The dependence of rainfall on elevation has frequently been documented in the scientific literature and may be relevant in Italy, due to the high
degree of geographical and morphological heterogeneity of the country. However, a detailed analysis of the spatial variability of short-duration
annual maximum rainfall depths and their connection to the landforms does not exist. Using a new, comprehensive and position-corrected rainfall
extreme dataset (

The spatial patterns of rainfall depth statistics are known to be affected by the geomorphological setting (Smith, 1979; Basist et al., 1994; Prudhomme and Reed, 1998, 1999; Faulkner and Prudhomme, 1998). The impact of orography on daily, multi-daily and annual precipitation events can generally be attributed to the so-called “orographic enhancement of precipitation”, i.e., an increase in rainfall depth along the windward slope of a relief and a decrease on the lee side, due to the lifting and the consequent drying of the air mass (Smith, 1979; Daly et al., 1994; Frei and Schär, 1998; Napoli et al., 2019). In a complex landscape, this effect can also entail significant precipitation values on the lee side, due to landforms that cause a delay in the hydrometeorological formation of precipitation and falling raindrops (Smith, 1979).

The impact of the orography on extreme rainfall depths and the complicated atmosphere–orography interactions for large areas are still not
sufficiently understood for sub-daily rainfall events. In a country like Italy, characterized by a high degree of morphological heterogeneity (Fig. 1),
these relations assume an evident importance, considering the significant exposure to Mediterranean storms (Claps and Siccardi, 2000). The focus of
this study is the entire Italian territory (

Elevation data with the boundaries of the 20 Italian administrative regions. Source: Shuttle Radar Topography Mission (Farr et al., 2007).

Most of the existing studies in Italy have focused on limited areas (Allamano et al., 2009; Caracciolo et al., 2012; Pelosi and Furcolo, 2015; Furcolo et al., 2016; Furcolo and Pelosi, 2018; Libertino et al., 2018; Formetta et al., 2022), and the only attempt to deal with sub-daily data covering the entire nation was made by Avanzi et al. (2015).

These studies suffered from a lack of a comprehensive and quality-assessed national database for sub-daily extremes. Several of them analyzed the
Italian Alpine area. For instance, Frei and Schär (1998) focused on the entire European Alps region and showed that foothills enhance monthly and
seasonal precipitation, while inner valleys produce an orographic shielding effect on rainfall. Nevertheless, they did not find a unique
precipitation depth–elevation relationship that could be considered valid for the entire Alps and attributed the observed variability to the effects
of slope and shielding. Allamano et al. (2009) investigated the dependence of sub-daily annual maximum rainfall depths on elevation over the Italian
Alpine region. They found a significant decreasing trend for increasing elevations and a nonuniform slope coefficient over the longitude range. The
slope of the rainfall depth–elevation regression was shown to decrease for event durations that increased from 1 to 24

Other regional works that attempted to identify orographic effects in the Mediterranean part of Italy are available for Campania and Sicily. Pelosi and Furcolo (2015) and Furcolo et al. (2016) analyzed the daily annual maximum rainfall depths over Campania (see Fig. 1 for the geographical location) and attempted to explain systematic variations as being the result of the presence of orographic barriers, identified through the application of an automatic geomorphological procedure (Cuomo et al., 2011). Their results showed a link between orographic elements and a local increase in rainfall depths and allowed orographic elements that produced enhanced variability of extreme rainfall to be identified. The same group later worked on sub-daily annual maximum rainfall depths (Furcolo and Pelosi, 2018) and proposed a power-law amplification factor of rainfall over three mountainous systems.

Caracciolo et al. (2012) found, in Sicily, that the longitude, latitude, distance from the sea and a concavity index are the variables that govern the spatial variability of rainfall depths. However, these authors found that no linear relationship between sub-daily annual maximum rainfall depths and elevation was significant at a 5 % level over the entire island of Sicily.

All of the previously mentioned analyses refer to an analytic relationship that connects annual maximum rainfall depths of various durations, i.e., the
average depth–duration (ADD) curve of the simple-scaling approach, which is usually represented by a power law:

Avanzi et al. (2015) analyzed the spatial variability of the ADD curve parameters,

On the basis of the described background and the significant improvements offered by a new, up-to-date rainfall dataset, i.e., the Improved
Italian-Rainfall Dataset,

Searching for models that allow the index rainfall to be estimated for various durations in any location in Italy is the first, important, necessary step toward addressing the building of depth–duration–frequency curves over the entire country. For this purpose, simple (Sect. 2) and multiple (Sects. 3 and 4) national-scale regression models were first investigated. We then introduced four geomorphological classifications to perform local-scale regression analysis in order to tackle the evident spatial clustering of the regression residuals (Sect. 5). The comparisons made between the results obtained from the wide-area and the local regressions allowed the role of the morphology in rainfall variability to be discussed, as shown in Sect. 6. Some conclusions are drawn in Sect. 7.

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

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

In this paper, we show and discuss only the results related to the shortest and the longest of the five durations (1 and 24

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

Parameter

For comparison purposes, Avanzi et al. (2015) obtained

The fitting of the four models is reported in Supplement no. 1.

As already mentioned, parameter

Comparison of national-scale error statistics related to the estimates performed with our data and those of Avanzi et al. (2015). The results were obtained with Eqs. (

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

In this regard, we mapped differences

Residuals of the estimations of the 1

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

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

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:

slope (

obstruction (OBS, in degrees), defined as the maximum angle needed to overcome the highest orographic obstacles in the eight main
cardinal directions (i.e., the maximum of the angles subtended by the line that connects the rain gauge with the highest orographic peak within a
15

barrier (BAR, in

maximum slope angle (MSA, in degrees), i.e., the angle with the greatest slope needed to overcome obstacles within a 15

maximum slope angle distance (MSAD, in

openness (OP, in radians), defined as a mean angular measurement of the relationships between the surface of the relief and the horizontal distances, in the eight main directions (Yokoyama et al., 2002).

Representation of the MSA, MSAD, OBS and BAR morphological variables.

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

Multiple linear regression models were built, based on the following relationship:

In order to select the best model equation, the number

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,

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

Considering the three

Regardless of the number of the variables considered, and despite the marked increase in the corresponding value of

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.

Error statistics of the multiple regression models at a national scale and for the four macro-regions described in Sect. 4.1, for

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

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

As expected, the

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

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

The second classification – GC2 – is the

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

The results for

Slope coefficients of the regression between the mean 1

The spatial distribution of the light gray zones is an important piece of information: no trend can be assumed over these areas because the

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

The same analysis was conducted on

National-scale error statistics for the 1

National-scale error statistics for the 24

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

A constant value of the index rainfall computed as the spatial average of

The different regression models used in this work to investigate the role of morphology in the spatial distribution of sub-daily annual maximum rainfall depths produced results deserving some comments. First of all, it must be mentioned that a nationwide multiple regression model that includes morpho-climatic attributes represents a significant step forward with respect to the simple regression model, as the error statistics show. In this approach, working at a national scale and given the elongated shape of the Italian peninsula, geographic location was expected to play a major role in the spatial distribution of extremes, even though this evidence was not mentioned in similar national-scale analyses (see, e.g., Faulkner and Prudhomme, 1998, for the UK, and Avanzi et al., 2015, for Italy). The role of geography progressively weakened while seeking further improvements, in terms of MAE and RMSE, through the application of distinct multiple regressions to four macro-regions, i.e., the Alps, peninsular Italy and the main islands (Supplement no. 5).

Our findings show that while the 24

While the multivariate regression can be a good tool to express geographic dependence, and on 24

The better suitability of the application of multiple regressions on four regions is confirmed by the increase of the adjusted coefficient of
determination (

Error statistics for the 1 and 24

The subsequent investigations undertaken in Sect. 5 descend from the above considerations; i.e., the building of regressions in morphological regions
that are a fraction of the whole area is an attempt to overcome the highlighted lack of regularity in the dependence between rainfall and
geography. Among all the considered geomorphological classifications, the selection of rain gauges for the model application is more effective in the
case of GC4 (Alvioli et al., 2020a), which also embeds hydrographic information. The GC4 model behaves reasonably well for both the 1 and 24

Analyzing the error statistics computed globally at the national scale, it seems that the four-region multiple regression approach is the most
precise. However, this is not necessarily true at a local scale. In order to clarify the drawbacks that large-scale regression models can produce, for
the 1

The 1

An additional comparison was undertaken to investigate the local bias. In this case, we computed the bias for each subdomain of GC4. We compared the
bias values obtained using the following four conditions: (1) the national-scale simple regression model (Eqs.

The results are illustrated in the maps of Fig. 6, which shows the best regression model for each area in different colors: Fig. 6a and b are related
to

From Fig. 6, it is possible to conclude that the morphological subdivisions allow a set of simple linear regression models to be built that can perform better almost everywhere than the other wide-area models in terms of local bias.

Absolute bias assessment for all the regression models used for the 1

In this paper, we have analyzed the role of orography and morphology in short-duration annual maximum rainfall depths, taking advantage of a new and
comprehensive database for Italy,

The results described in this paper show that a national-scale simple regression model of the precipitation vs. elevation presents some weaknesses (high residual values, high local- and national-scale bias, and low NSE coefficient, etc.) and therefore needs to be improved.

The use of multiple regression models introduces some benefits, such as a reduction of MAE and RMSE at the national scale; nevertheless they were not successful in reducing the local bias.

Considering the necessity of working on smaller domains, we analyzed several geomorphological classifications which are able to preserve the intrinsic value of the statistically significant landscape variables that emerge in regression models. Four different geomorphological classifications available in literature were used to provide criteria for the identification of homogeneous regions. We applied simple linear regression models over these homogeneous domains and compared the performances at both a national and a local level. Among all the considered classifications, the selection of rain gauges for the model application was found to be more effective in the case of GC4 (Alvioli et al., 2020a), which embeds hydrographic information.

The best approach was selected by evaluating the error statistics for the bias at both a national and a local scale and at a national scale for
MAE, RMSE and NSE. The obtained results have shown that using simple linear regression applied to the GC4 model produces better results
than all the others, in the areas in which the GC4 model is statistically significant, in terms of bias. As far as national statistics are concerned,
considering the mean rainfall depths in the gray areas in Fig. 5b does not significantly affect the performance of GC4, in terms of MAE,
RMSE and NSE, in particular for the 1

This work has led to the following conclusions. The relationship between precipitation and elevation is not meaningful in all the areas in Italy, as
already pointed out by Caracciolo et al. (2012) for the island of Sicily. In this work, this concept has systematically been extended to the whole
country, and significant relationships have only been obtained for 45 % of the area for

Details regarding the model based on GC4 and numerical values of the regression parameters are provided in the Data availability section.

The Iwahashi and Pike geomorphological classification (GC1) is available at

The rainfall data were obtained from the

The model based on GC4 and the numerical values of the regression parameters are available in the Supplement.

The supplement related to this article is available online at:

PC, IB, PM and MA conceptualized the paper and the methodology. PM conducted the analysis and curated the data under the supervision of PC, IB, and MA. Funding was acquired by PC and IB. PM wrote the first version of the paper, and PM, PC, IB, and MA reviewed and edited it.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors acknowledge the regional agencies involved in the management of the rain gauge networks that provided the rainfall measurements included in

This paper was edited by Efrat Morin and reviewed by Francesco Marra and one anonymous referee.