For climate change impact assessment, many applications require very high-resolution, spatiotemporally consistent precipitation data on current or future climate. In this regard, stochastic weather generators are designed as a statistical downscaling tool that can provide such data. Here, we adopt the precipitation generator framework of

Precipitation is a major component of the hydrological cycle. With global warming, the hydrological cycle is expected to intensify, and the risk associated with extreme events will increase

In Austria, where a large part of the country is covered by mountains, the local hydrological cycle depends heavily on temporal and spatial variations in precipitation. Tourism and agriculture are among the main drivers of Austria's economy, and the accessibility of water resources for human consumption and ecosystems is largely contingent on the spatiotemporal distribution of precipitation. In the Austrian Alps, studies on the observed and projected impact of climate change have shown changes in the availability of snow cover and water flux (e.g.,

A vast variety of WGs have been developed based on different approaches. The most widely used WGs are founded on a rather simplistic approach in which the sites are mutually independent in space and time. Such WGs are generally referred to as “single-site” WGs. Among the single-site WGs, the most popular are the parametric models based on

The major drawback of single-site WGs is that they are only focused on a single location; this can generate realistic data at a location, but it lacks a spatially correlated structure in the generated data. Obtaining a spatially and temporally consistent dataset – which is more realistic – from single-site models is impossible. Thus, over the past 2 decades, the focus has moved towards the development of spatiotemporal WGs, also known as “multi-site” WGs. For precipitation with its uneven nature of occurrence and intensity, it is even more challenging to model it under the condition of maintaining its spatiotemporal structure. In particular, in complex topography, such as the Alps, this task is even more challenging.

Numerous approaches have been proposed to generate spatially and temporally correlated precipitation data.

However, most of the aforementioned approaches simulate precipitation only at the locations where observations are available, and such multi-site WGs have been implemented for the Alps; for example,

Here, we propose an extension of the framework of

This article is organized as follows: Sect.

At a location

For a set of locations

As our goal is to use the model in complex topography, we introduce anisotropy in the covariance function (Eq.

Elevation dependence in the covariance structure is a natural assumption in complex terrain. In the literature (e.g.,

At the base of this model is the single-site precipitation generator based on a GLM framework (e.g.,

Note that we use a logit link function instead of a probit link function, as was the choice in the original model. The parameters

In order to spatialize the model to obtain gridded data, the estimated regression parameters at observation locations must be interpolated at grid locations. The Gaussian process allows for a spatial interpolation method called kriging; this method allows one to interpolate the model parameters

The variogram for the regression parameter associated with each of the covariates is estimated using the MLE. The nugget gets close to zero. Once the parameters of the logit model are interpolated, precipitation occurrence can be modeled at each grid point. However, the generated gridded field of precipitation occurrence, which is correlated in space, must also be correlated in time. Hence, the time-dependent covariance

Note that Gaussian process modeling is considered a nonparametric method. In a parametric model, the number of parameters remains fixed with respect to the amount of data available (i.e., number of stations in our case), whereas the number of parameters grows with the number of data points when using nonparametric methods.

We also compare the results of our model with a simulation using ordinary kriging (OK) instead of KED in our model as well as with the original isotropic model using OK and KED. This will be discussed in Sect.

To simulate spatially correlated fields of precipitation, another Gaussian process

At an individual location, the amount model is the gamma GLM with a logarithmic link function as given by

The scale and shape parameters (

The mean function

The model is implemented in a small region comprising highly complex terrain (ranging from 256 to over 3500 m a.s.l., meters above sea level) in the Austrian Alps. The area surrounds the catchment of the Oetz River, mainly in the federal state of Tyrol but also including a part of the Autonomous Province of Bolzano in northern Italy. The reason for selecting this region is that the catchment of the Oetz River is a widely researched area (e.g.,

The study area used in this work showing

In the northern part of the region, we have a dense network of stations, whereas the southern part has relatively fewer stations. The average inter-station distance between two locations is 28.15 km, the maximum inter-station distance is 72.84 km, and the minimum inter-station distance is 1.25 km. The average altitude difference between two stations is 0.605 km, while the maximum altitude difference is 2.272 km. The locations of the 29 stations are shown in Fig.

List of the 29 meteorological stations whose data are used in the study. The names in bold are the three representative stations used to illustrate the results.

The mean annual precipitation observed in the lowlands is approximately 780 mm over an average of 150 wet days per year, whereas the highest mean annual precipitation of 1320 mm over an average of 176 wet days per year is observed at the high-mountain station of Dresdner Huette. The highest number of mean annual wet days is 220 at St. Martin in the Passeier Valley in South Tyrol, with an annual average of 887 mm of precipitation.

Due to strongly different topography, a large variability in both space and time is observed in the dataset. Of the 29 stations, Prutz has the most distinct climatological characteristics. For example, Prutz has the largest variability in almost all months. Moreover, the most extreme precipitation (156.5 mm) in a day is recorded at Prutz (in July 2009), while the second highest amount of precipitation amongst the remaining 28 stations was recorded on the same day at Dresdner Huette (35.1 mm). Apart from Prutz, only Dresdner Huette recorded a daily precipitation amount as high as 120.4 mm during the 30 years of record. At the St. Leonhard im Pitztal location, there are two stations operated by two different service providers: one by the Austrian Hydrographic Service (St. Leonhard im Pitztal-1; see Table

To reduce the sampling uncertainty and increase the robustness of the observations, we increase the sample size of the observed data by considering a 7 d window centered on the day of interest. Thus, the chance that a particular date had, for example, 30 dry days by random choice is minimized, thereby avoiding the probability of a dry day being 1.0 (rather than a value such as 0.98), which is a problematic model setting.

We generate

For the Northern Atlantic Oscillation Index (NAOI) (see Sect.

Note that the observed data are from different service providers; therefore, the time of the data collection may differ, which may affect the results.

We allow several covariates in the model so that the model can capture a realistic structure of precipitation patterns over the region. This includes the day-to-day time dependence, the seasonality, and the influence of large-scale circulation. As the first covariate, we select the occurrence of precipitation on the day prior (

List of the covariates included in the occurrence and amount models (Eqs.

“

For the selection of the covariates in the model, we use both the Akaike information criterion (AIC)

The three most important covariates for the occurrence model at the majority of stations are

For the precipitation amount, we also consider all of the possible covariates described for the occurrence model. Furthermore, as above, we select the covariates using both the AIC and BIC for the amount model. Additionally, selecting the same seven covariates as in the occurrence model at the majority of stations, the second-order harmonic of sine is also selected by both the AIC and BIC (17 stations by the BIC and 16 stations by the AIC). Thus, we allow a total of eight covariates in the amount model (see Table

The correlations for the precipitation amount in the model are computed only for days when precipitation was observed.

Although the model produces daily fields of precipitation, before evaluating the model for gridded data, we first evaluate it at the individual locations for which observations are available. It is common practice for the validation of WGs that first the input statistics must be reproduced. From the simulated gridded data, the 30-year time series of daily precipitation at the nearest grid point to the observation locations is extracted from each of the

In the next step, we evaluate the model with respect to its ability to reproduce spatial statistics. Thus, gridded observed data are required. We use the Alpine Precipitation Grid Dataset (APGD)

In order to assess the interpolation accuracy, we perform a holdout cross-validation in which one or more stations are withheld from the model fitting process. We withhold the same three stations that were selected for the illustration of the results: Oetz, Pitztal Glacier, and Prutz. The model should be able to reproduce the observed statistics accurately at the withheld stations. For cross-validation, we also generate

For the uncertainty estimation in the

To quantify the model performance, along with various error metrics, we also take correlation coefficients (CCs) and coefficients of determination (

An extensive evaluation of the model-generated data is carried out here.

Daily conditional probability of a dry day following a dry day (

The same as Fig.

First, we assess the performance of the occurrence model with respect to the daily conditional probabilities on which the model was trained. These are important statistics, as they are associated with the ability of the model to reproduce dry and wet spells. Figure

The same as Fig.

At Prutz, the model performs badly for

Cross-validation produces similar results at all three sites for

To further examine the ability of the model to reproduce the observed climatology of wet days, we next consider the unconditional daily occurrence probability of wet days (

The performance metrics for daily occurrence probabilities are shown in Fig.

Another important feature of the model is its ability to simulate long sequences of wet and dry days. Here, we examine the ability of the model to simulate wet and dry spells of different lengths at individual stations. Figure

The frequency of wet spells of different lengths at three selected stations:

The same as Fig.

Figure

With respect to wet and dry spells, the model performance is mostly similar for both events (Fig.

Monthly mean precipitation (mm d

Performance metrics for the model at individual stations for the

Quantile–quantile (Q–Q) plot showing the percentiles of daily precipitation at each of the 29 individual stations (Table

An important aspect of the precipitation generator is its ability to reproduce the amount of precipitation observed at the stations. As the model for amount is the gamma distribution at the observed locations, the mean, which is the product of the shape and scale parameters of the gamma distribution, should be well reproduced. Figure

Here, we examine the Q–Q plots at each individual station for which observed data are available (Fig.

We further examine the distributions of the generated data for each month at each of the 29 stations using a Kolmogorov–Smirnov test and a Wilcoxon–Mann–Whitney test. The results are shown in the Supplement (Figs. S1, S2). As revealed by the Q–Q plots, the worst performance is observed at Prutz, St. Martin, and St. Leonhard im Pitztal-2, which have distinct climatologies.

The frequency of the areal spells of different lengths (see Sect.

Performance metrics of the gridded model for reproducing areal statistics.

In this section, we focus on evaluating the spatial dependence structure in the simulated data, which is an important aspect for many hydrological applications. Here, we present the results of the validation – first for the spatial statistics related to the occurrence and then for the occurrence and amount combined. More details on this aspect of validation are given in Sect.

One of the most challenging features of the gridded precipitation generator is its ability to reproduce the areal spells of wet and dry days of different lengths. This is one of the sought-after features in agricultural and hydrological applications. We define an areal wet (dry) spell as the number of consecutive days on which 95 % of the study area is wet (dry). Figure

For the areal spells, the performance metrics are shown in Table

The spatial distribution of the simulated probabilities of wet days (

Spatial distribution of the occurrence probability (

To test the model performance at each grid point, we upscale the simulated data from 1 to 5 km, and the difference between the observed and the simulated data is shown in Fig.

Figure

Next, we consider the mean wet-day daily precipitation in order to assess the ability of the model to reproduce the observed climatology of the precipitation amount over the region. We consider the mean daily precipitation on wet days in each season over the 30 years of the simulated 30 ensembles, i.e., 900 years of data at each grid point (Fig.

Spatial distribution of the mean wet-day daily precipitation amount for four seasons (mm d

Another important aspect to note with respect to the observed APGD data is that the probabilities of wet days in the southeastern part of the region in the colder seasons are low (Fig.

Figure

Performance metrics for the gridded model for the

Next, we assess the ability of the precipitation generator to provide an areal climatology of the precipitation amount. This is also one of the desired characteristics for impact modeling. Figure

Monthly mean areal precipitation (mm d

Here, we assess the ability of the model to reproduce extreme precipitation amounts. We consider the annual maximum daily precipitation at 29 sites. The daily sum of the observed daily precipitation at 29 sites is obtained, and the maximum of the daily sum in every year is shown as a box plot in Fig.

The annual maximum daily precipitation sum at the 29 sites for the observed and simulated data. The sum of daily precipitation at the 29 sites is obtained, and the maximum in each year is presented as a box plot. Thus, the box plot for the observed data is based on 30 values, whereas is based on 900 values (30 realizations of 30 years) for the simulated data.

Here, we compare the results of our simulations, i.e., using KED in the anisotropic model (Aniso–KED) with three different model setups: (i) considering OK in the interpolation of the parameters of the anisotropic model (Aniso–OK), (ii) using the original isotropic model that utilizes OK for the interpolation of the parameters (Iso–OK), and (iii) using the isotropic model with KED (Iso–KED). We examine the results for the monthly sum of the areal mean daily precipitation in these four simulation cases (experiments) with the observed APGD gridded data. Figure

Monthly sum of the areal mean daily precipitation (mm) for each month in the 30 years (1979–2008) of observed APGD data and the simulated 900 years (30 realizations of 30 years) of data for the four experiments: Aniso–KED, Iso–KED, Aniso–OK, and Iso–OK (see the text for specifications).

The model performance of Aniso–KED and Iso–KED is mostly similar in all months, whereas the model performance of Iso–OK and Aniso–OK is similar in all months. The model performance varies greatly from month to month for all four experiments, but both experiments involving KED (Aniso–KED and Iso–KED) outperform those using OK in the vast majority of months. It is evident that allowing elevation as a covariate in the kriging interpolation for prediction at each grid point appreciably improved the amount of precipitation in most months.

The median for both of the KED experiments is overestimated in February, March, May, and September, whereas the median is underestimated in the rest of the months. The IQR in Fig.

The differences in the observed and simulated median and IQR in each month for each of the four experiments are listed in Table

Difference (simulated

As for the occurrence model, the covariance structure has a slight influence on the model performance. Figure

The frequency of areal spells (

In this study, we have tested two extensions of the original isotropic framework of

The major improvement in the results from our model comes from the KED interpolation, rather than the included anisotropy in the covariance structure. This suggests that there is no strong directional dependency in the precipitation simulation. Although there are minor differences in model performance using the isotropic and anisotropic covariance functions, it can be concluded that an isotropic covariance function is sufficient, even for small-scale topographic variability such as that in the present study in the European Alps. However, the topographical details must be included in the interpolation of the parameters of the model. Similar results can be expected for complex terrain in other mountainous regions.

At individual locations with observations, the model satisfactorily reproduces various observed statistics and the overall distribution of precipitation. The model is also able to capture spatial and temporal variability over the entire region reasonably well. It is capable of simulating the dry-day statistics over the whole region very well; however, an underestimation is observed for the wet-day statistics. The frequency of the areal dry spells of 1 or 2 days is strongly overestimated. The model uses the previous day's occurrence as a covariate, which creates a first-order two-state Markov chain at an individual location.

The model captures the month-to-month variability in the monthly sum of precipitation very well due to the use of the time-dependent harmonics of sine and cosine as covariates in the modeled spatial covariance structure. However, the inter-annual variability is largely underestimated, mainly for the colder months. Even if we adopt the NAOI as a covariate to alleviate the well-known problem of overdispersion in this type of model, overdispersion remains an issue. One reason for this is the tendency of the model to underestimate large daily precipitation amounts. This is because the model generates the precipitation amount using a transformed Gaussian process that reduces to a gamma distribution at individual locations. The gamma distribution is not a heavy-tailed distribution and is, therefore, not well suited to reproducing heavy precipitation. However,

In complex mountainous terrain, individual stations can exhibit precipitation characteristics quite distinct from those of neighboring (or more distant) stations with more typical characteristics. A station-by-station evaluation (Sect.

However, if a distinct station is located in a data-sparse area (such as St. Martin in our study area), it dominates the entire neighboring region and destroys the spatial structure. Thus, for a spatial precipitation generator in complex terrain, the stations should not only be selected according to data availability (and quality) but also based on their precipitation characteristics. If they have distinctly different precipitation characteristics from the majority of the stations in the region, they should not be included in the training dataset, and if one is explicitly interested in such a station, one should use a single-site approach.

Another limitation of this model is its inability to realistically simulate autumn and winter precipitation. This is because there are systematic differences in the characteristics of weather types between various seasons. In autumn and winter, westerly currents are stronger and the associated precipitation patterns are more pronounced than during spring and summer. The precipitation pattern in winter is associated with dynamically active synoptic-scale weather systems (fronts and low-pressure systems specifically from the North Atlantic Ocean and the Adriatic Sea) in combination with orographic enhancement, whereas the pattern in summer is related to convective activity that is either embedded in frontal systems or generated locally. Our model does not account for the influence of wind on precipitation. This could be the reason for the model not being capable of capturing the spatiotemporal patterns in autumn and winter. The convective season in Austria usually starts in May and lasts until September, and the model successfully captures the spatiotemporal patterns during these months.

The covariance function used in the model is assumed to be stationary, which may not be a realistic assumption. Detecting spatial nonstationarity and modeling it is beyond the scope of this article and will be explored in future research. However, it is possible that the model performance may be improved by considering a nonstationary covariance function (e.g.,

A multi-site gridded precipitation generator that provides high-resolution two-dimensional fields of precipitation in complex terrain using historical observations from a network of meteorological stations is developed, implemented, and evaluated. The precipitation generator is an extension of the original framework of

The model is tested in a small region (about

The main findings of this study can be summarized as follows:

At individual stations where observations are available, the model reproduces the observed statistics realistically well, including annual cycles of daily probabilities of precipitation occurrence and monthly means of precipitation, dry and wet spells of different lengths, and the overall distribution of precipitation amount.

The model has a great capability to capture the spatiotemporal statistics in complex terrain, including the spatial distribution of occurrence probabilities and amount, areal dry and wet spells of different lengths, monthly mean areal precipitation, and monthly sum of areal daily mean precipitation.

The proposed extensions considerably improve the simulation of the spatiotemporal fields of precipitation – mainly due to the incorporation of elevation in kriging.

The use of an isotropic or anisotropic covariance function in the mountainous region is equally good, with marginal trade-offs for some of the statistics.

The performance of the model varies greatly from month to month, being the best in summer and the worst in autumn.

Intra-seasonal and inter-seasonal variabilities are well reproduced, whereas inter-annual variability is largely underestimated in autumn and winter.

At a few of the 29 stations, where the observed precipitation statistics and in particular their seasonality were distinctly different from all of the other stations, the model performance is markedly compromised.

The underestimation of large amounts of precipitation remains a problem.

Reproducing the spatiotemporal fields of precipitation in a region characterized by complex terrain like the Alps is a challenging task, especially at locations where no observations are available. However, this is an essential requirement for hydrological modeling, as hydrological models are driven by spatially and temporally coherent precipitation data. The proposed model can respond to this need to some extent; nevertheless, further improvement is required, as discussed in the article, to employ the model for downscaling purpose.

The data used in this study are available from the corresponding author upon request.

The supplement related to this article is available online at:

All of the co-authors contributed to developing the idea. HPD developed the code, prepared the input data, performed the simulations, carried out the analysis, prepared the visualizations, and interpreted the results. MWR supervised the work, and MO provided vital suggestions. The manuscript was prepared by HPD and finalized by all co-authors.

The contact author has declared that none of the authors has any competing interests.

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

The authors are grateful to the following data providers: the Austrian National Weather Service (ZAMG – Zentralanstalt für Meteorologie und Geodynamik), the Institute of Atmospheric and Cryospheric Sciences – University of Innsbruck, the Austrian Hydrographic Service, TIWAG (Tiroler Wasserkraft AG), and the Hydrographic Service of the Autonomous Province of Bolzano. The authors acknowledge the partial financial support provided by the University of Innsbruck to the first author.

This research has been supported by the Provincia autonoma di Bolzano – Alto Adige (CYCLAMEN).

This paper was edited by Nadav Peleg and reviewed by Dongkyun Kim and two anonymous referees.