Calibration of a conceptual distributed model is challenging due to a number of reasons, which include fundamental (model adequacy and identifiability)
and algorithmic (e.g., local search vs. global search) issues. The aim of the presented study is to investigate the potential of the variational
approach for calibrating a simple continuous hydrological model (GRD; Génie Rural distributed involved in several flash flood modeling applications. This model is defined
on a rectangular 1

In the variational approach one looks for the optimal solution by minimizing the standard quadratic cost function, which penalizes the misfit between the observed and predicted values, under some additional a priori constraints. The cost function gradient is efficiently computed using the adjoint model. In numerical experiments, the benefits of using the distributed against the uniform calibration are measured in terms of the model predictive performance, in temporal, spatial and spatiotemporal validation, both globally and for particular flood events. Overall, distributed calibration shows encouraging results, providing better model predictions and relevant spatial distribution of some parameters. The numerical stability analysis has been performed to understand the impact of different factors on the calibration quality. This analysis indicates the possible directions for future developments, which may include considering a non-Gaussian likelihood and upgrading the model structure.

Flash flood prediction remains a challenging task of modern hydrology due to a number of reasons. First, the heavy precipitation events (HPEs) leading
to flash floods are difficult to forecast due to complexity of the processes involved (deep convection triggered by orographic lifting, low-level wind
convergence and/or cold pools)

In order to better predict flash floods and reduce their potentially devastating impact, warning systems have been developed or are currently under
development

As noticed in

Methods of data assimilation (DA) have been engaged for several decades in geosciences, including meteorology, oceanography, river hydraulics and
hydrology applications. These methods are used for estimating the driving conditions, states and/or parameters (calibration) of a dynamical model
describing the evolution of natural phenomena. The estimates are conditioned on observations (usually incomplete) of a prototype system. Some early
applications of DA in hydrology are described in the review paper of

In variational estimation, one looks for the minimum of the cost function dependent on the control vector (i.e., vector of unknown model inputs) using
a gradient-based iterative process. The cost function itself represents the maximum a posteriori (MAP) estimator, which turns into the standard 4D-Var (variational) cost function

Using the variational estimation involving the adjoint is very common in atmospheric and oceanographic applications. But, in hydrology, only a very
few cases have been actually reported. In particular, in

The aim of our study is to assess a set of parameters which may represent the spatially varying hydrological properties of a chosen watershed; thus the distributed model parameters have to be calibrated over a very long assimilation window (i.e., several years). With this purpose we develop a variational-calibration method using the adjoint applied on a simple fully distributed model (GRD; Génie Rural distributed), involving a conceptual cell-to-cell routing scheme. This scheme has been designed keeping in mind the differentiability requirement. The adjoint is obtained by automatic differentiation and manually optimized to provide the capacity to work for long time periods (up to several consecutive years) over large spatial areas, with fine resolution. This requires a memory efficient and fast code. The distributed parameters of the GRD model are calibrated over a French Mediterranean catchment, the Gardon d'Anduze, using rainfall radar data and the discharge data from the outlet gauge station. The discharge data from other gauge stations available in this catchment are used for cross-validation (10 years, being split in two periods). Thus, the major questions addressed in this paper are: (a) can we, in principle, benefit from considering the spatially distributed set of coefficients given by the method instead of the uniform (homogeneous) set of coefficients, and, if so, to what extent? In particular, does it help to improve the discharge prediction over the catchment area including “ungauged” locations? (b) What are the major difficulties associated with this approach (insufficient data, structural deficiency of the model, identifiability issue, etc.), and what could possibly be done to improve the model predictive performance?

The paper is organized as follows. In Sect.

The GRD model is a conceptual distributed hydrological model designed for flash flood prediction

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)

General outlines of the GRD model:

Let

The effective rainfall

The total discharge (

For the sake of simplicity we describe the routing model in the 1D setting. The total discharge from node

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

Let us consider the rainfall and potential-evapotranspiration fields

We use additional constraints in the form

Matrix

Minimization of Eq. (

The background value

Parameter bounds are defined for each step. Numerical and physical considerations enforce the lower bounds so that

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

The main properties of the Gardon d'Anduze are described in

Characteristics of the five gauging stations on the Gardon watershed. QIX2 and QIX10 stand for the quantile discharge, respectively, for 2- and 10-year return periods.

This watershed is well gauged: at least five stations with continuous data collection are operational here (see Fig.

The Gardon watershed at Anduze: hydrographic network (blue) and gauging stations V7124015, V7124010, V7135017, V7135010 and V7144010 (red).

The variational algorithm described in Sect.

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

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

“Statistical” distribution of the NSE criterion comparing distributed vs. uniform calibration and the corresponding validation results: calibration of experiment 1

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

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

“spatial validation” for all validation catchments over the calibration period

“spatiotemporal validation” for all validation catchments over the validation period.

In addition, the current MPP is analyzed for the eight major flood events for periods

The percentage peak discharge

The synchronous percentage of the peak discharge

The peak delay

Selection of major floods for each period

The results associated with experiments 1 and 2 are presented in Fig.

Figure

Figure

To get a more detailed view of the results of experiment 2, Fig.

Considering the results obtained with uniform parameters (blue), the NSE values are always better (larger) when calculated over the
period

Considering the results obtained with distributed parameters (red), better results are always obtained when the NSE values are calculated
over

Comparing the NSE values for the uniform (blue) and the distributed (red) calibration, we notice that for the latter we obtain better
results, except in two cases: in spatial validation over

NSE criterion calibration and corresponding validation results for experiment 2 (one upstream gauging station), by stations and periods:

MPP criteria (NSE, KGE,

MPP criteria (NSE, KGE,

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

Flood events at those stations which occur during the period

In contrast, the prediction of flood events which occur during

Spatiotemporal predicted discharges (with the distributed and uniform set of parameters) and observed discharges during the height major events issued from period

Figures

Maps of the calibrated coefficients (experiment 1 and 5-sta):

Maps of the calibrated coefficients (experiment 3 and 1-sta):

Optimal uniform set of parameters for experiments uniform-5-sta and uniform-1-sta.

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. (

where

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.

Initial and final values (separated by a forward slash) of the cost function of Eq. (

The stability analysis (experiment 3) has been performed for

One can see that all minimization processes for a chosen assimilation period converge to a similar value of the cost function: for

Maps of stability measure

The spatial distributions of

The validation results presented in Sects.

It seems that the routing velocity

Calibration of the distributed hydrological models is a difficult task, with the data information content issue, the data interpretation criteria
issue and the model adequacy. Some of these issues will be addressed in the near future. For example, the use of the Gaussian likelihood which
leads to the quadratic cost function of the type in Eq. (

It is evident that, because of its conceptual nature and simplicity, the GRD model has some structural limitations. Looking for a simple structural
upgrade, which may help to improve the adequacy without increasing noticeably the dimensions and computational costs, is an important future
task. Another one is to provide better priors. For instance, one can use the nonuniform priors. In particular, for the capacities

In this paper, the model has been calibrated using data from different time periods, and the cross-validation experiments have helped to select the
best optimal set of the distributed parameters. However, using a longer calibration period may not necessarily improve the model predictive
performance due to the likely presence of very different hydrological regimes over the extended period, including some extreme cases. As a way
forward, one could consider calibrating the model independently for different hydrological regimes. This approach is coherent to the idea of the
“pooled analysis”. Dependent on the calibration strategy, different parameter sets could be used for the prediction purpose. In the operational flood
forecasting context, the model parameters are likely to be fixed, whereas the real-time update of the model initial state should be considered
instead. In that case, the initial states of the distributed reservoirs will serve as a control vector, and the assimilation will be performed over a
relatively short assimilation window (comparable to the characteristic time of the system). Finally, a running ensemble of models with different
calibrated sets of parameters could be considered. In the framework of a flash flood warning system design, the latter approach could be combined with
an automatic predictive control strategy such as the tree-based model

In summary, the variational approach based on the adjoint model has proved its great computational efficiency and relevance for solving the calibration problem involving the distributed hydrological model GRD. Technically, this problem can be solved over long time periods and for large spatial areas. The difficulties discovered in this process are the fundamental issues of calibration, not related to the chosen method. The answer to the main research question formulated at the beginning of this paper is positive: it is possible and beneficial to calibrate and then use distributed parameters rather than uniform parameters. A variational algorithm involving adjoint sensitivities has proved to be an efficient tool for such calibration. The calibration quality is expected to be improved by using a more appropriate cost function and by enhancing the model structure. Overall, this means that the suggested research and hydrological forecasting tool development direction is quite promising.

Meteorological and streamflow data used in this study can be obtained from Météo-France and the French ministry in charge of environment, respectively. The simulation data that support the findings of this study have been obtained using the SMASH (Spatially distributed Modelling and ASsimilation for Hydrology) platform coded in FORTRAN. They are available from the second author on reasonable request.

MJA performed the computational work. PJ, IG and PA supervised this work. All the co-authors collaborated, interpreted the results, wrote the paper and replied to the comments from the reviewers.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Hydrological cycle in the Mediterranean (ACP/AMT/GMD/HESS/NHESS/OS inter-journal SI)”. It is not associated with a conference.

This research has been done in the framework of the 2010–2020 HyMex program. It contributes to the French national PICS (Prévision immédiate intégrée des impacts des crues soudaines) project. The authors wish to thank Etienne Leblois (INRAE Lyon) for having provided the flow direction map used in this study. The editor and the four anonymous referees are also thanked for their constructive comments and suggestions.

This research has been supported by the “Agence nationale de la recherche” (ANR–17–CE03–0011), the Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA, now INRAE), and by the French ministry in charge of the environment.

This paper was edited by Véronique Ducrocq and reviewed by four anonymous referees.