The GRD model is a conceptual distributed hydrological model designed for flash flood prediction . Since March 2017, it has been used operationally by the national French flash flood warning system called “Vigicrues Flash” . The model version used in the present study has been specially developed for testing the potential of distributed calibration using the variational approach. It is defined on a regular 1 km2 grid and runs continuously at an hourly time step. For each time step the model input includes the gridded rainfall and potential evapotranspiration, and the output is the discharge field defined at the routing scheme nodes.

Our model incorporates some features from the GR (Génie Rural) model family, which include several lumped and semi-distributed bucket-style continuous models developed over the last 30 years at INRAE Antony (Institut national de recherche pour l'agriculture, l'alimentation et l'environnement). Those models have been extensively tested and have demonstrated good performance in various conditions and for different time steps .

Let us consider a 2D spatial domain (basin, catchment and watershed) Ω covered by the rectangular grid. For each cell (pixel), the model involves the production and transfer reservoirs, characterized by capacities cp and ct, correspondingly, and the discharge generated within each cell is routed between cells with local routing velocity v (see Fig. ). It means that only three parameters need to be defined for each cell. The integral flow, without distinction between surface, subsurface or groundwater flow, is simulated by the production and transfer reservoirs at the cell level and by the routing scheme at the inter-cell level. Obviously, such model conceptually describes the overall hydrological process rather than its physically meaningful components.

Calibrating a distributed model is often difficult due to a number of reasons. First, the total number of the sought parameters can be quite large (high dimensionality). This strictly limits the choice of suitable inference methodologies. Second, there is an identifiability issue given the sparsity of observations in space, the information content of the test signal (rainfall variability) and, possibly, the chosen model structure. The first two can be partially compensated by increasing the observation period or observation frequency to better analyze the system dynamics.

For distributed models the variational-estimation algorithm could be a natural choice, since it is scalable, i.e., it works efficiently for a practically unlimited size of the control vector. That is why this method (branded as 4D-Var) is commonly used in meteorology and oceanography for operational forecasting and reanalysis . The method provides the exact mode of the posterior distribution by minimizing the cost function defined over the given observation window. The key element of the method is the adjoint model, which provides the precise gradient of the cost function with respect to all elements of the control vector in a single run . This allows for the fast-converging gradient-based minimization methods to be used, such as the BFGS (Broyden–Fletcher–Goldfarb–Shanno) or the Newton type. Quite often, the need for development of the adjoint model becomes an obstacle for the practical implementation of this method. Heuristic methods such as the Nelder–Mead algorithm do not require the gradient to evaluate the descent directions but converge slowly and are not suitable for solving problems in high dimensions. The same is true as for the general-purpose statistical methods such as the Markov chain Monte Carlo (e.g., Metropolis–Hastings algorithm), as well as for the methods specially designed for hydrology applications, such as SUFI-2 (Sequential Uncertainty Fitting) .

Let us consider the rainfall and potential-evapotranspiration fields P(x,t) and E(x,t), ∀x∈Ω. We represent the hydrological model in Sect.  as an operator A mapping the input fields P(x,t) and E(x,t) into the discharge Qk(t) at the gauged nodes xk∈Ω,k=1,Ng: 14Qk(t)=A(P(x,t′),E(x,t′),h(x,0),θ(x),t),∀x∈Ω,t′∈(0,t), where h(x,t)=(hp(x,t),ht(x,t))T is the state vector, which includes the states of production and transfer reservoirs for all cells at time t; θ(x)=(cp(x),ct(x),v(x))T is the parameter vector, which includes the corresponding capacities and the routing velocities at all routing nodes; and Ng is the number of gauged routing nodes, i.e., where discharge observations are available. If the observation period is much longer than the characteristic time of the system (which is the case for calibration or reanalysis), one can use the trivial initial state h(x,0)=0, but consider the observation window t∈(t∗,T), where t∗ is the relaxation period. Given the observed inputs P∗(x,t′),E∗(x,t′),t′∈(0,t) and the output Qk∗(t), the calibration cost function can be defined as follows: 15J(θ)=∑k=1Ngak-1(t∗)∫t=t∗TA(P∗,E∗,0,θ,t)-Qk∗(t)2dt+α‖B-1/2(θ-θ∗)‖L22, where B is the background error covariance; θ∗ is a prior guess on θ, which comes from special measurements, land expertise or a modeling; α is the regularization parameter; and ak are the scaling constraints. If we consider ak(t∗)=∫t=t∗T〈Qk∗〉-Qk∗(t)2dt, where 〈Qk∗〉 is the temporal mean of Qk∗(t), then for each k the misfit term becomes 1-NSE, where NSE stands for the Nash–Sutcliffe efficiency criterion widely used in hydrology. In essence, Eq. () is more or less a standard quadratic cost function similar to the one used in variational data assimilation (4D-Var), where the weight α is introduced to mitigate the uncertainty in θ∗ and B. Let us note that this paper is focused on the parameter calibration problem involving long time series of observations.

We use additional constraints in the form 16θmin⁡≤θ≤θmax⁡, where θmin⁡ and θmax⁡ are the bounds which come from the empirical knowledge or physical considerations. Thus, the optimal estimate of the parameters θ^ is obtained from the condition 17θ^=arg⁡min⁡θJ(θ), given constraints of Eqs. () and ().

Matrix B can be represented in the form B=σθ⋅ICσθ⋅I, where σθ is the vector of mean deviations of θ, C is the correlation matrix, I is the identity matrix and “⋅” stands for the element-wise (Hadamard) product. Next, the scaling of parameters is introduced such that θ=θmin⁡+θ̃(θmax⁡-θmin⁡) to ensure that 0≤θ̃≤1. Then, the penalty term in Eq. () takes the form α‖(θmax⁡-θmin⁡)⋅σθ-1⋅IC-1/2(θ̃-θ̃∗)‖L22. Assuming that (θmax⁡-θmin⁡)⋅σθ-1=const, the cost function of Eq. () reads as follows: 18J(θ̃)=∑k=1Ngak-1(t∗)∫t=t∗TA(P∗,E∗,0,θ)-Qk∗2dt+α‖C-1/2(θ̃-θ̃∗)‖L22, given 19θ=θmin⁡+θ̃(θmax⁡-θmin⁡),0≤θ̃≤1. The results presented in this paper correspond to the simplest approach to regularization: we assume that C=I, and the regularization parameter is chosen a priori as a small value (α=10-4) to ensure the formal well-posedness of the calibration problem. More sophisticated approaches for regularization (nontrivial correlation matrix C and the optimal choice of α using the L-curve approach) have been utilized but are not presented in this paper for the sake of simplicity. In practice, the simplifications mentioned above lead to significantly more oscillating parameter fields, which does not seem to have a critical influence on the predictive performance of the model (in the open-loop forecasting, at least).

Minimization of Eq. () given constraints of Eq. () is performed by LBFGS-B (limited-memory Broyden–Fletcher–Goldfarb–Shanno bound-constrained; ). The minimization process can be written in the form 20θ̃i+1=θ̃i+γH-1(θi)R[Jθ̃′(θi)],i=0,1,…,θ̃0=θ̃∗, where J′(θi) and H-1(θi) are the gradient (with respect to θ̃) and the limited-memory inverse Hessian of Eq. () at point θi, respectively, γ is a descent step, i is the iteration number, and R is the gradient projection operator to account for the box constraints. Let us note that H-1(θi) is directly computed inside the minimization algorithm in such a way that its norm is always bounded. This serves as an additional regularization; thus the solution θ^ is always bounded, even for α=0 in Eq. (), i.e., even without the penalty term. The gradient J′(θi) is obtained by solving the adjoint model. This model has been generated by the automatic differentiation engine Tapenade , then manually optimized and, finally, verified using the standard gradient test.

The background value θ∗ is used both as a starting point for iterations and in the penalty term. Given the fact that the information content of the test signal (rainfall) and observations (discharge) may not be sufficient to uniquely identify the distributed coefficients, evaluating an appropriate θ∗ becomes an important issue. Thus, the overall calibration process involves two steps. In the first step we consider a uniform approximation: cp(x)=c‾p, ct(x)=c‾t and v(x)=v‾, ∀x∈Ω. In this case (referred as “uniform calibration”) the sought vector θ‾=(c‾p,c‾t,v‾)T consists of just three elements. For such a low-dimensional problem, obtaining the globally optimal estimate θ‾^ (i.e., the one which corresponds to the global minimum of Eq. ) is feasible by a variety of methods. In particular, we use a simple global-minimization algorithm, the steepest descent method summarized in . In the second step we estimate the distributed parameters using the uniform estimate as a background, i.e., θ∗=θ‾^. Here, three unknown parameters for each cell are estimated using the variational algorithm described above. This two-step algorithm is referred as “distributed calibration”.

Parameter bounds are defined for each step. Numerical and physical considerations enforce the lower bounds so that (cp,ct,v)>0. For the uniform calibration, the upper bounds are chosen to preserve the model dynamics. For example, 5 ms-1 is used as the velocity upper bound, since above this value the flow delay does not decrease significantly. For the production and transfer reservoirs the upper bounds are set to 5000 and 2000 mm, respectively, since the higher values do not noticeably change the model dynamics (reservoir states remain almost constant in time). For the distributed calibration, bounds are recomputed as θmax⁡=Kθ‾^ and θmin⁡=θ‾^/K. In particular, K=4 is used for results presented in this paper. Further increasing K does not seem to have any influence on the estimates.

A French watershed, the Gardon d'Anduze, has been considered for testing our model and calibration algorithm. Located in the western Mediterranean region, this catchment and its surrounding have been deeply studied in the framework of the HyMeX program to understand the processes leading to flash floods. For instance, some studies exploited a number of very detailed field measurements during severe storm events . Others tested hypotheses using physically distributed models, for instance the MARINE model or the CéVeNnes (VCN) model . A few conceptual distributed models were also considered in this area, such as those implemented into the ATHYS (Atelier Hydrologique Spatialisé) platform .

The main properties of the Gardon d'Anduze are described in . In brief, this is a steep mountainous watershed with a dense hydrographic network spreading over 540 km2 in the eastern part of the Cévennes mountains (France). The difference in levels between the highest elevation point and Anduze is about 800 m, and the slope reaches 50 % in the upstream part. A metamorphic (schist) but fractured geological formation dominates the watershed. Water infiltrates very quickly (the saturated hydraulic conductivity is greater than 100 mmh-1). The water circulation appends mainly underground, but with very short response times (less than 12 h). This area is governed by a transitional Mediterranean–oceanic climate with warm and dry summers, alleviated by the oceanic influence, followed by recurrent short, intense but persistent heavy rainfalls in autumn and winter, known as épisode méditerranéen, generating flash floods.

This watershed is well gauged: at least five stations with continuous data collection are operational here (see Fig.  and Table ). For numerical experiments, the discharge data have been extracted from the HYDRO database of the French ministry in charge of the environment. Rainfall gridded data have been provided by Météo France. It is a pseudo-real-time reanalysis (ANTILOPE – ANalyse par spaTIaLisation hOraire des PrEcipitations – J+1, i.e., available 1 d after the current date), merging radar rainfall estimation with in situ rain gauges. The gridded potential evapotranspiration is computed using interannual air temperature values and the formula developed by . The temperature data were provided by the SAFRAN reanalysis . All the time series have been processed at an hourly time step over the continuous 2007–2018 period. All the gridded data are defined at a 1 km × 1 km spatial resolution, over the same 46 km × 40 km domain overlapping the watershed. Furthermore, the flow direction (eight directions) map and the flow accumulation map have been carefully checked, in order to ensure that every cell is connected to the correct downstream cell.

The variational algorithm described in Sect.  is applied to the hydrological model presented in Sect. , using the Gardon d'Anduze watershed as a benchmark. The number of “active” cells (i.e., included into the watershed) is 540, so the total number of parameters to be calibrated is 3 ⋅ 540.

The calibrated-model validation step consists of checking the model predictive performance (referred as MPP) using the data not involved in calibration. That is, the full set of observations Qk∗(t),k=1,…,Ng,t∈(0,T) is divided in two complementary subsets: a calibration subset and validation subset. Since Q∗ depends on k (defining the spatial distribution of sensors) and t, we distinguish the temporal, spatial and spatiotemporal validation, following the split sample test defined by . In particular, we divide the whole period in two parts: P1 from 1 January 2008 to 1 January 2013 and P2 from 1 January 2013 to 1 January 2018. Each period of P1 and P2 can be considered as a calibration or validation period. A model warm-up of 1 year is performed before starting the simulations. We assume that 1 year is enough, according to recommendations by .

If data from a station are used in calibration, the corresponding catchment is called the “calibration catchment”; otherwise it is called the “validation catchment”.

Both the calibration quality and the MPP in validation are measured using the NSE criterion and the Kling–Gupta efficiency (KGE) criterion .

The spatiotemporal measured discharge data are partitioned into calibration and validation complementary sets. Then, we refer to: a.

“temporal validation” if MPP is evaluated for all calibration catchments over the validation period

In addition, the current MPP is analyzed for the eight major flood events for periods P1 and P2, listed in Table . Three other criteria from are used, which compare the magnitude and the synchronization of the modeled and observed flood peak.

The percentage peak discharge PD:21PD=100×Qmax⁡Qmax⁡∗,where Qmax⁡=Q(tm) and Qmax⁡∗=Q∗(tm∗) are the predicted and the observed maximum discharges, respectively. These maximum values are achieved at different time instants tm and tm∗, within the chosen time period.

The results associated with experiments 1 and 2 are presented in Fig.  in a “statistical” form (as a distribution of the NSE criterion ranked in increasing order), without relation to the period or gauge station.

Figure a shows the results of the “uniform” and “distributed” calibration from experiment 1. All five gauging stations are involved in calibration over two periods; thus we have 5⋅2 calibration points. One can see that the distributed calibration allows for a much better approximation of the observed discharge than the uniform one. This result confirms that the data assimilation procedure works. It also suggests that hydrological properties of the catchment are not uniform and may be captured using distributed parameters. On the other hand, the mismatch between the predicted and observed discharges remains significant. Since the model looks overparameterized and the regularization parameter in Eq. () is very small, this indicates that a few issues, either alone or in combination, may be presented: corrupted data (the data used are not synthetic), a deficient model structure (such as an unaccounted sink term, for example) or a local minimum is reached instead of the global one. Some of these issues will be later discussed. Figure b shows the same results for experiment 1, but they are this time in temporal validation. One can see that the distributed calibration allows for achieving a better “global” temporal MPP than the uniform one.

Figure c shows the results of spatial validation of experiment 2 (i.e., calibrating only with data from the Anduze gauging station). The remaining four gauging stations are involved in validation for two time periods; thus we have 4⋅2 spatial validation points. The calibration results for the Anduze station are not presented graphically, but the NSE values achieved are (for P1 and P2, respectively) uniform of 0.82 and 0.78 and distributed of 0.93 and 0.88, whereas the KGE values are uniform of 0.81 and 0.71 and distributed of 0.89 and 0.69. One can see once again that the distributed calibration allows for clearly achieving a better global spatial MPP than the uniform one. Finally, Fig. d shows the results of spatiotemporal validation (experiment 2), evaluated over the period not used for calibration. As before, only data from the Anduze gauging station are used for calibration, but data from all five gauging stations from a different time period are used for validation, giving 5⋅2 validation points. Two of them (in “×”) show the corresponding temporal validation results at the Anduze gauge station; others (shown in “⊗”) are the spatiotemporal validation results at the remaining gauge stations. One can see that the spatiotemporal MPP is generally better if the distributed calibration is used.

To get a more detailed view of the results of experiment 2, Fig.  relates each NSE value to the particular time period and gauging station. Here, Fig. a presents the NSE values calculated over the period P1, with parameters optimized over the same period (calibration); Fig. b presents the NSE values calculated over P1, with parameters optimized over P2 (validation). Conversely, Fig. c contains the NSE values calculated over the period P2, with parameters optimized over the same period; Fig. d presents the values calculated over P2, with parameters optimized over P1. All panels present results for the uniform and distributed parameters (blue and red, respectively). The similar figure has been obtained for experiment 1 but not presented here, since the results are also similar. Based on this figure the following observations can be made: 1.

Considering the results obtained with uniform parameters (blue), the NSE values are always better (larger) when calculated over the period P1, whatever case is considered: calibration (+), spatial validation (o), temporal validation (×) or spatiotemporal validation (⊗).

To complete the analysis, the MPP obtained in spatiotemporal validation at Mialet (V7124015) and Saint-Jean (V7135017) – two stations identified as “particular” just above – have been studied in more detail, for 2⋅4 flood events described in Table , corresponding to the four biggest floods over each period. The corresponding hydrographs are plotted in Fig. , and the resulting MPP criteria of NSE, KGE, PPD, SPPD and PD are presented in Tables  and . Based on these results, the following conclusions can be drawn. 1.

Flood events at those stations which occur during the period P1 are well simulated using both a uniform and distributed set of parameters calibrated over P2 (NSE‾>0.70 and KGE‾>0.60). In particular, the prediction of the event 3/P1 is noticeably improved for the distributed calibration. The simulated flood peak is well synchronized with the observed one, though slightly shifted when the distributed calibration is used. Note that these minor shifts are not critical for the peak discharge prediction, as PPD and SPPD remain similar.

An inverse problem is well-posed if a solution to the problem exists, is unique and is continuous with respect to the input data. Let us consider a mathematical model describing a certain physical phenomenon and assume that some variables of this phenomenon are partly observed. If the model is perfect (adequate) and observations are exact then, given the input, the model's output has to match observations. This implies that the minimum of a cost function penalizing the mismatch (mismatch functional) equals zero. Parameter calibration problems are often ill-posed in terms of the uniqueness (equifinality): there may exist a set of solutions for which the mismatch functional equals zero. However, the “true” value of parameters does exist as one element of this set. If the model involved is nonlinear with respect to its parameters, the mismatch functional may contain additional minimum points where its value is not zero. While the corresponding parameters do not belong to the set of solutions, in the local-search minimization methods such points are interpreted as solutions. Adding a penalty (regularization) term to the mismatch functional allows one particular solution, which is the “closest” to the prior in terms of a chosen norm, to be defined. This makes the problem formally well-posed but, in practical terms, transforms the non-uniqueness issue into the issue of choosing the prior. Thus, investigating the stability of the estimates with respect to the priors is an important step in the design and validation of a calibration algorithm. In particular, such an analysis allows for the parts of the solution dominated by observations and by the prior to be distinguished.

A straightforward approach implies solving an ensemble of calibration problems involving random (or quasi-random) uniform priors subjected to box constraints of Eq. (). Let us remember that the priors are used in two ways: as starting points for minimization and to define the penalty term. The influence of the latter has been neglected by choosing small α in Eq. (). Let θ^|θ‾l be the parameter estimate conditioned on the uniform prior θ‾l, l=1,…,L from the ensemble of priors of size L. Then, the stability for each element of the parameter vector is measured by the standard deviation (SD) given as follows:

24σθ=sign(a)|a|,a=1L∑l=1Lθ^|θ‾l-〈θ^〉2-θ‾l-〈θ‾〉2,

where 〈θ‾〉 and 〈θ^〉 are the ensemble average of priors and estimates, respectively. One can see that the negative values of σθ correspond to the case when the solution tends to approach the same value for all priors; i.e., it is dominated (or stabilized) by observations, whereas the positive values of σθ are for the case when the solution is dominated by the prior.

In reality, the conceptual GRD model used in this study is a fairly crude approximation of the hydrological phenomenon. Thus, no solution can be considered as a “truth”, but it is rather as an interpretation of data, given the model and the judgment criteria (cost function). Moreover, the measured data (test signal and observations) are not perfect. These imperfections result in a “generalized observation error”. The stability analysis described above can also be applied in this case, though understanding of results is more difficult. For example, the ensemble average of the mismatch functional values (achieved in minimization) can be considered as a reference level. The parameter estimates which correspond to the values around this level can be considered as possible solutions, whereas the outliers must be discarded.

The stability analysis (experiment 3) has been performed for L=16 configurations of uniform priors c‾p and c‾t (see Table ) with the same routing velocity v‾ = 5 ms-1. The table shows the values of the cost function J(θ) of Eq. () for θ before and after minimization.

One can see that all minimization processes for a chosen assimilation period converge to a similar value of the cost function: for P1 the mean value is 0.055, and the SD is 0.002, i.e., about 3.6 % of the mean; for P2 the corresponding numbers are 0.129, 0.004 and 3.1 %. No obvious outliers have been observed, which indicates that the cost function surface is sufficiently regular and the issue of “local search vs. global search” is not critical. The latter is not surprising, since, considering the operators involved, the model seems to be mildly nonlinear.

The spatial distributions of σθ are presented in Fig. , where the stable areas of the estimated parameter fields, i.e., those influenced by observations, are shown in blue. The purpose of this analysis is to understand why the estimates associated with periods P1 and P2 are so different. In particular, the question rises concerning the high capacities obtained for P1; see red areas in Fig. a–c for cp and dark-yellow areas for ct. In contrast, the capacities obtained for P2 are the lowest in the corresponding areas. However, comparing Figs.  and , we notice that these “questionable” areas in Fig.  largely coincide with the blue (stable) areas in Fig. . This clearly indicates that the corresponding estimates are determined by the observed data used in each case. Thus, one may suspect here the data information content issue, the data interpretation criteria (cost function) issue and/or the model adequacy issue, rather than algorithmic issues. This is the main conclusion drawn from the stability analysis. One can also notice that the estimates associated with period 2 (P2) are generally more stable than those associated with period 1 (P1), whereas the ensemble-average cost function value achieved for P2 is larger than for P1, i.e., 0.129 compared to 0.055. While a better minimization result has been achieved for P1, it looks like more error has been assimilated in this case.