The quantile regression forest (QRF) (Meinshausen, 2006) is a non-parametric regression technique that is based on random forests (RFs) and can be classified as a machine-learning approach. It learns from the training data by growing ensembles of regression trees on random subsets (bootstrap samples) of the training data (Francke et al., 2008a; Schmidt et al., 2023). In each regression tree, the data are recursively partitioned based on splitting rules, where both RF and QRF randomly select the predictors used for splitting. In contrast to random forests, QRF keeps all observations within a node (whereas RF only keeps the mean), which allows us to construct prediction intervals and to assess uncertainty (Schmidt et al., 2023).

The advantages of QRF include that it can handle multiple input variables; makes no assumptions on distributions; and can deal with interactions, nonlinearity and non-additive behavior. In terms of limitations, it does not allow for easy interpretation of the effects of single predictors, and model predictions will always be within the range of observations, i.e., if the predictors in the period of application exceed the range represented in the training data set (hereafter called out-of-observation-range (OOOR) data points), we can expect overestimations or underestimations of the target variable if the respective predictor has a continuing monotonic effect in this range (Schmidt et al., 2023).

With respect to modeling suspended sediment concentrations, studies have shown that QRF is very well suited to modeling sedigraphs and estimating annual SSY (Francke et al., 2008b, a; Zimmermann et al., 2012) and that it performs favorably compared to sediment rating curves and generalized linear models (Francke et al., 2008a; Schmidt et al., 2023). On a related note, RF (which QRF is based on) outperformed support vector machines and artificial neural networks (Al-Mukhtar, 2019) in modeling suspended sediment concentrations.

In a previous study (Schmidt et al., 2023), we trained QRF models at daily resolution on data of the two gauges Vent and Vernagt using the limited available time series of turbidity (4 and 15 years) and long records of the primary predictors, namely discharge (Q), precipitation (P) and air temperature (T) (Fig. 2). These can be seen as drivers or proxies for processes and catchment conditions crucial to sediment dynamics in high-alpine areas: e.g., discharge determines sediment transfer and erosion within the channel; precipitation is key for runoff formation and hillslope erosion, hillslope–channel coupling and the triggering of mass movement events; and air temperature controls the activation of sediment sources (e.g., sub- and proglacial sediments and their transport by glacier meltwaters or hillslope destabilization by permafrost thaw) and whether precipitation occurs as rain or snow. In addition to these primary predictors, we derived ancillary predictors to describe antecedent conditions and the cumulative effects thereof: e.g., longer-term discharge behavior may exhaust sediment sources or lead to sediment storage, long warm periods may deplete snow cover and accelerate glacier melt associated with increased subglacial sediment transport, and high antecedent moisture conditions may amplify surface runoff or promote mass movements in response to precipitation events. To capture these antecedent conditions while keeping the correlation between the derived predictors as low as possible, we computed sums of the primary predictors in non-overlapping windows of increasing sizes (e.g., 24, 24–72, 72 h to 7 d and 7 to 20 d ahead of each time step) (Schmidt et al., 2023; Zimmermann et al., 2012). The lengths of the considered time periods differ between the models as we optimized them for the highest model performance using the Nash–Sutcliffe efficiency (Schmidt et al., 2023). Additionally, we used the day of the year to capture the seasonality and the rate of change in discharge as predictors (Francke et al., 2008b). The models were validated at both gauges. At gauge Vernagt, we trained validation models on the two available 2-year training data sets (2000–2001 and 2019–2020) and compared the model estimates to measurements from the respective period not included in the training data (NSE of 0.73 and 0.6 for daily SSY; for details, see Schmidt et al., 2023). At gauge Vent, a 5-fold cross-validation demonstrated the superior performance compared to sediment rating curves, especially in periods containing threshold effects due to extreme events. For the past 5 decades, OOOR data points (see Sect. 2.4.2) were rare, which strengthened the notion that the available training data covered the majority of typical situations.

The two studied gauges Vent Rofenache (hereafter Vent, operated by the Hydrographic Service of Tyrol) and Vernagt (operated by the Bavarian Academy of Sciences and Humanities) are located in the Rofental in the Ötztal Alps, Austria (Fig. 3). The two corresponding nested catchments of 98.1 and 11.4 km2 span elevations ranging from 1891 m a.s.l. at gauge Vent and 2635 m a.s.l. at gauge Vernagt to 3772 m a.s.l. The area is characterized by a relatively warm and dry climate (for this alpine setting), with a mean annual temperature of 2.5 ∘C at gauge Vent (Strasser et al., 2018) and average annual precipitation as low as 660 mm at gauge Vent but a strong precipitation gradient with elevation (Schmidt et al., 2023). Both catchments are heavily glacierized (28 % and 64 % glacier cover in 2015 (Buckel and Otto, 2018)), but accelerating glacier retreat has been observed since the beginning of the 1980s (Escher-Vetter, 2007; Braun et al., 2007; Abermann et al., 2009). Apart from the glaciers, land cover at high elevations is dominated by bare rock or sparsely vegetated terrain, whereas mountain pastures and coniferous forests occupy lower elevations. Geology is dominated by biotite-plagioclase, biotite and muscovite gneisses, variable mica schists, and gneissic schists (Strasser et al., 2018).

The river Rofenache is a tributary stream of the Ötztaler Ache, one of the largest tributaries to the river Inn. The glacial-to-nival hydrological regime shows a pronounced seasonality, with almost 90 % of discharge occurring during snow and glacier melt from April to September (Schmidt et al., 2023). Mean annual suspended sediment concentrations at gauge Vent were the highest in an Austria-wide comparison (Lalk et al., 2014). Annual suspended sediment yields in Vent averaged 1500 t km-2 a-1 with an even more pronounced seasonality compared to discharge (99 % of the annual SSY transported from April to September) (Schmidt et al., 2023).

To analyze model performance and identify the limits of the applicability of the presented QRF modeling approach, we verified the modeled SSYs against measurement data in the overlap period (2007–2020) (Sect. 2.4.1); assessed the frequency of OOOR data points, including by how much the observation range of the predictors is exceeded in the projections; and analyzed whether the modeled SSY are sensitive to changes in these predictors (Sect. 2.4.2).

We analyzed the model results, i.e., estimated annual yields in the projection period for trends, as well as shifts in seasonality. To assess trends, we used two methods implemented in the R package FUME (Santander Meteorology Group, 2012): the Mann–Kendall test, which is a non-parametric tool to detect linear trends (specifically, we used a version that was modified to detect trends in serially correlated time series; Madsen et al., 2014; Yue et al., 2012) and Sen's slope estimator (Sen, 1968) to assess trend magnitude. Further, we compared the estimated yields in three time slices (near future: 2011–2040, intermediate future: 2041–2070, and far future: 2071–2100; see also Fig. 2), comparable e.g., to Jacob et al. (2014) and Hanzer et al. (2018). To assess changes or shifts in the seasonality of sediment export, we compared the mean monthly yields of the observations and the projections.

To assess whether the detected trends are sensitive to the potential underestimation of yields on OOOR precipitation days, we multiplied the daily yields estimated by our QRF model on these days by a factor of 5, i.e., assuming a very severe underestimation in the original estimates (Fig. 4). We chose this factor as it is close to the most severe exceedance extents in precipitation at both gauges, which are 456 % at gauge Vernagt and 442 % at gauge Vent (see also Sect. 3.3). We then compared the trends in annual SSY of the altered time series to the trends in the original QRF estimates. All analyses were conducted with the statistical software R (R Core Team, 2018).

The bias-corrected discharge data yield more adequate representations of measured monthly discharge amounts and their seasonal distribution (Fig. 6), as well as mean annual discharge volumes (Table 3). At gauge Vernagt, for example, maximum mean monthly Q in the observations and the bias-corrected data is in August, whereas the original AMUNDSEN simulations suggested a maximum in July. Nevertheless, some underestimation of August discharge remains at gauge Vernagt. At gauge Vent, the original AMUNDSEN simulations substantially overestimated discharge amounts in April to July, i.e., the snowmelt period, which was successfully corrected by the bias correction.

We find good agreement between observed and modeled annual SSY at both gauges (Fig. 7), and the Kolmogorov–Smirnov test does not yield significant differences between the observations and model results in mean annual sediment yields. Nevertheless, years with extreme annual yields (both lower and higher) occur in the model results, especially under RCP4.5 and RCP8.5 (e.g., for Vent, max. 3250 t a-1 in RCP4.5 vs. 2120 t a-1 in the observations), likely due to the higher sample size in the projections (42 or 196 years in the projections compared to 14 years in the observations; see also description of Fig. 7).

Similarly, the seasonality of sediment export is well represented in the overlap period of the projections (Fig. 8), and the Kolmogorov–Smirnov test does not yield significant differences in comparison to the seasonal distribution of the measurements. Monthly SSYs tend to be slightly lower in the projections in August at gauge Vernagt and in July and August at gauge Vent. Similar patterns had already become apparent in the comparison of mean monthly discharges at gauge Vernagt (Fig. 6).

The presented SSY projections in the Ötzal, Austria, suggest an overall decrease in annual SSY (Fig. 10). This is consistent across emission scenarios (i.e., with respect to the ensemble means), as well as all individual projections (i.e., based on the 31 different RCP–GCM–RCM chains). This is linked to shrinking of glaciers and thus decreases in glacier melt as temperature is projected to increase in all models (between +1.1 (RCP2.6) to +3.8 ∘C (RCP8.5) in the annual mean) (Hanzer et al., 2018). Accordingly, snow cover is projected to decline, especially in low elevations. With respect to precipitation, there is no clear general trend with respect to annual sums; both decreases and increases are projected by individual models (-14 % to +24 %), and multi-model averages are close to zero for all scenarios, yet with a general shift in precipitation from summer to winter

The modeled decrease in SSY is much more distinct at gauge Vernagt, where a trend analysis in the previous study (Schmidt et al., 2023) showed significant positive trends in the period until 2020. At gauge Vent, significant positive trends were detected if all data points since the 1970s were considered (Schmidt et al., 2023). However, if only the years after the distinct increase around 1980 were considered, the trend was slightly negative (Schmidt et al., 2023). This suggests that peak sediment has already been reached or is underway at both gauges and occurs simultaneously with peak water.

These findings match the expectations of conceptual models that sediment yield from deglaciating basins will initially increase (due to increases in glacial erosion, sediment supply accessibility, transport capacity and occurrences of extreme floods) and subsequently decrease as glacier masses decline, meltwater volumes and freeze–thaw weathering decrease, and vegetation colonizes (Antoniazza and Lane, 2021; Zhang et al., 2022). It is expected that peak sediment may lag behind peak meltwater, with a lag that can be up to decades or centuries (Delaney and Adhikari, 2020). This lag is hypothesized to be scale dependent – i.e., it will be shorter for areas closer to the glacierized regions – and to depend on the changes in erosive precipitation: if erosive precipitation decreases, peak sediment occurs simultaneously with peak water, while increasing or stable erosive rainfall scenarios are associated with a lag (Zhang et al., 2022). Indeed, for the study area of this study, a decrease in summer precipitation sums (i.e., June to August, which is the time of minimum snow cover and thus maximum erodibility) is projected (Hanzer et al., 2018). At the same time, heavy precipitation events are projected to become more intense (and only slightly more frequent; see Fig. A2 in the Appendix). However, the negative trend in discharge appears to prevail as our estimates suggest that peak sediment coincides with peak meltwater (Fig. 11).

Sediment export projections differed only slightly (if at all) between emission scenarios; i.e., the spread between projections within one emission scenario is much larger than differences between ensemble means of the three RCPs (Fig. 10). This is due to irreducible internal climate variability, which has been acknowledged in climate modeling (Deser et al., 2012), as well as in erosion modeling (Hirschberg et al., 2021). It should be noted that comparisons to RCP2.6 need to be treated with care as it comprises fewer GCM–RCM combinations (only 3 as compared to 14 in the higher-emission RCPs). Nevertheless, the absence of major differences between RCPs is in accordance with findings by Gobiet and Kotlarski (2020) that “until the middle of the 21st century […] it is projected that climate change in the Alpine area will only slightly depend on the specific emission scenario”. Accordantly, Hanzer et al. (2018) projected glacier volumes to decline by 60 %–65 % until 2050, “largely independent of the emission scenario”.

Despite the overall decrease in SSYs, our results suggest that high annual SSYs are possible, especially at gauge Vent and towards the end of the century (Fig. 10). Additionally, yields on days with heavy precipitation may increase at both gauges – in absolute terms and in relation to the annual export (Fig. A2 in Appendix). This is reasonable, given that increases in heavy precipitation intensity and/or frequency in the European Alps have been detected in measurement data from the past (e.g., Hiebl and Frei, 2018; Scherrer et al., 2016; Gobiet and Kotlarski, 2020), as well as future projections (Gobiet and Kotlarski, 2020; Jacob et al., 2014; Kotlarski et al., 2023) – despite the overall decrease in summer precipitation mentioned above. As a result, we can expect an increase in sediment-related harmful events triggered by heavy precipitation, such as flash floods and gravitational mass movements (i.e., debris flows, landslides) (Huggel et al., 2012; Savi et al., 2020; Gobiet and Kotlarski, 2020). Similar expectations, i.e., increasing high-magnitude transport events in the context of an overall decrease, have e.g., been expressed with respect to bedload in South Tyrol (Coviello et al., 2022). Such a development would have important implications e.g., for sediment management in hydropower production and flood hazard management as the flood risk can increase if cross-sections are reduced after sedimentation, and potential backwater effects need to be considered (Nones, 2019) as much of the damage is associated with transported solids rather than the water itself (Badoux et al., 2014).

Our findings suggest a shift in sediment export seasonality since the highest mean monthly SSY shifts from August to July (or even June) due to substantial reductions in sediment export in July, August and September at both gauges (Fig. 12). This is linked to the projected distinct reductions in glacier melt (Hanzer et al., 2018) and appears to be reasonable given that glacier melt has so far been the dominant transport medium of suspended sediments at these gauges (Schmidt et al., 2023). These results are not sensitive to potential underestimations of SSY on days with very heavy precipitation. Such shifts in seasonality and the concomitant overall reduction in fluvial sediment transport will likely have severe effects on biodiversity, i.e., flora and fauna of glacier-fed streams (Milner et al., 2009, 2017; Gabbud and Lane, 2016; Huss et al., 2017).

As a potential limitation to the presented quantile regression forest approach, out-of-observation-range data points in the predictors can lead to underestimates in SSY on the affected days. Yet, the analysis of such incidents in synopsis with the results of the sensitivity analysis showed that, before 2070, the effect on annual yield estimates is ≤ 3 % Fig. 9). This is very small given the overall high variability in SSC (Vercruysse et al., 2017; Delaney et al., 2018; Schmidt et al., 2023). On a similar note, even assuming rather generous increases in yields on days with OOOR precipitation altered the trends only marginally, which shows that underestimations on individual days with OOOR precipitation have little effect on long-term annual averages. However, we have less confidence in the model results after 2070 for two reasons. First, more frequent and severe OOOR incidents occur during this time, especially in the high-emission scenarios, and fewer projections are covered by the assumptions of the sensitivity analysis. We can therefore expect a higher uncertainty in the model results. Specifically, the effect of underestimation for single large events will aggravate. Second, more than a few glacier simulations suggest that glaciers could have disappeared almost entirely by 2070 (Hanzer et al., 2018), which implies a major shift in the hydro-geomorphic functioning of these catchments. While our QRF models were able to model threshold effects better than sediment rating curves (Schmidt et al., 2023) (likely because they are not bound to linear or monotonous relationships), this is only true for effects that are represented in the training data. Thus, the results for the period after 2070 need to be treated with caution. We have indicated this in the presentation of our results by using transparency or dashed lines instead of solid lines.

As a more general limitation, there are several other factors with the potential to substantially alter and influence sediment dynamics in the study area, which we cannot consider in our models. This concerns geomorphological changes, such as increased paraglacial erosion: debuttressed slopes may trigger landslides and rockfalls, and indeed, increased debris flow and rockfall activity have been shown in response to warming in other areas, likely associated with intensified alpine permafrost thaw (Savi et al., 2020; Hartmeyer et al., 2020; Huggel et al., 2012). Additionally, sediment availability and accessibility increase as erodible landscapes expand (Li et al., 2021a), and subglacial sediment availability might also increase (more subglacial sediment can be accessed by meltwaters as the equilibrium line altitude retreats upslope) until the glacier size becomes smaller than a critical size (Delaney and Adhikari, 2020; Zhang et al., 2022). Although these processes are likely already partially reflected in the observations used for the model training, their intensity may still be too low to be sufficiently learned by the model. Thus, future intensification of these processes could lead to higher sediment export rates than our estimates suggest and might thereby affect the estimated location of peak sediment. Notwithstanding, there are also several factors that could lead to decreases in sediment export, such as decreases in connectivity (such as the formation of supra-, sub- or proglacial lakes or outwash fans which act as sediment traps) or decreasing glacial erosion as glaciers recede (Zhang et al., 2022). Additionally, freeze–thaw weathering may decrease (Hirschberg et al., 2021), and it is not clear how quickly the deglaciating landscapes will stabilize, e.g., through eluviation of fine materials and fluvial sorting of sediment, which progressively increases the resistance to entrainment and transport (Ballantyne, 2002; Lane et al., 2017), or vegetation colonization (Haselberger et al., 2021; Altmann et al., 2023; Musso et al., 2020; Eichel et al., 2018). Many of these processes are ultimately governed by temperature and/or precipitation, and we have chosen the predictors to act as proxies (e.g., antecedent moisture and temperature conditions could be crucial for mass movements). While this is out of the scope of the presented study, we encourage future studies to work towards including more advanced proxies for geomorphological changes.

The presented results are associated with uncertainties, which are a combination of uncertainties inherited from the underlying climatological and hydrological projections and uncertainties inherent in the QRF approach. Climate model uncertainty represents a combination of uncertainties in assumptions of future anthropogenic greenhouse gas emission, GCM uncertainty (different GCMs produce different responses to the same radiative forcing) and RCM uncertainty (different RCMs forced by same the GCM produce different regional responses) (Evin et al., 2021; Gobiet et al., 2014). It has been found that EURO-CORDEX simulations may be biased towards “too cold, too wet, too windy”, but these uncertainties are mostly within the observational uncertainties, and simulations “reproduce fairly well the recent past climate despite some biases” (Vautard et al., 2021). To address this, it was recommended to carry out bias correction, which has been performed by means of quantile mapping for the precipitation and temperature projections (Hanzer et al., 2018). The hydrological model results are also associated with uncertainties, such as the tendency to overestimate spring runoff, winter snow accumulation and glacier mass balances. We have addressed this through bias correcting the discharge projections, which resulted in a more adequate representation of discharge seasonality and volumes. Certainly, bias correction methods such as quantile mapping in turn introduce uncertainties, e.g., by assuming that the biases are stationary, i.e., do not change over time (Gudmundsson et al., 2012). Hydrological simulations that do not show the necessity for this correction could eliminate this issue. Uncertainties in the QRF approach have been addressed in a previous study and include the tendency to underestimate rare, high-magnitude daily SSYs (albeit with small effects on the respective annual yields); the underestimations on days with OOOR values (which had small effects until 2070, as discussed in detail above); and the choice of temporal resolution (i.e., daily compared to hourly resolution involves some loss of information, e.g., on precipitation intensities, but the effect was also found to be small) (Schmidt et al., 2023). Since QRF is a data-driven approach, the quality of the estimates hinges on the underlying training data set, as well as the choice of predictors, i.e., a large and varied enough data set in combination with predictors that meaningfully represent the most important processes improves the quality of the estimates (Schmidt et al., 2023). Thus, future studies are recommended to explicitly sample extreme events and/or verify the representativity of the training data set.