Table  summarizes the 13 state-of-the-art macro-scale water sector models and two data products utilized in this study. Among these, six models belong to ISIMIP2a, while seven are part of ISIMIP3a. All models were obtained from the ISIMIP water sector data repository (https://data.isimip.org/, last access: 19 November 2024). For each model, their total runoff (i.e., sum of surface and subsurface runoff) was extracted for river routing (Sect. 2.2), and the key SMR indices were subsequently calculated for evaluation (Sect. 2.3).

These models were categorized into three groups (see Table  and references therein): i.e., six global hydrological models (GHMs: PCR-GLOBWB, VIC, CWATM, H08, HYDROPY, and WATERGAP2-2E), six land surface models (LSMs: DBH, MATSIRO, CLM40, JULES-W2, MIROC-INTEG-LAND, and ORCHIDEE-MICT), and one dynamic global vegetation model (DGVM: LPJML). We categorized the models into these groups primarily because GHMs tend to focus on water balance representation, LSMs are generally more advanced in simulating energy-exchange processes, and DGVMs are better suited for capturing vegetation ecosystem dynamics. Thus, comparative analyses across different model categories may provide insights into their relative strengths and limitations. We also included two observation-constrained datasets for evaluation – GRADES, a global runoff product based on VIC model simulations followed by bias correction using gauge-extrapolated information, and GRDR, a recently released global runoff product that enhances accuracy by assimilating river-width data from Landsat into a model-based discharge simulation framework. Incorporating these two discharge products allows for discussions on the gains brought by observational constraints.

Because parameter calibration can influence runoff simulation, especially for runoff magnitude and peak flow, we further documented the calibration status of the evaluated models based on the available model documentation and calibration information. Five models, including PCR-GLOBWB, LPJML, HYDROPY, CWATM, and WATERGAP2-2E, were classified as calibrated models, whereas MATSIRO, DBH, CLM40, MIROC-INTEG-LAND, VIC, ORCHIDEE-MICT, JULES-W2, and H08 were grouped as uncalibrated models because no explicit calibration entry was available in the original model information.

All models were run at a spatial resolution of 0.5° with daily time steps. Models with similar meteorological forcing and simulation scenarios were purposefully chosen such that our comparisons more directly focus on process diagnostics, although other uncertainty sources such as calibration differences, forcing uncertainty, and model-specific implementation choices cannot be fully eliminated. All models and datasets were matched to the same geospatial framework (i.e., the MERIT-Basins river network; ) to ensure consistency. Further details are provided in Sect. 2.2.

To ensure that all models are comparable under the same geospatial framework, we first routed the gridded runoff of all ISIMIP2a and ISIMIP3a models through the same river routing model, RAPID (the Routing Application for Parallel computatIon of Discharge; ). RAPID is an efficient vector-based river routing model that enables intercomparison of discharge at global scales , making it an ideal choice for routing ISIMIP runoff.

The river network used for routing is MERIT-Basins , a high-resolution, vector-based hydrography dataset constructed from the MERIT-Hydro DEM . We use the area-weighted mapping technique to map the gridded runoff (0.5°) onto the vectorized hydrography to determine lateral inflows, with the connectivity of the river-network topology pre-specified as input. RAPID employs the Muskingum method, which requires two parameters: a weighting factor x and the flood-wave travel time k. Since x is less sensitive in the Muskingum method, it is typically set to 0.3 globally. In contrast, k plays a key role in routing performance and is estimated using river-specific characteristics. Specifically, k is calculated for each river reach by combining channel length with flow celerity estimated from Manning's equation, as expressed in Eq. (): 1k=3ln⁡5S0wd2d+w2/3 Where l is the length of the river channel, n is Manning's roughness coefficient and is typically set to 0.035 for natural rivers . S0 is the channel slope, and w and d are the river width and depth estimated by multi-year average runoff using a long established power-law equation . Note that the choice of river routing model and its parameters can influence both the timing and magnitude of SMR, particularly for Qmax and CTQ, which are more sensitive to runoff concentration pathways and travel-time estimates than Qsum. In addition, a scale mismatch exists between the 0.5° gridded runoff outputs and the finer MERIT Basins river network used for routing. Although the area weighted mapping method and the consistent RAPID routing framework improve comparability across models, they cannot fully resolve subgrid heterogeneity or the allocation of coarse grid runoff to fine river reaches. This mismatch may introduce uncertainty into the absolute evaluation of basin scale runoff characteristics. Nevertheless, because our analysis focuses primarily on long-term mean SMR characteristics and applies the same routing procedure to all models and datasets, we expect this uncertainty to have a limited influence on the relative inter-model comparison. In summary, we standardized the forcing data, geospatial framework, and routing scheme as much as possible to improve inter-model comparability and to focus the evaluation more directly on differences in land model parameterizations and process representations. Nevertheless, forcing and routing-related uncertainties may still affect the simulated SMR characteristics, particularly Qmax and CTQ, and thus should be considered when interpreting both the absolute model performance and inter-model differences.

As briefly discussed in Sect. 1, diagnosing processes related to SMR can be challenging due to the many processes involved (Fig. a). To simplify this, we focus on three first-order characteristics–total runoff (Qsum), peak flow (Qmax), and the centroid timing (CTQ) of runoff in the snowmelt period. Qsum is closely linked to water availability, Qmax determines flood hazard potential, and CTQ provides essential information for water resource management. More importantly, these metrics are directly linked with snow accumulation and melt processes (Fig. b). Specifically, rainfall–snowfall partitioning controls the precipitation phase and the magnitude of snow accumulation, while snow interception and sublimation regulate snow redistribution and loss. Together, these processes determine the amount of meltable snow, thereby influencing both Qsum and Qmax. In comparison, the melt process, being more closely linked to energy dynamics, governs the timing and rate of SMR and exerts a stronger control on CTQ (Fig. b). Compared with a full time-series analysis, our focus on these key runoff characteristics offers a more direct and physically interpretable evaluation of model performance.

To obtain these metrics, we first defined the snowmelt period using snow water equivalent (SWE) data from the fifth-generation atmospheric reanalysis of the European Centre for Medium-Range Weather Forecasts (ERA5; ). Specifically, the snowmelt period was defined as the interval from the date of maximum SWE to the date when SWE dropped below 1 mm (Fig. c). This definition was designed to identify the dominant seasonal snowmelt period in a consistent manner across all basins, rather than to capture every short-term snowmelt-related runoff event. Therefore, it may not fully represent complex melt dynamics such as multi-peak melt seasons, intermittent melt–refreeze cycles, or rain-on-snow events. Here, ERA5 SWE was used mainly to identify the timing window of the main snowmelt period rather than to evaluate SWE magnitude. We acknowledge that ERA5 SWE contains uncertainties, especially in complex terrain where snow accumulation and melt are affected by elevation gradients, slope, aspect, vegetation cover, and sub-grid snow redistribution processes. However, their influence on the extracted SMR characteristics is expected to be limited, because this study focuses on the dominant seasonal snowmelt signal rather than short-term melt events or daily SWE variability. Moreover, Qsum, Qmax, and CTQ were derived as long-term mean characteristics across multiple years, which reduces the sensitivity of the results to occasional errors in snowmelt-period boundaries. Therefore, although ERA5 SWE uncertainty should be considered, it is unlikely to substantially alter the main runoff characteristics used in this global scale evaluation.

After this, Qsum is defined as the total runoff in the snowmelt periods. Qmax is the maximum discharge in the snowmelt periods. CTQ refers to the calendar date on which cumulative runoff reaches 50 % of the total runoff in the snowmelt period, which measures the timing of concentrated runoff during snowmelt, reflecting both the onset and the rate of melt. These definitions are conceptually consistent with previous studies that examined annual or cold-season runoff , but they are further restricted to the snowmelt period to increase the SMR signals. We also introduced a few filtering steps to ensure the SMR signals are the dominant ones, which will be introduced in Sect. 2.2.2.

We obtained daily streamflow data from 1455 gauges (Fig. d) for evaluation. The gauge selection and associated catchment information were based on the Global Streamflow Characteristics, Hydrometeorology, and Catchment Attributes dataset (GSHA; ), which covers 21 568 gauges. We note that the public GSHA release provides annual and monthly streamflow indices, while the daily discharge time series were obtained through following the GSHA data retrieving packages, and the data were used to calculate Qsum, Qmax, and CTQ. The gauges were then filtered to include only those above 30° N (i.e., mid- to high-latitude regions) and those with long-term snow water equivalent (SWE) ≥1 mm in at least one month of the cold season (i.e., during October to March). Additionally, to minimize the impact of other processes such as human activities, glacier melt, and permafrost thaw on runoff, we excluded basins with urban land cover >5 %, degree of regulation (DOR) >10 %, a combined fraction of urban and cropland areas >10 %, and glacier or permafrost coverage >5 %. To identify snowmelt dominated basins, we estimated the relative importance of SMR during the snowmelt period using a residual water-balance proxy (Eq. ). Only basins with SMRratio>0.5 were retained, indicating that snowmelt signals are likely to dominate runoff during the defined snowmelt period. 2SMRratio=SMRQ where SMR was approximated using a residual water-balance approach, following Eq. (): 3SMR=Q-P+ET where Q was derived from gauge observations in GSHA , P from Multi-Source Weighted-Ensemble Precipitation , and ET from the Global Land Evaporation Amsterdam Model .

We note that this residual estimate is used only as a pragmatic screening proxy for identifying snowmelt dominated basins, rather than as a rigorous quantification of the exact snowmelt runoff contribution. This approach does not explicitly account for changes in catchment storage, delayed groundwater release, or rainfall–snowmelt interactions, all of which may affect the estimated SMR contribution. Therefore, SMRratio should be interpreted as an indicator of dominant snowmelt influence during the snowmelt period.

Above all, only basins with more than 10 years of runoff observations during 1979–2019 were included. For each year, the proportion of missing daily records had to be <10 %, and missing values were filled using linear interpolation. Via the above constraints, potential interferences on basin-scale SMR were minimized, ensuring the robustness of the SMR assessment.

As illustrated in Fig. , the key characteristics of snowmelt runoff exhibit pronounced spatial variations across the Northern Hemisphere. In the western coastal and mountainous regions of the United States, the northeastern United States, northern Europe, and northern Japan, SWEmax values are relatively high (Fig. a). This corresponds to later melt onset (Meltstart) and melt completion (Meltend) in the season, as well as higher Qsum and Qmax values. The CTQ date is correspondingly later (Fig. a–f). In comparison, the central United States, the eastern coastal United States, western Europe, and northeastern China generally exhibit lower values of these snowmelt characteristics, indicating relatively less snow, earlier melt timing, and reduced meltwater contributions.

To quantify the impact of land surface complexity on SMR simulation, it is essential to first identify the primary factors that may challenge model performance. Based on existing literature , we focus on two key dimensions of basin complexity: topography and vegetation. Topographic complexity affects SMR simulation through several mechanisms. Mean elevation influences temperature, precipitation phase, snow accumulation, sublimation, and snow–radiation interactions, whereas topographic relief introduces finer-scale variations in surface energy balance, snow redistribution, melt timing, and runoff generation. Vegetation complexity also affects SMR-related processes through canopy density and vegetation composition. Higher canopy density can enhance snow interception, sublimation, and canopy shading, while greater diversity in plant functional types increases landscape heterogeneity in energy exchange and hydrological responses .

Correspondingly, we selected four metrics to represent these two major dimensions of basin complexity: mean elevation (denoted as DEM), topographic variability (denoted as DEMstd), mean leaf area index (denoted as LAI), and the entropy of plant functional type (denoted as PFTh). Specifically, DEM represents the mean elevation background and associated climatic controls, while DEMstd reflects within-basin terrain heterogeneity. LAI represents vegetation density and its effects on canopy–snow interactions, whereas PFTh describes vegetation-type diversity and the heterogeneity of vegetation-related hydrological and energy-exchange processes. Together, these metrics form a four-dimensional vector describing the overall topographic and vegetation complexity of a basin, where higher values generally denote more challenging conditions for SMR simulation.

To synthesize this multi-faceted complexity, we introduced a basin Complexity Index (CI) by summing the normalized values of these four metrics (Eq. ). This index is designed to provide a simple and physically interpretable representation of the major topographic and vegetation controls on SMR simulation difficulty: 4CI=DEMnorm+DEMstdnorm+LAInorm+PFThnorm.

We acknowledge that these four components may not contribute equally to hydrological complexity and that dependencies may exist among them. Therefore, we further examined the correlations among the CI components (Appendix Fig. ). This additional analysis shows that, although some dependence exists among the components, especially between DEM and DEMstd, the four variables capture complementary topographic and vegetation controls on SMR simulation.

To further clarify the physical meaning of CI, we examined the spatial distributions of both the individual components and the integrated CI (Fig. ). The four components show distinct but complementary spatial patterns, indicating that basin complexity is jointly shaped by elevation background, topographic variability, vegetation density, and vegetation-type heterogeneity. The integrated CI highlights several regions with relatively high basin complexity, including the western United States, western Europe, Japan, and northeastern China, where complex terrain and/or vegetation conditions may pose greater challenges for SMR simulation. These spatial patterns support the use of CI as a synthetic descriptor of basin complexity and provide a basis for interpreting subsequent model-performance changes along the CI gradient.

Consequently, we consider a model to be robust in representing SMR processes if it not only performs well in basins with low basin complexity, but also maintains its accuracy as basin complexity increases. Our analysis therefore examines model performance against each complexity factor individually, and then assesses model performance along the integrated CI gradient.

As described above, a robust model should accurately simulate SMR across diverse land-surface conditions. In this study, robustness is assessed by examining how model bias changes along the basin complexity gradient. Specifically, we developed a Robustness Index (RI) from two complementary perspectives: (1) the overall magnitude of model bias across all basin-complexity levels, hereafter referred to as performance stability; and (2) the rate at which model bias changes as basin complexity increases, hereafter referred to as adaptability.

Before calculating RI, all basins were ranked according to their integrated CI and divided into ten equal-sized groups along the CI gradient. Each group therefore represents one basin complexity level, from the least complex to the most complex basins. For each group, the representative CI value was defined as the median CI of all basins within that group, and the model bias was summarized using the median bias across basins in the same group. This grouping strategy reduces the influence of uneven basin distributions along the CI gradient and allows model performance to be evaluated consistently across different environmental complexity levels.

First, to quantify the overall bias magnitude, we calculated the Stratified Mean Absolute Bias (SMAB). SMAB represents the average absolute bias across the full CI gradient and is used here as a measure of performance stability. Compared with a direct average over all basins, this stratified calculation avoids potential distortion caused by uneven basin distributions along the complexity gradient and provides a more balanced estimate of model performance across different basin complexity levels. SMAB is calculated as follows (Eq. ): 5SMAB=1CIn∑i=1n-1Biasi+Biasi+12CIi+1-CIi where i denotes the index of the complexity groups, ranging from 1 to n-1, Biasi represents the median model bias within the ith complexity group, and CIi indicates the representative CI value of that group.

Second, to quantify the sensitivity of model performance to increasing basin complexity, we regressed the absolute bias against CI using simple linear regression. The regression slope S was adopted as a measure of adaptability. A larger positive slope indicates that model bias increases more rapidly with basin complexity, implying stronger performance degradation and weaker adaptability under complex land-surface conditions. The regression is expressed as follows (Eq. ): 6|Biasi|=S×CIi+b where S is the slope of the absolute bias along the CI gradient, and b is the intercept.

For comparison across all models, both SMAB and S were normalized to a [0,1] scale. The two normalized metrics describe complementary aspects of model robustness. Specifically, SMABnorm reflects the overall magnitude of model bias across the full basin complexity gradient and therefore represents performance stability. Snorm reflects the rate of bias increase with basin complexity and therefore represents adaptability to complex basin conditions. A robust model should have both low SMABnorm and low Snorm, indicating low overall bias and weak performance degradation as basin complexity increases.

Together, these two components form a two-dimensional metric space, represented by the vector (SMABnorm,Snorm). In this space, the origin point (0,0) represents an ideal model with zero average bias and no degradation trend along the CI gradient. Following previous studies that use distance-based approaches to integrate multiple performance dimensions , we adopted the Euclidean distance from this ideal point to define RI. A larger distance indicates a greater deviation from the ideal condition and therefore lower robustness. The final RI is calculated as one minus this distance (Eq. ): 7RI=1-SMABnorm2+Snorm2

A higher RI value indicates that a model is closer to the ideal origin point, reflecting both lower overall bias and stronger adaptability to increasing basin complexity. In this formulation, models with large overall bias are penalized through SMABnorm, whereas models whose errors increase rapidly along the CI gradient are penalized through Snorm. Therefore, RI provides an integrated measure of model robustness by jointly accounting for performance stability and adaptability. We note that RI is used here primarily as a relative robustness metric for inter-model comparison, rather than as an absolute measure of model performance.

To clarify the interpretation of RI, we added a conceptual illustration in the Appendix Fig. . The schematic shows that a model with low SMAB and low slope corresponds to high robustness, whereas a model with either large overall bias or strong bias increase along the CI gradient receives a lower RI. Meanwhile, to further examine whether the RI-based model ranking is strongly affected by the specific aggregation method, we conducted a set of sensitivity experiments using alternative RI formulations (Table  and Fig. ). These experiments included weighted Euclidean-distance formulations and weighted linear formulations with different relative weights assigned to SMABnorm and Snorm. The results show that most models exhibit limited changes in both RI magnitude and relative ranking across the sensitivity experiments. In particular, models with consistently high or low robustness remain generally stable, indicating that the identification of robust and non-robust models is not strongly affected by the specific weighting scheme. Based on these sensitivity results, and considering that Euclidean-distance-based metrics have been widely used in previous studies to integrate multiple performance dimensions , we retained the equal-weight Euclidean-distance formulation as the baseline RI calculation. This formulation provides a simple and physically interpretable way to jointly represent performance stability and adaptability, while avoiding the need to impose subjective preference on either component. Therefore, RI is used here as a relative and integrated measure of model robustness for inter-model comparison.