As the first step of the analysis, we investigated the influence of elevation on the spatial distribution of the average of annual maximum rainfall depths. We calculated the ADD curve parameters for all the stations of the I2-RED dataset (Mazzoglio et al., 2020a) to compare them with previous studies (mainly Avanzi et al., 2015). Parameters a and n (cf. Eq. 1) were initially obtained by means of a linear regression of the logarithm of the average of all the available extremes over the 1 to 24 h durations. We computed the median values of these parameters, for all over Italy, to compare them with those of Avanzi et al. (2015), who grouped the stations into elevation ranges of 50 up to 1000 ma.s.l. and into intervals of 100 m for higher elevations. We then plotted both series of medians (a and n) against the median elevation of each interval (to consider that the distribution of the rain gauges in each elevation interval was skewed) and fitted regression models.

We studied the differences between the measured and estimated rainfall statistics to assess the effectiveness of the regression models, considering the observed averages of the extremes over 1, 3, 6, 12 and 24 h. We obtained performance indices for each station using the estimation errors Δd: 2Δd=havg(d)-a^⋅dn^, where havg(d) is the sample average of the extreme rainfall depth for duration d, while a^ and n^ are the estimates of parameters a and n.

In this paper, we show and discuss only the results related to the shortest and the longest of the five durations (1 and 24 h), as they can be considered the most representative of the different classes of rainfall events (convective and stratiform, respectively). The corresponding dependent variables are called h‾1 and h‾24 in the following.

The error statistics that were computed are the bias, the mean absolute error (MAE), the root mean square error (RMSE) and the Nash–Sutcliffe model efficiency (NSE) coefficient (Nash and Sutcliffe, 1970; Wasserman, 2004). Among all the statistics, particular attention was dedicated to spatial bias, i.e., the bias evaluated as the difference between the spatial mean of the observations over a generic area, and the corresponding values predicted by the model.

By applying the procedure described in Sect. 2.1, we obtained results that are in agreement with those of Avanzi et al. (2015):

Parameter a decreases linearly with elevation (R2=0.89), through the following equation:3a=30.61-0.0060⋅z,which is comparable with the equation obtained in Avanzi et al. (2015):4a=29.17-0.0062⋅z.

As already mentioned, parameter a is roughly equivalent to h‾1. Its overall inverse dependence on elevation is somewhat counterintuitive, even though other authors have confirmed this dependence (e.g., Allamano et al., 2009; Marra et al., 2021).

The error statistics computed on the two sets of residuals, in this work and in that of Avanzi et al. (2015), are listed in Table 1. The results show that the increase in the number of stations and the recording length achieved in I2-RED have led to an improvement compared to the results of Avanzi et al. (2015). This result is not surprising, but more insights can be derived from the observation of the spatial distribution of the residuals, which were not discussed explicitly in the previous literature.

In this regard, we mapped differences Δ1 and Δ24 to investigate where the under- and overestimations show spatial coherence. The maps, reported in Fig. 2, clearly show that clusters of residuals with high residuals of the same sign emerge in various areas of the country: for instance, many coherent errors larger than 3 times the MAE are present in the Liguria region (see Fig. 1 for the geographic position) for both durations. Therefore, despite the high R2 values, significant spatially correlated errors can undermine the practical validity of these general relationships.

On the basis of these results, the need for a more detailed spatial analysis of these variables became evident. A set of new analyses, aimed at reducing the local bias and increasing the reliability of the results, was therefore introduced.

In an attempt to improve the evaluation of the relationships between rainfall and topography, we undertook an analysis of the relationships between rainfall and several geomorphological (and climatological) parameters, which may complement the explanatory power of elevation. Unlike what was done in Avanzi et al. (2015), multivariate models were used in the literature to relate rainfall statistics and various morphological variables, both of which were evaluated at the same location. In these approaches, no aggregated or median spatial statistics of rainfall were considered. Prudhomme and Reed (1998, 1999), for instance, identified meaningful geographic and morphological attributes of each location as good explanatory variables of the daily rainfall maxima in Scotland. They showed that obstruction indices, derived from the orography, and the distance from the coastline are able to define how morphological barriers influence the characteristics of the extremes. These appear to work better than the EXPO variable used by Basist et al. (1994) and Konrad (1996).

Basist et al. (1994) defined EXPO as the distance between a rain gauge and an upwind barrier whose elevation is at least 500 m higher than the station. Konrad (1996) suggested an elevation of the barrier at least 150 m higher than the station. Prudhomme and Reed (1998) also tried to use this variable, setting the elevation difference at 200 m, but concluded that the definition of EXPO has several drawbacks, as it is based on arbitrary thresholds and is defined assuming a specific and subjective direction.

Introducing new variables with omnidirectional meaning, as the distance from the sea, the obstruction and the barrier, which are evaluated in the eight main directions, Prudhomme and Reed (1998) were able to explain a much larger percentage of variability in the annual maximum daily rainfall than that explained by the EXPO variable.

Caracciolo et al. (2012) applied this latter approach on the island of Sicily (south of Italy): they found that the longitude, elevation, a barrier obstruction index and the distance from the coastline are able to represent the spatial variability of parameter a for the whole island, while the longitude, elevation, a concavity index and the slope are able to satisfactorily describe the variability of exponent n. They also noticed that different descriptors became significant when analyzing smaller portions of the island.

Based on the above considerations, in this work we followed the approach suggested by Prudhomme and Reed (1998, 1999), considering two groups of variables computed for each station location: a.

Geographic and climatic variables. These do not require computation and do not depend on the landscape forms, that is, longitude (LONG, expressed in the WGS84 UTM32N reference system, in m), latitude (LAT, expressed in the WGS84 UTM32N reference system, in m), elevation above sea level (z, in m), minimum distance from the coastline (C, in km) and mean annual rainfall (MAR, taken from Braca et al., 2021, in mm); the latter represents a very robust climatological variable, which is seldom used as ancillary information but easily available throughout the world thanks to the presence of various rainfall databases (Schneider et al., 2011; Fick and Hijmans, 2017; Muñoz Sabater, 2019).

The values of all of these variables depend on the landscape forms and can vary according to the resolution of the used DEM. In our case, after thorough consideration, we adopted the Shuttle Radar Topography Mission (SRTM) DEM, which has a resolution of 30 m (Farr et al., 2007). However, the openness required to be evaluated on the SRTM DEM resampled at a resolution of 500 m due to computational limitations. This computation was conducted with the SAGA “Topographic openness” module, using a radial limit of 5 km. This value was obtained after testing different radial limits and selecting the one that presents the best correlation with the mean rainfall depth.

Multiple linear regression models were built, based on the following relationship: 7Y=X⋅β+δ=∑i=1NβiXi+δ, where the dependent variable Y is related to the matrix of the independent variables X or covariates. β in Eq. () is the vector that contains the regression model coefficients, and δ is the vector of the residuals.

In order to select the best model equation, the number i of covariates can be increased as necessary, according to the criteria of statistical significance of the estimated parameters. Caracciolo et al. (2012), for instance, used a stepwise regression approach. In this paper, we have preferred to use a generalized multiple regression approach, whereby increasing the number of covariates to i+1 does not necessarily preserve the descriptors that were the most significant at step i. This approach entails always considering all the possible combinations of two, three or four covariates until the “best” model is found. Other tests made using five or more variables did not lead to significantly higher Radj2 values.

The best regression model was selected on the basis of an analysis of the regression residuals, favoring models with the highest adjusted coefficient of determination, Radj2. Student's t test was used to quantify the significance of the independent variables. We checked for the possible presence of multicollinearity for each model in which all the covariates passed Student's t test, as this could lead to the formulation of an unstable model. Multicollinearity was measured using the variance inflation factor (VIF), which is determined by placing the jth independent variable as the dependent variable and calculating the coefficient of determination Radj2 of the multiple regression performed on the remaining p-1 independent variables (Eq. ). 8VIFj=11-Rj2

Values of VIF greater than 5 were associated with an unacceptable level of multicollinearity, and the corresponding model was discarded (Montgomery et al., 2012).

The equations of the best regression models (built using two to four variables) are reported in Eqs. (9) to (11) for h‾1 and in Eqs. (12) to (14) for h‾24. The Radj2 values in Eqs. (9) to (11) are 0.46, 0.52 and 0.54, respectively. The coefficients of determination are higher for h‾24, i.e., 0.66, 0.67 and 0.68 (Eqs. 12 to 14, respectively). 9h1=20.3163-0.0080⋅z+0.0117⋅MAR10h1=-21.6293-0.0061⋅z+0.0134⋅MAR+26.2682⋅OP11h1=-10.6928-0.0051⋅z-0.0273⋅C+0.0131⋅MAR+19.8449⋅OP12h24=16.1392-0.0937⋅C+0.0712⋅MAR13h24=33.1529-1.8574⋅10-5⋅LONG-0.1319⋅C+0.0701⋅MAR14h24=127.2773-2.9498⋅10-5⋅LONG-1.9130⋅10-5⋅LAT-0.0971⋅C+0.0735⋅MAR

Considering the three h‾1 models (Eqs. 9 to 11), it is possible to notice the negative slope coefficient associated with elevation. This confirms what was discussed in the previous section. On the other hand, it is remarkable that the best models found for h‾24 do not include the elevation: this outcome can be explained by considering the fact that MAR is always significant, regardless of the number of variables. Models in which MAR was excluded actually present z as a significant covariate but with less relevance than the regression models for h‾1.

Regardless of the number of the variables considered, and despite the marked increase in the corresponding value of Radj2, compared to the simple regression, we found that the residuals of the multivariate regressions were still characterized by spatial clustering and high local errors, basically all in the same areas in Fig. 2. In other words, resorting to additional variables but keeping a uniform relationship between each variable and precipitation over all of Italy does not produce a decisive reduction in the bias for large areas of the country. Thus, we decided to deconstruct the modeling approach and to look for clues of distinct generating mechanisms in distinct areas of Italy.

In this section, an additional paradigm is introduced into the models for the spatial variability of precipitation to reduce the spatial bias, namely the selection of limited areas to build “local” regression models, as an alternative to using data for the whole of Italy. Such an attempt was already made by Caracciolo et al. (2012), who borrowed the subdivision criterion from previous regional frequency analyses. In this work, we have focused on the role of geography and morphology in the spatial variability of annual maximum rainfall depths.

To better understand how to move from national-scale relationships to relationships valid for smaller areas, we started by considering the Alpine area separately from the Apennine region along the entire peninsula, and from the two main islands (Sicily and Sardinia; see Fig. 1 for the geographic positions), as a first approximation. We then built four different multivariate models: (1) the Alpine region (i.e., from Piedmont, including the western part of Liguria, eastward up to Friuli Venezia Giulia, delineated using the SOIUSA classification, as suggested by Accorsi, 2016); (2) the Apennine region, including peninsular Italy; (3) Sicily; and (4) Sardinia. We evaluated the best regression models for these four regions, as described in Sect. 3.1, using up to four covariates.

The new set of models built for the four regions were tested by computing the error statistics over the entire country. The obtained results indicated that they provided higher Radj2 than for the national case and better error statistics (see Table 2 for a comparison with the previous multivariate approach). The better results achieved in terms of RMSE, MAE and NSE at the national scale are due to the improvements obtained for the two main islands (Sicily and Sardinia). More insights are provided in Sect. 6.

It is interesting to compare the results obtained for the individual Alpine region with those of Allamano et al. (2009), who analyzed almost the same area. In that case, the ADD curve parameters appeared to be related to elevation and longitude. For the different durations Allamano et al. (2009) also estimated a regression model by linear regression between rainfall depth, elevation and longitude. The dependence of short-duration rainfall on elevation and longitude was found to be statistically significant for all the time intervals, except for the 1 h duration: in this case, the longitude was not statistically significant. In our application, the best relationships found for h‾1 and h‾24 are those of Eqs. (15) and (16) (characterized by Radj2=0.75 and 0.76, respectively): 15h‾1=60.9365-1.6664⋅10-5⋅LAT-0.0046⋅z+0.0148⋅MAR+25.1825⋅OP16h‾24=59.0632-7.2955⋅10-5⋅LONG-0.2223⋅C+0.4306⋅OBS+0.0822⋅MAR.

As expected, the h‾1-z relationship has a negative slope, and the Eq. (15) does not include the longitude as covariate, in agreement with Allamano et al. (2009). The same negative relationship is found in a 24 h equation that includes z (which produces an Radj2=0.74, that is, lower than that of Eq. 16). These findings confirm the results of Allamano et al. (2009), who found a general decrease in rainfall depth for increases in elevation for all the durations (up to 24 h). Equation (16) also confirms that, although h‾1 decreases systematically with elevation over the whole alpine region, the dependence of h‾24 on z decreases as the longitude increases, i.e., moving westward.

The full set of equations used for the four regions is provided in Supplement no. 2, together with the Radj2.

Although the improvements achieved with multivariate models over the simple regressions are evident, we found that they were not decisive in providing a homogeneous spatial distribution of the errors. We in fact observed that, even with the best model, we were not able to reduce the clustering effect shown in Fig. 2 for the peninsular region (see also Supplement no. 3). We believe that a model capable of describing the observed spatial variability of the index rainfall simultaneously at a national and a local level requires additional insights, which can be obtained using a finer spatial segmentation of Italy.

On the basis of the considerations presented above pertaining to the spatial clustering of residuals, we examined the possibility of obtaining a meaningful segmentation of large areas in subdomains that could be used to obtain “local” relationships between annual maximum rainfall depths and terrain properties. The main reasoning behind the segmentation is that some macroscopic morphological differences can determine markedly different behaviors of the relationships between rainfall and elevation (or other local variables). One example concerns what happens in the windward and leeward sides of mountain ridges, which represent transversal obstacles to the humid masses coming from the sea. Accordingly, we considered some general geomorphological classifications of the landscape that delineate homogeneous areas based on the homogeneity of the macroscopic land properties, such as convexity and texture.

We considered four geomorphological classifications (GCs) and denominated them as GC1 to GC4, according to their diversity and success in the geomorphological literature (see the Data availability section for more information).

The first considered classification, called GC1, was proposed by Iwahashi and Pike (2007); they classified the Earth's surface into 16 topographic types, at a 1 km resolution, based on slope gradient, local convexity and surface texture. We vectorized the raster map, which is available on the European Soil Data Centre website, and then, to reduce the presence of small areas, which could have an extent of just a few square kilometers, all the areas covered by fewer than 10 pixels (10 km2) were merged with the adjacent class. Among the four different classifications that were used, this is the only one that has a worldwide coverage, as all the other classifications are available at a national scale. A detailed description of the methodologies used by the authors is available in the related references, thus allowing all the classifications to be reproduced over other nations.

The second classification – GC2 – is the Carta delle Unità Fisiografiche dei Paesaggi italiani (“Map of the physiographic units of Italian landscapes”) and is included in the Carta della Natura (“Map of nature”; Amadei et al., 2003). A vector map, which was obtained by means of a visual interpretation of satellite images, aided by the analysis of further land cover maps and morphological–lithological characteristics, was available at a 1:250000 scale.

The third classification – GC3 – was proposed by Guzzetti and Reichenbach (1994). It was obtained, in vector format, by combining an unsupervised three-class cluster analysis of four properties of altitude (altitude itself, slope curvature, frequency of slope reversal and elevation–relief ratio) with a visual interpretation of morphometric maps and an inspection of geological and structural maps.

The fourth classification – GC4 – is the one that delineates areas with the greatest detail, as it is based on local morphometric properties of the landscape. It was proposed by Alvioli et al. (2020a), who considered a set of 439 watersheds, covering the whole of Italy, grouped into seven clusters on the basis of the various properties of the slope units within each basin, e.g., a distribution of slope unit sizes and aspects. In this work, adjacent watersheds of the same class were collapsed (GIS Dissolve), thus producing a total of 178 areas. Geomorphologically homogeneous terrain partitions were defined as “slope units” that were delimited by drainage and divide lines and delineated with a method that was first introduced by Alvioli et al. (2016) and which is widely used in the literature for geomorphological zonation purposes.

An additional geomorphological classification, which was proposed by Meybeck et al. (2001) and which has a worldwide coverage, was also considered. It is based on a combination of a relief roughness index and elevation and in principle could have been a good fifth candidate. However, it was not included in this analysis because, except for a very large geographical zones, the resulting delineated areas often contained very few rain gauges, which would have made it impossible to perform the desired statistical analyses.

Coherently with the aim of addressing connections between terrain properties and rainfall at a more local level, we built a set of linear regression models between elevation and index rainfall for all the classifications, considering an individual model for each outlined geomorphological zone. Only the internal rain gauges in each of these homogeneous areas with a minimum of five available stations that had to ensure at least 100 m of difference in elevation were considered for the regressions. There were four possible outcomes of the applications: (a) a positive and significant correlation (at the 5 % level), (b) a negative and significant correlation; (c) a nonsignificant correlation and (d) an insufficient number of stations or an insufficient difference in elevation.

The results for h‾1 are presented hereafter, while the details on h‾24 are available in Supplement no. 4. The results obtained for each geomorphological zone are mapped in Fig. 4a–d. Blue areas denote geographical zones, where h‾1 increases together with elevation, while the red palette applies to zones where rainfall decreases with elevation. The color intensity is proportional to the respective slopes. The light gray color denotes zones in which the linear regression is not statistically significant (at a 5 % level), while dark gray denotes insufficient data (case d). A comparison of the maps (Fig. 4a–d) clarifies that the more detailed the geomorphological zonation is, the less likely it is to satisfy the requirements necessary to build a significant regression. On the other hand, if one applies regression models to finer geomorphological classifications, it is possible to see that the regression sign is not uniform over the entire country. For example, with regard to 1 h data, it is possible to clearly recognize the presence of zones with a positive rainfall depth versus elevation trend for pre-hill/plain morphology in both GC3 (Fig. 4c) and GC4 (Fig. 4d).

The spatial distribution of the light gray zones is an important piece of information: no trend can be assumed over these areas because the p value is greater than 0.05. Consequently, h‾1 can be considered constant over these areas. Finally, the occurrence of the dark gray zones is directly connected to the kind of classification: the smaller the areas delineated by the classification are, the more likely it is that the requirement of having at least five rain gauges with at least 100 m difference in elevation is not satisfied. In this regard, we can observe that the above requirements are not met for the elevation difference, i.e., in plain areas, and it is necessary to assume a constant h‾1 in the area as being the most reasonable value.

The maps in Fig. 4a–d show that the availability of more detail in the spatial analysis of the relationship between rainfall depth and elevation has a remarkable effect on both the sign of the regression and the slope of the regression line in several areas. In addition, even the quality of the relationship can improve, as can be seen from a comparison of Fig. 4e–h: far more areas with high R2 can be seen in Fig. 4h than in Fig. 4e. This allows us to conclude that lower values of R2 are obtained in wider areas.

The same analysis was conducted on h‾24, where all of the above outcomes were confirmed, except for the sign of the precipitation vs. elevation relationship (Supplement no. 4).

To test the reliability of the regression models built over the GCs, the linear equations found in each geomorphological zone were applied to all the rain gauge positions, to obtain errors that could be examined at the country scale. The global indices computed for the GC areas in which the regressions were statistically significant are reported in Table 3 for h‾1 and in Table 4 for h‾24. These results clearly show a lower performance of the GC1 than the national-scale regression model. On the other hand, the error statistics in Tables 3 and 4 show that GC4 produces the smallest errors, and this geomorphological subdivision therefore presents the best performances. It is possible to understand this result by considering that GC4 uses watershed units, while the other classifications are based on the automatic processing of digital terrain data.

A constant value of the index rainfall computed as the spatial average of h‾d was adopted over any areas where the morphological regression model was not statistically significant. In this way, it was possible to compute the statistical indexes at the whole country scale. The error statistics obtained with this application are reported in the last rows of Tables 3 and 4.