Statistical downscaling models (SDMs) have been widely used to generate local weather scenarios for past or future climates from outputs of climate models e.g., and to produce local weather forecasts from outputs of numerical weather prediction models e.g.,. For recent years, they have also been used to reconstruct past weather conditions from atmospheric reanalysis data e.g.,.

Among the different SDM approaches presented over the last decades seefor a review, perfect prognosis SDMs make use of the physical relationships that exist between some large-scale atmospheric parameters and local weather variables. Local weather scenarios can then be produced for any prediction day, conditional on the large-scale atmospheric configuration observed or simulated for this day, where the “prediction day” refers here to some future, past, or present simulation day, depending on the application context at hand (e.g., forecasting, simulation, reconstruction). Perfect prognosis SDMs include transfer functions, weather-type-based models, and methods based on atmospheric analogs. In the latter case, atmospheric analog days of the current prediction day are searched for on the basis of some atmospheric similarity criterion in the historical database. The weather variables observed for the most similar day, for one similar day chosen randomly, or for a selection of the k-most similar days are then used as a weather scenario for the prediction day of interest e.g.,.

Transfer functions mainly consist of regression models where the expected value of the predictand for time t is expressed as a linear or non-linear function of a set of predictors. For precipitation, the regression can be achieved with multiple linear regressions or generalized linear models (GLMs) which extend the linear regression to non-Gaussian data e.g.,. Transfer functions can also make use of classification and regression tree (CART) e.g., artificial neural networks or least squares support vector machines e.g.,.

The downscaling relationship used in transfer functions is usually established empirically between a selection of large-scale predictors and the predictand (e.g., precipitation occurrence) from a set of observations available for recent decades. As physical processes driving surface weather vary in time, the most relevant predictors and the downscaling link are however expected to vary in time too. When inferred from all observations available for a given period, the downscaling relationship – which is thus likely inferred from a heterogeneous ensemble of weather configurations – is consequently likely to be sub-optimal. To reduce this potential limitation, the parameterization of the relationship is often estimated after some data stratification. In the usual calendar stratification, one parameter set is for instance optimized for each calendar month or season e.g.,. The stratification can also be based on some weather-type information. In this case, a set of parameters is usually estimated for each weather type of a given pre-established weather-type classification e.g.,. Often applied, this weather-type-based approach is expected to allow for a better identification of the most important driving large-scale variables and consequently for a more relevant downscaling. An obvious limitation however remains for prediction days that do not clearly belong to one specific weather type (e.g., prediction days that are close to the “weather frontiers” delimiting two or more weather types). Those days are indeed likely to be rather dissimilar to the weather configurations that each weather type is expected to characterize, making the downscaling relationships to be used not suited anymore or, at least, sub-optimal.

A smoother weather-type-like approach consists in defining the weather type from all atmospheric situations that are similar to the situation of the prediction day. The ensemble of days from which the downscaling link can be identified is thus expected to be rather homogeneous and to rather well inform the large- to small-scale link sought for the considered prediction day. This is in turn expected to make the link stronger and to improve the prediction e.g.,. Such an approach can actually be achieved by identifying for each prediction day the transfer function from atmospheric analogs of that day. To our knowledge, this two-stage downscaling approach, combining in turn the two popular analog and transfer function methods, has only been explored in a few previous studies. In , it was found to improve the probabilistic prediction of local surface temperature in the Spanish Iberian Peninsula. The multiple linear regression of the regression stage, estimated from the 150 most similar atmospheric analogs of the prediction day of interest, uses forward and backward stepwise selection of predictors from a set of four potential predictors (thickness of the air column and three temperature indexes of previous days). For precipitation, the authors did not test the potential of the two-stage combination, building directly the predictions from the precipitation observations of the 30 most similar atmospheric analogs. In the deterministic approach presented by , incorporating the regression stage (with 79 potential atmospheric predictors) was found to allow a clear though not overwhelming improvement of precipitation prediction over the simple analog-based predictions. A multiple linear regression model was also applied here for the regression stage.

In the present study, we present a two-stage analog/regression downscaling model for the probabilistic prediction of small-scale daily precipitation: for each prediction day, the statistical downscaling link between some large-scale atmospheric predictors and small-scale precipitation is estimated from large-scale and local-scale observations available from an ensemble of days which are atmospheric analogs to the prediction day. The analog model (AM) used for the analog stage is based on developments from different studies initially focusing on the probabilistic quantitative precipitation forecasts in southern France e.g., and extended to the prediction of precipitation on larger spatial domains e.g.,. The statistical distribution of daily precipitation is strongly non-Gaussian with a non-negligible mass in zero (corresponding to the probability of a dry day), and a skewed distribution for non-zero daily amounts. For the regression stage, we thus do not use a multiple linear regression model as in previous studies, but a two-part GLM approach where the probability of precipitation occurrence and the distribution of wet-day amounts are modeled separately following and . Conversely to the work of , this allows prediction of the full distribution of precipitation, including the probability of a wet day. In this two-stage analog/regression approach, the analogs change from one prediction day to the other. This makes the statistical downscaling link potentially adaptive; i.e., the predictors and the regressions parameters are likely to vary from one day to the other.

As mentioned above, SDMs are used for the simulation of local weather scenarios in different contexts, e.g., local weather forecasts, reconstructions, or climate impact studies. No specific context is considered here and the two-stage model could be further considered for either forecasting, reconstruction, or future projections. Depending on its intended use, some specific issues would obviously apply, calling for specific focused analyses and developments. For instance, the large-scale atmospheric parameters to be considered as predictors would depend on the dataset considered (e.g., atmospheric reanalyses, climate models, or numerical weather prediction models) as a result of their intrinsic quality e.g.,. The development of climate projections would require one to check the temporal transferability of the model in a modified climate context and would thus likely also condition the selection of the predictors as highlighted by . These context-specific issues are not considered here. Our main objectives are to present the principles of the two-stage analog/regression approach developed for the prediction of small-scale precipitation, to assess its predictive power for both precipitation occurrence and amount, and to give some insight into its adaptive behavior and thus into the temporal variability of the downscaling link. For this, we explore the model skill and behavior for the prediction of daily precipitation for a large number of sites in France.

The paper is structured as follows: Sect.  describes the data and Sect.  the two-stage downscaling model. Section  presents the skill of the model for the prediction of both precipitation occurrence and amount. The adaptive behavior of the model is considered in Sects.  and .

The predictand is the daily small-scale precipitation estimated for the 1982–2001 period over 8981 grid cells of 8×8 km2 covering the continental French territory. The predictand is “local” precipitation, i.e., precipitation at a given grid cell. Each of the 8981 grid cells is thus considered in turn in the following independently of the other cells. In other words, the predictions do not target precipitation fields. Small-scale precipitation data are obtained from the SAFRAN analysis produced for several surface variables at an hourly time step by MeteoFrance . SAFRAN precipitation estimates are obtained each day from the closest measurement stations. They are considered as pseudo-observations.

Atmospheric predictors are taken from the European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis ERA-40,. This global meteorological re-analysis is available on a 1.125∘×1.125∘ grid with a 6-hourly temporal resolution.

For the analog stage, predictors are the 1000 and 500 hPa geopotential height fields over a large spatial domain (roughly lat =10∘, lon =8∘) centered on the target location. These predictors have been found to be the most informative large-scale predictors to be used in this context for France e.g.,. They also correspond to the best large-scale predictors of daily precipitation for different regions in Europe with contrasted meteorological regimes .

For the regression stage, 22 other predictors were also considered. The selection gathers most predictors considered in previous studies over Europe e.g.,. They include predictors characterizing the thermal state of the atmosphere, its dynamics, the atmospheric water content, and its thermo-dynamical instability (see Table ). As potential predictor, we also consider the occurrence of precipitation on the previous day. All predictors here are scalar variables. Atmospheric predictors are estimated on a daily time step (mean of the four values available at 06:00, 12:00, 18:00, and 24:00 UTC) from the four ERA-40 grid cells surrounding the prediction grid cell (inverse distance interpolation).

To avoid the multi-colinearity in the predictors for the regression, we identified a subset of uncorrelated predictors. The cross-correlations between all predictor pairs were first estimated on an annual basis from all available data. The correlation structure can however differ from one atmospheric configuration to the other. The set of uncorrelated predictors could thus differ from one prediction day to the other. We thus repeated the correlation analysis for each prediction day, using for this estimation the predictor values observed for the 100 nearest atmospheric analogs identified for this day. The main features of the inter-variable correlations were found to be roughly independent of the day (not shown). The final subset of uncorrelated predictors is highlighted in Table . These predictors are tested for the prediction of both precipitation occurrence and amount.

A large number of different possible predictor sets can be built from these predictors. In the present work, for the sake of robustness, we consider that a maximum of four predictors can be integrated into a given regression model. Predictors are obviously expected to be both day and location specific. In the present work, for the sake of simplicity and readability, we select them from a unique set of four potential predictors. This allows us to reduce the degrees of freedom in the model and to better highlight its skill and adaptive behavior.

For each predictand, the set of the four potential predictors was selected as follows. For 12 SAFRAN grid cells uniformly distributed over the French territory, we first identified with a standard iterative forward/backward algorithm the four-predictor set which leads to the best prediction skill for the all-days configuration. From the 12 different sets obtained, respectively, for the 12 grid cells, we finally retained the set which leads on average to the best prediction skill for the 8981 SAFRAN grid cells.

For precipitation occurrence, this best four-predictor set is constituted from the relative humidity R700, the helicity H, the vertical velocity of the air at 700 hPa, W700, and the precipitation occurrence Occ-1 of the day before the prediction day. For precipitation amount, the best four-predictor set is similar except that the occurrence of the previous day Occ-1 is replaced by the 700 hPa air temperature T700. Note that the selection of predictors R700, W700, and T700 is consistent with results of several past studies in the region e.g.,.

Predictors considered for the analog and regression stages obviously inform about different features of the atmosphere state for different scales. Geopotential fields, by their spatial extent, characterize the large-scale atmospheric circulation configuration (the spatial domain of several thousands of kilometers includes a part of the northeastern Atlantic and covers France and a part of the neighboring countries), whereas scalar predictors used in the regression stage are descriptive of a more local (and mostly thermodynamic) state of the atmosphere (the spatial domain of several hundreds of kilometers is roughly centered above the target location).

As illustrated in Fig. , the cumulative distribution function (cdf) FY of precipitation Y at a given site (grid cell) can be expressed for any given day as the composition of the no-precipitation occurrence probability 1-π and the cdf FQ of the precipitation amount Q for non-zero precipitation: FY(y)=(1-π)+π⋅FQq=y, where π is the precipitation occurrence probability, and y and q correspond to the precipitation value with regard to the whole precipitation distribution and to the non-zero precipitation distribution, respectively.

In the present work, the cdf of precipitation is modeled for each grid cell and each prediction day with GLMs , estimated for this specific day from atmospheric analogs of the day. The probability of precipitation occurrence and the cdf of the non-zero precipitation amount are modeled separately.

In the following, we first describe the AM used to identify atmospheric analog days (Sect. ) and the GLMs applied in the regression stage (Sect. ).

As discussed later, one can face prediction days where the regression stage fails, i.e., where the regression parameters are not significantly different from zero at the chosen significance level (α=5%). For such days, we use the analog model as a backup prediction model. The backup model can be used for precipitation occurrence probability, for non-zero precipitation amount, or for both predictands simultaneously.

The way these different models are combined to finally give, for the current prediction day, a probabilistic prediction of precipitation, is presented in Sect. . In the following, this two-stage analog/regression model is further referred to as SCAMP (SCAMP stands for Sequential Constructive atmospheric Analogs for Multivariate weather Prediction and refers to the model presented by , for the multivariate prediction of precipitation/temperature/radiation/wind).

The two-stage model is used for the probabilistic prediction of small-scale precipitation over the continental French territory for each day of the 1982–2001 period. We here present the prediction skill obtained for occurrence and amount with the two predictors sets presented in Sect. . As discussed later in Sect. , the four predictors of each set are not necessarily all used; the predictors which have some predictive power for the considered predictand vary from one day to the other.

The sets of potential predictors used in SCAMP for the prediction of precipitation occurrence and amount have been listed in Sect. . For each variable, the number of potential predictors is here equal to four. All four predictors are not necessary retained for the GLM. For a given prediction day, a GLM with a single predictor or a combination of several predictors among the four can be selected. Fifteen regressive structures plus the backup AM25 model are possible in our context (Table ).

For a given prediction day, the regressive structures selected by SCAMP for precipitation occurrence or for precipitation amount are supposed to include the best information for the prediction. In the following, we assess how often each structure has been selected. This allows for some insight into the atmospheric information really used for the regression stage and how this information varies in time.

Figures  and present the percentage of times that the 15 regressive structures and the backup AM25 are used for the prediction of precipitation occurrence and amount, respectively. As in Fig. , gray cells indicate that the regression structure has been retained less than 35 times over the 20-year evaluation period. For both occurrence and amount, the selection frequency of the structures is also rather region dependent and strongly influenced by topography.

For occurrence (Fig. ), the most often selected structure is Str. no. 1, which is only based on R700 (more than 25 % of the time for the whole of France). R700 was actually found to give the highest predictive power when used in a single predictor configuration. Another structure which is also often selected (more than 15 % for a high number of grids) is Str. no. 7 which combines R700 with Occ-1. Secondary structures – as for example Str. no. 6 and no. 13 combining R700 to W700 and Occ-1 – can be selected more than 10 % of the days for some given regions. Other structures are seldom selected and some of them (Str. no. 8, no. 11, no. 14, and no. 15) are almost never selected.

For precipitation amount, the most frequently selected structures are Str. no. 3 and Str. no. 1, both based on one single predictor, W700 or R700, respectively. These structures are selected more than 25 % of the time (or more than 15 %) for a high number of grids. The secondary structures (Str. no. 6, no. 8, and no. 13) are selected from 5 up to 20 %, depending on the region. They always include W700 in combination with some other predictor (R700,H, or R700+T700). Str. no. 9, no. 12, no. 14, and no. 15 are almost never selected. The others are selected less than 10 to 5 % of the time.

Note that for the selection of the best regression structure for a given prediction day, all 15 of these regressive structures have been tested in turn. The results above suggest that this systematic test is not necessary and that it could be reasonable to consider only the few structures which are frequently retained or which are retained a “reasonable” fraction of the days. However, the selection frequency of a given structure actually varies with the seasons and/or the encountered synoptic situation, and some secondary regressive structures can be retained frequently for specific situations. This is illustrated in Fig.  for a cell located in northwestern France. The figure presents how the selection frequency of each regression structure differs in different seasons and weather patterns WP, defined in Table , from the selection frequency obtained for the all-days situation.

For precipitation occurrence (Fig. a), the selection of the main regressive structures (i.e., Str. no. 1 and no. 7, respectively, based on R700 and R700+Occ-1) is up to 15 % more frequent (less frequent) for WP3 (WP5) compared to the all-days situation. For precipitation amount (Fig. b), the selection frequency of the main regressive structures (Str. no. 1 and no. 3 based on R700 and W700, respectively) can similarly change up to ±10 %. The reduced selection of a main regressive structure for a given season or WP can lead to preferential retention of some secondary regressive structure. For instance, the regressive Str. no. 8 based on W700+H is selected 10 % more frequently for WP2 than for the all-days situation (Fig. b).

The preferential (or conversely reduced) selection of some regression structures for given WTs or seasons was estimated for all grid cells of France. In most cases, the preferential (or reduced) selection was found to present a noticeable spatial coherency. Different configurations are observed as illustrated in Fig.  and discussed below.

The preferential selection of some regression structures can first be observed over large to very large regions. As an example, the preferential selection of Str. no. 3 for the prediction of precipitation amount for days in WP7 (more than +15 % compared to usual) is obtained for all grid cells in France. Whatever the location, the vertical velocity W700 seems thus required in this specific weather pattern. Another example is that of WP8 which corresponds to an Anticyclonic situation. Whatever the location, no precipitation is really expected for this configuration. No predictor is thus required in addition to geopotential heights used in the analog stage. This configuration logically leads to a large preferential selection of the backup AM25 model.

For a given weather pattern, the preferential selection of a regressive structure can also vary from one region to the other. For WP2 for instance, the structures based on W700 or on W700 and H are selected much more often along the Atlantic coast and in the north of France. The backup AM25 model is conversely more selected in the southeast, on the Mediterranean coast especially. For this weather regime, the southeast is actually protected by the Massif Central mountain and thus usually does not receive precipitation cf. Fig. 3 of.

The preferential selection of a regressive structure can also be obtained for rather small and specific regions. In Fig. b, the regressive Str. no. 8 based on W700+H is more frequently selected for WP7 (around +15 %) in the Cevennes–Vivarais region (southeastern part of the Massif Central) and in the pre-Alpine mountains (western part of the Alps). The combination of W700 and H seems thus to be very informative in those configurations for this really rare WP (4 % of the 20-year period).

Whatever the configuration, the preferential selection of regression structures presents some spatial coherency, at small or large regional scales. This obviously also suggests the spatial robustness of the informative predictors to be retained for given large-scale weather configurations.

The relevance of a two-stage analog/regression model has been explored in this study for the probabilistic prediction of precipitation over France. Atmospheric analogs of the prediction day are identified to estimate the parameters of a two-part regression model further applied for the prediction. The regression model consists of a logistic GLM for the prediction of precipitation occurrence and a logarithmic GLM for the prediction of precipitation amount. The prediction obtained with this two-stage approach updates the predictive distribution that would have been achieved directly from a one-stage analog model based on atmospheric circulation analogs. The two-stage approach makes the downscaling model adaptive: as the analog days are identified for each prediction day, the predictors and regression coefficients of the regression models can vary from one day to the other.

The regression stage allows a non-negligible prediction skill gain compared to the reference analog model (gain up to 0.1 skill score points for both the BSS and the CRPSS). The CRPSS gain is mainly achieved due to the regression model estimated for the precipitation amount. The introduction of local-scale predictors such as relative humidity is obviously crucial there. The adaptive nature of the model and thus the possibility of tailoring the downscaling relationship (both predictors and regression coefficients) to the current prediction day seems to be decisive as well. The CRPSS gain obtained with the two-stage approach is actually 2 times larger than the one obtained by with a two-level analog model where a unique and same second-level analogy variable (namely humidity) is considered for all days.

The prediction skill and adaptability of this two-stage approach was illustrated for the prediction of both the precipitation occurrence and amount in a simplified configuration where four predictors, selected in a preliminary analysis from a large ensemble of potential predictors, are used in the regression stage. The predictors used for precipitation occurrence are the relative humidity and vertical velocity at 700 hPa, the helicity integrated from 1000 to 500 hPa, and the occurrence of the previous day. A similar set of predictors is used for the precipitation amount (the occurrence of the previous day is replaced by the 700 hPa temperature). Most of the time, the final regression model only includes one or two predictors. It also very often includes the relative humidity R700 for precipitation occurrence and R700 or the vertical velocity W700 for precipitation amount. Some combinations of predictors, almost never used in general, appear to be more frequently retained for some specific weather patterns and/or locations in France, revealing their potential interest for these situations.

For the sake of simplicity and to limit the degrees of freedom in our analysis, we considered a unique set of four potential predictors for all SAFRAN grid cells. This obviously leads to a sub-optimal prediction configuration. The main meteorological processes driving precipitation in France obviously differ from one region to the other. The most informative predictors are thus expected to be region-dependent and the set of predictors to be considered in the regression stage could be refined on a regional basis. This is expected to improve the skill of the prediction. The same would apply for an application of SCAMP to other regions worldwide.

A number of atmospheric variables have been considered as potential predictors in similar downscaling studies. The predictors found to be of interest are most often few. They are roughly the same than those considered in the preliminary analysis of the present work. However, as in the present work, the analyses usually carried out to identify these informative predictors are potentially misleading. The selection of a variable is indeed often based on its predictive power, estimated with some prediction skill score in an all-days evaluation framework. As highlighted in the present work however, some predictors are likely to be informative for very few meteorological situations. An all-days evaluation is expected to reveal robust predictors. It however very likely misses important situation-specific predictors. The two-stage approach here estimates the statistical downscaling link from a homogeneous set of days, with respect to their large-scale atmospheric circulation configuration. Those days are moreover atmospheric analogs to the prediction day. This two-stage approach has thus the potential to reveal the predictive power of very specific predictors, suited for very specific meteorological configurations. It leaves very likely room for significant improvements of the prediction skill for such unusual configurations. It gives likely also the opportunity to better understand the atmospheric factors under play in a number of non-frequent and atypical meteorological situations. Notwithstanding the technical limitations that may hamper such analyses, a broader exploration of a much larger diversity of predictors, possibly non-conventional ones, would be thus definitively worth in this context.

Both the predictors and the regression coefficients were shown in our work to depend on the analog days identified in the analog stage. This is the reason for the adaptability of the downscaling discussed above. Besides the adaptability, we ideally expect that for a given prediction day the predictor selection and the associated regression coefficients will be robust. Further analyses should explore this issue. An interesting work would be for instance to check that the predictors and their related coefficients do not significantly change when the set of analog days considered for the estimation is modified as a result of a different setup of the analog model (e.g., when one changes the archive period or the archive length).

Results of our work depend on a number of choices and assumptions. They for instance likely depend on the database used for the large-scale atmospheric predictors. The day-to-day behavior of such an analog/regression approach (and the skill of the prediction) likely depends on the database and especially on the quality of the predictors. An atmospheric reanalysis with a higher spatial resolution would for instance likely allow for a better description of the shapes of geopotential fields and for a more relevant simulation of regional/local thermodynamic processes. It would likely lead in turn to higher-quality variables for some atmospheric parameters such as air instability. This may allow for a better identification of the daily specificity in the downscaling relationship and for the most informative predictors to be used each day. The reverse may occur when using lower-quality predictors, for instance lower-quality data from reanalyses available for the 20th century or lower-quality data from climate or numerical weather forecasting models. The quality of the predictors is thus obviously also an important issue to be further considered. It may lead to different informative predictors, depending on the intended use of the model (forecast, simulation, or climate impact studies).

SCAMP was used here for the prediction of small-scale precipitation at individual grid cells. The prediction of precipitation fields, obviously required for a number of impact studies, is also a challenging issue e.g.,. Different adaptations of SCAMP would be worth investigating in this context. SCAMP could be for instance applied for the prediction of mean areal precipitation over the whole targeted spatial domain and some spatial disaggregation process could be further used to generate the required fields e.g.,. As highlighted by , the prediction skill of SCAMP is expected to increase with the size of the spatial domain targeted for the prediction, which makes such an approach rather appealing. Another possible strategy for spatial predictions would be to rely on the advantages introduced by the analog stage of SCAMP. indeed showed that for a given prediction day the same set of analog days can be used over rather large domains (up to a few 100s of kilometers) for a quasi-optimal prediction of local-scale precipitation. The precipitation field of each analog day (which is thus spatially coherent because already observed) could thus be used as a first-guess precipitation field scenario for the considered region. The field could be next updated at each location with day- and location-specific coefficients obtained from the regression stage of SCAMP. This spatial prediction issue will be considered in future works.