The CARRA dataset is produced by the convection-permitting numerical weather prediction model system HARMONIE-AROME (HIRLAM–ALADIN Research on Mesoscale Operational NWP in Euromed–Application of Research to Operations at MEsoscale) . The dataset covers the area shown in Fig. , with a grid point spacing of 2.5 km. As we focus on the snow component of the reanalysis, we guide the reader to the full system documentation for details concerning the remaining components.

The CARRA system uses the externalized surface (SURFEX) as lower boundary in the model integration. The soil and snow is modelled by the force restore version of Interactions between Soil, Biosphere and Atmosphere (ISBA) , where two soil layers represent rapid (hourly) and slow (daily) variations in temperature to provide fluxes back to the atmospheric component . The snow is represented by a single layer scheme . This snow model represents the snowpack with snow water equivalent, snow density and albedo as prognostic variables.

In the CARRA system, in situ snow depth measurements and a binary satellite snow cover product are assimilated to update snow water equivalent of the model. The in situ snow depth observations are converted to snow water equivalent using monthly climatological snow densities from a lookup table. The satellite snow extent data is converted to snow water equivalent pseudo observations following a set of rules. If the satellite reports “no snow”, the pseudo observation yields 0 kgm-2; if the satellite reports “snow”, the pseudo observation is set equal to the model background value in the observation point. However, if the model exceeds 25 kgm-2, “snow” observations are discarded, and if the model exceeds 100 kgm-2, “no snow” observations are discarded. Furthermore, the snow analysis is not allowed to add snow on snow free grid cells if the surface temperature is above the melting-point of water. Such rules can be considered to represent some confidence in the model state as the direct insertion method does not account for that. The pseudo observation snow water equivalent value is then treated as an in situ observation. The satellite product and assimilation method are reported in . The in situ measurements (including pseudo observations from satellite) are interpolated horizontally (from point measurements to a 2D field) based on distance dependent correlation functions with the optimal interpolation (OI) scheme . While the spatial correlation length is equal for in situ and pseudo observations from satellite, they are weighted differently relative to the background field with observation error ratios σo=1.0 and σo=1.6, respectively. The snow water equivalent is finally inserted directly into the model replacing the background field, snow density and snow albedo are not modified in the analysis but kept constant.

Production of the regional reanalysis presented in this study consists of two main components. A land surface model to cycle the model snow states between analysis times, and the assimilation scheme that ingests the observations into the model state variables. To model the time evolution of the snow state, we use the ISBA model within the SURFEX framework . The ISBA model is set up using two patches (low and high vegetation), the multi-layer diffusion soil scheme , explicit snow scheme and the explicit canopy option for the high vegetation patch.

We have also adopted some of the configurations in . In their study, they found that heat flux from the uppermost soil layer to the lowermost snow layer was too large in the case of low snow fractions leading to too rapid snow melt. They proposed a workaround by reducing the heat flux in these situations which we included in our system. We used soil clay, sand and organic carbon dataset from soilgrids .

Using two patches means that within a grid cell, two independent mass and energy budgets are computed, adapted to the vegetation type. The resulting soil and snow states thus represent different sub grid conditions.

We use the ISBA explicit snow model with the default 12 layers, adding information about the vertical structure of the snow compared to the single layer model. The prognostic variables are snow water equivalent, snow density, snow heat content, albedo and snow age.

The land surface model is driven by atmospheric variables including precipitation (snow and rain), radiation (direct shortwave, diffuse shortwave and longwave), air temperature and humidity, surface pressure and wind. In this work, the forcing data is obtained from CARRA . The CARRA dataset has proven to perform better compared to the ERA5 product for both temperature and wind particularly in areas with complex topography . The forcing data are interpolated spatially to the model grid using bilinear interpolation and SURFEX internally interpolates the hourly values to the 10 min model time step using linear interpolation.

Data assimilation aims to bring the model state closer to the true unknown state based on observations. The corrected state, called the analysis (xa), can be expressed as the original state, often referred to as the background state (xb), and a correction term, 1xa=xb+δx.

Here the correction (δx), referred to as the increment, is typically a function of the innovation vector (yo-yb) where yo represents the observations, and yb=h(xb) is the model state in observation space, with h being a nonlinear observation operator.

According to , the most commonly used method for snow data assimilation in numerical weather prediction systems is the optimal interpolation (OI) method e.g.,. OI uses predefined structure functions to correlate observations with grid point values based on horizontal and vertical distances, often modelled with Gaussian curves . When snow depth measurements are used to correct snow water equivalent and snow density, the solution is not unique. It is common practice to assume that the snow density is constant and only update the snow water equivalent (see references above). With more advanced snow models, like ISBA explicit snow, the state vector becomes significantly larger compared to single layer models. With a 12 layer configuration, the full control vector consists of 49 prognostic variables at every grid point patch (snow water equivalent, snow density, snow age and snow heat, each times 12 layers plus snow albedo). Distributing the analysis increments between all the different control variables can thus be challenging. chose to distribute the snow depth increments proportional to the background layer thicknesses, thus assuming the profile of the snow is unchanged. However, so called flow-dependent data assimilation schemes can objectively optimize the analysis state by taking the current conditions (errors of the day) into account and thus narrow down the possible solution space and potentially updating the layers that actually are most likely to explain the observed condition. The EnKF uses an ensemble of model realizations to represent the background error covariance matrix and it is widely used in snow data assimilation . In the Kalman Filter, the increment is written 2δx=K(yo-yb) where K is called the Kalman gain matrix, and is defined as 3K=PHT(HPHT+R)-1 where P is the background error covariance matrix, H the linearized observation operator and R is the observation error covariance matrix. Furthermore, show that the EnKF gain matrix is written 4K=cov[x,h(x)][cov[h(x),h(x)]+R]-1.

Here the error covariance matrices are computed implicitly through the ensemble covariances.

In geophysical applications, the state vector has high dimensionality. For example, the CARRA domain has 1000 × 800 grid points which for a model with 12 vertical levels and 5 physical variables results in a control vector of size ∼ 5 × 10⁷. Large ensembles can thus be computationally expensive. Consequently, small ensembles have to be used and sampling noise is inevitable. The sampling noise could lead to spurious correlations between variables . Essentially, the spurious correlations result in increments based on unrealistic relationships between the observation and the control vector. This behaviour is avoided by limiting the impact of observations outside their representative area, referred to as localization. Localization can be applied by tapering the background covariance matrix, which means that the covariances between distant grid points are forced to zero. Another technique is the so called local analysis , where the data assimilation is split into smaller regions and only nearby observations are used.

proposed a localized version of the EnKF, solving the filter equations individually at each grid point. Compared to the example above, the control vector is now only 12 × 5 = 60 elements. At each point, relevant observations (typically within a radius) are selected and used in the assimilation and inflation of observation errors as function of distance is often used to smoothly reduce observation impact over increasing distances. Due to the manageable use of computational resources and the flexibility regarding observation types, we have chosen their implementation of the local ensemble transform Kalman filter (LETKF). The LETKF has shown promising results in both atmospheric and surface data assimilation systems . The LETKF update equations for a grid point are: 5xa=x‾b+Xbwa6wa=Wa+w‾a7w‾a=P̃a(Yb)TR-1(yo-y‾b)8Wa=[(k-1)P̃a]1/29P̃a=[(k-1)β-1I+(α∘(Yb)TR-1)Yb]-1 where x represents the ensemble control vector, X=x-x‾ the ensemble anomalies (or perturbations), bar represents ensemble-mean, a and b indicate analysis and background, respectively, w is the transformation weights between the background and the analysis, Yb represent the ensemble perturbations of the observation equivalent. P̃ and R are the error covariance matrices. β controls the inflation of the background error covariance matrix. α is a location dependent weight vector used to inflate R using element wise multiplication. Each element corresponds to an observation in y. In line with structure functions used in the CARRA OI scheme, we use a Gaussian curve as function of distance for α, 10α=exp⁡-2dhrh2-2dvrv2, where dh and dv are horizontal and vertical distances, respectively, between observations and the grid point, rh and rv are prescribed impact distances. Note that y holds the observations selected for the grid point. Despite that the assimilation is independent in each grid point, the same observations could be used in several close grid points depending on the specified impact distances and localization function for R. Close grid points using the same observations thus obtain the same weights (w) only modified by the localization weights α. The resulting increments are connected to neighbouring points through the spatial structure of the background ensemble perturbations. This is illustrated in Fig.  f) where the observation is marked by a star and the correlations (indicated by colour shading) are proportional to the increments for the surrounding area. For spatially homogeneous ensemble perturbations, the spatial structure of the increment would be Gaussian centred on the station.

shows that the analysis equation for the ensemble-mean x‾a can be written as 11x‾a=x‾b+(k-1)-1XaYaTR-1(yo-y‾b).

This is the common form of the Kalman Filter where (k-1)-1XaYaR-1 corresponds to the gain matrix K mapping the innovations (yo-y‾b) in observation space to increments (xa-xb) in model space. Investigating the elements of K is important for understanding the filter performance. We recognize that XaYaT is proportional to the ensemble correlation between the model space state and the observation space state. The ensemble correlation between a model variable (i) and an observed quantity (j) is given by 12rij=XiYjTXiXiT+YjYjT.

In situ observations of snow depth are usually placed in flat open areas away from high vegetation. Since the surface model used in this study explicitly represents low and high vegetation one could choose to assume that the observation was more representative for the patch with low vegetation. However, there are some factors that complicates this approach. First, the low vegetation patch is not defined for all grid points. Second, not all observations are placed sufficiently away from high vegetation to only represent that patch. We consider this to be out of scope for this work. To compute the model state in observation space, we use the following observation operator: 13yb=h(xb)=∑i2∑j12γiwi,jbρi,jb where w and ρ represent the snow water equivalent and snow density at the closest model grid point, respectively and γi is the patch (open land or forest) fraction. The index i represents the model patch and j the snow model level.

For ensemble based data assimilation methods, the ensemble perturbations X and Y are used to represent the uncertainty of the model estimate. They also describe the relationships between locations and between variables which map innovations to increments as shown above. It thus require care when generating the ensemble to ensure realistic relationship between the variables. The uncertainty can be sampled by perturbing forcing data, model state variables, model parameters or a combination of these . The benefit of only perturbing the forcing data is that the resulting model states are consistent with the model physics and the relationship between variables is as realistic as the model. However, processes that are unresolved by the model, for example redistribution by wind in our model, will not be represented by the ensemble by only perturbing the forcing data. Since we in this work introduce a multivariate control vector using the univariate observation vector, we only perturb the forcing data (rain, snow, longwave radiation, shortwave radiation, air temperature) to reduce the chance of introducing sampling noise. We follow the above studies and use cross-correlation between forcing variables to ensure realistic forcing conditions, temporal auto-correlation of the noise and spatially correlated noise . The details about the perturbation parameters are found in Table . The cross-correlations shown in Table  are computed from one year of forcing data. Spatial and temporal correlation lengths are provided in Table .

Precipitation forcing is usually perturbed by multiplying the field with noise from a log-normal distribution to avoid positive precipitation bias in the case of no precipitation (see references above). This method is not able to represent the possibility of precipitation if the original dataset has zero precipitation. used an advanced error model for satellite precipitation products to approach this issue and saw good results for soil moisture fields. However, it requires calibration with satellite and a radar reference which is not generally available. In this work we introduce a novel approach to improve the spatial representation of precipitation uncertainty. By assuming that precipitation fields in the forcing data have an error in horizontal placement, we can introduce uncertainty in areas surrounding such fields. We refer to the method as remapping and is based on advection using a random vector field. The approach builds on ideas from , where precipitation fields are remapped to better fit observations. An example of the remapping method is shown in Fig. . In this case we show a precipitation field from the CARRA dataset (Fig. a and several realizations after remapping Fig. b–d). The ensemble correlations shown in Fig. f) shows how an observation at the location indicated with a star can influence the surrounding area based on the current conditions. A detailed formulation of the method is found in Appendix . We evaluated the method by comparing occurrences of snowfall at observation locations. For snowfall greater than zero the unperturbed snowfall dataset had a hit rate of 83 %, while the ensemble had 91 %. For snowfall greater than 5 cm (snow depth) over 24 h the deterministic hit rate was 21 % while the ensemble hit rate was 56 %.

In the SURFEX model, snow water equivalent is defined in all model points. In grid points with no snow, the snow water equivalent is zero and the other prognostic variables (density, age, albedo and heat) are undefined. In order to include these grid points in the analysis and potentially add snow to snow free members, undefined values need to be initialized. While some predefined values can be used, e.g. corresponding to freshly fallen snow, such simplifications can quickly introduce nonlinear relationships between ensemble members. For example, during melting season, one member becomes snow free while the other members still have old and very dense snow. A different approach is to use a sample from the members containing snow to initialize zero snow members. Even though this method might not produce a representative spread for the snow, it avoids the filter to produce unrealistic analyses. In this work we initialized zero snow members with the ensemble-mean of members with snow. We use the ensemble-mean because it will have a neutral impact on the analysis ensemble spread. If all members are snow free, the filter is unable to add snow.

After the data assimilation is performed, the analysed state is the best estimate given that the assumptions of the filter hold. However, it might not be physically consistent, nor within the model bounds. The result can thus cause instabilities in the preceding model integration For example, negative snow water equivalent can in theory occur in the analysed state. In the data assimilation system, we use the ensemble-mean approach as mentioned above for all snow related variables given that a member violates any of the conditions in Table . This method prioritizes stability of the model integration over optimal performance of the filter.

In situ snow depth observations used in the data assimilation are obtained from the Meteorological Archival and Retrieval System (MARS) of the European Centre for Medium-Range Weather Forecasts (ECMWF). The observation dataset is not identical to the one used in CARRA, as CARRA uses additional local observations and a different quality control. Thus, we can bin observations into three sets. Let A^ be the observations used in CARRA-Land-Pv1 and B^ the observations used in CARRA. Further let C=A^∩B^ be the observations used in both systems, A=A^-B^ the stations only used in CARRA-Land-Pv1 and B=B^-A^ the stations only used in CARRA. In the following we will denote dataset A as “OBS-ONLY-Pv1”, B as “OBS-ONLY-CARRA” and C as “OBS-BOTH”. In addition, snow water equivalent from 6 snow pillow stations are used to evaluate the unobserved (by the assimilation system) snow water equivalent state. In CARRA, satellite derived snow cover maps are converted to in situ pseudo observations and assimilated in a similar way as real in situ snow depth. We considered to assimilate satellite snow cover in this study, but chose to focus on point observations. We suggest that satellite snow cover products are included in a follow up study.

Our experiments are set up on the eastern domain of the CARRA reanalysis, covering northern Scandinavia and Svalbard with 2.5 km grid spacing. The simulations are initialized on 1 September 2015 after a 2 month spin up without data assimilation, and cover the next four consecutive winters. Snow data assimilation is performed daily at 06:00 UTC using in situ snow depth measurements. The assimilation control vector includes snow water equivalent, snow density and snow heat. Snow albedo and snow age are not included since we focus on snow depth observations in this study. However, albedo and snow age might change after the analysis if snow is removed entirely or added on snow free grid points. In line with and references therein, the number of ensemble members was chosen after comparing shorter experiments with different ensemble sizes. Compared to using 20 members, a 10-member ensemble gave no considerable degradation in performance, we therefore opted for a 10 member ensemble. Additionally, an unperturbed control member was run without data assimilation for evaluation purposes. Hereafter, we refer to the unperturbed run as CTRL and the analysed ensemble simulation as CARRA-Land-Pv1 (Prototype version 1). The purpose of CARRA-Land-Pv1 is to provide a more realistic description of the snowpack covering the region.

We focus on vertically aggregated values, summarizing the values in the 12 layers of the model and define 14dtot=∑i12diwtot=∑i12wiρtot=wtotdtot for snow depth, snow water equivalent and snow density, respectively, where i indicates the model layer of the snow. The increments presented below are computed from the aggregated variables. For example, the total snow depth increment is written δdtot=dtota-dtotb, where a denote analysis and b denote background.

Figure  shows maps of the time averaged increments for snow depth δdtot (Fig. a), snow water equivalent δwtot (Fig. b), snow density δρtot (Fig. c) and their respective standard deviations in Fig. d–e. Snow depth and snow water equivalent increments are consistently positive over the domain, while snow density increments are mainly negative. The mean snow depth increments corresponds to several millimetres each day which roughly accumulates to a metre over 1 year. For snow water equivalent, about 200 mm of water is added each year through the increments. The time averaged snow depth innovations (yo-y‾b‾) are shown together with the average snow depth increments (Fig. a). Some observation sites are surrounded by increments with non-zero mean, particularly in the inland areas. The patterns of these increments are similar to the Gaussian localization function (Eq. ) limiting the impact of the observations and shows the impacted area from each station. The systematic positive increments are related to positive innovations, indicating too little snow in the model. Similar patterns but with opposite sign are not seen around stations with negative innovations (indicating too much snow in the model). The stations with mean negative innovations are situated in mountainous areas and have small impact on the analysis. This is due too the large vertical distance between model and observation and thus the increment is dampened by the localization (not shown). For snow density, the stations are not as pronounced as for snow water equivalent and snow depth, and the largest values are found in mountain areas. The maps of standard deviations indicate larger increment magnitude over mountain areas for all variables, and smaller over the inland areas. Inland, the pattern around stations seen in mean increments are not as strong, indicating that the areas suffer from a systematic underestimation. On the other hand, the mountain regions are more exposed to random errors which cause larger increments, but with less systematic signal. The standard deviations of snow density increments (Fig. f) show fine spatial structure compared to the smoother snow water equivalent (Fig. e). This suggests that the topography and thus temperature forcing might play a bigger role than precipitation for the density uncertainty.

Time series of domain averaged increments are shown in Fig.  for the same variables as Fig. . Largest positive snow depth increments are seen during spring where snow water equivalent increments are positive and density increments are negative. The early accumulation phase is also dominated by positive snow depth increments, but with smaller density increments. During the winter, snow depth increments decrease and in some years they become slightly negative before the melting phase begins. The figure suggests that snow is melting (loosing mass and compacting) too rapidly during spring, and that there is a slight underestimation of snowfall in the early accumulation phase.

We continue by investigating the structures of the Kalman gain matrix K through the ensemble correlations according to Eq. (). Figure  presents ensemble-mean snow profile time series with ensemble correlations between observation and model variables. Large positive correlations, as seen for snow water equivalent, indicate that a positive innovation (model snow depth is too small) will cause a positive increment to the control variable (increasing snow water equivalent). We note that the upper and lower layers have weak correlations for deep snow, this is a result of the vertical discretization of the snow scheme. The thickness of these boundary layers are limited to resolve the diurnal energy transfer between snow and air and snow and soil. For snow density (Fig. b), there is a positive correlation between snow depth and density in the lower layers during accumulation. This can be explained by the compaction of the lower layers in the snowpack due to increased mass in the upper layers during snowfall. The upper layers have negative correlations and strongest in the melting phase. Given a positive innovation in snow depth, the negative correlations would cause the analysis to decrease the density (increase snow depth) to compensate for too high compaction in the model simulation. The introduction of density in the control vector allows the snow depth to be adjusted (1) without changing the snow mass (negative correlations between total snow depth and snow density) and (2) larger change in snow mass compared to only updating mass in the case of positive density correlation.

Figure  shows the evolution of domain average snow depth (Fig. a), snow water equivalent (Fig. b) and snow density (Fig. c) for the CARRA-Land-Pv1, CTRL and CARRA. In terms of snow water equivalent, CTRL and CARRA-Land-Pv1 have higher values at the yearly maximum compared to CARRA. In general, there are relatively small differences between CARRA-Land-Pv1 and CTRL. In CTRL and CARRA-Land-Pv1 the evolution of snow water equivalent is smoother compared to CARRA. CARRA tends to have lower snow depth than CARRA-Land-Pv1 and CTRL during the accumulation phase and higher during the melt phase. These characteristics are consistent for all years. The density evolves quite differently in the two snow models (CARRA vs CTRL and CARRA-Land-Pv1). In the single layer snow scheme used in CARRA, the maximum snow density is 300 kgm-3 while the explicit snow scheme (CTRL and CARRA-Land-Pv1) has an upper limit at 973 kgm-3. The snow density reach high values in the melting phase in the multi-layer snow scheme. In CARRA, snow density is increasing faster in the accumulation phase, and stabilises earlier in the winter. During the melting phase the snow density in CARRA decreases compared to the density in the multi-layer models. There is a discrepancy between the CARRA snow density and the climatological values used in its snow analysis. On average, this climatological density is less than the CARRA density before April and greater after May. This coincides with the snow depth differences described above. Comparing CARRA-Land-Pv1 and CTRL shows a slightly increased density in the assimilation experiment.

Time series of mean errors in total snow depth averaged over all observation sites (assimilated and independent) are shown in Fig. . All datasets have negative bias in snow depth and the largest bias is seen in CARRA. CARRA-Land-Pv1 has smallest negative bias indicating larger snow depth compared to both CARRA and CTRL.

Time series of continuous rank probability score (CRPS) (for ensemble) and mean absolute error (MAE) (for deterministic) over the three different observational sets are shown in Fig.  and Table . The CRPS is equal to the MAE for a single member ensemble, or deterministic dataset. Largest errors are seen for CTRL in all sets and yearly maximum varies between 15 and 30 cm. Comparing the three observation sets, the largest errors overall are in OBS-ONLY-CARRA (Fig. c). CARRA has larger errors than CARRA-Land-Pv1 in OBS-ONLY-Pv1 and OBS-BOTH (Fig. a and b), with values ranging from 10 to 20 cm and 5 to 10 cm, respectively. For CARRA-Land-Pv1 a degradation compared to CARRA is seen for OBS-ONLY-CARRA (Fig. c) but it is consistently better than CTRL for all observation sets. We also note a significant drop in errors in CARRA during spring, this is seen for all years. This drop coincides with the transition from March to April and thus new values for climatological density used in CARRAs snow data assimilation From Fig.  (c), April is the month where the climatological densities fit best with the model density in CARRA.

We have included both CRPS and MAE of the ensemble-mean to highlight the benefit of CARRA-Land-Pv1 being an ensemble product. The CRPS values are consistently lower than the MAE of the ensemble-mean. However, relative to the other datasets, the CRPS and MAE of ensemble-mean are quite close. When CRPS for CARRA-Land-Pv1 and MAE of the other products are compared, the advantage of using the ensemble is only marginally contributing to the differences. Furthermore, the largest differences between MAE of ensemble-mean and CRPS are seen in the OBS-ONLY-CARRA. This shows how the uncertainty is increased in regions with few observations.

Maps of the difference in CRPS between CARRA-Land-Pv1 and CARRA for the three observation sets are shown in Fig. . For the stations in OBS-BOTH (Fig. a) and OBS-ONLY-Pv1 (Fig. b), CARRA-Land-Pv1 performs better than CARRA/CTRL (CTRL not shown) for most station points. This is also the case for the OBS-ONLY-CARRA set (Fig. c), except for a number of stations along the Norwegian mountains and a few scattered inland stations. A closer investigation of the points where CARRA has smaller errors, reveals the following characteristics. (1) The station is isolated, through the localization, from other stations, (2) the point is close to another (assimilated) station with different local conditions and systematic differences. In one case we found up to 50 cm difference in snow depth between observations at neighbouring grid points (not shown). We also note that the spatial distribution of observations in each set differs. In OBS-ONLY-Pv1, more observations are found in the southern part of the domain compared to OBS-ONLY-CARRA.

Summary scores are presented in Table . To assess the statistical robustness of the values we use bootstrapping with 1000 samples to compute the 95 % confidence interval. The intervals did not overlap between the models which indicates statistical robustness of the presented values. Compared to CTRL, CARRA-Land-Pv1 improves on all scores in all observation sets. Relative to CARRA, scores are worse over OBS-ONLY-CARRA, except for bias which is improved by 85 %. Over OBS-ONLY-Pv1 and OBS-BOTH, CARRA-Land-Pv1 performs better than the other datasets.

In the following comparison, we compute the MAE between CTRL, CARRA and CARRA-Land-Pv1 (ensemble-mean) and the six snow water equivalent observation stations available over the domain. Figure  shows differences in MAE (CARRA-Land-Pv1 minus CTRL (Fig. a) and CARRA-Land-Pv1 minus CARRA (Fig. b)) for each station. The dots indicate that snow water equivalent is improved through snow depth data assimilation in four stations and has neutral impact in two. Compared to CARRA, CARRA-Land-Pv1 has improved snow water equivalent in three, neutral in two and degraded in one station. The areas surrounding the stations with reduced errors compared to CARRA are associated with larger snow water equivalent values. The degraded station is on the other hand surrounded by negative snow water equivalent differences. In inland areas, the snow water equivalent differences are smaller in magnitude, but mostly positive.