We use daily time series of precipitation and temperature from the global historical climatology network dataset (GHCNd, version 3.30-upd-2023080717) . Because of known biases in the temperature data within the SNOwpack TELemetry Network (SNOTEL) dataset , which is part of GHCNd, we replace the SNOTEL data in GHCNd with the Bias Corrected and Quality Controlled (BCQC) SNOTEL Data . For better coverage of the European continent, including the European Alps, we also use time series from the European Climate Assessment and Dataset (ECA&D, https://www.ecad.eu/ last access: 3 July 2023) , and from the MeteoSwiss data portal IDAWEB https://opendata.swiss/ (last access: 17 August 2023) . If the mean daily temperature is unavailable, we use the daily minimum and maximum to compute the daily mean as the midpoint between the two. A total of 34 536 stations are initially available. We apply a simple gap-filling and quality control procedure (see Supplement Sect. S1) and select stations with at least 5 years of gap-free data from 1950 to 2023, although the period available does not need to be continuous. A total of 14 881 stations with temperature and precipitation time series are still available after these filters are applied.

We use the Northern Hemisphere daily snow water equivalent time series from the NH-SWE dataset . A total of 11 071 quality-controlled and gap-filled stations are initially available in the dataset. We match the locations of the stations in the NH-SWE dataset with the locations of the stations with temperature and precipitation time series. We select those that are separated by less than 0.01° latitude and 0.01° longitude (ca. 1 km distance) from each other, with an elevation difference not larger than 10 m, and which contain at least 5 years of overlapping gap-free data with the temperature and precipitation time series. A total of 4736 stations from the NH-SWE dataset are matched with 3349 stations from the GHCNd, 766 from the BCQC SNOTEL, 593 stations from ECA&D, and 28 stations from MeteoSwiss (Fig. ). This results in a large range of stations across elevation, latitude and longitude gradients in the Northern Hemisphere (Fig. ). As the NH-SWE data contains only stations with at least 40 d of continuous snow cover on average, warmer climates and ephemeral snow climates are generally not included. The length of the remaining time series available shows the long-term nature of this study, with 50 % of stations having at least 30 years of available data (Fig. S1 in the Supplement).

Our study adopts the spatial scale of the Northern Hemisphere despite key data gaps in highly snow-dominated areas such as High Mountain Asia, including the Tibetan Plateau , Himalayas , and high mountains in Central Asia , because we did not find data meeting the requirements of our study in these regions.

We use the same snow season definitions as , as well as two temperature-based indices from to define the characteristic climate of each station, and two more climate indices based on temperature and precipitation (Table ).

We simulate daily time series of snow water equivalent, SWE, using the daily time series of temperature and precipitation and a simple temperature-index model as follows: 1dHSWE(t)dt=A(t)-M(t), where HSWE(t) [mm] stands for SWE at time step t [d], A(t) [mm d⁻¹] is the snow accumulation rate at time step t and M(t) [mm d⁻¹] is the snowmelt rate at time step t. We use an explicit time-stepping scheme to solve Eq. (), i.e.: 2HSWE,t=HSWE,t-1+Δ(At-1+Mt-1), where Δ [d] is the time step length (in this study, daily), At-1 is the snow accumulation rate during time step t-1 [mm d⁻¹] and Mt-1 the snowmelt rate [mm d⁻¹] during the preceding day t-1. At is computed as follows: 3At=PtifTt≤Ta0ifTt>Ta where Pt [mm d⁻¹] is the daily precipitation, Ta [°C] is the threshold temperature for snow accumulation, and Tt [°C] is the temperature on day t. Mt is computed as follows: 4Mt=α×(Tt-Tm)if Tt≥Tm0if Tt<Tm where α [mm (°C d)⁻¹] is the melt factor, which we assume to be constant in time, and Tm [°C] is the threshold temperature for snowmelt. Snowmelt Mt cannot exceed available snow, i.e. ΔMt≤HSWE,t+ΔAt.

We empirically derive and then estimate values for two parameters of the temperature-index model for all stations in our study: the snow accumulation temperature threshold (Ta) and the melt factor (α). We also evaluate whether a Tm=0 °C threshold is a reasonable assumption for the snowmelt temperature threshold. We estimate all parameters by comparing the station-based SWE time series with co-located time series of observed temperature and precipitation.

As the SWE time series from the NH-SWE dataset are based on snow depth observations (not on direct observations of snow water content), a daily increase in SWE (ΔSWE>0) can be attributed to actual snow accumulation with high confidence (i.e. incoming snowfall exceeding any potential melt during that time step), except in rare cases of measurement noise, which are negligible at the scale of this study. We can, therefore, empirically derive the snow accumulation temperature threshold by analysing the distribution of observed daily temperatures on days with ΔSWE>0. We set, for each station, the snow accumulation temperature threshold equal to the 80th percentile of the distribution of observed temperatures on days with ΔSWE>0 (i.e., 80 % of snow accumulation days occur at or below this temperature threshold). This choice is a compromise between capturing the majority of snow accumulation events while avoiding excessive near-freezing precipitation to be classified as snow. Based on these snow accumulation thresholds empirically derived from observed data for each station, we build a simple a multilinear regression model via Ordinary Least Squares that estimates the snow accumulation threshold (Ta,e) based on the climate variables from Table  (T‾ and ΔT). The parameter estimation model yields a coefficient of determination of R2=0.68 and the following coefficients: 5Ta,e=0.210×T‾-0.319×ΔT+1.834 where T‾ is the mean annual temperature of a station, and ΔT is the amplitude of the annual temperature cycle at a station (Table ). However, we set the minimum threshold value to 0 °C, as the model estimates snow accumulation thresholds below 0 °C for the colder climates, which could classify too much snow as rain. We use the empirically derived and estimated parameters at each station to run the temperature-index model simulations in Sect. .

Unlike for snow accumulation, the NH-SWE time series cannot confidently distinguish whether a daily decrease in SWE (ΔSWE<0) is due to snowmelt or due to an alternative loss mechanism such as sublimation or snow redistribution by wind, or due to a model inaccuracy if the maximum snow density parameter is reached too quickly . The minimum temperature at which ΔSWE<0 occurs cannot reliably indicate a melt threshold, as the decrease in SWE at that temperature may not be due to snowmelt but to other processes. Nevertheless, analysing the distribution of observed temperatures on days with ΔSWE<0 allows us to evaluate whether a Tm=0 °C is a reasonable assumption, and is also useful to gain insights into temperature-index modelling limitations as well as NH-SWE data limitations (see Sect. 4.1.2).

To empirically derive the daily melt factor (α^, in mm (°C d)⁻¹, from the NH-SWE time series, we divide the daily absolute magnitude of ΔSWE<0 by the corresponding observed daily temperature: 6α^=|Δ(HSWE,t-HSWE,t-1)|Tt Two observations are important here: (1) computed daily melt factors can become very large, or even infinite, if daily SWE decrease is small or temperature is close to zero. (2) computed melt factors can become negative (physically not possible) if ΔSWE<0 occurs on days with negative temperatures. We, therefore, remove daily melt factor values higher than 20 mm (°C d)⁻¹ (4 % of all data), as these are rare in the literature , and we also remove negative values (23 % of all data). After removing these values, we define the annual melt factor as the median of all daily melt factors in a year, and the melt season melt factor as the median of daily melt factors during the snowmelt season only (as defined in Table ). At each site, one final empirically derived melt factor (αd) is obtained by computing the median of seasonal melt factors.

Given the variability of the melt factor with different sources of snowmelt energy, it is interesting to check whether a multilinear regression model can be used to estimate the melt factor (αe) based on climate and geographic variables. Among the variables tested (T‾, ΔT, mean annual snowfall, latitude and elevation) and following a stepwise linear regression, the best model discarded ΔT (Table 1) as a predictor for the melt factor and shows, however, a very low coefficient of determination, R2=0.16, suggesting such an approach offers very low predictive skill. The model equation reads as follows: 7αe=9.6-0.00083×z-0.0868×θ-0.117×T‾ where α is the melt factor [mm (°C d)⁻¹], z [m] is the elevation above sea level, θ [decimal degrees] is the latitude, and T‾ [°C] is the mean annual temperature. We use the empirically derived and estimated parameters (despite the low predictive power) at each station to run the temperature-index model simulations in Sect. .

Note that, for the empirically derived parameter set, we use the first half of the available gap-free years at each station to derive the parameter set, and the second half of available data to evaluate the model performance in Sect. . For the estimated parameter set based on climate variables, we use a random 2/3 subset of the available stations for the multilinear regression model, and the other 1/3 of stations to evaluate model performance in Sect. .

We simulate three daily SWE time series for each station, with three different parameter sets (Table ). The first simulation is run with a common parameter set for all stations across the Northern Hemisphere, where the single common snow accumulation temperature threshold is arbitrarily set to 0.5 °C as the mid-point between 0 and 1 °C which is the typical range of values used in the literature. The common melt factor corresponds to the median of all derived melt factors across the Northern Hemisphere. The second simulation is run with the empirically derived snow accumulation threshold and melt factor from observations at each station individually (using half of the available data at each station), as described in Sect. . The third simulation is run with the estimated parameters based on the multiple linear regression models built for the snow accumulation threshold and the melt factor, which are built using 2/3 of the available stations (see Sect.  and ). For all simulations, the threshold temperature for snowmelt is set to 0 °C, but we also analyse the sensitivity of the results to a different melt threshold (see Sensitivity analysis subsection). An overview of the workflow to obtain the three parameter sets and to set up the model simulations is shown in Fig. , and the parameter values used are shown in Table .

We evaluate the performance of the temperature-index model for important indicators of snow season dynamics, namely: accumulation season onset, melt season onset, end of snow season, peak SWE, number of snowmelt days, and snowmelt rate (Table ). Timing errors are measured in days, with negative values indicating an early bias, and positive values indicating a late bias. Errors in peak SWE and snowmelt rates are expressed as relative percent error. We use these variables as indicators because they quantify important features relevant to our understanding of snow hydrology and because they provide a more comprehensive analysis of SWE time series performance than the typically used root-mean-square-error and the bias of the entire SWE time series. Note that for the empirically derived parameter set, performance is evaluated over the half of the available data at each station that is not used for deriving parameters. For the estimated parameter set, performance is evaluated over the 1/3 of stations that are not used for parameter estimation.

We provide a sensitivity analysis of temperature-index model performance to the parameter choices and dataset splitting choices.

First, we test different values of the fixed temperature threshold for snow accumulation, and also a smooth temperature threshold for the rain-snow phase transition, as suggested in the literature . We do this by varying the temperature threshold for snow accumulation from -0.5 to +1.5 °C in 0.5 °C steps for fixed thresholds. For smooth thresholds, we take a ±1 °C range around each fixed threshold, where precipitation is 100 % snow at the lowest end of the range and linearly decreases to 0 % snow at the highest end of the range. For each tested fixed and smooth threshold we repeat all model simulations and analyse model performance for total snowfall, peak SWE, and accumulation onset timing. Second, we analyse the sensitivity of the results to varying the snowmelt threshold from -0.5 to +1 °C in 0.5 °C steps. We then repeat all model simulations and analyse model performance for melt season onset timing and the end of snow season timing, as these are expected to vary with melt initiation thresholds. These sensitivity tests are applied to the first set of model runs (see Fig. ).

Furthermore, we apply sensitivity tests for the second set (temporal split per station) and third set (spatial split for parameter fitting and validation) of simulations. For the temporal split of stations, we test whether splitting the station time series in the middle or randomly selecting half of the available years has an impact on the model performance. For the spatial split of stations, we check whether repeating the random split of stations (2/3 for model fitting and 1/3 for model evaluation) has an impact on model performance.

Our analysis highlights the range of temperatures at which snow accumulation events occur and shows that most accumulation days occur at daily average temperatures below freezing. This is broadly in accordance with previous studies , but we now provide this evidence at much larger spatial and temporal extents. This highlights the potential of a simple snowfall air temperature threshold to capture most snowfall days successfully, especially for cold, continental climates with shallow snowpacks (e.g., North/East Asia, continental Canada). However, our results also highlight that some relatively large snow accumulation days at daily average temperatures close to, but just above freezing temperatures, are difficult to capture. The risk of missing snow accumulation days can be higher in warmer maritime climates with deeper snowpacks (e.g. western North American mountains, the European Alps, or Scandinavia) at near freezing temperatures. While a 0 °C threshold captures most events, regional adjustments or wet-bulb thresholds are preferable in climates with frequent near-freezing precipitation. Therefore, where snow accumulation days above freezing temperatures are critical to capture, we emphasise the importance of using a different precipitation phase method, such as the use of wet bulb temperature , if available. Alternatively, snow accumulation temperature thresholds may be adjusted regionally, as our analysis in Figs.  and  shows that warmer climates generally have higher snow accumulation temperature thresholds.

While a temperature-based precipitation phase threshold may miss snowfall days above freezing, they are well captured by snow observations. Conversely, snowfall days are well captured at temperatures well below the snowfall threshold. It is important to note that time series of snowpack observations may also fail to register precipitation days if they are small (<3 mm), which is typical of cold climates, such as Northe/East Asia and continental Canada. Additionally, our analysis suggests that rain-on-snow days during the melt season are more frequent and widespread than previously recognised. Undercatch of solid precipitation in rain gauges is also observed (ca. 10 % of fraction of precipitation days below freezing). It is important to note that the daily time scale nature of our approach limits the possibility to capture sub-daily variability, and the co-occurrence of snow accumulation and snowmelt within one day. However, our analyses show that these are small in frequency and magnitude and do not materially impact our results and conclusions. In the context of long-term performance of a temperature-index model at the daily scale, it is important to emphasise that our data clearly demonstrates that no snowfall threshold, fixed or smooth, consistently improves all metrics or reduces interquartile uncertainty ranges. This suggests that, although the temperature-threshold approach for snow accumulation is satisfactory on average, further performance improvements are challenging to achieve, highlighting the inherent limitations of the temperature-index approach.

We have tested the validity of a simple temperature threshold of 0 °C to capture snowmelt using an observation dataset at the largest spatial and temporal scale to date. The data clearly indicate that snowmelt occurs almost exclusively at daily average air temperatures above 0 °C, as evidenced by the sharp decrease in the frequency of days with no change in SWE as temperatures rise above this value and the linear increase of snowmelt magnitude with temperature (Fig. ). Temperature-index snow models often choose a value of 1 °C , 0 °C or even -1 °C as the threshold for snowmelt initiation. Overall, our results highlight snow accumulation and snowmelt days reach an equal frequency of occurrence at air temperatures around 0 °C, suggesting this is an appropriate air temperature threshold for most data in a large-scale study. However, we also find that to accurately predict the onset of the melt season, a slightly higher air temperature threshold may be required (Figs.  and S6). This is likely because zero degree snowpacks still need to overcome the latent heat of fusion to initiate snowmelt , and thus air temperatures may be slightly higher than zero during actual melt initiation. However, this degrades the predictive accuracy of the end of the snow season (Fig. S7), which is more sensitive to daily temperatures near 0 °C once melt has already begun. It must also be noted that the NH-SWE time series used for validation exhibited a slight delay on average in melt season onset , thus it is possible to speculate this could reduce the overall uncertainty in melt onset for the temperature-index model results presented here.

As with snow accumulation thresholds, these results highlight trade-offs, whereby adjusting thresholds improves performance for some variables at the cost of others. Interquartile ranges remain broadly similar across threshold choices, suggesting that model performance is relatively insensitive to the precise value of the threshold within the ranges we tested.

We have shown that melt factors vary substantially across spatial and temporal scales. Seasonal scale melt factors across the Northern Hemisphere ranged from 1 to 12 mm (°C d)⁻¹, in accordance with ranges estimated in the literature from individual sites at smaller spatial scales . While these literature sources find most common melt factors to range between 2 and 6 mm (°C d)⁻¹, our results further refine this range and show there is a clear dominance of melt factors between 3 and 5 mm (°C d)⁻¹, particularly in regions with deeper snowpacks and more temperate climates. This key finding suggests that melt factors tend to stabilise within these specific climatic conditions, which also tend to have warmer melt seasons. Furthermore, the low interannual variability of these melt factor values suggests that climatic conditions with deep snowpacks and temperate climates are conditions that are well suited for temperature-index modelling. For these deeper snowpacks, the dominant snowmelt energy sources are probably similar year to year, making the melt factors less variable and easier to predict. In contrast, it may be harder to robustly estimate melt factors for colder, continental climates with shallower snowpacks, as they typically exhibit higher melt factors with higher interannual variability. For these shallower snowpacks, it is likely that the partition of snowmelt energy sources varies from year to year, making the melt factors more variable. In addition, our results suggest other loss mechanisms, such as sublimation or snow redistribution by wind, are more common in these colder, shallower snowpack climates. As a result, temperature-index melt modelling will likely provide a poorer representation of SWE dynamics in these conditions, as other accumulation or loss mechanisms that are typically more important during the accumulation season may not be captured by the temperature-index approach.

The median performance of the temperature-index model across various indicator variables for a wide range of climate conditions across the Northern Hemisphere is surprisingly good given the model's simplicity and that parameters were only estimated from data, and not calibrated to improve model performance. Evaluation of snow season timing variables across the Northern Hemisphere showed a median error of 0 d for the accumulation onset, 4 d early for the melt onset, and 1–2 d early for the end of the snow season. This shows the long-term time series dynamics are, on average, well captured by the temperature-index approach. Nevertheless, our climate clustering analysis suggests that accurate snow cover timing dynamics are best captured for shallow (peak SWE <300 mm) and mid to low elevation snowpacks (<1500 m), while deeper (peak SWE >300 mm) or higher elevation (>1500,m) snowpacks show a worse performance. This is probably linked to an increased complexity of temperature and precipitation processes and their measurements over alpine areas .

Regarding snow accumulation and melt magnitude variables, a median relative error of -10% was found for peak SWE, a median relative error of +10% for the number of snowmelt days, and a median relative error of -22% for the snowmelt rate. These results highlight the excellent performance that the temperature-index models can achieve if the input data is of high quality and if most snowfall days occur at temperatures below 0 °C. However, the temperature-index model underestimates peak SWE especially for deeper snow climates (e.g. western North American mountains, the European Alps, and the Scandinavian mountains), where a negative bias is observed for up to 75 % of stations (Cluster 3). This is consistent with our analysis and highlights a possible combination of undercatch of solid precipitation, the inability of the temperature threshold to capture snow accumulation days above freezing, the potential underestimation of the snowmelt initiation temperature threshold, climate station measurement errors, or residual uncertainties with the validation dataset.

It is challenging to point out the exact cause of errors in peak SWE as it can be a combination of factors relating to accurately capturing accumulation and melt during the accumulation season. At the stations in Cluster 2 (shallow, long lasting snowpacks in the coldest climates), the peak SWE overestimation is linked to too much precipitation as snow accumulating on the ground compared to the observations (Fig. S8c). In these cold and more continental climates it is likely that drier low density snow is subject to redistribution processes not captured by the temperature-index model, leading to an overestimation of peak SWE. At the stations in Cluster 3 (deep, mostly high alpine snowpacks) the peak SWE overestimation is linked to not enough precipitation as snow accumulating on the ground during the accumulation season (Fig. S8e), which could be linked to increased measurement complexity in alpine areas, such as undercatch but also missing snow accumulation events at mean daily temperatures just above freezing not captured by the temperature-index model, as discussed in Sect. .

During the accumulation season, the temperature-index model shows more melt than observed for all clusters (Fig. S8b, d, f), but this could also be because of the already discussed limitation that the NH-SWE dataset may not capture small melt events during the accumulation season. Our analysis therefore suggests that inaccurately capturing precipitation is the main culprit for errors in peak SWE. Furthermore, feedbacks to other metrics arising from errors in peak SWE may occur. For instance, a temperature-index model underestimation of peak SWE may, in turn, result in an underestimation of the number of melt days, but an underestimation of the snowmelt threshold temperature may overestimate the number of days with melting conditions and compensate for the underestimation of melt days.

Contextualising these results is not straightforward because a direct comparison to other models is not possible at this scale, due to the lack of input data and information on appropriate parameter ranges. However, we can offer a useful benchmark by comparing reported performances from recent model evaluations. report peak SWE biases from −50% to +35% across ESM-SnowMIP models . found that land surface models underestimated peak SWE by 268 mm on average at SNOTEL sites. In comparison, our TI-based approach underestimates peak SWE at SNOTEL sites by 114 mm on average, and the timing of peak SWE is 23 d early vs. 36 d early reported by . While such comparisons are not one-to-one, they demonstrate that TI models, when interpreted cautiously, can yield useful constraints. Even for the worst performing climate clusters, most stations showed a reasonable performance, showing that simple approaches can be applied across climates, with known limitations described in this paper.

The temperature-index model performance analysis suggests that the estimated melt factors are relatively robust. Snowmelt rates, which should be the model performance indicator most sensitive to melt factor accuracy, have smaller relative errors than peak SWE and the number of melt days. However, it may also suggest that despite the challenges in estimating melt factors for colder and shallower snowpacks (as outlined above), melt rate model performance is not overly sensitive to melt factor accuracy, as melt rate estimates are consistently robust even under conditions where the melt factor experiences high interannual variability. In fact, the mean melt rate is more strongly linked to the mean melt season temperature than the mean melt factor (Fig. S9), suggesting that accurate temperature data and a reasonable melt factor value can yield robust model performance across the Northern Hemisphere. In any case, an important implication is that melt rates can be more confidently estimated using temperature-index models than some of the other key snow metrics (such as peak SWE and melt onset) evaluated here. However, it is important to note that any trend introduced by climate change impacts may complicate the robust estimation of melt factors .

A striking result of our study is the small difference between the three parameter sets used for model simulations, and in particular that the simulations with the single common parameter set perform comparably, and better in some metrics, than the simulations with the empirically derived and estimated parameter sets (as seen in Fig. ). There are a variety of reasons leading to small differences in model performance between the three parameters sets. First, the melt temperature threshold is the same for all sets of simulations, which limits the possible differences in results. Second, the snow accumulation temperature threshold varies per station but only minimally, as we set a minimum value of 0 °C. For the empirically derived and estimated parameter sets, 79 % and 86 % of stations have a threshold value of 0 °C, respectively. For the rest of the stations, values are mostly below 2 °C (Fig. S10). The final factor contributing to variability in model performance across parameter sets is the melt factor. The empirically derived and estimated melt factors are weakly correlated (R2=0.13, Fig. S11), showing that differences are big between parameter sets. However, the model performance metrics most sensitive to the melt factor (the melt rate and the timing of the end of the snow season) exhibit strong correlations between parameter sets (R2=0.70 and R2=0.62, respectively, Fig. S12). This indicates that model performance is not very sensitive to model parameters, at least when evaluated over long-term time series, and suggests that for long-term studies using temperature-index modelling, it is not necessary to use complex parameter estimation methods, as the use of commonly applied values in the literature will lead to similar model performances.