 19 summarize the results of cross validation with the two methods: the so-called reference method [kriging (interpolator using only local knowledge of the field for interpolation)] and the method using the relief (regression with relief and kriging of regression residuals). The means of the different fields are generally better represented by kriging than by the method including the relief, which always overestimates them. Similarly, the standard deviations of the fields are more underestimated by the second method than by kriging. However, the study of determination coefficients *R*^{2} between estimated and real values gives a better evaluation of the respective performances of these two methods (see Fig. 19). Irrespective of the variable studied (10- or 100-yr rainfall), the determination coefficients *R*^{2} always decrease with the time step up to 12 h, with either method. At 24 h, the field is always well interpolated, which is explained by the density of rainfall information for this time step. However, the results of the cross validation show that adding information on the relief always improves the quality of interpolation for the short time steps (i.e., up to 3 h). There is a gain in variance of up to 10% at 1 h. With longer time steps, this is not the case. As mentioned before, the relationships between rainfall parameters and topographic parameters were poor for the longer time steps (see Fig. 13).

Last, for the method including relief, it was decided to distinguish between the contribution of the “relief” component and that of the “regression residuals” component. Cross validation was therefore carried out without adding the residual field. It was applied to the fields obtained by multivariate regression with relief. This study shows that the residual field does not improve the estimation of the parameters for time steps other than 24 h. Indeed, the differences between the *R*^{2} vary from 0 to 0.03 for time steps equal to or shorter than 12 h (see Fig. 20). In contrast, for 24 h, the percentage of explained variance increases by 15% for the 10-yr rainfall because of the addition of the regression residuals.

## Conclusions

This study attempts to explain the characteristics of heavy rains in the French Alps (10- and 100-yr rainfall) by topography in order to be able to map rainfall risks.

### Relationships between heavy rainfall and topography

Good results were obtained when parameters characterizing the environment of a site (its elevation, exposure, steep-sidedness, and slope) and its general situation in the French Alps are used. Generally speaking, the 10- and 100-yr rainfall are more closely linked to regional variables than to local variables, with short time steps (up to 3 h) more satisfactorily explained than longer ones. Indeed, the multivariate linear relationships give a good representation for short time steps: 77% of the variance of 100-yr hourly rainfall is explained, whereas only 57% of the variance of 100-yr daily rainfall is explained. These results could not be validated for high altitudes of more than 2000 m because of the lack of rain gauges.

Up to 3 h, the same topographic parameters explain heavy rainfall. These parameters are regional ones that represent the general shape of the Alps in the form of a crescent, the influence of the sea, and the possibility for fluxes to reach the observation point. It is a combination of these effects that explains the spatial distribution of short-duration, heavy-rainfall events. In the case of time steps longer than 3 h, the heavy rains are due either to westerly influences or to Mediterranean influences, depending on the position within the domain. Furthermore, the steep-sidedness of the site is important, because the local variables introduced into the equations are always tangents.

### Mapping method based on relationships with relief

Based upon relationships with relief, a method for mapping the heavy rainfall fields is defined. The fields may be mapped by applying the regression equation corresponding to the variable at each point of the domain. A study of the spatial variability of the regression residuals then shows that they can be mapped by kriging. This field is added to the previous one to obtain a final map of the rainfall field. The map thus obtained provides the known value at each measuring point. Last, the efficiency of this mapping method is compared by cross validation with a simple interpolation technique: kriging of heavy rainfall. It is demonstrated that using relief to describe heavy rainfall improves the results for time steps of 3 h or less. For example, there is a gain in variance of up to 10% at 1 h.

### Conclusions concerning the spatial distribution of heavy rainfall in the French Alps

For all time steps, an area of low heavy-rainfall values is situated in the center of the Alps in a region including the Tarentaise valley and the northern part of the Briançonnais area. Heavier rains occur in the north, west, and south. In particular, the heaviest rainfall is recorded in the south (Var and Alpes Maritimes) for all time steps, which can be explained by the influence of the Mediterranean. The maps therefore reveal an increasingly pronounced shelter effect as one penetrates into the Alpine massif. As the duration increases, there are noticeable areas of heavier rainfall, and at 24 h, several mountain chains may be distinguished as areas of high intensity: Jura, Chartreuse, Dévoluy, and Baronnies.

## Acknowledgments

The authors thank the EDF–DTG, Cemagref, and Météo France for providing the rainfall data used in this study. The study was funded by the Plan-Etat-Région concerned with “natural risks in mountains” and the 1994 Programme National des Risques Naturels implemented by the Institut National des Sciences de l’Univers. It was also partly funded by the Grenoble Natural Risks Unit. The authors thank the two anonymous reviewers for their constructive criticism and their helpful suggestions.

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Locations of the 90 recording rain gauges and 463 daily rain gauges used in this study

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Locations of the 90 recording rain gauges and 463 daily rain gauges used in this study

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Locations of the 90 recording rain gauges and 463 daily rain gauges used in this study

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Distribution of daily rain gauges and recording rain gauges used in the study as a function of altitude

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Distribution of daily rain gauges and recording rain gauges used in the study as a function of altitude

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Distribution of daily rain gauges and recording rain gauges used in the study as a function of altitude

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

(a) Distribution of length of the rain gauge records used in the study and (b) representation of 100-yr hourly rainfall with the 80% confidence interval

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

(a) Distribution of length of the rain gauge records used in the study and (b) representation of 100-yr hourly rainfall with the 80% confidence interval

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

(a) Distribution of length of the rain gauge records used in the study and (b) representation of 100-yr hourly rainfall with the 80% confidence interval

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Locations of the 25 altitudes chosen to characterize site G

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Locations of the 25 altitudes chosen to characterize site G

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Locations of the 25 altitudes chosen to characterize site G

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Measurement subnetworks used for cross validation (left: two subnetworks of rainfall recorders; right: two subnetworks of 24-h gauging stations)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Measurement subnetworks used for cross validation (left: two subnetworks of rainfall recorders; right: two subnetworks of 24-h gauging stations)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Measurement subnetworks used for cross validation (left: two subnetworks of rainfall recorders; right: two subnetworks of 24-h gauging stations)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} as a function of time step for heavy-rainfall parameters in the case of regression with“AURELHY”-type variables

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} as a function of time step for heavy-rainfall parameters in the case of regression with“AURELHY”-type variables

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} as a function of time step for heavy-rainfall parameters in the case of regression with“AURELHY”-type variables

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and some local topographic variables as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and some local topographic variables as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and some local topographic variables as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and regional topographic variables as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and regional topographic variables as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and regional topographic variables as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and altitudes ZFS as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and altitudes ZFS as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Determination coefficients *r*^{2} between 100-yr rainfall and altitudes ZFS as a function of time step

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} (biased) plotted as a function of time step for different cases of linear multivariate regression: example of 100-yr rainfall

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} (biased) plotted as a function of time step for different cases of linear multivariate regression: example of 100-yr rainfall

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} (biased) plotted as a function of time step for different cases of linear multivariate regression: example of 100-yr rainfall

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} plotted as a function of number of topographic variables introduced into the regression equation for 100-yr rainfall

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} plotted as a function of number of topographic variables introduced into the regression equation for 100-yr rainfall

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Multivariate determination coefficient *R*^{2} plotted as a function of number of topographic variables introduced into the regression equation for 100-yr rainfall

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Relationships between heavy-rainfall and topographic parameters: multivariate determination coefficient *R*^{2} as a function of time step for heavy-rainfall parameters

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Relationships between heavy-rainfall and topographic parameters: multivariate determination coefficient *R*^{2} as a function of time step for heavy-rainfall parameters

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Relationships between heavy-rainfall and topographic parameters: multivariate determination coefficient *R*^{2} as a function of time step for heavy-rainfall parameters

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Experimental and theoretical variograms of regression residuals with relief for 100-yr rain

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Experimental and theoretical variograms of regression residuals with relief for 100-yr rain

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Experimental and theoretical variograms of regression residuals with relief for 100-yr rain

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Application of the “regression with relief and kriging of regression residuals” method to the case of 100-yr rainfall lasting 1 h (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Application of the “regression with relief and kriging of regression residuals” method to the case of 100-yr rainfall lasting 1 h (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Application of the “regression with relief and kriging of regression residuals” method to the case of 100-yr rainfall lasting 1 h (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Application of the “regression with relief and kriging of regression residuals” method to the case of 100-yr rainfall lasting 24 h (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Application of the “regression with relief and kriging of regression residuals” method to the case of 100-yr rainfall lasting 24 h (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Application of the “regression with relief and kriging of regression residuals” method to the case of 100-yr rainfall lasting 24 h (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Maps showing 100-yr rainfall lasting (left) 1 h and (right) 24 h obtained by kriging (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Maps showing 100-yr rainfall lasting (left) 1 h and (right) 24 h obtained by kriging (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Maps showing 100-yr rainfall lasting (left) 1 h and (right) 24 h obtained by kriging (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Maps of 100-yr rainfall lasting 3 and 12 h obtained by (left) kriging and by (right) the “regression with relief and kriging of residuals” method (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Maps of 100-yr rainfall lasting 3 and 12 h obtained by (left) kriging and by (right) the “regression with relief and kriging of residuals” method (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Maps of 100-yr rainfall lasting 3 and 12 h obtained by (left) kriging and by (right) the “regression with relief and kriging of residuals” method (computation step: 5.25 km)

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Results of cross validation: determination coefficients between known and estimated values

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Results of cross validation: determination coefficients between known and estimated values

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Results of cross validation: determination coefficients between known and estimated values

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

(left) Determination coefficients *R*^{2} between real and estimated values for the method including relief. (right) Differences between determination coefficients obtained (i) for the regression method with relief and kriging of residuals and (ii) for the same method without adding the residual field

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

(left) Determination coefficients *R*^{2} between real and estimated values for the method including relief. (right) Differences between determination coefficients obtained (i) for the regression method with relief and kriging of residuals and (ii) for the same method without adding the residual field

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

(left) Determination coefficients *R*^{2} between real and estimated values for the method including relief. (right) Differences between determination coefficients obtained (i) for the regression method with relief and kriging of residuals and (ii) for the same method without adding the residual field

Citation: Journal of Applied Meteorology 40, 4; 10.1175/1520-0450(2001)040<0720:TEOSCO>2.0.CO;2

Statistical values of heavy rainfalls in millimeters per time step (90 recording rain gauges with time steps ranging from 1 to 12 h and 553 daily rain gauges)

Local variables that describe the close environment of a point

Definition of regional variables (km)

Multivariate linear regression: final results for 100-yr rainfall (=millimeters per time step, *σ*_{ε} is residual standard deviation, *n* is number of observations); *F* values must be compared with critical values: *F*5% = *F*(4, 83) = 2.48 (1 h to 12 h) and *F*5% = *F*(4, 546) = 2.37 (24 h)

Regression coefficients *b _{i}* using standardized centered variables and associated standard deviations

*σ*

*for 10- and 100-yr rainfall*

_{bi}Characteristics of variograms of regression residuals with relief for 100-yr rainfall (exponential models)

Regression residual statistics (millimeters per time step): example of 100-yr rainfall (551 observations for 24 h, 88 others)

Statistical comparison of real and estimated values for 100-yr rainfall—*m*: mean and *s*: standard deviation (values in millimeters per time step; computation step: 5.25 km)