The heterogeneity of available studies posed several methodological challenges, e.g. the heterogeneity in projected regions and elevations, emission scenarios and examined variables.

Reported variables ranged from decreases in seasonal means for different periods , season lengths for different thresholds in snow depths , monthly values or seasonal evolutions of either SWE or snow depth. Since the relative reductions in SWE and snow depth were comparable across these studies , we treated the reduction values as interchangeable.

Tables  and in the Appendix summarize the examined variables. These studies were generally divided into two groups: Literature-Fit and Literature-Validation (see comments in Tables  and ). The data in Literature-Fit provided daily or monthly snow depths, both for the reference period and the future projection, which could be used to train the reduction curves in Sect. . The data in Literature-Validation reported seasonal reduction values or decreases in season length and was used for validation only (Sects.  and ).

We analyzed the seasonal snow depth reduction curves for the Literature-Fit data, by first linearly interpolating reported values to daily data. If reduction values were not provided, we computed those from reference and future snow depth or SWE data.

Figure  (top) shows reference and future snow depth at 1350 ma.s.l. under a +2.4°C scenario from . Peak snow depth decreased from 116 to 52 cm, with a reported reduction in season length of approximately 20 d. In most of the Literature-Fit data this shortening is asymmetric: for example, Fig.  (top) indicates a slight delay of a few days in future accumulation, whereas ablation occurs approximately two weeks earlier. In contrast, the corresponding future relative snow depths (Fig. , bottom), which are equivalent to relative reduction curves, appear symmetric around the value b. The reduction curve peaked shortly before the seasonal maximum (b=144DOWY), and approached 0 % toward both ends of the season. This shape implies both a delayed onset and an earlier end of the season, reflecting this general shortening of the snow duration. All relative reduction curves consistently showed this behavior without any systematic asymmetry, therefore, reduction curves fred(x) could be well approximated by a quadratic function: fred(x)=-100+a-ac2⋅(x-b)2,2with f(x)≥-100,

To express the relative future snow depth f(x)=fred(x)+100 directly, this can be rewritten as: 3f(x)=a-ac2⋅(x-b)2, with f(x)≥0, where:

x is the day of water year (DOWY, ranging from 1 to 366, with DOWY 1 = 1 September),

In Fig. , the maximum future snow depth a=47 % and the reduction curve peaks on b=144 DOWY (corresponding to 23 January), hence the peak of the reduction curve is 41 d prior to the peak in reference snow depth (Δb=-41). The total snow duration decreases from 216 to 195 d, resulting in a relative decrease of Δc=90%.

Finally, we compared all reduction parameter (a, Δb, and Δc) from the Literature-Fit data with temperature scenario and elevation (see Sect. ) and trained a linear regression model. To avoid overfitting, we used linear terms of temperature scenario and elevation and their interaction terms (elevation ×ΔT), which were scaled before training. While the reduction parameters a, Δb, and Δc can be computed for any given elevation and temperature scenario, these parameters were trained on data with elevations ranging from 750 and 2750 m and temperature scenarios from +1.1 to +4.8°C and should be treated with caution outside these ranges.

We applied the reduction curves to retrieve future snow depths for four measurement stations in Switzerland: Weissfluhjoch (WJ, 2540 ma.s.l.), in the eastern Swiss Alps, Maloja (MA, 1810 ma.s.l.), in the southern Swiss Alps, Saanenmöser (SM, 1390 ma.s.l.), in the western Swiss Alps, and Engelberg (EN, 1023 ma.s.l.), in the central Swiss Alps. These stations provide daily manually measured snow depth data from winter season 1991–2020 (30 years). We computed daily median, as well as the 5th and the 95th percentile for those stations, as our reduction curves were trained on data, which does not account for extreme events.

We chose a temperature scenario of ΔT=+2°C for the projections and then computed the reduction parameters a, Δb, and Δc for the given elevations. As our reference period 1991–2020 experienced a mean annual temperature increase of +0.5°C compared to the period 1981–2010 , this temperature scenario of ΔT=+2°C refers to the climate period 2043–2072 for the RCP8.5 scenario of .

Climate projections show uncertainties in temperature scenarios of around ±1°C for each RCP-scenario between years 2000 and 2100 . Therefore, we applied the same uncertainty range for the snow projections as follows:

Temperature scenario ΔT for median snow depths (here: ΔT=+2°C)

To apply the reduction curve (Eq. ) to climatological evolutions we have to compute b and c from Δb and Δc, respectively. To this end, we first determined DOWY(HSref,max) and the reference season length len(HSref>0cm). We suggest smoothing reference snow evolutions prior to determining DOWY(HSref,max) using a running mean of 30 d, to smoothen temporal variability in the data. Then b and c can be computed as follows: 6b=DOWY(HSref,max)+Δb7c=len(HSref>0cm)×Δc2=len(HSfut>0cm)2

Finally, using the reduction curve (Eq. ) and reference snow depths HS(x)ref, future snow depths HS(x)fut can be computed as follows: 8HS(x)fut=f(x)⋅HS(x)ref

After synthesizing all seasonal decreases from the Literature-Fit and Literature-Validation dataset into NDJFMA-decreases (Sect. ), we compared these Literature values to the NDJFMA-decreases from our projections of the four stations using Eq. ().

Furthermore, we also looked at the relative reduction in season lengths by counting the days for which a certain snow depth was reached, both in the reference len(HSref>x) and for the projections len(HSfut>x). The relative reduction in season length (HS>x)-decrease was then computed as follows: 9(HS>x)-decrease=len(HSfut>x)-len(HSref>x)len(HSref>x)⋅100

The Literature-Validation dataset contains decreases in season lengths for the following thresholds: >5, >30, >50, and >100cm. As most values were reported for the former two thresholds, we chose the following variables for validation:

Relative decrease in mean November–April snow depth (NDJFMA-decrease).

Projected relative decreases were calculated for the median snow depths and 5th–95th percentiles.

Figure  (left) presents the reduction parameter a, which corresponds to the maximum future relative snow depth. As expected, a decreases with increasing temperature scenarios, indicating less snow under warmer conditions. Furthermore, a increases with elevation, indicating that the decrease of future snow depth is more pronounced at lower elevations.

Although no consistent trend was observed for the parameter b itself, Δb was predominantly negative (Fig. , middle), suggesting a shift in the timing of peak snow depth towards earlier in the season. Notably, this shift becomes more pronounced with elevation, as Δb decreases with higher elevation. Δb positions the peak of the reduction curves relative to the peak in snow depth, which peaks later in the season for higher elevations as the accumulation period is longer.

The relative change in season length Δc is shown in Fig.  (right). All studies indicated shorter snow seasons in future scenarios, with the reduction in season length being more substantial at lower elevations and under higher temperature scenarios.

The following regression formulas were derived to describe the relationships of the reduction parameters with temperature and elevation (lines in Fig. ): a=83.51239-23.89164ΔT+0.01085⋅h10+0.00463ΔT⋅h, with a∈[0,100]Δb=16.28888-1.61312ΔT-0.02390⋅h11-0.00094ΔT⋅hΔc=115.18564-16.41657ΔT-0.00595⋅h12+0.00570ΔT⋅h

Snow projections under a ΔT=+2°C temperature scenario indicate a decline in snow depth across all sites (Fig. ). At Weissfluhjoch, the peak median snow depth decreases from 215 to 171 cm, while in Saanenmöser it drops from 64 to 36 cm. In Engelberg, the median snow depth during the reference period never exceeded 30 cm, and in 5 % of winters, snow depth remained at 0 cm throughout the entire year.

All projections also indicate shorter snow seasons in the future. For example, at Weissfluhjoch, the snow season (with HS>0 cm) is projected to begin approximately two weeks later and to end nearly two weeks earlier on average.

We validated the projections by comparing the projected decreases at all four study sites with values reported in the literature (Fig. ). Projected decreases shown as bars represent the 5th and 95th percentiles, while black lines indicate the decrease in median snow depth. Since literature values were not always based on the exact same temperature scenarios or elevations, comparisons were quantitatively made based on the range of values rather than exact matches.

Both the Literature-Fit and Literature-Validation datasets show similar trends (percentage changes) for the NDJFMA-decrease, the (HS>30 cm)-decrease and the (HS>5 cm)-decrease: the relative decreases are stronger with higher temperature scenarios and at lower elevations (Fig. ). The projections align well with the reported ranges and replicate the expected elevation-dependent trends. Both, literature data and projections show weaker (HS>5 cm)-decreases compared to the (HS>30 cm)-decrease.

Uncertainty ranges in the projections, showing the relative decrease in the 5th and the 95th percentile, seemed to be occasionally larger than uncertainty ranges from literature values. This is due to the methodology (see Sect. ), as we are projecting median and 5th–95th percentiles rather than individual years: For instance, the 5th percentile snow depth at Engelberg (EN) is zero at each calendar day in the reference period (see Fig. ). This indicates that there is currently not one period (day) throughout a year, where snow on the ground can be guaranteed in Engelberg. As such the projected decrease in Fig.  was set to -100 %. On the other hand, we want to highlight that this does not imply that 5 % of the future winters will be entirely snow-free.