The constraining of SWE reconstructions through streamflow can be framed as an inverse problem, where the known output of a system (streamflow) is used to infer an unknown internal state (SWE). Prior knowledge on snow physics, topographic controls, and meteorological inputs reduce the solution space. Still, the inversion remains ill-posed: we aim to retrieve the space-time evolution of gridded SWE (3-dimensional aspect) from a catchment-integrated streamflow signal (single dimension).

We denote the time series of observed streamflow with Qobs, and the spatio-temporal SWE field as HSWE. In a Bayesian framework, we seek the posterior distribution: 1P(HSWE∣Qobs)∝P(Qobs∣HSWE)⋅P(HSWE).

The prior distribution P(HSWE) reflects our initial uncertainty about SWE, and the likelihood P(Qobs∣HSWE) quantifies how well a given SWE realization explains the observed discharge. Since HSWE is not a free variable but the result of snow model simulations, we rather define P(HSWE) as the result of the finite sampling of the informative prior distributions of parameters θ as follows: 2HSWE(i)=fsnowM;θmeteo(i),θsnow(i),  with θmeteo(i)∼P(θmeteo),θsnow(i)∼P(θsnow), where M is the meteorological forcing (precipitation and temperature), θmeteo are meteorological parameters (e.g., precipitation scaling, lapse rates, phase partitioning), and θsnow are snow model parameters controlling melt rates and snowpack dynamics. Repeating this for i=1,…,Nprior yields an ensemble that approximates the prior distribution P(HSWE).

To be able to compute the likelihood, the resulting SWE and the meteorological forcing are passed to a runoff generation model frunoff: 3Qsim(i)=frunoffHSWE(i),M;θmeteo(i),θrunoff(i),  with θrunoff(i)∼P(θrunoff) where Qsim is the simulated streamflow and θrunoff governs surface and subsurface runoff generation, soil storage, and evaporation. The model thus maps each parameter set Θ={θmeteo,θsnow,θrunoff} to a streamflow simulation Qsim, and the inverse problem becomes one of estimating the posterior distribution: 4P(Θ∣Qobs)∝P(Qobs∣Θ)⋅P(Θ).

While it is difficult to compute this posterior distribution analytically, it can be approximated with numerical methods that generate samples of the posterior distribution, such as Importance Sampling and Markov Chain Monte Carlo methods . These methods repeatedly sample parameter sets from their prior distributions, use them to run a simulation model, and evaluate their likelihood against observations. Parameters sets with a high likelihood have more chances of being considered as samples from the posterior (e.g., ).

Both formal and informal methods exist in hydrological parameter inference literature: formal methods use a well-defined likelihood function based on an assumed error distribution and combine this with the prior to obtain a well-defined posterior distribution . Informal methods do not necessitate a formal likelihood function and instead obtain a heuristic approximation of the posterior distribution using performance metrics as proxies for likelihood . We opt for an informal approach where we select a fixed percentage of the best-performing members among the prior ensemble as the heuristic posterior ensemble. This informal approach has the main advantage that the size of the posterior ensemble remains constant across experiments, which is helpful in assessing whether the posterior ensemble indeed contains the most realistic SWE realizations. Section  presents the sampling strategy, while Sect.  introduces the posterior ensemble selection and the performance metric used for streamflow evaluation.

Figure  illustrates the streamflow-constrained SWE reconstruction framework implemented in this study. Meteorological forcing M (Sect. ) is used to drive a snow model fsnow (Sect. ), producing gridded SWE and snowmelt estimates. These are combined with rainfall inputs and routed through a runoff model frunoff (Sect. ) to generate simulated streamflow Q.

For each year between 2001 and 2022, 5000 model realizations are generated by randomly sampling parameter sets from uniform prior distributions Θ={θmeteo,θsnow,θsoil} using Latin Hypercube Sampling (Sect. ). The resulting prior ensemble of simulated streamflow Qsim is compared to observed streamflow Qobs using a performance metric d(Qsim,Qobs), and the 1 % best-performing members are selected as the heuristic posterior ensemble (Sect. ).

To test the methodology in a controlled environment, we evaluate it in two synthetic experiments: a fully synthetic case (FS; Sect. ), which eliminates all modeling chain uncertainty, and a semi-synthetic case (SS), which adds meteorological and snow model structural uncertainty. The lower panel of Fig.  outlines the anticipated challenges for real-world applications, where additional uncertainty sources, particularly in the runoff model and streamflow observations, further complicate the inversion (Sect. ).

To evaluate the constraining potential of streamflow for SWE reconstruction, we perform two synthetic experiments, both use the same prior ensemble of 5000 parameters described above.

We use the Nash–Sutcliffe Efficiency NSE, as the streamflow performance metric to quantify agreement between model output and observations (denoted as EQ-NSE). An NSE of 1 indicates perfect agreement, while an NSE of 0 implies no improvement over using the observed mean as a predictor. We calculate NSE over the snowmelt season (March to July) to focus on snowmelt-driven discharge. Although NSE can give inflated values in catchments with strong seasonality, such as the Dischma , our focus is on relative differences in NSE, reflecting variations in squared error magnitudes.

We adopt a rank-based heuristic posterior selection. All prior ensemble members are evaluated against observed streamflow using NSE, and the top 1 % are selected as the posterior ensemble, yielding a posterior size of Nposterior = 50. The quality of this posterior ensemble is then evaluated on different SWE metrics (Sect. ).

The inferred posterior parameter ensembles do not consistently align with the true parameter values in the FS experiment (Fig. a and b). Among all parameters, csnow is the most sensitive. Its annual posterior values generally reflect the imposed artificial bias fluctuations csnow∗, but still span a wide range. For example, in 2001 where csnow∗ = 1, posterior values range from approximately 0.9 to 1.3, implying that both a 10 % underestimation and a 30 % overestimation of total snowfall can result in high streamflow skill. The remaining parameters are considerably less sensitive and generally span most of their prior ranges, equally indicating that a wide range of SWE realizations can produce similarly high-performing streamflow responses and suggesting compensating behavior among parameters.

In the SS experiment, of all true parameter values Θ∗ only crain∗ is imposed. The true value of the remaining parameters is unknown, as the reference SWE is an external product. Figure c shows that crain is consistently overestimated, which is compensated by an annual underestimation of catchment-wide SWE accumulation of 6.6 ± 4.4 %. The posterior csnow values vary widely across the prior range, suggesting annually varying biases in the snowfall forcing and confirming the need for annual rather than multiannual inversion. The values of Tthresh, m, r and Cret are generally on the lower edge of the prior range, while γTthresh and βcv are generally on the higher edge. This suggests slower melt taking place preferentially at higher elevations, lower water holding capacity, and faster snow cover depletion of SWEref,SS (i.e. OSHD-TI product) compared to our prior assumptions.

In the previous section, we showed that we cannot reliably recover the true parameter values from streamflow alone, with NSE as the streamflow performance metric. To better understand this result, we analyze the model performances associated with the best ranked parameter sets. Figure  shows the EQ-NSE results and posterior ensemble selection (Fig. a and c) and the subsequent evaluation of this selection on EGRIDaccumulation (Fig. b and d), for both FS and SS. We show EGRIDaccumulation results as it is arguably the most relevant performance metric for gridded SWE reconstructions. The results for other target SWE metrics are presented in Figs. S1–S10 in the Supplement.

The EQ-NSE results confirm strong agreement between simulated and synthetic streamflow in both experiments, with an overall mean posterior NSE of 0.99 ± 0.01 for FS, and 0.94 ± 0.03 for SS (Fig. a and c), compared to an overall mean prior NSE of 0.67 ± 0.11 for FS and 0.56 ± 0.16 for SS. EGRIDaccumulation results reach maximum scores of near 0 % in FS in some years, while not exceeding 10 % in most other FS and SS years. This is likely due to a combination of high sensitivity of the highest and lowest elevation grid cells to the γsnow and γTthresh parameters, and the fact that in the SS experiment a perfect approximation of the external SWE product is generally not possible.

Across both experiments, posterior members generally occupy the lower-error portion of the prior distribution, indicating that streamflow provides a meaningful constraint on gridded SWE accumulation. However, many prior members outperform the posterior ensemble, demonstrating that high streamflow skill does not uniquely translate into high spatial SWE accumulation skill. The strength of this constraint varies between years, with some years showing a narrow spread and others showing a wide spread among the posterior ensemble.

Across both experiments, EAGGmelt is the most strongly constrained metric among all SWE metrics, indicating that streamflow most effectively constrains catchment-scale melt dynamics (Fig. ). In FS, Rpost,median values lie close to the perfect-constraint limit in most years, whereas in SS they are substantially higher and more variable, reflecting reduced identifiability under added forcing and model uncertainty. In contrast, EGRIDmelt is only weakly constrained in both experiments, indicating that while streamflow constrains integrated melt production, it provides limited information on its spatial origin.

Snowfall-related metrics are more weakly and inconsistently constrained. In FS, both EAGGsnowfall and EGRIDsnowfall show moderate constraint, likely benefiting from the same parameter sets that favor melt performance. In SS however, both metrics are weakly constrained, confirming the limited ability of streamflow to inform snowfall dynamics when accumulation and melt biases differ.

Accumulation metrics show intermediate constraint. EGRIDaccumulation is more strongly constrained than EAGGaccumulation in both experiments, suggesting that the spatial distribution of snow accumulation is equally or better constrained by streamflow than the total catchment-wide accumulation. Note, however, that a different streamflow performance metric than NSE (e.g. seasonal streamflow bias) might favor the constraint of catchment-aggregated accumulation more (Sect. ).

Among timing metrics, melt-out dates are relatively well constrained, particularly in AGG mode. This is consistent with their physical link to the cessation of snowmelt-driven streamflow. In contrast, SWE onset dates are weakly constrained across experiments and scales.

Overall, constraints are systematically weaker and more heterogeneous in SS than in FS. Rpost,median increases by 989 on average across all metrics, corresponding to 20 % of Nprior, while its standard deviation increases on average by 414. This confirms that the added structural and input uncertainty in SS reduce the ability of streamflow to constrain SWE. Additionally, the increased spread in Rpost,median across different SWE performance metrics in SS indicates that members performing well on one metric no longer consistently perform well on others. This suggests a decoupling of performance among metrics and growing trade-offs between competing aspects of SWE reconstruction under added uncertainty. Nonetheless, except for EGRIDsnowfall in SS, most median ranks remain above the no-constraint threshold.

To assess the complementarity of the retained SWE metrics, Fig  shows the Spearman rank correlation between all SWE performance metrics and streamflow NSE across the full prior ensemble for each year. Consistent with previous results, EAGGmelt is the SWE metric most strongly associated with EQ-NSE in both experiments (ρFS = 0.89, ρSS = 0.63). Overall, correlations are systematically stronger in FS than in SS, and catchment-aggregated metrics generally correlate well with their gridded counterparts. While EAGGmelt correlates the strongest with EQ-NSE, correlation of other SWE metrics with catchment-aggregated melt does not translate to strong correlation with EQ-NSE as well. It is the case for Emelt-out, but not for EGRIDmelt. These results suggest that, in general, streamflow contains information about different SWE metrics independently. In other words, the posterior members performing well on one SWE metric do not necessarily perform well on others. This supports the use of multiple, complementary SWE metrics and cautions against drawing conclusions from any single metric.

The fully synthetic experiment confirms that streamflow-constrained SWE inversion works in theory, but near-perfect constraint is only achieved for catchment-aggregated melt. Other SWE properties generally remain well-constrained, but far from perfectly constrained. We hereby demonstrate that even under highly idealized conditions, streamflow does not consistently identify the best-performing SWE scenarios across all performance metrics. This finding is mainly explained by physical non-uniqueness in the SWE-streamflow relationship : different SWE and rainfall scenarios can lead to equivalent streamflow responses. Catchment-aggregated melt being better constrained than distributed melt is a first indication of this, by showing that biased spatial melt distributions can lead to an accurate aggregated melt output. A second indication is given by the imperfect constraint on catchment-aggregated accumulation: the best-performing streamflow performance can be achieved with biased catchment-wide SWE accumulation estimates. Finally, we show that multiple distinct SWE and rainfall combinations can yield similar streamflow responses (Fig.  and Sect. ).

A second source of uncertainty in the FS experiment is structural non-uniqueness or equifinality , whereby multiple parameter sets yield similar SWE outcomes. However, since this study focuses on SWE performance rather than parameter convergence, such equifinality is not of major concern. A third potential source is parameter estimation uncertainty, i.e., the failure to identify optimal parameter combinations by the sampling algorithm. Yet this is also of minor importance, as the posterior simulations already achieve high streamflow skill, and further optimization or a different Nposterior/Nprior ratio would not affect the SWE-streamflow relationships central to our analysis.

The semi-synthetic experiment shows that adding meteorological and snow model uncertainty significantly reduces the ability of streamflow to constrain SWE across all performance metrics. This reduction suggests a mismatch between our meteorological forcing and snow model versus the OSHD reference that the current inversion framework is unable to correct. Alongside physical non-uniqueness, uncertainties throughout the modeling chain thus present an additional barrier to accurately identifying realistic SWE scenarios from streamflow. These added uncertainties also introduce stronger trade-offs between SWE metrics: even when the best-performing ensemble members for one SWE metric are correctly identified, they are less likely to perform well on other SWE performance metrics.

Under real-world conditions, constraining SWE reconstructions using streamflow presents additional challenges beyond the idealized setup explored in this study. These challenges include both substantially increased uncertainty and reduced opportunities for performance evaluation (Fig. ). One major source of uncertainty not addressed here is runoff model uncertainty. This encompasses both uncertainty in static catchment properties and uncertainty in the representation of water transport processes through the catchment . Such uncertainty can introduce persistent timing biases in the translation of snowmelt into streamflow, thereby complicating efforts to infer SWE dynamics from streamflow observations . Additional uncertainty arises from streamflow observations, including errors in stage measurements, discharge gauging, rating curve estimation , and ice-related effects , which can propagate into both event-scale timing errors and biases in seasonal water balance estimates. Finally, the meteorological and snow model uncertainties imposed in the semi-synthetic experiment are likely to be conservative relative to real-world conditions. In practice, meteorological forcing errors are expected to be larger, not least due to the addition of evaporation estimation uncertainty, and snowpack dynamics are more heterogeneous and complex than represented by the OSHD model. Taken together, these additional sources of uncertainty are expected to further diminish the constraining potential of streamflow on SWE reconstruction beyond the reduction observed here between the fully synthetic and semi-synthetic experiments.

An additional challenge in real-world applications is the lack of long-term, temporally continuous and catchment-scale SWE observations against which to evaluate inversion results . Unlike in our synthetic experiments, real-world evaluations forcibly rely on spatially or temporally incomplete observations, making it inherently difficult to assess whether the SWE inversion was successful. At present, the best evaluation dataset is arguably the biweekly gridded SWE product of the Airborne Snow Observatory , available for a limited number of catchments and years in the Western US. The lack of evaluation data equally implies a lack of training data for data-driven methods, thereby limiting the potential of machine learning methods as an alternative link between SWE and streamflow in inverse hydrological SWE reconstruction. A practical way forward is to continue refining idealized experiments by further adding controlled sources of uncertainty, such as runoff model and streamflow observation errors, thereby approximating real-world complexity while retaining the ability to assess inversion effectiveness quantitatively.

Several elements of the inversion framework proposed here may require adaptation under real-world conditions, where uncertainty is higher and evaluation opportunities are more limited. One key limitation is the use of NSE as the streamflow performance metric. NSE is highly sensitive to timing errors, potentially penalizing simulations that reproduce melt events with small temporal shifts more strongly than simulations that miss them entirely. This sensitivity is particularly relevant given our finding that streamflow most strongly constrains catchment-aggregated melt, which is inherently timing-dependent. In addition, as a residual-based metric, NSE may preferentially favor parameter sets that perform well on residual-based SWE metrics, while disadvantaging those that perform better on bias-based SWE metrics. Alternative streamflow metrics targeting hydrological signatures, such as variability or seasonal volume , may therefore provide more robust constraints under real-world uncertainty. Consequently, the results presented here should be considered strictly in light of the use of NSE as the streamflow performance metric. A full analysis of the streamflow and SWE performance metric interactions is outside the scope of this work, but is recommended for future research.

Given the increased spatial and temporal variability of meteorological forcing in real-world applications, the simple bias-correction factors used here (θmeteo) are likely insufficient. In reality, snowfall, rainfall and melt biases vary at sub-seasonal and event time scales, and elevation dependencies are often non-linear in complex terrain. Allowing for greater temporal flexibility and spatial heterogeneity in the correction of meteorological biases may therefore improve the identifiability of relevant SWE processes from streamflow observations, albeit at increased computational cost.

Finally, the choice of snow and runoff models is likely to influence inversion performance. While relatively simple snow models can perform well at the catchment scale , more physically based formulations may be better suited where meteorological data are abundant . In contrast, the inversion could benefit from decreased complexity in the runoff model. Since the primary function of the runoff model in this framework is to translate spatial melt into streamflow, semi-distributed or lumped models could reduce computational costs and allow for larger ensembles compared to the fully distributed runoff model used here. More broadly, employing multiple snow and runoff models within the inversion framework could enhance robustness by better accounting for structural model uncertainty and increasing the likelihood of capturing realistic SWE evolution and snowmelt runoff.

Under real-world conditions, streamflow alone may fail to reliably distinguish biased from unbiased SWE simulations. In the absence of reliable SWE evaluation data, the streamflow-based selection of biased SWE simulations might even go undetected. This implies that streamflow may, in some cases, not provide added value compared to simply running a snow model with uncorrected meteorological forcing. Several factors determine whether streamflow is likely to provide added value for SWE reconstruction. First, the quality of meteorological observations is crucial. Low meteorological biases result in low biases in SWE reconstructions, reducing the need for streamflow to constrain them. Secondly, the size, shape, and climate of the target catchment play a role. Smaller, elongated catchments (e.g. the Dischma catchment of this study) exhibit lower non-uniqueness than large, round catchments , while snow-dominated catchments offer better identifiability of snowmelt than snow-scarce catchments . Dry spring and summer climates particularly benefit streamflow-assisted SWE inversion as they limit the confounding between rainfall and snowmelt signals . The same logic likely also applies to inter-annual variability within each catchment. In years with higher snowfall fractions and less spring rainfall, streamflow likely has greater constraining potential on SWE reconstructions. The above factors favor the application of streamflow-assisted SWE inversion as far back as streamflow observations allow, as meteorological forcing products have become less biased , and snowfall dominance has decreased with time . They also favor its application to meteorologically under-observed mountain regions such as the Himalayas and the Andes, where forcing products equally tend to be more biased and SWE evaluation is scarcer . While this study focuses on mountainous catchments, it would be valuable to assess the constraining potential of streamflow in lower-relief, less topographically complex environments such as boreal catchments, where different controls on snow accumulation and melt may lead to different identifiability of SWE dynamics in streamflow.

In this study, we isolate the constraining potential of streamflow alone. However, streamflow is likely most effective when used in combination with other sources of snow information. These can be direct observations of different snow properties , but we particularly encourage future work to explore the use of recurring spatial patterns of snow dynamics . Such patterns represent catchment-specific prior knowledge that, once characterized, can be reused across time, including periods predating satellite observations but overlapping streamflow observations. Prior SWE ensembles could be defined as scaled versions of present-day observed spatial SWE patterns, thereby reducing the number of parameters to be inferred and mitigating part of the physical non-uniqueness identified in this study. Alternatively, the recurring patterns could be used to filter posterior SWE members based on physical plausibility. So far, these recurring patterns have been characterized in a limited number of catchments, but recent advances in automated UAV and LiDAR technologies are likely to increase this number in the near future . Combined with long streamflow records, this opens the possibility of extending SWE reconstructions back in time by decades while maintaining realistic spatial SWE patterns.

Regardless of the accompanying information source, streamflow remains a unique source of snow information in its ability to capture catchment-integrated SWE dynamics, most notably the timing and total volume of snowmelt runoff. Our finding that streamflow most effectively constrains catchment-aggregated melt supports its potential role in this context. In light of results by , who showed that many SWE products systematically misrepresent average melt rates in mountainous terrain, streamflow is the only observational source capable of directly constraining such errors at the catchment scale. We therefore propose that future studies investigate the integration of streamflow with other snow data sources to constrain SWE reconstructions as much as practically possible.