According to the classic convolution integral, streamflow Q is computed as the convolution of precipitation J with the stationary hydrologic response IUH, which in continuous time is expressed as: 1Q(t)=∫0∞J(t-τ)IUH(τ)dτ

Here we use a discrete-time approximation to Eq. (), generalized to allow the IUH to vary with time: 2yt=∑T=0Tmaxxt-Tht-T(T)ΔT where y is the output flux (i.e., streamflow) at time t, x is the input flux (i.e., precipitation) T time steps earlier (i.e., at time t-T), and ht-T(T) is a time-variable and non-unitary response function that reflects the streamflow response to precipitation falling at time t-T, as a function of lag time T. The dependence of h on the precipitation time t-T incorporates any dependence on internal and external forcings, such as precipitation intensity and wetness conditions at the time that rain falls.

At this stage we make very few assumptions about the shape that h can take. It is not a probability density function, meaning that its area can be smaller or larger than one. While in principle h can take negative values (if this is what the system under consideration does, and is reflected in its data), we will assume that h is always non-negative (see Sect. ). By design, h refers to the response to precipitation falling over a specific time step t-T. Any two time steps are generally expected to initiate different responses, but Eq. () is obviously ill-posed because the array ht-T(T) contains many more unknowns than can be constrained by the vectors yt and xt-T. Thus it is necessary to evaluate ht-T(T) as an average over one or more ensembles of time steps (for example during which the precipitation intensity or antecedent wetness is within a given range). In particular, the ensemble responses introduced by can be readily obtained in a post-processing step. Given an ensemble of time points E, the Runoff Response Distribution (RRD, units of 1/T) is the average response weighted by precipitation intensity h over the selected time points E: 3RRDE(T):=∑t′∈Ext′ht′(T)∑t′∈Ext′, (where t′=t-T represents the time that precipitation falls), while the Nonlinear Response Function (NRF, units L/T²) is the average response multiplied by the corresponding precipitation intensity: 4NRFE(T):=1|E|∑t′∈Ext′ht′(T).

GAMCR is a machine learning model that estimates transfer functions from flux data. GAMCR models the catchment's response to any single precipitation event as a weighted sum of spline basis functions. The time-varying coefficients of these basis functions are estimated using machine learning techniques, specifically Generalized Additive Models. As a result, we use more technical language in this section and the next, drawing terminology from the data science literature.

The problem of learning time-dependent transfer functions from rainfall-runoff data is ill-posed, meaning that considering a too large model class might result in zero training loss but with poor test error. In the machine learning community, the standard approach to cope with such badly conditioned inverse problems is to exploit prior knowledge on the structure of the studied system to either shrink the class of target functions or to regularize the optimization problem . Following this approach, GAMCR is built on three core principles.

First, GAMCR specifies a set of features that are assumed to be the main drivers of the catchment response to a given precipitation event. These features can be modified by the user if needed and should typically include information characterizing the catchment condition and the precipitation event considered.

With these guiding principles, we model the transfer functions as follows: 5ht′(T)=∑ℓ=1Lgℓ(zt′)bℓ(T), where bℓℓ∈[L] are B-splines constructed by considering an irregular spacing of knots, zt′ is a feature vector describing both the catchment conditions and the precipitation event at time t′ and gℓ is a GAM. The basis functions bℓℓ∈[L] are illustrated in Fig. , highlighting that the knot density is much higher for shorter lags, while the knots become more spaced out for longer lags. This design enables the model to capture the large variability of the transfer functions at short lags, while still accounting for potentially long recessions. The feature vectors zt′ used in GAMCR are the intensity of the precipitation event at time t′, the weighted averages of both the past precipitation and the past evapotranspiration over different time windows, and the sine and cosine of the fractional year.

Since we model the functions gℓℓ∈[L] using GAMs, one can write 6gℓ(zt′)=ξt′⊤γℓ, where (ξt′⊤)t′ is the design matrix of the GAM. Each entry of ξt′ corresponds to one of the spline basis functions evaluated at a given feature (i.e. a specific entry of zt′). We have: 7yt=∑ℓ=1L∑T=0Tmaxxt′bℓ(T)ξt′⊤γℓ=∑ℓ=1LWt,ℓ,:γℓ=w→t⊤γ→, where γ→:=vec(γℓ,ℓ∈[L]) and w→t:=vec(Wt,ℓ,:,ℓ∈[L]) given by W∈Rn×L×ds such that: 8Wt,ℓ,:=∑T=0Tmaxxt′bℓ(T)ξt′. Here, ds is the number of features resulting from the GAM formulation.

With GAMs we can encode prior knowledge and control overfitting by using penalties and constraints during training. In our case, we consider two smoothness-inducing penalties. The first one promotes the smoothness of the functions gℓ by penalizing the second order derivative, as commonly done in the GAM literature (see ). This penalty ensures that the coefficients of the transfer functions in the basis (bℓ)ℓ smoothly evolve with respect to the catchment features zt′. This penalty acts on the L time-dependent coefficients of the transfer functions in the basis (bℓ)ℓ independently. The second regularization term promotes the smoothness of the transfer functions globally by adding a similar penalty on the model coefficients.

The final optimization problem considered is: 9min⁡(γℓ)ℓ∈[L]≥01n∑t=1n(yt-∑ℓ=1LWt,ℓ,:γℓ)2+λ1∑ℓ=1Lγℓ⊤P1γℓ+λ2∑1≤ℓ,k≤L1n∑t′=1n(ξt′⊤γℓ)[P2]ℓ,k(ξt′⊤γk), which can be equivalently written using a vectorized formulation as: 10min⁡γ→≥01n∑t=1n(yt-w→t⊤γ→)2+γ→⊤λ1P1′+λ2P2′γ→, where, denoting by ⊗ the Kronecker product between two matrices we have defined 11P1′:=IdL⊗P1,andP2′:=P2⊗1n∑t′=1nξt′ξt′⊤

Provided that the hyperparameters λ1 and λ2 are not both zero, the optimization problem Eq. () has a strongly convex objective function with convex constraints. As a result, it admits a unique optimal solution, and the projected gradient descent algorithm is guaranteed to converge to this solution provided that the learning rate is set small enough (cf. ). For all experiments, we used λ1=10-3 and λ2=1 (we refer to Sect.  for further details). In practice, the parameters γ→ are initialized by solving the unconstrained version of the problem, which involves computing the minimum L2-norm solution via the pseudoinverse of a matrix. This initial solution is then projected onto the positive orthant, after which the projected gradient descent algorithm is applied. The learning rate starts at a large value (namely 10-1) and is gradually and adaptively reduced throughout the iterations to ensure a strict decrease in training loss at each step.

The matrix W is precomputed offline prior to running the projected gradient descent algorithm, and parallel computation can be employed to obtain W quickly. This precomputation significantly accelerates the training process by eliminating redundant calculations.

The model developed in Sect.  has been implemented in the Python language as the software GAMCR v1.0. The code is publicly available on Zenodo , and a detailed online tutorial https://quentin-duchemin.github.io/GAMCR/tutorials/ (last access: 30 September 2025) is provided to guide users through usage and reproducibility steps. To use GAMCR, the user must provide time series of precipitation, streamflow, potential evapotranspiration and the corresponding dates and times at equally spaced time intervals. The software operates in a series of steps to ensure accurate and efficient analysis. First, users can use a pre-defined notebook to ensure that their data has the proper format (e.g. column names that conform to the software's requirements). Next, a predefined script is run to perform key precomputations, including the calculation of the matrix W, which significantly enhance the efficiency of the model training process. These precomputations are completed within a few seconds to a few minutes on a standard laptop for a decade of hourly data. Once these precomputations are completed, users can proceed to train the GAMCR model on their dataset. Let us stress once again that the number of basis functions L used by the model is automatically computed based on the maximum lag Tmax⁡. With Tmax⁡=5, 10 or 15 d the model uses 6, 7 or 8 basis functions, respectively. After the model has been trained, users can launch another predefined script to compute key statistics of interest, such as the NRFs over predefined ensembles (such as different precipitation quantiles) and the RRD. These results are automatically saved for further analysis. A detailed tutorial is provided in the online documentation of the GAMCR package, where users can reproduce the results of this paper for the Euthal catchment. The tutorial offers a step-by-step explanation of each stage, equipping users with the necessary tools to apply GAMCR effectively to their own datasets. Overall, GAMCR can be efficiently used on personal laptops, with model training on 20 years of hourly data typically taking around 30 min for Tmax⁡=10 d.

We compiled a 15-year record (2005–2019) of measured, hourly precipitation-runoff data from six Swiss watersheds (Fig. ). The data from the first 13 years were used for model training, while the final two years of data (2018 and 2019) served as the test period. Streamflow time series were provided by the Federal Office for the Environment (FOEN). Precipitation data were sourced from the `CombiPrecip' product, developed by MeteoSwiss (). Potential evapotranspiration time series were computed based on air temperatures provided by MeteoSwiss, through the Hargreaves method from (implemented through the Python Pyeto package https://github.com/woodcrafty/PyETo, last access: 30 September 2025) and then uniformly distributed across each day at hourly intervals. At each site we also extracted key catchment attributes and hydrological statistics (Tables  and ) and computed the mean monthly precipitation, streamflow, potential evapotranspiration over the study period (Fig. ), and the flow duration curve for the snow-free period (Fig. ). We selected these sites because of their diversity in hydrological regimes, elevation and soil depths, which we expect will be reflected in substantially different hydrologic responses. Additional criteria included minimal glacier influence, natural flow regimes (no dams or major abstractions), and complete, reliable data records. The sites have comparable size (between 34–185 km²), which classifies them as small to small/medium basins. Further catchment characteristic analyses appear in Sect. S1 in the Supplement.

As snow introduces complexities in catchment response, such as delayed runoff generation and temperature-driven melt rates, we focused our analysis on snow-free periods only. We considered as snow-free periods the months from May to October, inclusive, except for two basins at the highest altitudes (Euthal and Lavertezzo, with maximum and mean elevations above 2200 and 1300 m a.s.l., respectively), for which we assumed that first snow-free month is June (Fig. ).

Figure  shows the Flow Duration Curves (FDCs) of streamflow during the snow-free months. The curves highlight clear contrasts among catchments: Chiasso and Salmsach are dominated by low flows typical of pluvial regimes, while Lavertezzo and Euthal display broader distributions with higher discharges, reflecting steep topography and rapid runoff. Magliaso and Sonceboz fall in between, balancing frequent low flows with moderate events. These patterns confirm that the selected basins capture a meaningful gradient of hydrological responses, even when snow-driven processes are excluded.

Key hydrological statistics (Table ) further quantify these differences. Q10–Q90 ranges emphasize the variability between rainfall-dominated and more responsive basins, with Chiasso and Salmsach showing low median flows and narrow ranges, and Lavertezzo and Euthal displaying much higher values. Runoff ratios vary widely (0.30–0.82), while the center of timing (CT) spans from day 184 to 221, marking earlier peaks in pluvial catchments and later ones in higher-elevation sites. Together, these indicators confirm the diversity of regimes and support the suitability of these basins for testing the proposed methodology.

While our model is designed to estimate the hydrologic response to each precipitation event, we are primarily interested in the model's ability to reproduce the ensemble responses (RRD or NRF) over given conditions of precipitation intensity or antecedent wetness. Therefore, the model will be tested over ensemble responses. This also offers the opportunity to estimate the hydrologic response – and its main statistics – with ERRA and assess the consistency between GAMCR and ERRA.

We tested the need for optimization of the hyperparameters λ1 and λ2 through initial (and computationally expensive) cross-validation experiments. Since we obtained only minor improvements over the default values λ1=10-3, λ2=1, we consistently used the defaults across all applications. Since we are only interested in the evaluation of the hydrologic response up to a few days after precipitation, we kept the hyperparameter Tmax=24×10 h. The positions of the knots to get the B-splines basis functions bℓ follow an exponentially increasing sequence, starting at 0 with an initial step of 1. After each step, the step size doubles, leading to a pattern where knots are densely spaced at the beginning and become increasingly sparse as values grow. Following this procedure, the value of Tmax⁡ automatically sets the number of basis functions to L=7 in our case. All models were trained for the full set of predefined epochs using gradient descent with adaptive learning rate, ensuring a strictly decreasing training loss.

In the observed data series, the response to very small rainfall events may be easily hidden by measurement noise and other processes. While these events are not particularly relevant for the hydrologic response, they may corrupt the training phase. Hence it is convenient to set a precipitation intensity threshold Jth and train the model only for events that exceed Jth. We trained GAMCR using Jth=0.05 mm h⁻¹.

The results from ERRA were obtained using the R scripts accessible at the following repository: 10.16904/envidat.529, as specified in . The RRD curves were computed considering a maximum lag of 40 h. Initial estimates of precipitation bins were automatically generated by the algorithm, invoking six approximately even ranges, while ensuring a minimum threshold of 40 nonzero values in each precipitation bin. To improve comparability across models, the same precipitation ensembles were used to average the true transfer functions and the GAMCR estimates. Using a coarser input data resolution is beneficial to ERRA when the hydrologic response is long relative to the input temporal resolution (because in such cases, it can be difficult to separate the overprinted effects of input signals at closely spaced lag times). Using a coarser time step helps clarify these impacts. For this reason, after some initial testing, the flashy, base, and damped synthetic input time series are aggregated into 2, 3, and 6 h time steps, respectively.

The model was trained (Sect. ) on the synthetic data (Sect. ), which consists of three cases: the reference response, a flashier response, and a more damped response. We validate the model by first computing the hydrologic response in the form of NRF over six quantiles of precipitation intensity, and comparing it against the ERRA estimates and the benchmark generated directly from the model Fig. .

As a result of the different aggregation of the input time series for the three synthetic data sets, their precipitation intensities (and thus the bins used in Fig. a–c) appear different, although the original hourly input data are the same.

Figure  shows that GAMCR accurately estimates the transfer functions on synthetic data, particularly in the flashy and damped scenarios, where their curves nearly overlap with the benchmark. In the base case, (panel a) the peak value and tail of the response are well captured, but the peak timing is systematically early compared to the benchmark. Overall, in these three cases characterized by very different responses (Fig. d) ERRA and GAMCR provide generally consistent estimates.

We also computed the peak height, peak lag and runoff volume of the NRF, and explored their relationship with precipitation intensity (Fig. ). The results highlight GAMCR's ability to accurately estimate key quantities related to the magnitude of the catchment's response (peak height and runoff volume). These statistics are also very consistent with those estimated by ERRA. For both approaches, the flashy case remains the most sensitive for estimation, with GAMCR underestimating runoff volume for intermediate precipitation values (from 10 to 25 mm h⁻¹) but accurately capturing peak height. In the base case, GAMCR slightly underestimates runoff volume while maintaining accurate peak height estimates. For the damped scenario, it closely matches ground truth values for peak height and produces nearly overlapping runoff volume estimates. Overall, both approaches show a strong consistency in their peak height and runoff volume estimates across different scenarios.

Despite the models' strong performance in estimating the magnitude of the catchment response, both face challenges in predicting peak lag, though in opposite ways. ERRA tends to produce more variability across the NRFs (as shown by dashed lines with triangles in all the right panels). By contrast, GAMCR tends to produce lag values that are much less variable than the benchmark across different precipitation ranges.

When applying GAMCR to observed data it is not possible to validate its accuracy in estimating hydrologic response, because the true response is not known. However, it is instructive to compare the modeled vs measured streamflow series, for both the training and test periods. Since the model was not developed for the purpose of reproducing streamflow, its performance should not be compared to hydrologic models that are designed to maximize fit, but the simulated hydrograph serves as a valuable diagnostic tool. For example, periods where the modeled hydrograph deviates significantly from the measurements could be flagged as unreliable and excluded from the analysis. To support this evaluation, Fig.  shows the streamflow predictions generated by GAMCR.

The agreement between training and test set results varies across basins. Sonceboz and Magliaso show the strongest consistency, with only minor over- or underestimations. Lavertezzo follows a similar pattern but has some overestimated values on training points. Salmsach and Chiasso, by contrast exhibit considerable dispersion and overestimation on the test set, suggesting lower predictive performance. Performance at Euthal is intermediate between these two groups, with overestimation of low test streamflow values. These results suggest that the performance of GAMCR in reproducing streamflow is not directly correlated with the hydrological characteristics of the basins. This is even more visible when looking at the model performance aggregated over streamflow quantiles (Fig. S2), where the fit is consistently good across sites and only a minor underestimation of the lowest flow conditions stands out. Timeseries plots for the test period (Fig. S3) indicate that the temporal dynamics of the predicted hydrograph are appropriate and there are no periods that should be flagged and removed from the analysis.

Figure  presents the weighted average RRDs and the peak heights of the NRFs estimated by ERRA and GAMCR for the six sites in the observed dataset. Computations consider all events whose precipitation intensity exceeds 0.5 mm h⁻¹. The differences in basin response are also evident in the range of runoff coefficients derived from ERRA, which vary from 0.10 in Sonceboz to 0.64 in Euthal. The results align well with the hydrological regimes and characteristics of the basins (see Table  and Figs.  and ). The Sonceboz basin, in particular, shows a very flat runoff-response distribution, which is consistent with the relatively low mean slope, large area, and elongated shape of its basin. These features, along with its moderate permeability and location in the Jura's pluvio-nival region, contribute to the basin's very damped runoff response, reflected in the low runoff coefficient (0.10). A slightly flashier response is observed in the Salmsach basin, which has low mean slope, low permeability, and a pluvial hydrological regime. This results in a damped response, though less damped than Sonceboz's, consistent with its higher runoff coefficient (0.19). The Chiasso and Magliaso basins exhibit similar peak values, but with different response shapes. Despite similarities in altitude and mean slope, Chiasso is larger than Magliaso and has lower permeability, consistent with the larger area under its RRD (i.e. the runoff coefficient, 0.26 versus 0.23 for Magliaso). The flashier response in Magliaso is consistent with its high mean slope, in common with Lavertezzo and Euthal, where the flashiest responses (at least for GAMCR) are observed. Lavertezzo and Euthal are characterized by the highest altitudes, highest annual precipitation and lowest annual potential evapotranspiration. The higher RRD peak for Euthal compared to Lavertezzo is consistent with the lower permeability in the Euthal basin and is also reflected in its larger runoff coefficient (0.64 versus 0.40). Overall, the weighted average RRDs provided by both GAMCR and ERRA are broadly consistent with the distinctive characteristics of each basin. The results shown in Fig.  demonstrate the general consistency between the GAMCR and ERRA approaches across the different sites. Only two basins exhibit some discrepancy: the Chiasso and Lavertezzo basins (purple and brown curves, respectively). In both these cases, GAMCR estimates a more pronounced RRD peak than ERRA within the first 7 h, and a slightly lower tail after 10 h. The estimated NRF peaks for different precipitation intensities for these sites (Fig. d and f) are consistent between ERRA and GAMCR for most precipitation bins, but deviate slightly for the highest one. Overall, the responses estimated by GAMCR and ERRA are broadly similar, and since the models work very differently, consistency in their estimates increases our confidence in both approaches.

GAMCR can be used to investigate how variations in precipitation intensity and antecedent wetness affect the hydrologic response. Here we explore such effects at the six study sites. To characterize precipitation intensity, we use the same six precipitation intervals defined in Sect.  above. As a proxy for antecedent wetness, we use the values of streamflow during the timestep prior to the precipitation event under consideration, which we separate into five ranges. We then aggregate the individual response (RRD curves) over each class of precipitation intensity or antecedent wetness. Results are shown in Fig. , where we plot the RRD peak height (not to be confused with the peak of the NRF shown in Fig. ) against precipitation intensity and antecedent wetness.

As Fig. a shows, the RRD peak heights do not vary systematically with precipitation intensity. By contrast, Fig. b demonstrates clear increasing trends in RRD peak heights with increasing antecedent wetness. Nearly all sites exhibit at least a threefold increase in peak heights across antecedent wetness levels, with the exception of Lavertezzo, which shows a rise in peak heights just for only the last two bins of antecedent wetness. Chiasso, in particular, displays the highest variability, with peak heights spanning almost an order of magnitude (from 0.006 to 0.05 h⁻¹). Notably, for each site, the highest two antecedent wetness levels are widely separated, leading to a marked increase in RRD peak heights. These findings highlight a clear nonstationary response of the six catchments, strongly influenced by their antecedent wetness.