ADAMONT is primarily a quantile mapping adjustment method . In general, quantile mapping is considered to be one of the most efficient bias adjustment methods available . It consists of adjusting the quantiles of the simulated historical distribution based on the quantiles of the observed distribution. The main issues with quantile mapping relate to the assumption of time-invariant model biases, the fact that temporal properties of the RCM are untouched by the adjustment method and that the spatial and intervariable consistency is not guaranteed. Moreover, showed that for mid-latitude climates, such as in Morocco, quantile mapping adjustment can vary for different weather regimes, because model biases vary in different regimes. Similarly, demonstrated the sensitivity of quantile mapping adjustment to circulation biases over the alpine domain. Additionally, the frequency of weather regimes may change in a changing climate . To improve the stationarity of our method in a changing climate, weather regimes are thus taken into account, i.e. quantile adjustment functions are computed and applied depending on the weather regime.

Assuming the availability of a gap-free meteorological observational dataset at hourly time resolution consisting of one or several geographical locations considered sharing similar large-scale meteorological conditions, and daily RCM model outputs covering the geographical domain of interest, the statistical adjustment method ADAMONT consists of the following steps:

RCM grid point selection: for each observation point, a RCM grid point is selected by minimising the following distance:(Δx)2+(Δy)2+(N×Δz)2,

where Δx, Δy and Δz represent the longitudinal, latitudinal and vertical distances (in km) between the observation point and the RCM grid points, and N is referred to as the elevation factor. Values of 0, 50 and 100 were tested, but 0 (N0) and 50 (N50) are reported in this study. The factor N is a scaling factor between horizontal and vertical distances, allowing us to take into account the strong dependence of meteorological variables (mainly precipitation and temperature) on altitude e.g..

Although the ADAMONT method is highly generic and can be applied using any hourly-resolution observational dataset, in the following we focus on the use of ADAMONT using the SAFRAN reanalysis data as an observational dataset. We first describe SAFRAN, then we present specific features of the ADAMONT method when using SAFRAN as the observational dataset.

The SAFRAN system is a regional-scale meteorological downscaling and surface analysis system , which provides hourly data of temperature, precipitation amount and phase, specific humidity, wind speed, and shortwave and longwave radiation for each mountain region (or “massif”) in the French Alps (23 massifs, as illustrated in Fig. ) but also in the French and Spanish Pyrenees and Corsica. Unlike traditional reanalyses, SAFRAN does not operate on a grid but on French mountain regions subdivided into different polygons known as massifs. Massifs correspond to regions ranging approximately between 500 and 2000 km2 for which meteorological conditions are assumed to be spatially homogeneous but vary with altitude. SAFRAN data are available for elevation bands with a resolution of 300 m, i.e. altitude levels 600, 900, 1200, 1500 m etc. are typically considered, making it possible to extract meteorological information at these altitude levels, or in-between using altitude interpolation. It was used by to create a meteorological reanalysis over the French Alps by combining the ERA-40 reanalysis with various meteorological observations including in situ mountain stations, radiosondes and satellite data. It was complemented after the end of the ERA-40 reanalysis (2002) by large-scale meteorological fields from the ARPEGE analysis so that it now spans the period from 1959 to 2016, making it one of the longest meteorological reanalyses available in the French mountain regions.

When the ADAMONT method is applied using the SAFRAN reanalysis, only one geographic coordinate is used for each massif, corresponding to the centre of the massif (see Fig. ). However, for each massif several altitude levels are considered, which means that depending on the N factor considered different RCM grid points may be selected for a given massif and altitude. Also, in order to maximise the consistency between massifs after the adjustment process, the dry/wet analogue day criterion used for the time disaggregation of RCM-adjusted variables into hourly variables is computed generally for the entire SAFRAN dataset, here in the 23 French Alp massifs. This means that a day is considered dry when the average of all daily precipitation data is below 1 kgm-2day-1 and wet if it falls above the threshold for all massifs and all altitude levels (from an observational perspective), and for all corresponding adjusted RCM grid points (from an adjusted RCM perspective).

Crocus is a detailed snowpack model within the SURFEX externalised surface module . It enables the computation of the exchanges of energy and mass between the snow surface and the atmosphere (radiative balance, turbulent heat and moisture fluxes, etc.), but also between the snowpack and the ground underneath. Similarly to most land surface models, it requires sub-diurnal (ideally hourly) meteorological forcing data including air temperature, humidity, incoming longwave and shortwave radiation, wind speed, and rain and snow precipitation. The one-dimensional multilayer physical snow scheme Crocus is able to simulate the evolution of the snowpack over time by accounting for several processes occurring in the snowpack, such as thermal diffusion, phase changes, metamorphism, etc. The SAFRAN-Crocus model chain has been operationally used for more than 20 years for avalanche hazard forecasting and extensively evaluated over the alpine domain in particular with snow depth observation stations . Here we apply the Crocus model using either the SAFRAN reanalysis itself or adjusted fields from a given RCM using the ADAMONT method, in order to compute and compare snow conditions using either driving data. This is both a proof-of-concept of the applicability of the ADAMONT method to generate data appropriate to driving land surface models and a means to assess the intervariable consistency of the ADAMONT output given that Crocus is simultaneously sensitive to all meteorological fields and potentially disturbed by inconsistencies in the forcing dataset.

To evaluate the ADAMONT method, it was applied to the Météo France ALADIN RCM forced by ERA-Interim over the time period from 1980 to 2010. This RCM was run at 12.5 km resolution and we use the daily time resolution output, which is consistent with typical output from EURO-CORDEX RCMs. This simulation was then adjusted against the SAFRAN reanalysis. The spatial domain (2200km×2200 km, centred on France; see Fig. ) is deliberately smaller than EURO-CORDEX (5000km×5000 km domain covering all of Europe; Fig. ) although both are on the same order of magnitude in order to place more emphasis on the method skills than on the output of the RCM itself, especially in terms of chronology. Indeed, the smaller the domain, the more it is constrained by its driving large-scale model (be it a GCM or a reanalysis) .

Performance indicators described below were computed for temperature and total precipitation but also for the snow depth, which integrates all the meteorological variables considered in the ADAMONT method. Focus was hereby placed on evaluating the ability of the method to correctly represent integrated outputs computed using SURFEX/ISBA-Crocus from meteorological variables adjusted independently of each other. This is often applied to river discharge for downscaling methods used for hydrological applications e.g..

The method was applied for all 23 massifs of the French Alps and all elevation bands (Fig. ), totalling 187 massif/altitude configurations. Performance indicators, described below, were either computed spanning all configurations or focusing on a given altitude level (1200 and 2100 m) and/or a subset of massifs (the Vercors massif was taken as an example, and computations were also performed separately for the northern and southern Alps, respectively).

We specifically tested the following aspects of the method:

RCM grid point neighbour selection techniques (N=0 or N=50).

The following indicators were analysed for temperature, total precipitation and snow depth:

the seasonal average time series from 1980 to 2010 in the SAFRAN and the adjusted RCM datasets;

These indicators are classically employed e.g. to assess the following:

the ability of a model or method to reproduce the statistical characteristics of the observed meteorological variables (through the RMSE, the mean bias, the ratio of standard deviation values, the duration and persistence of precipitation events and the cumulated PDFs) and their spatial variability (through the mean values at each elevation band and the analysis of different massifs);

When available, we compare the indicators with the same criteria applied to analog-resampling-based or transfer-function algorithms by and , and for other downscaling and adjustment methods by and .

Table  outlines the input and output variables of Crocus. Table  presents a summary of the different configurations used for the evaluation.

This section provides the evidence needed to assess the performance of the ADAMONT method applied to a RCM driven by a global reanalysis (ERA-Interim) using the SAFRAN meteorological reanalysis as the observational dataset in the French Alps. Adjusted RCM data are compared to SAFRAN itself. Adequate performance of the method is attained when the two datasets match best.

Figure  presents the location of the Vercors massif and its average temperature, precipitation and snow depth for each elevation band, for the evaluation period in the SAFRAN/Crocus reanalysis as well as adjusted RCM. The shape of the mean altitudinal evolution of all three variables is well represented compared to SAFRAN, which is also the case for other massifs (see the Supplement). The computed temperature values are very similar to the ones in SAFRAN. It is less so for precipitation, with over- or underestimation depending on the learning period (Fig. ) and the massif considered (see the Supplement). Despite the differences in the magnitude of average precipitation in the adjusted RCM compared to SAFRAN, the magnitude of average snow depth in the different adjusted RCM simulations is remarkably close to the results obtained using the reanalysis as meteorological input, with slight differences depending on the massif (see the Supplement). For all variables and all massifs, the difference between simulations using the two RCM grid points neighbour selection techniques (N=0 or N=50) is smaller than the difference induced by using different learning periods.

Figures – display the mean bias and the RMSE for each raw and adjusted RCM simulation compared to SAFRAN, for temperature, precipitation and snow depth for the Vercors massif. Additionally, Table  presents the corresponding scores on the annual time scale compared to mean values, for the adjusted RCM L. 1980–2010 simulation, for each massif in the French Alps and for the northern and southern Alps at 1200 and 2100 ma.s.l. This highlights the large biases and RMSE values obtained when using raw RCM simulations compared to adjusted simulations, a feature common to all massifs (Figs. – and the Supplement).

For temperature, biases of the adjusted RCM simulations vary with elevation and for the different massifs (Fig. , Table  and the Supplement), but lie always within 1 K. Biases are generally smaller in autumn (SON) than for other seasons. RMSEs also vary with elevation and massifs, and can differ significantly between simulations using the two different RCM grid point neighbour selection techniques. For elevations above ≈2100 ma.s.l., stronger biases and higher RMSEs are found for simulations using the selection technique accounting for altitude differences (N=50), especially in summer (JJA) than for other seasons. Temperature biases and RMSE values also depend on the learning period considered, the longer learning period 1980–2010 generally presenting smaller biases and RMSEs (Fig.  and the Supplement).

For precipitation, biases generally vary with altitude (Fig. , Table  and the Supplement), but less than for temperature (Fig. , Table  and the Supplement). Biases of the adjusted simulations remain smaller than 150 kgm-2 per month in absolute value, corresponding to up to 90 % depending on the massif and altitude, and are generally stronger in summer. Smaller autumn and winter precipitation biases lead to a good agreement between the magnitude of average snow depth in the different adjusted RCM simulations and the results obtained using the reanalysis as meteorological input (as noted in Fig. ). RMSE values generally increase with altitude. Using different RCM grid point neighbour selection techniques has less impact on precipitation scores than for temperature, except that the N=50 configuration yields more variability in scores with altitude. This is due to the choice of different grid points for different altitudes of a single massif, because precipitation is spatially more variable than temperature. The influence of the learning period on scores is also visible.

For snow depth, the biases never exceed 50 cm, which corresponds to up to 50 % depending on the altitude and the massif (Fig. , Table  and the Supplement). The biases are smaller in autumn than for other seasons, similar to temperature (Fig.  and the Supplement). Summer biases at high altitudes are almost always negative, which cannot always be explained by a combination of positive biases in temperature and/or negative biases in precipitation, indicating the possible impact of other variables on snow depth (such as longwave radiation for example). RMSE values increase with altitude due to the effect of increased snow accumulation with altitude. Using the N=50 configuration generally degrades scores at high elevations, similar to the effect on temperature.

For precipitation and snow depth, simulations without the final quantile mapping on snowfall and rainfall are also presented (by definition it has no impact on temperature). It is clear from Figs. – and the Supplement that without this final correction (no corr), biases in precipitation and snow depth are much stronger and RMSEs much higher than when this correction is applied.

Figure  represents the ratio of the standard deviation values between each adjusted RCM simulation and SAFRAN for temperature, precipitation and snow depth and as a function of the integration window (from 1 day to several years) over the learning period. Ratios are displayed for the Vercors massif, for altitudes of 1200 and 2100 ma.s.l. If this ratio is lower than 1, it means that adjusted RCM simulations have a smaller standard deviation (i.e. variability) than SAFRAN. For temperature, the ratio of standard deviation is very close to 1 for integration windows of 1 day to a few months. It varies more for longer integration windows of 1 year or more. The differences between the two altitudinal levels considered or between massifs are limited (Fig.  and the Supplement). Similarly, choosing different learning periods or different grid point neighbour selection techniques has little effect on the ratio of the standard deviation values. For precipitation, ratios of the standard deviation values are also close to 1 (generally between 0.8 and 1.2) for integration windows of 1 day to 1 month. This result is similar to ratios of variance between daily RCMs adjusted with a cumulative distribution function transform and observations for the Mediterranean region in . For integration windows of 1 month or more, the ratios vary more with under- or overestimation of variance depending on the massif, the learning period and the grid points neighbour selection technique considered (Fig.  and the Supplement). For snow depth, the ratio does not vary until 1 month of integration approximately (Fig.  and the Supplement), and shows larger variations for higher values. Some differences can be noted for different altitudes, and different massifs, but also for different learning periods and the two grid point neighbour selection techniques considered.

Figure  presents the cumulated PDFs of daily temperature, precipitation and snow depth at 1200 and 2100 ma.s.l. for the Vercors massif. The distributions of daily temperature for adjusted RCM simulations are remarkably close to the distribution of SAFRAN (Fig.  and the Supplement). The agreement is better than the one observed in and between the different configurations of analog-based and transfer function algorithms and SAFRAN for the Durance basin (see Fig. F.2 in , and Fig. 5 in ). A similar agreement was observed in between two configurations of a distribution-based scaling method and observations in Finland. Only small differences are observed for different altitudes or different massifs (Fig.  and the Supplement), and the choice of the learning period or the grid point neighbour selection technique has almost no impact on the PDF. For precipitation, the PDFs of adjusted RCM simulations are also very close to the PDF of SAFRAN with a slight overestimation or underestimation of moderate to high precipitation depending on the learning period, occurring for most massifs (Fig.  and the Supplement). This result is similar to that observed in for the Durance basin (see Fig. 11.7 therein). As for temperature, altitude and massif location have only a small impact on the distribution as well as the grid point neighbour selection technique considered. The distribution of snow depth, however, depends more on the massif considered and the altitude (Fig.  and the Supplement). As for precipitation, the moderate to high snow-depth values seem to be slightly overestimated or underestimated for most massifs, depending on the learning period. The choice of the grid points neighbour selection technique has also slightly more impact on snow depth PDFs than for temperature and precipitation. The fact that PDFs for temperature and precipitation are very close to the ones of SAFRAN is a logical consequence of using a quantile mapping approach. That it is also true for snow depth indicates that even if they are treated separately, the intervariable consistency of the meteorological fields generated using our method is, in general, appropriate.

The capacity to reproduce the duration and persistence of precipitation events is shown in Fig. . The ratio between the number of dry days and the number of rainy or snowy days is very correctly reproduced for every massif and altitude (Fig.  and the Supplement), the relative error on the probability of a dry day being lower than 5 %. This feature was also observed by in his study of the Durance basin (see Fig. 11.10 therein). The persistence of dry and rainy/snowy events is generally underestimated (up to about -30 %), which was also the case in , even though the error depends on the massif and the altitude considered. In general, errors on the persistence of precipitation events are larger in massifs of the southern Alps than the northern Alps (see the Supplement). Using different learning periods and different grid point neighbour selection techniques has an impact on scores, but this is small compared to the influence of the massif or the altitude.

Figure  represents the mean annual cycle of temperature, precipitation and snow depth for the different adjusted RCM simulations vs. the SAFRAN/Crocus reanalysis, for the period 1980–1995 and 1995–2010 for the Vercors massif at 1200 and 2100 ma.s.l. The mean annual cycle of temperature is very well reproduced for every massif and altitude (Fig.  and Supplement). Using different grid point neighbour selection techniques has a limited impact on the mean annual cycle. For precipitation, the mean annual cycle is relatively well reproduced (Fig.  and Supplement). The choice of grid point neighbour selection technique can have slightly more influence on the results than for temperature. For snow depth, the annual cycle is remarkably well reproduced, with peak snow depth in the core of winter (JFM) and no snow or reduced amounts in late summer months (JAS) (Fig.  and Supplement). As for temperature, the impact of the grid point neighbour selection technique is very limited.

The chronology of time series of seasonal averages of temperature, precipitation and snow depth from 1980 to 2010 is shown in Figs. –, for the Vercors massif at 1200 and 2100 ma.s.l. in SAFRAN and the adjusted RCM. Temperature RCM time series are similar to SAFRAN, with an interannual variability which is well reproduced (Fig.  and Supplement). Some significant differences appear when using different learning periods, as already noted in Sect. . Using different grid points neighbour selection techniques has an impact on the time series of temperature which is generally smaller than the influence of the learning period. However, as already noted in Sect. , the agreement between observed and simulated time series is degraded for high altitudes under the spatial and altitudinal (N=50) grid point neighbour selection technique. The interannual variability in precipitation is also well reproduced for most massifs and altitudes (Fig.  and Supplement), especially given that the only forcing of the RCM comes from ERA-Interim reanalysis at the boundaries of the RCM domain. It is slightly less well reproduced in summer (JJA), as observed by for the analog-resampling-based transfer function algorithm DSCLIM and the Durance basin (see Fig. 10.1 therein). Differences between simulations using different learning periods mostly appear in summer (JJA). The use of different grid point neighbour selection techniques has a rather limited impact on time series of precipitation, whose magnitude depends on the massif and the altitude (Fig.  and Supplement). For snow depth, the interannual variability is well reproduced in winter (DJF) and correctly reproduced in intermediate seasons (MAM and SON). Summer snow depths are generally underestimated, as already noted in Sect. , but represent a small portion of the annual snow accumulation. Likewise, adjusted data using the spatial and altitudinal (N=50) RCM grid point selection technique can be degraded at high altitudes, similarly to temperature.

Figure  displays the temporal correlation between each adjusted RCM simulation and SAFRAN over the evaluation period for temperature and precipitation and as a function of the integration window (from 1 day to several years). Correlations are displayed for the Vercors massif, for altitudes of 1200 and 2100 ma.s.l. Additionally, Table  presents the same correlation values at the same altitudes, for an integration window of 1 year, and for the adjusted RCM L. 1980–2010 simulation only, for every massif of the French Alps and for the northern and southern Alps. Snow depth values were not included because of their cumulative nature. Correlations for temperature are very high (always above 0.8) for all massifs and altitudes until an integration window of a few months to 1 year (Fig. , Table  and Supplement), as found by (see Fig. F.21 therein). The differences between learning periods are negligible. As already observed in Sect.  and for the time series above, the correlation is clearly degraded for high altitudes (above ≈2100 ma.s.l.) in simulations using the N=50 grid points selection technique. Precipitation also yields satisfactory correlation values (always above 0.4) until a few months integration window, which vary depending on the massif considered (Fig.  and Supplement). Correlations are generally similar or even better than the ones observed in for various statistical downscaling models and different configurations of the ALADIN RCM (see Fig. 12.10 therein). The use of the N=50 grid point neighbour selection technique increases or decreases correlation values depending on the massif and the altitude considered. The choice of learning period has a limited effect on correlation, at least up to integration windows of a few months. Correlations are higher on the scale of the northern and southern Alps than on the massif scale (Table ). This scale dependence of precipitation downscaling skill was also illustrated by and .

Scores for the detection of precipitation events are presented in Fig.  for the Vercors massif for altitudes of 1200 and 2100 ma.s.l. The scores vary depending on massifs and altitude but a general pattern emerges (Fig.  and Supplement). The POD is the highest with values between 0.55 and 0.8, very similar to (see Figs. 11.14 and 12.8 therein). The FAR is rather low (always below 0.5) as is the POFD (below 0.2). TSSs are generally better for massifs of the northern Alps (0.25 to 0.6) than the southern Alps (0.1 to 0.4, Supplement), where PODs are lower and FARs much higher. Such results indicate that the method performs well in detecting precipitation events. Using different learning periods has a rather limited impact on the detection of precipitation. The choice of the grid point selection technique has a limited influence at low- to mid-altitudes, which increases above ≈2100 ma.s.l.

The temporal transferability of the ADAMONT method, i.e. its capacity to apply adequately to a period which is different from the learning period, can be evaluated from results in Sects. , and .

Figures – and the Supplement reveal some significant differences when using different learning periods. This feature is generally most visible in summer. It denotes a limit in the temporal transferability of the ADAMONT method, which was also the case in for the analog-based and transfer function algorithms (see Figs. 11.11 and 11.12 therein). Using the longer learning period of 1980–2010 yields better results, most probably due to the fact that, in this case, the learning and evaluation periods are the same, but also the fact that the learning period is longer.

There are some limits in the conclusions which can be drawn from this transferability assessment. First, reanalysis data used here as forcing for the RCM (ERA-Interim) or for statistical adjustment and evaluation purposes (SAFRAN reanalysis) are heterogeneous in time . These heterogeneities are especially marked in summer in the SAFRAN reanalysis, when most observations from mountain stations are not available . Secondly, variations which will occur in the future climate are expected to be much stronger than the variations which can be tested in our evaluation period. Issues related to the time transferability of the adjustment approach may be amplified when applied in the context of climate projections, but their relative impact will probably be lower than shown here given the magnitude of the expected changes.

The impact of the RCM grid point selection technique is illustrated in Sects.  and . Indeed, Figs. –, – and the Supplement show a clear degradation of scores for elevations above ≈2100 ma.s.l. using a selection criterion explicitly accounting for the altitude difference (N=50). This is linked to the scarcity of high-altitude grid points in ALADIN compared to SAFRAN, resulting in grid points being selected several tens of kilometres from the centre point of most SAFRAN massifs (see Fig.  and the Supplement for the location of selected grid points). The impact of this issue depends on the location of massifs relative to high-altitude grid points in ALADIN. For example, most southern Alps massifs are affected, except the southernmost massifs of Ubaye, Alpes Azur and Mercantour (see the Supplement), which are located less than 15 km from high-altitude points. This shows that, although it seems appealing to select RCM grid points at elevations matching the elevation of the observation dataset rather than using RCM grid points with a potentially large elevation difference (hence leading to stronger adjustment requirements), in practice the results are far more homogeneous and quantitatively generally equivalent or better when concentrating only on the horizontal distance between the RCM grid points and the observation dataset.

The lack of explicitly enforced intervariable consistency in the quantile mapping method can be a major disadvantage. As we focus on a mountainous region for the evaluation and future use of the method, the consistency between temperature and precipitation phase is crucial. The impact of this final correction is assessed in Sect. . Figures – and the Supplement show that without this final correction (no corr), biases for precipitation are much stronger and RMSEs much higher than with this final correction, highlighting its importance.

The intervariable consistency of the ADAMONT method is indirectly assessed by applying the evaluation metrics described above to an integrated output of the Crocus model, the snow depth, which is computed from meteorological variables adjusted independently from each other. As mentioned above, snow depth results are generally satisfying, which tend to indicate a good intervariable consistency. Performance indicators for snow depth are often consistent with temperature and precipitation indicators, even though they cannot always be explained by these two variables alone (for example the analysis of biases in Sect. ), indicating the probable influence of other variables not directly analysed here such as longwave radiation.

The spatial consistency of the ADAMONT method has not been evaluated other than by using spatial averages. In future studies, it would be necessary to test it by evaluating spatial correlations for example using metrics described in or by using integrated variables requiring spatial variability such as snow cover area or river discharges.

In this study, we evaluated the method using only the ALADIN-Climate RCM. However, showed that the choice of RCM could have a significant impact on the evaluation of the performance of the adjustment method. Evaluation using another RCM could thus prove useful, even though we would have to use RCM outputs run on the same spatial domain as the ALADIN-Climate RCM in order to compare them.