Snow interception is the amount of snow captured in a forest canopy. As much as 60 % of the cumulative snowfall may be retained in evergreen coniferous forests . In deciduous forests in the southern Andes as much as 24 % of the total annual snowfall may be retained . Due to the sublimation of intercepted snow, a large portion of this snow never reaches the ground , and the interplay of interception and sublimation creates significant below-forest heterogeneity in snow accumulation. estimated that 20 % of the seasonal snow cover in the Northern Hemisphere is located within forested areas. As such, the mass balance of solid precipitation in forested regions, characterized by strong spatial variability of snow accumulation, is a large contributor to the global water budget. Accurately modeling the spatial distribution of snow in forested regions is thus necessary for climate and water resource modeling over a variety of scales see. Furthermore, intercepted snow can drastically change land surface albedo values in forested regions. Previous studies observed large albedo differences (a range of 30 %) between snow-free and snow-covered forest stands e.g.,. Thus, in mountainous areas where forested and alpine regions coexist, accurate estimates of forest albedo play a key role in correctly modeling the surface energy balance. Due to the connectivity between interception and albedo, formulations of surface albedo over forested areas necessitate estimates of intercepted snow e.g.,.

To date, direct snow interception measurements have only been retrieved from weighing trees. These measurements are limited to the point scale, are resource-intensive sampling, and only allow for the analysis of small- to medium-sized trees or tree elements . However, there are indirect techniques that allow for estimations of interception over larger spatial scales. Indirect measurements that compare snow accumulation between open and forest sites allow for a larger spatial sampling but may be affected by other forest snow processes, such as unloading of the intercepted snow. As such, sample timing of snowstorm conditions needs to be evaluated e.g.,. Until recently, snow interception could not be characterized over length scales on the order of several tens of meters. However, at these scales snow interception can spatially vary due to canopy heterogeneity. The extensive data set of indirect snow interception measurements in evergreen coniferous forests (further referred to as coniferous forest) in eastern Switzerland collected by is likely the first data set that allows for a thorough spatial analysis of snow interception.

Several statistical models for forest interception of snow water equivalent (ISWE) have been suggested using a variety of canopy metrics and functional dependencies for the rate and amount of storm snowfall e.g.,. Though these models have been demonstrated to perform well, they often rely on detailed forest canopy density and structure metrics that are either not readily available or cannot easily be upscaled, thus limiting functionality in models where the mean of model grid cells over several hundreds of meters to a few kilometers is required, which potentially reduces validity in large-scale modeling efforts.

Traditional forest metrics used to parameterize snow interception include leaf area index (LAI), canopy closure (CC), and canopy gap fraction (GF) or sky view. These are mainly derived from hemispheric photographs (HP) taken from the forest floor looking upwards. However, these indices can also be estimated from synthetic hemispheric photographs (SP). The SP mimic HP but are generated from aerial light detection and ranging (lidar) data. This requires the inversion of lidar to a ground perspective and conversion from a Cartesian to a polar coordinate system . Prior work has also used return density ratios of lidar, which is computationally faster but less accurate than SP . Canopy structure, or the position of a canopy element relative to the surrounding forest canopy, has also been used to model snow interception. However, as pointed out by , some forest structure metrics such as LAI and CC are highly cross correlated. Therefore, expanded on prior interception models (which mostly rely on the highly cross correlated traditional forest density parameters of LAI and CC) by introducing uncorrelated novel forest structure metrics. Their empirical interception model utilizes the total open area, mean distance to canopy, and CC. While the latter parameter was derived from SP , the first two parameters were directly computed by a digital surface model (DSM). The total open area is defined as the total open area in the canopy around a point, and the mean distance to the canopy defines how far away the edge of the canopy is from a point. Recently extended the mean distance to the canopy vertically by deriving it for 1 m horizontal slices that were normalized with the corresponding elevation above the ground.

Due to the difficulties in measuring snow interception, previous empirical snow interception models were not validated in different snow climates, forest types, or at varying spatial scales. During the Snow Model Intercomparison Project Phase 2 (SnowMIP2) 33 snow models were validated at individual forested and open sites and many models used the snow interception parameterization from . This interception model was one of the first that used canopy metrics (LAI and CC), although a snow interception model for larger scales also requires the greater canopy structure. Overall, SnowMIP2 showed that maximum snow accumulation predictions had large errors compared to observed values in most models, but the snow cover duration was well estimated. Furthermore, a universal best model could not be found because model performances at forest sites varied. This may explain why there is still no common ground with several snow-related variables in land surface models , which led to the current Earth System Model-Snow Model Intercomparison Project (ESM-SnowMIP) showing overall larger errors in simulated snow depth on forest sites than on open sites . Recently validated three snow interception models developed for coniferous forests with observed point snow interception values in a deciduous southern beech (Nothofagus) forest of the southern Andes. All three empirical models required recalibration, with the recalibrated model showing the overall best performance. Similarly, model simulations of largely overestimated the observed accumulated snow depth in a spruce forest at Col de Porte in the southeastern French Alps. They attribute this to errors in the processes linked to the snow interception model based on due to an underestimation of the melt of intercepted snow. In a maritime climate previous snow interception models also failed to accurately model snow interception . While successfully modeled snow interception in a maritime climate, their model consistently underestimated snow interception in a continental climate forest. Overall, this demonstrates the need for more robust parameterizations of the processes affecting snow under forest, which is an important challenge for global snow modeling.

When modeling at resolutions greater than the point scale, accurate implementation of forest snow processes necessitates not just the mean of a grid cell but the standard deviation within a grid cell or model domain. However, to our knowledge, the standard deviation of snow interception has not yet been quantified. In this paper, we propose empirical parameterizations for the spatial mean and standard deviation of snow depth interception (IHS and σIHS) derived from indirect interception measurements at sites with length scales on the order of several tens of meters. We analyzed an extensive data set consisting of several thousand interception measurements collected immediately after storm events in a discontinuous coniferous forest stand in the eastern Swiss Alps . From a lidar DSM with elevations z , we derived the following two canopy structure metrics: (1) the standard deviation of the DSM (σz) in order to represent the spatial heterogeneity of surface height in a forested model domain; and (2) the spatial mean sky view factor (Fsky), which roughly represents the spatial mean canopy openness but is derived here on the DSM from geometric quantities that describe the received radiative flux fraction emitted by another visible surface patch (i.e., canopy patches) . These two metrics were correlated to spatial means and standard deviation of the indirect interception measurements. We validated the novel models with new indirect snow interception measurements from one site located in the Rocky Mountains of northern Utah, US, and from one site located at Col de Porte in the southeastern French Alps.

In this study we only used indirect snow depth interception measurements. Indirect snow interception data were obtained by comparing new snow depth accumulation on the ground between open and forest sites. As such, snow depth interception (further referred to as snow interception) leads to reduced snow depth on the ground at forest sites. This indirect measurement technique allows for a collection of snow interception data over a larger area and also for an investigation of spatial snow interception variability. We used three snow interception data sets as follows: one from the eastern Swiss Alps for the development of snow interception models and two data sets for the independent validation of the developed snow interception models. Of these, one data set was from the Rocky Mountains of northern Utah in the US and one from the southeastern French Alps. In each of the three data sets the snow interception was derived slightly differently, which is described in the following sections.

Subgrid parameterizations were derived for site means and standard deviations of the snow interception using forest structure metrics and open-site snowfall. We parameterize mean and spatial variability of snow interception for a model grid cell by accounting for the unresolved underlying forest structure (subgrid parameterization). Forest structure metrics are derived from DSMs to integrate both the terrain elevation and vegetation height.

We proposed two empirical models for spatial mean interception IHS to be employed in hydrological, climate, and weather applications. One model is a more compact model, Eq. (). This model uses a power-law dependency between IHS and accumulated storm precipitation PHS that is scaled by one forest structure metric: the standard deviation of the DSM σz. The other model, Eq. (), integrates a more complex parameterization by using a combination of a power law with an exponential dependence similar to the one suggested by for PHS and is scaled by two forest metrics, namely the sky view factor Fsky in combination with σz. For both IHS models, interception increases faster with increasing snowfall when forest density increases (i.e., larger σz). In the more complex model, increasing forest density is implemented by increasing σz and decreasing Fsky. Though Fsky can be precomputed and is temporally valid for many years (unless the forest structure changes due to logging, fires, insect infestations, or other forest disturbances), computing Fsky over large scales and/or with fine resolutions is more computationally demanding than for σz . A subgrid parameterization for the sky view factor of coarse-scale DSMs over the forest canopy would eliminate the precomputation of sky view factors on fine-scale DSMs. Such a subgrid parameterization for sky view factors over forest canopy could be similarly set up as previously done for alpine topography and would lead us towards a global map of sky view factors cf..

In general, more differences between the compact and more complex modeling approaches only displayed at the extremes. For instance, for small storm precipitation values (PHS=3 cm), the more compact parameterization performs slightly better, whereas for very large storms (PHS=43 cm) the more complex model displayed improved performance. The choice of one of these two models thus depends on the focus range of precipitation values and available computational resources.

Our choice for the functional form of PHS differs from previous parameterizations for snow interception solely using the sigmoid growth ∼1/(1+exp⁡(-k(P-P0))) e.g., or an exponential form ∼(1-exp⁡(-k(P-P0))) e.g., with increasing precipitation. While the base function of worked better for , a drawback of this relationship is that interception does not become exactly zero for a zero snowfall amount. To account for this, the model becomes complicated when applied to discrete model time steps . For this reason, selected the relationship proposed by for their parameterization of snow interception. However, the functional form of the model does not account for snow bridging or branch bending, thus modeling interception efficiency as decreasing through time. We also compared means and standard deviations over all the sites as a function of forest metrics and found that the use of storm means can introduce precipitation dependencies that might originate from an insufficient number of sites showing similar forest canopy structure parameter values for a given precipitation (cf. black line compared to colored dots in Fig. ). Based on the functional dependencies revealed by analyzing our data as a function of PHS and forest structure metrics, a simple power law was able to describe the spatial mean PHS dependency of snow interception (cf. Eq. ). The equation displayed that with increasing PHS, IHS increases. This is less pronounced with smaller σz or larger Fsky values (Fig. ). Very recently, a storm event power-law dependency was also found to best describe fine-scale SWE interception in a maritime snow climate . Our base functions for site means and standard deviations thus bear some similarity to previously developed fine-scale snow interception models. Despite an ongoing debate regarding the proper representation of interception, we believe that the interception models presented here have the advantage that they could be applied in various model applications for horizontal grid cell resolutions larger than a few tens of meters. Due to the lack of measurements over larger scales a validation remains impossible at the moment.

We derived just one empirical model for the standard deviation of snow interception σIHS that uses a power-law dependency on accumulated storm precipitation PHS scaled by one forest structure metric, namely the standard deviation of the DSM σz. We also tested a more complex model for σIHS using both forest metrics (Fsky and σz) that integrates a power-law dependency of PHS. However, model performances for the validation data set did not differ considerably from the ones for the more compact model. Therefore, we propose the more compact parameterization for σIHS (Eq. ) to facilitate broad model applications.

By using Fsky and σz derived from DSMs as forest structure metrics, we focused on the overall shape of the forest. This simplification is similar to the assumption by for solar transmissivity in forests under cloudless sky conditions. They assumed the fraction of solar radiation blocked by the canopy was equal to 1-Vf with Vf being defined as the fraction of the sky visible from beneath the canopy. Our simplification is also in line with previous suggestions. Primarily, to reliably describe interception by forest canopy over larger areas, the larger-scale canopy structure needs to be taken into account instead of only using point-based canopy structure parameters e.g.,. We proposed to calculate Fsky and σz on DSMs rather than on CHMs to account for terrain and vegetation height. This results from our correlation analysis for measurement data collected in rather flat field forest sites (Sect. ) and should be verified once spatial snow interception measurements become available in steeper terrain and over larger length scales.

The models for IHS and σIHS were statistically derived from measured snow interception data gathered in the eastern Swiss Alps. Naturally, empirically derived parameterizations can only describe data variability covered by the data set. However, even though the parameterizations were developed empirically, we could display that the parameterizations perform well for two disparate, independent snow interception data sets collected in geographically different regions, different snow climates, coniferous tree species, and prevailing weather conditions during collection of the validation data sets (French Alps and Rocky Mountains, US). For instance, in the French Alps, rather warm to mild winter weather conditions predominated, whereas rather mild to cold weather prevailed during the campaigns in the Rocky Mountains of northern Utah in the US. Though snow cohesion and adhesion are clearly temperature dependent, we did not observe decreases in overall performances under these differing weather conditions for our two IHS models, which do not include air temperature. In contrast, in a maritime (warm) snow climate correlations between air temperature and snow interception were recently found by . In addition to the spread in observed temperature conditions, our ranges of accumulated snowstorm PHS values of the development data set are fairly broad (e.g., PHS between 10 and 40 cm). The measurements of the validation data set are well within the range of the development data set values but also cover extremes, such as one very small (PHS=3 cm) and one very large snowfall (PHS=43 cm) (cf. Fig. ). It is thus reassuring that our models perform sufficiently well in varying climate regions; however, more validation data sets would be advantageous especially in regions experiencing extreme climates such as the cold arctic or warm maritime forests. Despite the existing variability in the data set, more spatial snow interception measurements would clearly help to increase the robustness of our empirical parameterizations.

To date, interception models have been created for SWE instead of snow depth and were mostly point models instead of spatial mean interception parameterizations. As such, a comparative assessment (beyond the independent validation sets in the body of this paper) of our models to absolute performance measures of previous interception models was difficult. However, we calculated relative error estimates for an intermodel comparison with two SWE interception models. We selected the empirical, recently developed SWE model from as well as the 50m×50m stratified SWE model (for 50m×50m grid cell size) from . The model utilized the same Swiss data set as this study, which is currently the largest set of spatial interception measurements in the world. The model error estimates were calculated for a subset of their data set which included three snowfall events and interception values acquired at three elevations under mild temperature conditions in the McKenzie River basin, Oregon, US for details on the data see Table 2 in. We estimated an NRMSE of 28.9 %, MPE of -5.7 %, and MAPE of 31.2 % for the three modeled and measured interception values. The model error estimates were calculated for a subset of the Swiss data set consisting of 34 spatial mean observed interception values (50m×50m) and 34 parameterized values. We estimated an NRMSE of 9.3 %, MPE of -16.5 %, and MAPE of 23.5 %. Compared to previous SWE interception models, we obtained improved performances (using means of error estimates over I(a) and I(b), respectively in Table ). The fairest comparison for our mean error estimates is with the stratified SWE model of . Thus, our mean error estimates reduce as follows for the more complex model (Eq. ) and the more compact model (Eq. ): by 9 % and 4 % in the NRMSE, by 60 % and 75 % in the MPE, and by 40 % and 50 % in the MAPE. Our improved model performances, when compared to prior interception models in tandem with a good performance for two distinctly different validation data sets, lend validity to improving coarse-scale climate and hydrologic (watershed and snow) model applications.

Despite the overall good performance of the models, we observed differences between the two validation data sets. The data set collected in France shows improved error statistics for snow interception IHS (e.g., for Eq. (): RMSE=0.35 cm, NRMSE=4 %, MAE=0.26 cm) when compared to the data set collected in the US (e.g., for Eq. (): RMSE=1.52 cm, NRMSE=14 %, MAE=1.4 cm). In France, intercepted snowstorm depth was measured as the difference of new snow depth in wooden boxes below trees and open-site new snowstorm depth. This was done in relatively short time intervals after a snowstorm. In the US, intercepted snow was inferred from total snow depth before and after a snowstorm event within forests and in an open site. Derived snow interception was often integrated over several storm events due to longer periods between measurement campaigns. Thus, these measurements were potentially influenced by other snow and forest processes such as snow settling, wind redistribution, sublimation, unloading, and melt and drip. Our interception models, however, only calculate how much snow is intercepted at any point in time, which provides the input for other forest snow process models such as for unloading, sublimation, and melt and drip. We thus assume that these processes will be addressed separately, as in all prior interception models . Despite some uncertainties in the validation data set from the US, it allowed for validation in a different snow climate than the French Alps and also covered a large spread in storm snowfall amounts (Fig. ).

Differences in model performances between the two validation data sets could also be attributed to the more accurate forest structure metrics for the French data set because of a higher resolution lidar DSM (higher point density of 24 returns m-2 returns and 17 last returns m-2) compared to the lidar flyover from the US (on average 7 returns m-2 and 5 last returns m-2). While it is clear that the higher the point cloud density, the greater the potential detail of derived DSMs, 1 m resolution DSMs computed from point clouds above 5 returns m-2 are usually quite consistent and are suitable for deriving coniferous canopy models that allow tree-level analyses . Current available or scheduled countrywide data sets are now around 1–5 returns m-2 e.g.,, and these densities can be expected to increase thanks to technical improvements in lidar sensors. Since fine-scale DSMs are the only input required to derive the forest structure metrics Fsky and σz, a global applicability of our snow interception models for coniferous forest would be possible with minimal required information.

To understand if the models would also work in other forest types or in disturbed forests, e.g., due to logging, fires, or insect infestations, more snow interception measurements in deciduous, mixed, and disturbed forests are required. Very recently showed that previously published snow interception models developed for coniferous forests from , , and required recalibration to match observed point snow interception observations in a deciduous southern beech Nothofagus stand of the southern Andes. We investigated the performance of our models for two measurement campaigns in a deciduous quaking aspen (Populous tremuloides) forest in our US field site. The measurement setup (20 m transects) was identical to the ones in the coniferous forest at this location (see Sect. ). Though overall the models compared well with the measurements, the model performance was not as good as for the coniferous forest. Because the lidar DSM was acquired in the summer, i.e., with leaves on the trees, the models naturally overestimated IHS and σIHS. For instance, using the more complex model for IHS (Eq. ) we obtained a mean bias of -6 cm, whereas when using the more compact model for IHS (Eq. ) we obtained a mean bias of -8 cm. For σIHS, the performance was slightly better overall with a mean bias of -3 cm (Eq. ). While this shows that the performance is clearly lower in such sites, we assume that the performance would be improved when the lidar is acquired in leaf-off conditions.

The lidar-derived DSM sky view factors do not account for small spaces between leaves or branches, which are well accounted for when sky view factors are derived from HP or LAI. In principle, sky view factors that are computed on DSMs represent, depending on the return signal used to create the DSM, a coarser view of the underlying forest canopy. While this increases the overall fine-scale error, we feel that the ability to calculate both our canopy structure metrics in the Cartesian DSM space, which allows an easy model application, far outweighs fine-scale resolution losses.

The statistical models for spatial mean and standard deviation of snow interception presented here are a first step toward a more robust consideration of snow interception for various coarse-scale model applications. They were built upon a very large data set, validated by two other data sets from different geographic regions and snow climates, and performed well for all three sites and under differing weather conditions. For spatial mean interception all NRMSEs were ≤10 % and for the standard deviation of interception all NRMSEs were ≤13 %. Compared to a previous model for spatial mean SWE at 50m×50m the presented models for spatial mean snow interception show improved model performances.

In our observed snow interception data sets, as much as 68 % and on average 43 % of the cumulative snowfall (accumulated snowfall of snowfall event in cm) was retained by coniferous forests (interception efficiency (snow interception/accumulated snowfall) of site means) and as much as 14 % and on average 11 % was retained by deciduous forests. These values compare well to previously observed snow interception in coniferous trees reaching up to 60 % of cumulative snowfall and up to 24 % of total annual snowfall in a deciduous forest in the southern Andes .

The empirical models integrate forest parameters that can be derived from fine-scale DSMs, which can be pregenerated and stored for large regions. One of the presented interception models only relies on the standard deviation of the fine-scale DSM, which is a very efficient way to integrate snow interception in coarse-scale models such as land surface models. This could greatly improve current forest albedo estimates and the subsequent surface energy balance for various model applications such as hydrological, weather, and climate predictions.

However, the presented parameterizations were developed and validated for spatial means and standard deviations over horizontal length scales of a few tens of meters. We can only hypothesize that the parameterizations are also valid at coarser length scales due to the use of nonlocal forest structure parameters. Representative nonlocal forest structure parameters require that a DSM of high enough resolution is available to represent the subgrid variability of forest structure in the coarse-scale model grid cell. However, there was and probably is, to date, no validation data available at large spatial scales. The investigated length scale matches current satellite resolutions (e.g., Sentinel and Landsat), which opens further cross validation and deployment opportunities with satellite-derived parameters such as surface albedos and fractional snow-covered area. With parameterizations for both the mean and standard deviation of snow interception by forest canopy, the distribution of intercepted snow depth in forests can now be derived whenever a sufficiently high-resolution DSM is available.