Spatial downscaling of precipitation from GCMs for climate change projections using random cascades: A case study in Italy



[1] We present a Stochastic Space Random Cascade (SSRC) approach to downscale precipitation from a General Circulation Models (GCMs), developed for the assessment of water resources under climate change scenarios for the Oglio river (1440 km2), in the Italian Alps. The snow-fed Oglio river displays complex physiography and high environmental gradient and statistical downscaling methods are required for climate change assessment. First, a back cast analysis is carried out to evaluate the most representative within a set of four available GCMs (R30, ECHAM4, PCM, HadCM3). Monthly precipitation for the window 1990–2000 from 270 gauging stations (one every 25 km2) in northern Italy is used and scores from objective indicators are calculated. The SSRC model is then tuned upon the Oglio river catchment for spatial downscaling (2 km2) of daily precipitation from the NCAR Parallel Climate Model, giving the comparatively best results for the area. Scale Recursive Estimation coupled with the Expectation Maximization algorithm is used for model estimation. The seasonal parameters of the multiplicative cascade are accommodated by statistical distributions conditioned upon the climatic forcing, based on a regression analysis. The SSRC approach reproduces well the spatial clustering, intermittency, self-similarity, and spatial correlation structure of precipitation fields, with relatively low computational burden. Downscaling of future precipitation scenarios (A2 scenario from the Parallel Climate Model) is then carried out and some preliminary conclusions are drawn.

1. Introduction

[2] Global warming greatly impacts the climate of mountain areas in temperate regions and the water resources distribution therein [see, e.g., Barnett et al., 2005; Solomon et al., 2007]. Hydrologists are therefore required to make accurate predictions of the impact of climate change on the intensity, amount, and variability of precipitation and their fallout upon streamflow regime [Kang and Ramírez, 2007]. For this purpose, the local evaluation of climatic trends is coupled with scenarios from climatic models [see, e.g., Drogue et al., 2004; Gangopadhyay and Clark, 2005; Kang and Ramírez, 2007] to provide the climatic input for medium to long term impact analysis on water resources [see, e.g., Bultot et al., 1992; Beniston, 2003; Hagg and Braun, 2005; Hagg et al., 2007], hydrological extremes [see, e.g., Burlando and Rosso, 2002a, 2002b; Boroneant et al., 2006], and on the habitats of animal and vegetal species [see, e.g., Gottfried et al., 1999; Keller et al., 2005].

[3] General Circulation Models (hereon, GCMs) and Limited Area Models (hereon, LAMs) are physically based tools presently used in predicting climate change effects [see, e.g., Bardossy, 1997; Bates et al., 1998]. GCMs deliver meteorological variables in a fine time resolution (30 min. to a few hours) but in a usually coarse spatial grid (50–500 km), while in LAMs a finer computational grid over a limited domain is nested within the coarse grid of a GCM. Although GCMs and LAMs perform reasonably well in simulating synoptic atmospheric fields, they usually reproduce poorly the statistics of historical records at the spatial scales of interest in the analysis of the impacts [see, e.g., Gangopadhyay and Clark, 2005] and proper tailoring is required for local use, before any accurate guess about hydrologic cycle can be ventured [see, e.g., Lammering and Dwyer, 2000; Burlando and Rosso, 1991, 2002a].

[4] Downscaling in space and/or in time of back calculations outputs from climatic models (i.e., typically rainfall and temperature) requires appropriate data assimilation schemes [see, e.g., Bocchiola, 2007; Kang and Ramírez, 2007]. Then future projections from the same GCMs can be downscaled, in the hypothesis that the downscaling rules are preserved in the near future [see, e.g., Stehlik and Bardossy, 2002]. Downscaling plays a major role in mountain areas, where snow precipitation is strongly controlled by topography, poorly represented in GCMs, and it has a considerable bearing upon the hydrological budget [see, e.g., Beniston, 2003] and in the dynamics of habitat formation for several plant and animal species [see, e.g., Theurillat and Guisan, 2001]. Downscaling of GCM-based precipitation data seems useful in climatologically driven hydrological simulation, because it may support the description of the hydrological fluxes at the daily scale [Lammering and Dwyer, 2000] and the use of remote sensing images for water balance of snow fed watersheds in temperate regions [see, e.g., Ranzi et al., 1999; Corbari et al., 2009].

[5] Most previous studies addressing streamflow formation via use of precipitation from GCMs used downscaling schemes that do not account for the intermittency and self-similarity properties of precipitation in space and time, which are key features of precipitation fields [see, e.g., Lovejoy and Schertzer, 1990; Over and Gupta, 1994, 1996; Harris et al., 2001]. A class of methods that accounts for these properties are those facing statistical downscaling via Stochastic Space Random Cascade approach (SSRC) [Tessier et al., 1993; Over and Gupta, 1994, 1996; Menabde and Sivapalan, 2000; Veneziano and Langousis, 2005; Veneziano et al., 2006]. SSRC approach was already used for downscaling of precipitation from GCMs for climate change projections [Kang and Ramírez, 2007], as well as for improvement of water balance estimation [see, e.g., Lammering and Dwyer, 2000].

[6] A considerable effort has been recently devoted toward the development of multiscale data assimilation schemes using scale recursive estimation, based upon the SSRC theory. This novel approach makes model estimation possible when information from multiple sources is available, which includes imperfect (i.e., noisy) measurements from the real world [Bocchiola and Rosso, 2006; Gupta et al., 2006; Bocchiola, 2007].

[7] This paper addresses the downscaling (2 km2) of precipitation for the 1440 km2 Oglio river basin (closed at Costa Volpino), in the Retiche Italian Alps (Figure 1). This study is carried out under the umbrella of the CARIPANDA project, funded by the Cariplo Foundation of Italy, and aims to evaluate scenarios for water resources in the Adamello Natural Park of Italy, which is located within the Oglio river watershed, and projecting out 50 years or so (until 2060). The Park also includes the Adamello Group, made of several glacierized areas (about 24 km2), including the largest Italian glacier, named Adamello, covering an area of about 18 km2 [see, e.g., Maragno et al., 2009].

Figure 1.

The study area: Oglio river basin closed at Costa Volpino, in the Retiche Italian Alps. Cells are PCM mesh, dots are rain gauges.

[8] First, one has to identify the GCM that best reproduces precipitation in the examined area. Four models (R30, ECHAM4, PCM, HadCM3) are selected here from the literature. To rank these models we introduce a back cast procedure based upon the comparison against the baseline. Monthly precipitation for the period 1990−2000 from 270 gauging stations (one every 25 km2) in Northern Italy is used for the purpose.

[9] We then focus on the Oglio river basin to downscale the precipitation input given by the chosen GCM (PCM from NCAR). For model estimation, we use a 10 year (1990−1999) series of observed daily precipitation data from 25 rain gauges within the watershed. The multiplicative bias between the PCM model and the local precipitation mean are estimated and accommodated using a seasonal statistical approach. Then, we use a multiplicative random cascade SSRC model with intermittence to mimic the spatial distribution of rainfall. Model estimation is performed using scale-recursive estimation coupled with the expectation maximization algorithm, henceforth referred to as SRE and EM respectively [Bocchiola and Rosso, 2006; Bocchiola, 2007]. SSRC models are usually estimated against spatially distributed (i.e., gridded) precipitation estimates, e.g., from ground radar and satellites [see, e.g., Over and Gupta, 1994; Kang and Ramírez, 2007; Bocchiola, 2007]. Seldom, however, are acceptably long and accurate series of observed precipitations available under this form. Here, no long term radar or satellite precipitation estimates are available, and the SRE-EM approach is innovatively used for SSRC model estimation from sparse rain gauge data.

[10] Statistical downscaling is generally carried out using the dependence of the model parameters on an external forcing, e.g., on atmospheric circulation [Katz and Parlange, 1993; Over and Gupta, 1994; Perica and Foufoula-Georgiou, 1996; Stehlik and Bardossy, 2002]. Over and Gupta [1994] found a dependence of the beta model parameter for intermittence of SSRC upon the external forcing given by the synoptic-scale precipitation. Perica and Foufoula-Georgiou [1996] performed statistical downscaling using wavelets, including the dependence of the (additive) weights against CAPE index. Here, the seasonal parameters of the multiplicative cascade are accommodated by statistical distributions conditioned on the precipitation intensity, based on regression analysis. We then use the model for downscaling future precipitation scenarios, necessary for water budgeting pending climate change.

2. Case Study Area

[11] The study area (Figure 1) covers the mountainous part of the Lombardia region, in the central Alps and pre-Alps. This includes the upper Oglio snow-fed watershed, which is about 1440 km2 in area. The Oglio river basin is the major tributary to the Iseo Lake and its emissary, the lower Oglio, is a left-hand tributary of the Po River. Elevation ranges from 186 m above sea level (asl) at Costa Volpino to 3.538 m asl of the Adamello peak.

[12] The Oglio river valley displays an alpine climate with very cold winter and moderate summer temperatures, considerable solar radiation, and a high frequency of clear sky conditions, especially during winter. Noticeable wind circulation is provided by orographic interaction with general circulation. Average annual precipitation in the area is about 1300 mmy−1. Snowfall is frequent from October to May, and the snow cover generally persists in time, particularly in the northeast area, due to the effect of cold air masses driven from the Pisgana glacier and the large snowfall amounts from northeast through the Tonale Pass. The precipitation regime according to the Köppen-Geiger climate classification [e.g., Peel et al., 2007] belongs to the temperate/cool continental class, featuring seasonal continuous snow cover above 1000 m asl or so and a maximum of precipitation toward the end of the summer and fall and a minimum during the winter. Runoff is mainly influenced therein by snow melt in Spring and by rainfall in early Fall.

3. Database

3.1. Data Sources and Investigated Periods

[13] Precipitation data from different sources for two different periods are used, namely (1) observed data from period 1990 to 2000, (2) historic control data from GCMs from 1990 to 2000 and (3) scenario data generated by the PCM model following the IPCC A2 scenario, covering the 50 year period from 2010 to 2060.

3.2. Observed Rainfall Data, 1990–2000

[14] We collected data on 11 years (1990−2000) of daily precipitation RG from 270 gauges in northern Italy. The database integrates the institutional database from the Regional Agency for Environmental Protection (ARPA) with those operated by private hydropower companies, namely A2A and ENEL, those by Adda and Oglio Irrigation Authority, and the data available from Annali Idrologici by the former SIMN (Servizio Idrografico e Mareografico Nazionale) of Italy. The observed precipitation data for the Oglio catchment are used in this study for a threefold purpose, namely, (1) to analyze the performance of the GCMs in depicting the present-days climate within northern Italy, (2) to calibrate the SSRC approach for the Oglio river catchment, and (3) to evaluate the projected future precipitation in the Oglio river basin, and to compare these against the already observed past trends in the area [e.g., Bocchiola and Diolaiuti, 2010]. We estimated the mean area precipitation, RGA, to evaluate the baseline, for benchmarking the rain rates from the GCM under exam, RGCM, during 1990−2000. To calculate the area averaged value RGA, we used simple average of the observed daily data. As expected, a preliminary analysis carried out using kriging and the input data from Global Precipitation Climatology Center (GPCC) showed in practice no significant difference (not shown here, reported by Groppelli and Pengo [2005]), because the mean areal precipitation at the monthly scale displays a low sensitivity to the estimation procedure.

3.3. Benchmarking of GCMs Outputs, 1990–2000

[15] In studying the regional impact of climate change, one needs to assess the GCM capability for describing the local meteorological conditions [Salathé, 2004]. For detailed applications like the present one involving statistical downscaling, the computational demand may imply the need of considering only a limited set of models and scenarios [Salathè et al., 2007], and one must select the GCM outputs which best depict the observed climate of the region under exam. We evaluated the performance of four GCMs in back casting mode, i.e., by comparing the models' estimation of precipitation, RGCM, against the observed rainfall series, RGA, for the period 1990−2000. Four different GCMs were selected, namely, (1) ECHAM4 by the Max Planck Institute for Meteorology (henceforth ECHAM4) [Roeckner et al., 1999], (2) HadCM3 by the Hadley Centre for Climate Prediction and Research (henceforth HadCM3) [Mitchell et al., 1998; Gordon et al., 2000], (3) Parallel Climate Model by the National Centre for Atmospheric Research (henceforth PCM) [Washington et al., 2000; Meehl et al., 2000], and (4) R30 by the Geophysical Fluid Dynamics Laboratory, USA (henceforth R30) [Delworth et al., 2002]. The IPCC Data Distribution Centre made the 1990–2000 outputs from these models available to the scientific community after the publication of Houghton et al. [2001].

3.4. Scenario GCMs Data, IPCC A2

[16] The Special Report on Emission Scenarios, SRES, by the Intergovernmental Panel on Climate Change [Nakicenovic and Swart, 2000] described four possible future storylines (A1, A2, B1, B2), each one referring to the effect of different potential causes of greenhouse gases (hereafter GHG) emissions and to their possible future dynamics. Four different scenarios were defined based on the storylines, considering possible demographic, social and economical evolution trends and technological developments as causes for future GHG emissions. Each SRES scenario family assumes one out of the four possible storylines and does not include additional climate initiatives, which means that no scenarios are included that explicitly assume implementation of the United Nations Framework Convention on Climate Change or emission targets defined by the Kyoto Protocol [Houghton et al., 2001]. We used the data generated via the IPCC SRES A2 Scenario, covering the 50 year period 2010−2060, as described by Beniston [2004, p. 90]: “A2 scenarios assume little change in economic behavior. In addition, rising population levels and relatively little international collaboration on resource and environmental protection exacerbate the problem of emissions; the A2 are sometimes referred to as ‘Business-as-usual,’ a phrase that was coined for one of the previous sets of IPCC scenarios”.

[17] We considered the period 2010–2060 to analyze the future climate and environmental conditions centered around 2050, i.e., to test the effects of potential climate change at the middle of the current century.

4. Methods

4.1. GCM Selection

[18] First, we carried out a qualitative analysis of the fitting of the average monthly precipitation from each GCM against the average monthly precipitation retrieved by averaging rain gauges observations (Figure 2). One notes that PCM and ECHAM4 reproduce more closely the observed bimodal precipitation patterns, displaying two maxima, in the fall (September−November) and in the spring (March−May). Extreme storms (and floods therein) in this area occur during fall, thus claiming for appropriate downscaling, to mimic rainfall variability in time. All GCMs also overestimate precipitation during winter (January−March).

Figure 2.

Average monthly precipitation from the chosen GCMs against the observed values from the rain gauge network for Northern Italy.

[19] We carried out a number of objective tests to assess the GCMs predictions. We first considered yearly total precipitation Py, i.e., the year-round sum of RGA and RGCM, Py(RGA) and Py(RGCM), respectively. We performed a Student's t (same or different variance) test for the mean of the observed and predicted values of Py, and Fishers' F test for the variance. We performed the Fisher's F test to discriminate between the two cases of same and different variance for Student's t test, and then we carried out the t test accordingly [Chow et al., 1988; Kottegoda and Rosso, 2008]. Here we used accumulated precipitation data P, obtained as a sum of the daily values. When using sums of independent variables, due to the central limit theorem [e.g., Kottegoda and Rosso, 2008] the distribution of these sums tends to a normal one, no matter what the initial distribution of the variables is. Therefore, we may hypothesize here that P is approximately normally distributed, as required for application of Student's t and Fisher's F tests. Notice that while the application of these tests to monthly precipitation seems reasonable in a relatively wet climate as here (i.e., with many rainy days, up to 50% or so), the convergence to a normal distribution may not equally hold in drier regions (i.e., with few rainy days).

[20] The variance of Py is well reproduced by all the GCMs equation image, but PCM and ECHAM4 only provide acceptable estimates of the mean, whereas HadCM3 and R30 overestimate and underestimate the observed annual mean precipitation, respectively. The results of the benchmarking of annual precipitation are shown in Table 1.

Table 1. Assessment of Yearly Accumulated Precipitation From GCMs, Py(RGCM), Against Its Observed Value Py(RGA)a
GCMsSD Py(RGA) (mmy−1)SD Py(RGCM) (mmy−1)Fisher's F TestMean Py (RGA) (mmy−1)Mean Py(RGCM) (mmy−1)Student's t Test
  • a

    Fisher's F and Student's t Test used. Italicized values are not significant results equation image.


[21] We further investigated monthly precipitation Pi, for i = 1−12. We considered here the sample mean and variance of the observed and predicted monthly values, i.e., the monthly sum of RGA and RGCM, Pi(RGA), and Pi(RGCM). The results are reported in Table 2. The performance in reproducing the observed fall precipitation is somewhat poor for all the investigated GCMs outputs. However, the best overall performance is provided by the PCM, which only fails in fulfilling the average rainfall test in three months (February, September, and October) out of twelve reported in summary of Table 2. The PCM and ECHAM4 models also provide the least monthly RMSE value on average, as reported in Table 2. The above analysis relies on a relatively small data sample (11 years), but still one realizes that GCM outputs are not very accurate in reproducing the observed monthly precipitation in the area under exam. Notice further that here we use the 11 year database to (1) evaluate the fitting between NCAR-PCM model and ground precipitation, and (2) calibrate the cascade model for downscaling purposes. So we do not perform here any analysis concerning modified climate, which would require a longer series. Indeed, we charge upon the GCM model the task of providing a credible projection of future climate, while the SSRC is used only for the purpose of downscaling. However, our benchmarking exercise indicated that the PCM model is somewhat better than the other model in representing the dynamics of precipitation in this area. Therefore, we selected the PCM model to produce precipitation scenarios to be downscaled here.

Table 2. Assessment of Monthly Accumulated Precipitation From GCMs, Pi(RGCM), Against Its Observed Value Pi(RGA)a
MonthSD Pi(RGA) (mmd−1)SD Pi(RGCM) (mmd−1)Fisher's F TestbMean Pi(RGA) (mmd−1)Mean Pi(RGCM) (mmd−1)Student's t TestcRMSE (mmol)
  • a

    Fisher's F and Student's t Test used. Italic values are not significant results equation image. Summary of Fisher's F and Student's t Tests (months with statistics significantly different) is reported below. Root mean square error RMSE of estimation of monthly accumulated precipitation by the GCMs is also provided. Monthly value is i=1:12.

  • b

    Summary for Fisher's F Test: R30, 2; ECHAM4, 2; HadCM3, 3. and PCM, 6.

  • c

    Summary for Student's t Test: R30, 7; ECHAM4, 4; HadCM3, 4; and PCM, 3.


4.2. SSRC Downscaling of Daily Precipitation From PCM

[22] The PCM data are available at a daily temporal scale, which is coherent with hydrological simulation in the Oglio watershed. However, the PCM provides estimates of average rainfall intensity over an area of about 200 km2, which is too coarse to provide reasonable inputs to hydrological models, so one needs a downscaling procedure. Because we focus here on the precipitation within Oglio river, we considered a subsample of precipitation data, i.e., from 25 of the most complete data sets of precipitation gauges among those available within the Oglio watershed (Figure 1).

[23] We here assume that a rain gauge value is representative for a 2 km2 area. This represents a tradeoff between the need to keep a small representative area, and the computational time required for EM algorithm and downscaling development. Also, by using this method each rain gauges would be laid within a different cell in our grid, and we could fully profit from the information from each station.

4.3. Bias of Daily Precipitation in the Oglio Watershed

[24] Here, we perform downscaling of daily precipitation within the Oglio river. First, we compute the baseline average areal precipitation in the Oglio watershed using the 25 rainfall gauges, namely RGAO. Visual analysis of the daily area average precipitation series from the PCM, RGCM, against the observed baseline, RGAO, shows quite different patterns. Baseline precipitation displays considerable intermittence, i.e., the presence of several dry days and of considerably long dry spells [Kottegoda et al., 2007] and high precipitation amounts during wet spells. The predicted precipitation RGCM displays instead much lower intermittence (i.e., many more wet spells), with lower rates.

[25] Because the preservation of the intermittence of rainfall in time (i.e., the sequence of dry-wet spells) is a fundamental issue in order to investigate hydrological cycle, one needs to introduce some additional analysis to correct daily values of precipitation RGCM, so as to obtain daily precipitation scenarios consistent with the baseline in term of intermittence and of average daily precipitation. The transition from the daily area average precipitation of GCMs, RGCM, to that observed upon the Oglio catchment, RGAO, is modeled here using a random multiplicative process including intermittence, termed BiasGAO

equation image

where BGAO, p0, and equation image are the model parameters to be estimated from data. The term BGAO is a constant, forcing the average daily value of RGAO to equate its sample value because of the GCM underestimation of precipitation during wet spells. The term B0 is a equation image model generator [see Over and Gupta, 1994]. It gives the probability that the rain rate RGAO for a given day is not zero, conditional upon RGCM being positive, and it is modeled here by a binomial distribution. The term W0 is a “strictly positive” generator. It is used to add a proper amount of variability to precipitation during spells labeled as wet. This approach is used for consistency with the SSRC approach, as described further on.

[26] Model bias may arise from various effects and may not necessarily indicate that the GCM cannot correctly capture the large-scale climatic signal. Albeit the general circulation influence upon precipitation is correctly brought about, the simplified depiction of rugged local topography may lead the GCM to mimic rainfall process as smoother than observed in truth [Salathè et al., 2007].

[27] The application of equation (1) to daily precipitation implies that daily values of BiasGAO are independent from each other. We carried out a preliminary analysis concerning the correlation structure of daily precipitation upon the area, RGAO. First, the duration of the observed dry and wet periods was investigated [e.g., Kottegoda et al., 2000]. Linear correlation analysis was carried out to evaluate the dependence between dry and wet periods duration, which resulted in negligible signature upon each other. Also, we investigated the dependence between the duration of each dry period (i.e., consecutive days of dry weather) and the subsequent rainfall RGAO in the first wet day. Then, we investigated the dependence between RGAO in two consecutive wet days. Both these analysis yielded no significant dependence arising from the data. Also, we found no significant linkage between the duration and the intensity of wet periods (which would imply correlation between the daily rainfall values within a wet spell). Therefore, one can conclude that using BiasGAO as defined here is consistent with the observed daily rainfall field.

4.4. Random Cascade

[28] The variability in space of the daily rain rates is modeled here using a homogeneous SSRC [Schertzer and Lovejoy, 1987; Gupta and Waymire, 1990, 1993; Over and Gupta, 1994, 1996; Deidda, 2000]. Spatial rainfall distribution is modeled as a branching tree structure [see, e.g., Bocchiola, 2007; Groppelli et al., 2008]. Each layer in the tree represents a lattice, where the size of the cells (or nodes) is coincident with the resolution (or scale) associated with the samples of the rainfall field obtained with some measurement device(s), or in some way estimated. The node at the coarsest resolution is called “root” node, while the nodes at the finest resolution are called “leaves.” Named R0 the average rainfall rate at the synoptic scale, the dimensionless rainfall rate in any cell, indexed by i, at a generic scale, s, namely equation image is

equation image

[29] X0 is X at a root node and the operator Y is a “generator” of the cascade at a given scale, with statistics

equation image

[30] Equation (3) yields the “average” conservation of the cascade mass (i.e., a “canonical” cascade), [see, e.g., Gupta and Waymire, 1990, 1993; Over and Gupta, 1994]. Rainfall processes display spatial intermittence, this meaning that the process has a finite probability mass at zero [Kedem and Chiu, 1987; Kumar and Foufoula Georgiou, 1994; Shimizu, 2003]. Based upon Over and Gupta [1996] we modeled the cascade generator here as the product of two independent generators, as

equation image

where Ws is a “strictly positive” generator, modeling the rainfall process for the rainy areas, and Bs is a equation image model generator, i.e., the probability that rain rate in a cell at scale s is non null, conditional on its parent being positive

equation image

where b is the branching number (i.e., number of “children” generated from one coarser scale to the finer one) and equation image is a parameter estimated from the data set [as by Over and Gupta, 1994]. In the rainy areas, one also needs to represent the pdf of W. This is modeled here as a scaled lognormal variable [see, e.g., Marsan et al., 1996; Over and Gupta, 1994; Tustison et al., 2003; Bocchiola, 2007] as

equation image


equation image

[31] One notes that the lognormal distribution is capable of representing the observed rainfall variability in space [see, e.g., Smith and Krajewski, 1993; Seo et al., 1999] and of accommodating the SRE procedure [Primus Gorenburg et al., 2001; Bocchiola, 2007].

4.5. Model Estimation

[32] The SSRC model used here is homogenous in space by construction, because the cascade weights' statistics do not depend upon the position in space but only upon the scale. Therefore, it may miss spatial variation of precipitation with topography (e.g., for different altitudes). However, as a preliminary we carried out an analysis aimed at evaluating the variability of observed precipitation with altitude. We evaluated the seasonal accumulated amount of precipitation within the available rain gauges and we tentatively highlighted the relationship of its second order statistics (mean, variance) against altitude (ranging from 200 m asl to 2600 m asl). Neither noticeable dependence upon the altitude was observed by a qualitative analysis, nor any significant regression coefficients were highlighted (5% confidence level). Albeit within storm variation with altitude may occur, on the average no significant drift of precipitation with altitude is seen in any season. We therefore rely upon the assumption of homogeneous precipitation in space. Notice that a downscaling approach using the SSRC and taking into account altitude drift may be setup with reasonable burden. This may be done by calculating the mean precipitation within each cell (with given altitude) by explicitly scaling the overall (i.e., upon the watershed) observed (or given by BiasGAO) mean value against altitude. Then, one can estimate the X values in equation (2) for each cell by dividing the observed rainfall therein for the average rainfall as expected at the altitude of the cell. This would result again into a homogeneously distributed cascade (of X dimensionless values), but based upon a support (dimensional values within the cells) which takes into account altitude. However, we decided not to use this procedure here, because no considerable drift was observed, as reported.

[33] To estimate BGAO and SSRC model parameters we used an analogous approach. BGAO in equation (1) was estimated as the mean value of the ratio of the average rain gauge observed daily precipitation, RGAO, to the PCM control run precipitation, RGCM, for the period 1990−1999. We only considered those days when non null rainfall was flagged by both the gauge network and the PCM. Whenever RGAO was null, BGAO was set to 0, thus indicating intermittence in time. When RGCM was 0 but RGAO was not (occurring in 5% of the cases when RGAO > 0), we did not use the data within BGAO statistics estimation. Because PCM displays on average a considerably larger number of storms (i.e., of wet days) than observed in actuality, it is more important to discriminate the case when PCM indicates a false storm event, than vice versa. This is because one needs to operate a correction upon GCM's output to decrease the projected number of wet days when analyzing future precipitation scenarios. The estimation of p0 is performed using the observed probability of non null rainfall upon the gauges, conditioned on GCM's non null rainfall, and one can use this estimated value for simulation. We did this seasonally, because a preliminary analysis showed that the average duration of dry spells depends upon the season.

[34] The variance of W0, namely equation image, was also estimated seasonally, using a maximum likelihood (henceforth referred to as ML) approach, performed using a modified version of the SRE-EM algorithm in time [Bocchiola and Rosso, 2006]. The EM algorithm was shown to be the most accurate among those used hitherto for model estimation aimed to SRE [e.g., Gupta et al., 2006; Bocchiola, 2007]. Here, we assumed that W0 is stationary in time during a given season, so we can use daily observations of RGAO and RGCM to increase the sample dimensionality for model estimation. The estimated parameters are reported in Table 3.

Table 3. BiasGAO Parametersa
SeasonBGAO (.)equation image (.)FWAG (%)FWAS (%)c (.)k (.)Rmax (mmd−1)p-value (.)
  • a

    Estimated parameter c (equation (8)) and related average spatial intermittence, observed, FWAG, a simulated, FWAS, and Regression Parameters p/RGAO (equation (9)). Italicized values are not significant results equation image.


[35] One notes that the SSRC model is usually tested and tuned against spatially distributed remotely sensed precipitation estimates, e.g., from ground radar and/or satellites [see, e.g., Over and Gupta, 1994; Bocchiola, 2007; Kang and Ramírez, 2007]. However, it seldom occurs that long and accurate series of observed precipitations are available from remote sensing devices, and we are not aware of similar databases for the catchment of interest. Here, the SRE-EM approach is used for SSRC model estimation from sparse rain gauge data, i.e., to evaluate the process noise equation image in equation (6) [Groppelli et al., 2009]. The SRE-EM procedure is carried out on a daily basis, to highlight storm to storm variability as suggested, e.g., by Over and Gupta [1996], in contrast to ensemble estimation.

[36] The SRE-EM approach is based on a scalar version of the Kalman filter, or RTS filter [see, e.g., Basseville et al., 1992]. It consists of a double sweep along the cascade tree [Daniel et al., 2000; Primus Gorenburg et al., 2001], that provides the estimates at each level, as well as the associated mean square error of estimation. The full set of equations was applied to precipitation fields by, e.g., Tustison et al. [2003], Gupta et al. [2006], Bocchiola and Rosso [2006] and full reference can be made to Bocchiola [2007]. The Expectation Maximization algorithm EM [Dempster et al., 1977] was used to evaluate the maximum likelihood ML estimate of equation image at each scale. We perform estimation of the equation image, making no assumptions concerning their scale structure (e.g., by using regular scale invariance, either bounded or unbounded, as, e.g., by Tustison et al., [2003]). Gupta et al. [2006] and Bocchiola [2007] demonstrated that the use of a purely data driven approach results into better process noise estimates (i.e., equation image). Consistently, downscaling was carried out using unconstrained weights. Although the statistics of W, i.e., equation image, under particular assumptions can be shown to define the scaling structure of precipitation (i.e., scale invariance [e.g., Over and Gupta, 1994, 1996]), since here equation image estimation is unconstrained, no hypothesis is done concerning such scaling structure. Accordingly, we do not perform any analysis of (regular) scaling of precipitation as resulting from estimation of the SSRC weights. Notice further that, given the sparse nature of the rain gauges network, no scaling analysis based upon sample moments is possible for reference, because resampling of the rainfall field at different scales, which may be performed when dealing with precipitation estimates upon a regular grid, e.g., using radar data, is unfeasible here. The SRE-EM accuracy in estimation of the process variances equation image, which in turn affects the accuracy of the final rainfall estimates, depends (inversely) upon sample size, as reported e.g., by Bocchiola [2007]. However, here in view of the considerable size of the adopted database, including more than 2000 storm events, we may assume that the distriubiton of equation image values necessary for downscaling is properly estimated.

[37] Suitability of use of the SSRC approach using lognormal weights W as here can be demonstrated by fitting of the observed (measured) rain rates to lognormal (LN) distribution [e.g., Primus Gorenburg et al., 2001; Bocchiola, 2007] at different scales. This should be done for each event, because process weights estimation is carried out daily whenever a storm occurs. Here, however, this is particularly critical because (1) few data are available in each event (nd ≤ 25), and (2) only the finest scale (i.e., rain gauges) can be investigated, and no resampling is easily feasible. Here, we proceeded as follows. We decided a least number of rain gauges (nd= 15) that should reasonably be available during an event for testing adaptation to LN distribution. So doing, we obtained 238 events (i.e., on average 24 events per year or so). One test we could reasonably use here in view of the paucity of data was the Kolmogorov-Smirnov test for distribution fitting (while other tests, e.g., the equation image, or Pearson test, as from Bocchiola [2007] seemed senseless). For each of the 238 events, we verified equation image that the empirical plotting position of the observed precipitation was consistent with the theoretical LN distribution (± the confidence bounds as from Kolmogorov-Smirnov nonparametric test), as estimated from the related sample moments. Only in two cases out of the 238 (0.8%) did we find improper fitting. On such basis, we concluded that the LN distribution may be taken as fit enough to provide accommodation of the observed precipitation fields. Because LN distribution is probably the most widely adopted one in modeling precipitation in space and time for several purposes [Simpson et al., 1988; Shimizu, 2003; Smith and Krajewski, 1993; Seo et al., 1999], it seems here that the SSRC approach based upon such distribution can be used with enough confidence.

[38] The SRE method can be used either as a smoothing algorithm, i.e., to provide smoothed (meaning, with measurement noise identifed and “removed” [e.g., Bocchiola, 2007]) estimates of precipitation from (multiple) measurement devices, or as an interpolation method, i.e., when sparse or clustered data are missing (as, e.g., in remote sensing devices, when a full stripe of data is missing, [see, e.g., Primus Gorenburg et al., 2001]). However, here the SSRC (plus EM) algorithm is used to provide optimal estimation for the SSRC model for downscaling purposes. We apply the SRE-EM approach upon the raw sparse precipitation data, without using an interpolated field (say, by kriging or other methods). In fact, because the interpolated values within the field would be (linear or not) a combination of the observed values, no information would be actually gained for use of the EM algorithm. We use here a variable branching number b in scales, namely b = 9, 9, 25, from the coarsest scale (90 km2) to the finest scale (2 km2). The estimation of beta generator B is generally based upon the calculation of scaling of the sample moments of order 0 at different resolutions is space [see, e.g., Over and Gupta, 1994]. Using of a sparse gauge network does not allow accurate evaluation of B. The observation of non null precipitation within a sparse rain gauges network may lead one to assume high coverage of rainfall area (Fractional Wetted Area (FWA)), whereas a considerable part of the catchment may display dry weather, undetected due to lack of measurements [see, e.g., Braud et al., 1992; Over and Gupta, 1994]. Here the rain gauge network is not uniformly distributed but rather clustered around the river network. Therefore, if precipitation is observed gathered around the rain gauges area, one is brought to incorrectly assume full coverage of the area, or FWAG, close to 100%. Also, the case of underestimating FWA can occur, when dry weather is detected by the network, and instead precipitation is occurring within unmeasured areas. We carried out some preliminary tests of calculation of the spatial intermittence by simply upscaling the observed rainfall at the rain gauges. So doing, we found an inconsistency between the spatial and temporal intermittence at rain gauges. In fact, we found that the rainfall time series at a given rain gauge indicated a rate of intermittence, i.e., a given number of wet spells during a storm on the area (i.e., when other rain gauges indicate rainfall) on the order of 50% or so on average. Conversely, for several storm events, all the rain gauges indicate precipitation. In such cases, one would assume full FWAG, whereas applying the 50% no rain probability as reported, likely 50% of the area (on average), should be dry. To overcome this issue we proceeded as follows. For every wet day, we used the SRE-EM to produce an interpolated (honoring observed data) rainfall fields at 2 km2 from the rain gauge observations. Then, we applied a threshold of precipitation, used to set to 0 low-estimated precipitation. In this way, areas where low precipitation is attained are cast as dry, obtaining an intermittent rainfall field. The threshold is dynamically evaluated for each day and depends upon the minimum observed rainfall intensity for that day RGmin, as

equation image

[39] A similar approach was already adopted to deal with intermittency within the framework of optimal estimation and downscaling of precipitation [Perica and Foufoula-Georgiou, 1996; Tustison et al., 2003] and it is used here accordingly. The critical value of c is cast so as to fulfill the criteria of average spatial intermittency as observed, i.e., of the average number of dry days for each rain gauge during a storm (i.e., circa 50% as reported). In fact, we observed that average spatial intermittency changes with season, so we evaluated c upon a seasonal basis. In Table 3, we report the so obtained values of c and the related average spatial intermittence, or FWAG and FWAS (from gauges and simulated). It is important to notice that spatial intermittence, FWA, changes from storm to storm, depending on the spatial distribution of rainfall. We used the so obtained zero field to estimate the B generator parameter (i.e., p in equation (5)), for each storm. We relied here on the assumption that p is constant with scale [see, e.g., Gupta and Waymire, 1993; Over and Gupta, 1994, 1996], also in view of the few available cells at the coarsest resolutions, making estimation of p less accurate due to sampling issues [Over and Gupta, 1994]. Notice that a break in scale of intermittency p may occur at a coarse resolution, in the order of 100 km2 or so [see, e.g., Gupta and Waymire, 1993; Over and Gupta, 1994]. Here we deal with a coarsest resolution of 90 km2, thus we may reasonably hold the constant p assumption.

4.6. Dependence of Cascade Parameters Upon Climate Forcing

[40] Following Over and Gupta [1994], we tentatively linked FWA, given by the parameter p, to an external scale forcing, as expressed by the average precipitation on the area, approximated here by RGAO. We used a functional dependence as

equation image

where Rmax and k are empirically estimated parameters. According to Over and Gupta [1994], the large-scale forcing drives spatial clustering of precipitation, and therefore the distribution of wet areas within the field in a way that FWA increases proportionally with RGAO. The equation image generator B in equation (5) is characterized by the rate 1-p at which zeroes (i.e., dry clusters) appear in the field, and therefore the FWA is strongly linked to 1-p. Therefore, equation (9), linking 1-p to RGAO gives reason of the increase of FWA with RGAO. Consistently with what was done in p estimation, we estimated Rmax and k with a seasonal approach, albeit no considerable difference was spotted between seasons. The parameters we retrieved are reported in Table 3. Qualitative assessment of the proposed equations indicates a similar behavior as in Over and Gupta [1994], reported in Figure 3. Also, the process variances of SSRC, equation image, were tentatively linked to climate variables, here using RGAO (similarly to, e.g., Perica and Foufoula-Georgiou [1996]). However, no significant dependence was found there. The equation image parameters displayed a limited storm to storm variability, particularly at the finest scale, and a somewhat similar behavior in seasons. The scatterplot of equation image at each scale could in practice be accommodated using seasonally valid LN distributions (not shown for shortness), the parameters of which are reported in Table 4.

Figure 3.

Dependence of intermittence parameter 1-p within SSRC against average precipitation RGAO. (a) Winter, (b) spring, (c) summer, and (d) fall.

Table 4. Parameters of LN Distribution of equation image
SeasonParameterFine Scale (2 km2)Medium Scale (10 km2)Coarse Scale (30 km2)
WinterMean equation image (.)0.9870.8950.897
WinterDev.St equation image (.)0.0190.1600.167
SpringMean equation image (.)0.9890.9030.735
SpringDev.St equation image (.)0.0080.1470.277
SummerMean equation image (.)0.9850.8950.703
SummerDev.St equation image (.)0.0210.1720.305
AutumnMean equation image (.)0.9850.8940.765
AutumnDev.St equation image (.)0.0250.1810.324

4.7. Downscaling Procedure

[41] For every day, we consider the GCM simulated precipitation, RGCM. If this is positive (i.e., if a wet spell is predicted), we correct this value by multiplying it times BiasGAO, according to equation (1). First, we consider the B0 process. We extract an uniformly distributed (0,1) random variable U1. If U1 > p0, rainfall for that day is set to 0, to preserve the observed intermittency, as reported. Contrarily, if, U1p0 we randomly extract another uniform variable U2, which we use to obtain a random value of W0, lognormally distributed as in equation (1). Then, we apply BGAO for overall mass conservation, so obtaining a predicted value of average daily area rainfall, RSA. We then downscale the so obtained value by recursively applying equation (2) to equation (7) from the full scale (90 km2) to the finest scale (2 km2). First, we use equation (9) to evaluate the expected FWA at each scale from the average precipitation RSA, i.e., though p. Then, we evaluate rainfall according to the SSRC structure, by scale recursive random simulation at each scale using equation (2) to equation (7). The path-wise use of the Bs model generator to model across scale intermittence (i.e., FWA) displays in fact a drawback upon the wet area distribution. Because Bs is spatially uncorrelated at each scale, the resulting FWA displays randomly placed dry areas, together with high rainfall patches nearby. This results in unlikely representations of precipitation fields, which in real world observations display more regular increase from null rainfall areas to wet areas, until storm center, i.e., clustering of precipitation (see Over and Gupta [1994] for a discussion on this topic). To overcome this issue, with the aim of providing a more realistic rainfall simulation, we combine the equation image model approach here with the use of a threshold method, as suggested within the available literature about downscaling of precipitation [e.g., Perica and Foufoula-Georgiou, 1996]. To do so, we simulated path-wise precipitation using equation (2) to equation (7), without intermittence (using cell by cell random extraction of a uniform variable as for W0 above). Then, we superimposed upon the so obtained field a map of zero rainfall obtained as follows. We iteratively chose a threshold (in mm) below which rainfall is set to zero, which is increased until a given percentage of null rainfall cell (i.e., a given FWA) is obtained, as required by the proper value of p. This procedure results into clustered dry areas because of the inherently clustered structure of precipitation as from the SSRC approach, which displays zones of low precipitation (i.e., those cells being downscaled from a father node displaying low precipitation). Further, rainy areas surrounding dry spots generally display low precipitation, whereas clusters with high precipitation are normally far away from dry spots. So doing, we profit from the more likely spatial intermittency pattern given by the threshold approach, while providing respect of the scaling properties of rainfall fields, including intermittence, as described by the B generator.

4.8. Model Validation: Back Casting of Control Data Series

[42] We validate the model by downscaling the GCM's control run for 1990−1999 and then comparing the related statistics against those from observed precipitation. Here, we tuned separately the BiasGAO part, which accounts for the average rainfall in time, and the SSRC part, which deals with the spatial distribution of precipitation in each event. Therefore, we need to assess the accuracy of the model in depicting the joint precipitation process in time and space. Also, the calculation of the fractional wetted area, FWA, within the SSRC depends upon external rainfall RGAO (equation (9)), given by BiasGAO, so its correct representation needs to be verified.

[43] We first validate BiasGAO by (1) comparing the sample fraction of wet days in the simulated series RSA, p0S, against its sample value from RGA series, p0, and (2) comparing the second-order statistics of the estimated daily area rainfall RSA = RGCMBiasGAO against those of RGAO. We evaluated the mean precipitation equation image, and the corresponding variances equation image, and compared these statistics against their sample values from RGAO series. We then validated the SSRC by (1) verifying the agreement of the simulated FWAS (depending upon pS from the downscaled series), with its sample value from the observed series FWAG, both on the catchment as a whole, and upon each single rain gauge (using the 2km2 cell nesting the gauge), (2) comparing the second-order statistics of the simulated yearly accumulated precipitation and of the daily precipitation upon single rain gauges (2 km2), RScumi, RSi against the observed series of RGcum and RG, and (3) comparing the spatial correlation of the simulated fields against that of the observed ones. This is to verify that the spatial rainfall as obtained from SSRC output correctly reproduces the statistics of the observed rainfall fields in the back cast series. The results are reported in Tables 5 through 8, and in Figure 4. In Figure 4, we report the sample values of the pairwise (station i, station j) correlation coefficient of daily precipitation within the network, equation image. The calculation of equation image is carried out seasonally (Figure 4a, winter and spring, Figure 4b, summer and fall), using the observed precipitation at the rain gauges during wet spells (to avoid high correlation as given by zeroes). We use binned values for six classes of distance (distance L ranging from 0−5, 5−10, 10−15, 15−20, 20−30, 30−40 km), and we take the mean of the observed values of equation image for (pairs of) stations within those ranges of distance from each other, to decrease the scatter of the sample values of equation image. We here report values of equation image until L = 40 km, because for larger distances the sample values of equation image would either become too low (in winter), or start increasing (other seasons), which seems unlikely. We therefore decided to limit the comparison for L = 40 km, because for all seasons this was the largest distance displaying reasonable behavior of equation image. In Figure 4 we also report the theoretical values of the pairwise (cell x, cell y) correlation coefficients as from the SSRC model, Cx,y [see, e.g., Over, 1995; Bocchiola, 2003]. The values of Cx,y decrease with distance according to a power law, depending upon the resolution, or scale, and upon the cascade weights, equation image. Here, we calculated Cx,y for the finest available scale (2 km2), using the average values of the seasonal cascade weights, equation image, as in Table 4.

Figure 4.

Seasonal correlation of precipitation against distance, observed equation image and modeled using the SSRC, Cx,y. For equation image binning of distance in 6 classes is used, and the average value, together with confidence limits (95%) is reported. (a) Winter and spring, and (b) summer and fall.

4.9. Projections of Future Precipitation Upon the Oglio Catchment

[44] We used the SSRC approach to downscale future A2 precipitation scenarios from PCM. We considered the period 2010–2060 to analyze future climate and environmental conditions centered around 2050 [Houghton et al., 2001], to evaluate the effects of the potential climate change at the middle of the current century. The 50 years of scenarios data were downloaded from the IPCC Data Distribution Centre. We applied the downscaling procedure as described in section 4.7, so obtaining spatial daily precipitation fields for the considered 50 years period upon the Oglio catchment. The average yearly accumulated rainfall projected by PCM A2 scenario during 2010−2060 in northern Italy is 1182.1 mmy−1. We here report the results concerning one sample decade, centered around 2050, i.e., 2045−2054, for consistency with the decadal observed period during 1990−1999. These results are summarized in Tables 5 through 8. Therein, we report the statistics of the simulated future rainfall upon the watershed, RSAF, together with the simulated intermittency, i.e., p0SF, i.e., the expected future dynamics of wet and dry spells (Table 5). Also, we report the expected future fractional wetted area, FWASF (Table 6). Because FWA depends upon the general forcing of precipitation, here RSAF, we expect a possibly different projected value of FWA in the future. Further, we report the statistics of the future daily (Table 7) and yearly (Table 8) accumulated precipitation for the considered rainfall stations, RSFi and RScumFi, to be compared against the control ones.

Table 5. Validation Statistics of Simulated Area Precipitation RSA Against Its Observed Value RGAOa
 StatisticsRGAO (1990–1999) (mmd−1)RSA (1990–1999) (mmd−1)p-value (.)RSAF (2045–2054) (mmd−1)
  • a

    Italicized values are not significant results equation image. Projected area precipitation RSAF (2045−2054) is also reported.

WinterMean (unit1)1.761.470.0422.08
WinterSD (unit2)4.173.25<10−53.48
Winterp0 (%)54%57%0.41140%
SpringMean (unit1)2.572.680.4563.71
SpringSD (unit2)4.274.67<10−55.24
Springp0 (%)40%41%0.82130%
SummerMean (unit1)4.714.920.3526.28
SummerSD (unit2)7.197.93<10−511.32
Summerp0 (%)19%20%0.81310%
AutumnMean (unit1)4.844.830.9515.17
AutumnSD (unit2)9.579.300.0159.31
Autumnp0 (%)34%35%0.84223%
Table 6. Station-Wise Comparison of FWAS Against Its Observed Value FWAGa
StationFWAGi (.)FWASi (.)p-value (.)FWASF (.)
  • a

    Station wise is the index of station i. All stations feature 10 years of data. Italicized values are not significant results equation image. FWASF of projected precipitation (2045–2054) using the SSRC approach is also reported.

L. Arno0.580.510.0100.51
L. Avio0.490.520.2000.52
L. Benedetto0.500.510.3100.51
L. Salarno0.450.510.0700.53
P. Avio0.450.510.0180.51
V. d'Oglio0.590.490.0010.51
Table 7. Station-Wise Comparison of s Order Moments of Simulated Daily Precipitation RS Against Observed One RGa
GaugesMean RGi (mmd−1)Mean RSi (mmd−1)p-value (.)SD RGi (mmd−1)SD RSi (mmd−1)p-value (.)Mean RSFi (mmd−1)SD RSFi (mmd−1)
  • a

    Station wise is the index of station i. All stations feature 10 years of data. Italic values are not significant results equation image. Statistics of projected precipitation (2045–2054) RSF using the SSRC approach are also reported.

L. Arno3.783.270.0110.7412.02<E-054.2719.27
L. Avio3.623.540.7210.1512.73<E-054.4922.66
L. Benedetto3.563.280.129.6910.75<E-055.2935.09
L. Salarno3.593.150.019.399.590.163.9013.77
P. Avio3.583.106E-0310.179.857E-033.9915.96
V. d'Oglio3.543.440.5810.0411.09<E-053.6412.09
Table 8. Station-Wise Comparison of s Order Moments of Simulated Yearly Accumulated Precipitation RScum Against Observed One RGcuma
GaugesMean RGcumi (mmy−1)Mean RScumi (mmy−1)p-value (.)SD RGcumi (mmy−1)SDRScumi (mmy−1)p-value (.)Mean RSFcumi (mmy−1)SD RSFcumi (mmy−1)
  • a

    Station wise is the index of station i. All stations feature 10 years of data. Italic values are not significant results equation image. Statistics of projected precipitation (2045–2054) RSFcum using SSRC the approach are also reported.

L. Arno1378.001220.450.24019.4018.620.6151607.0621.06
L. Avio1308.331294.540.85115.2718.790.1641639.8228.23
L. Benedetto1262.401198.680.40214.6715.770.6721933.0133.01
L. Salarno1074.311148.720.41316.2017.410.6701423.4220.60
P. Avio1305.861132.410.06616.0116.430.9501457.9619.07
V. d'Oglio1279.031252.920.75715.9016.900.7461329.2420.89

5. Results and Discussion

[45] First, we present the validation statistics for the back cast exercise. We report the p-values of the goodness of fit tests of the statistics of the simulated RSA to those of the observed RGAO. The fraction of wet days p0 is well described, and its yearly average simulated value is E[p0S] = 38.4%, against E[p0] = 36.7%, with p-value = 0.45. A similarly good agreement was found seasonally, as reported in Table 5. This indicates the suitability of the model to correctly depict the daily wet-dry spells sequence, tremendously important in hydrological balance exercise.

[46] Then, we investigate the relationship between the (year round) average of the daily simulated rainfall RSA, E[RSA] = 3.52 mmd−1 (with Dev.St[RSA] = 7.02 mmd−1) and the observed value RGAO, E[RGAO [= 3.47 mmd−1 (Dev.St [RGAO] = 6.81 mmd−1). Good agreement is therefore seen (p-value = 0.61 and p-value = 0.03, for mean and standard deviation respectively) and similar good fitting is observed in each season for the mean values, reported in Table 5. When dealing with seasonal standard deviations, albeit their simulated and observed numerical values are somewhat close, p-values for Fisher's F test are low. However, this may be due to the somewhat large number of seasonal daily data used here (circa 900 values in each season), making the goodness of fit tests very severe. Apart from the winter season, normally relatively close values of standard deviations are observed. Therefore, although the test provides indication of different statistics, we suggest that the model may be used with acceptable confidence.

[47] The average (year-round) simulated wetted area FWAS is then evaluated for the catchment as a whole, and compared against its sample value, FWAG. The model generally does well in preserving spatial intermittency and, at the finest scale, the yearly average simulated wetted area is E[FWAS] = 0.51 against E[FWAG] = 0.50 (p-value = 0.65). The (year-round) station wise comparison of FWAS against FWAG is reported in Table 6. We report here only those stations featuring a complete database for the 10 years in study, for robustness. Generally, acceptable agreement is observed, also seasonally (not reported for the sake of shortness).

[48] The second-order (year-round) statistics of simulated daily and yearly accumulated precipitation at the single rain gauges (2 km2 cell including the rain gauge), RSi, RScumi are reported in Table 7 and Table 8, respectively, together with their observed counterpart, RG and RGcum. Concerning daily precipitation, the mean values of RSi are reasonably correct. Comparison of RSi against RGi shows weaker results in term of standard deviation, with simulated values slightly overestimated. However, the large number of considered daily values (i.e., 3652 values) may result again here into considerably low p-values, while the absolute values are in practice very similar. Concerning the accumulated precipitation RScumi, the comparison against RGcumi displays satisfactory results for both mean and standard deviation (in Table 8). The same analyzes were carried out for the seasons and the results were similarly good (not shown for shortness).

[49] Analysis of precipitation spatial correlation in Figure 4 is of interest. The lowest degree of spatial correlation is observed during spring and summer, which seems reasonable in view of the mostly convective nature of storms in these seasons. Highest spatial correlation is observed during winter and fall, possibly witnessing more stratiform precipitation therein. Notice the acceptable representation of the average observed spatial correlation given by the SSRC approach. Because spatial correlation structure of precipitation is of interest in a number of applications, its correct representation seems an asset of the SSRC model.

[50] According to what shown, the BiasGAO plus SSRC seems capable of reproducing reasonably the single site (i.e., 2 km2 cell) precipitation field in the area. The daily simulated values show good depiction of the spatial intermittency and of the daily rainfall. The accumulated at site values are also well represented using the homogeneous SSRC as reported, in spite of a possible dependence of precipitation upon topography, not explicitly modeled here, as reported.

[51] These results may be sensitive to the representative area we assumed for the rain gauge measurements (2 km2), which represents as said a tradeoff between the computational time required for the SSRC procedure, and the spatial variability of precipitation. In the future, some sensitivity analysis will be necessary to assess the effect of such choice.

[52] Eventually, we can comment the statistics of the projected future rainfall RSAF upon the watershed for the period 2045−2054, i.e., centered around 2050, together with the projected dynamics of wet-dry spells, p0SF. We found E[RSAF] = 4.32 mmd−1, and Dev.St[RSAF] = 8.13 mmd−1 (against E[RGAO] = 3.47 mmd−1 and Dev.St[RGAO] = 6.81 mmd−1, during 1990−1999), and E[p0SF] = 25.5% (against E[p0] = 36.7%, 1990−99). The largest projected increase in precipitation RSAF on the area occurs during spring and summer (+44% and+ 33%, respectively), whereas during winter and especially fall slighter changes are foreseen (+18% and+ 6% respectively).

[53] Notice that the projected increase of the average rainfall upon this area is not an artifact of our model, but in fact it is brought about by the PCM model. The latter depicts a future situation where the yearly average precipitation upon the GCM footprint is 1182.1 mmy−1, whereas in the control decade it was of 950.8 mmy−1. Similarly, the PCM model projects a fraction of dry spells close to 1.6% for the future decade, whereas the control counterpart was approximately 18%.

[54] In Figure 5, as an instance, we report the projected sequence of the daily precipitation upon the Oglio river RSAF for 1 year (2050), compared against the observed series of RGAO for 1 sample year (1995). Qualitative analysis witnesses the likelihood of the projected series.

Figure 5.

Daily averaged precipitation RSAF upon the Oglio river area closed at Costa Volpino as obtained from downscaling of PCM output using BiasGAO in equation (1), for the year 2050, compared against the observed series of daily averaged precipitation upon the same area RGAO during 1995. Also, original daily precipitation for the same years from the PCM are reported.

[55] In Figures 6a and 6b we report two sample snapshots of the simulated future precipitation (namely for 14 July 2050 and 24 March 2053) to illustrate the concept of FWA and of precipitation clustering as reproduced using our approach. In Figure 6a, one event is shown with high spatial intermittency (i.e., low FWA). Notice the capability of the model to reproduce considerably clustered precipitation field, showing high spatial intermittence, as is frequently observed. In Figure 6b, a high FWA is seen instead, linked to higher average precipitation upon the area. In Table 6, we report the expected value of the future fractional wetted area, FWASF. In spite of the increased average precipitation RSAF, controlling p, and therefore FWASF via equation (9), no relevant increase seems apparent in the spatial clustering of wet areas. This is due to the power law relationship of 1-p against RGAO in equation (9). For instance, if one takes the average (yearly) values of RGAO and RSAF, as from Table 5, namely RGAO = 3.7 mmd−1 and RSAF = 4.31 mmd−1 (increasing by 25% or so), and the yearly average values of k = 0.135, and Rmax = 0.46 mmd−1in equation (9), the value of 1-p for the finest level (branching number b = 25) i.e., the increase in dry area from the 9 × 9 level to the 45 × 45 level is 1-p = 0.44, and 1-p = 0.42, respectively, with a slight decrease (2% less or so, only −3% in absolute value). Because 1-p is (inversely) linked to FWA, a non negligible increase of RSAF implies indeed a very slight increase of FWA itself.

Figure 6.

Spatial (2 km2) daily precipitation for two sample events from the projected rainfall series using SSRC. (a) The 14 July 2050 Event with high spatial intermittence (i.e., low value of p). (b) The 24 March 2053 Event with low spatial intermittence (i.e., high value of p).

[56] On the other hand, Tables 7 and 8 illustrate clear changes in the statistics of future daily and yearly precipitation for the considered rainfall stations, RSFi and RScumFi, during 2045−2054. The daily mean precipitation E[RSFi] is increased on average (upon the seven considered gauges) of 19% with respect to its observed value during 1990−1999. Much the same way, the average accumulated rainfall E[RScumFi] increases by 26%. The standard deviation of RSFi increases by 204% on average (and CVRSFi varies from 2.77 to 4.53), and the standard deviation of RScumFi increases by 48% (but CVRScumi varies from 1.27% to 1.47%, i.e., 16% bigger). Albeit the SSRC model provides slightly overestimated standard deviation of RSFi, the projected values here show considerable changes in their second-ordered statistics, way beyond such effect.

[57] Several studies are available within the present hydrological literature aimed to downscale projected future climate (i.e., precipitation) under the assumption that the downscaling approach remains stationary in time [e.g., Kang and Ramírez, 2007; Barontini et al., 2009; Bavay et al., 2009]. Our overall feeling is that when providing what if scenarios for future hydrological cycle this straightforward assumption is possibly the only reasonable one. Indeed, all the GCMs studied here provided somewhat poor performance in describing the monthly (and daily) amount and seasonality of precipitation in the investigated area. Therefore, the downscaling method and the assumptions therein play a fundamental role in accurately depicting the past local variability, and in providing credible projections for the future. This in turn raises the need for investigation concerning holding of these assumptions, e.g., using control series from the past to test the sensitivity to downscaling hypothesis.

[58] Notice that recent evidence was raised that the use of complex schemes like the SSRC may indeed help in accurately describing hydrological cycle [e.g., Lammering and Dwyer, 2000], so making an effort to employ such models for the future seems proper.

[59] As a benchmark for the present study, Bocchiola and Diolaiuti [2010], addressed the presence of climate trends within the Adamello Park of Italy, the northeast part of Oglio catchment here, using a series of locally measured precipitation at four rainfall gauges, for the period 1967−2007. They found a measurable (albeit nonstatistically significant) positive trend in the total precipitation therein, more pronounced since 1990 or so [see Bocchiola and Diolaiuti, 2010, Figures 2−3], together with considerable trading of snowfall for rainfall at the highest altitudes, also depending upon local temperature, and upon the general circulation, through NAO index. Also depending upon altitude (ranging in their study between 1820 and 2300 m asl) they found a variable rate of increase of precipitation, from +10 mmy−1 (lowest altitude) to -3.3 mmy−1 (highest altitude), with an average of +2.7 mmy−1. Here, for the period 1990−2060 we may conjecture a rate of +3.9 mmy−1 or so, qualitatively consistent with that previous study.

[60] Brunetti et al. [2006] investigated the presence of trends within precipitation in the greater alpine region, GAR, including the larger area investigated here (see Brunetti et al. [2006, Figure 1] as compared with Figure 1 here), using long-term observed series from 192 stations. They highlighted four different regions, with somewhat variable behavior. Particularly, Oglio river is laid along the boundary between their region NW (EOF-1 in Brunetti et al. [2006, Figure 4]), where they found clearly significant increasing total precipitation and their region SW (EOF-2 in Brunetti et al. [2006, Figure 4]), where they found less evidently decreasing total precipitation. However, evaluation of trends in total precipitation during the 20th century upon the GAR displays some variability, as it shows an increase in the northwestern areas and a decrease in the southeastern ones [e.g., Faggian and Giorgi, 2009], and the boundary between these two areas may be defined with some degree of uncertainty.

6. Conclusions

[61] We here illustrated a method for the statistical downscaling of precipitation from a particular GCM, and we showed its application within an Italian Alpine catchment. First, we used objective indicators to identify the most suitable GCM to interpret the variability of precipitation within the area from among four worldwide spread models. Then, we set up the downscaling procedure for the model so chosen. The method uses a data driven parameterization of a random cascade model, which is calibrated for the first time in our knowledge using rain gauge data, more widely available worldwide than any other source for precipitation study. The SSRC approach explicitly accounts for the scaling properties and the internal correlation structure of precipitation, including spatial intermittency, thus yielding realistically distributed rainfall fields. The validation we performed shows that the SSRC is capable of depicting acceptably well the statistics of past precipitation, starting from control GCM scenarios. The preliminary assessment of future precipitation upon the investigated area indicates that one may expect largest average daily and yearly precipitation, together with an increase of the number of wet spells, particularly during spring and summer, and with an enhanced variability of precipitation, as witnessed by the increase of the local coefficients of variation.

[62] Here, also in view of the relatively low average altitude of the gauge network, we dealt in practice with total precipitation. For prospective hydrological balance exercise we will need to account for changing snowfall patterns, that we may depict, e.g., by using projected temperatures against altitude. In the future, we will also test the use of the SSRC approach to downscale precipitation scenarios from other GCMs, with the aim to provide an ensemble approach to the production of precipitation scenarios. Eventually, the tentative use of the SSRC approach here provides encouraging results, and we plan to henceforth use this tool to evaluate hydrological and ecological implications of the expected future rainfall regime upon this area, and possibly upon the Alpine region.


[63] The present paper reports work carried out under the umbrella of the CARIPANDA project, funded by the CARIPLO foundation of Italy ( under the direction of the ADAMELLO park authority, which is here acknowledged also for supporting with logistic aid. ENEL Produzione Milano is acknowledged for providing snow and precipitation data from their stations and for helping with logistic aid. Antonio Ghezzi is kindly acknowledged for his contribution as a Cotutor of the first author in her MS Thesis. Eng. Alberto Pengo is also acknowledged for his contribution to the present work in fulfillment of his Master's Thesis. We acknowledge the international modeling groups (IPCC, Max Planck Institute for Meteorology, Hadley Centre for Climate Prediction and Research, National Centre for Atmospheric Research and Geophysical Fluid Dynamics Laboratory) for providing their data for analysis, for collecting and archiving the model data, for organizing the model data analysis activity and for technical support.