# Extra constraint on actual evaporation in a semi-distributed conceptual model to improve model physical realism

Shu-Chen Hsu, Alban de Lavenne, Charles Perrin, Vazken Andréassian

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Shu-Chen Hsu, Alban de Lavenne, Charles Perrin, Vazken Andréassian. Extra constraint on actual evaporation in a semi-distributed conceptual model to improve model physical realism. Hydrological Sciences Journal, 2025, 70 (7), pp.1143-1156. 10.1080/02626667.2025.2468846. hal-05443803

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# Extra constraint on actual evaporation in a semi-distributed conceptual model to improve model physical realism

Shu-Chen Hsu, Alban de Lavenne, Charles Perrin & Vazken Andréassian

**To cite this article:** Shu-Chen Hsu, Alban de Lavenne, Charles Perrin & Vazken Andréassian (19 Mar 2025): Extra constraint on actual evaporation in a semi-distributed conceptual model to improve model physical realism, Hydrological Sciences Journal, DOI: 10.1080/02626667.2025.2468846

**To link to this article:** https://doi.org/10.1080/02626667.2025.2468846

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# Extra constraint on actual evaporation in a semi-distributed conceptual model to improve model physical realism

Shu-Chen Hsu ID, Alban de Lavenne ID, Charles Perrin ID and Vazken Andréassian ID

INRAE, HYCAR Research Unit, Université Paris-Saclay, INRAE, Antony, France

## ABSTRACT

When evaluating the calibration of hydrological models, in some discharge-only calibration cases, we sometimes find that the selected parameter sets succeed in discharge estimation, while producing unreasonable simulations of other fluxes. In this paper, we hypothesize that the realism of the unmeasurable intercatchment groundwater flow (IGF) may be improved by introducing a constraint on the actual evaporation (AE). This paper evaluates three multi-objective calibration cases in which an extra constraint on AE is added to the classical objective function, which only focus on discharge efficiency. The results show that for the multi-objective cases (1) the discharge efficiency is not significantly compromised; (2) the spatial efficiency (SPAEF) metric with the reference AE data on the annual scale improves considerably; (3) an agreement with each other for the partition of annual water balance components, especially AE and IGF, is reached. These results support the improved realism of the AE and thus the IGF estimation.

## ARTICLE HISTORY

Received 2 July 2024
Accepted 24 January 2025

## EDITOR

K. Soulis

## ASSOCIATE EDITOR

M. Grodzka-Åukaszewska

## KEYWORDS

multi-objective calibration;
evaporation; semi-
distributed model; spatial
pattern

# 1 Introduction

In hydrological modelling practices, we traditionally evaluate model performance using discharge measurement data, selecting as the “optimal” parameter set the one providing the best simulation of the river flow over the calibration period. Other fluxes in the water cycle are also simulated by the models, but many of them are not well constrained and less often examined. In the work of Bouaziz et al. (2021), the simulation of the internal processes of 12 models is compared with remote-sensing data. This evaluation shows that the models disagree not only with the remote-sensing data but also with each other, even though they all have good performance in terms of discharge simulation. How, then, do we know whether we are “getting the right answer for the right reason” (Klemeš 1986b, Kirchner 2006)?

Hydrologists have been aware of the challenge of equifinality (Beven and Binley 1992, Freer et al. 1996, Zak and Beven 1999), in which different parameter sets can result in similarly good performance. These equifinal parameter sets imply dissimilar partitions of internal processes. A calibration against discharge only is more prone to equifinality since the model can avail itself of the compensation between internal processes to reach similar performance criteria on discharge. This compensation problem can be exacerbated if the simulation is carried out in a spatially distributed model. On the other hand, using spatially distributed models allows accounting for spatial heterogeneity of climate inputs (Smith et al. 2004, Lobligeois et al. 2014) and catchments' physical characteristics (Dembélé et al. 2020a), while bringing the inevitable challenge of the high dimension of parameter search space. Different strategies have been developed for calibrating spatially

distributed models, to facilitate the calibration of the large number of parameters, and to ensure the realism of simulated fluxes. They can be classified into two groups: “constraint on parameter estimation” and “constraint on model state variables.”

Methods of constraining parameter estimation include (a) application of the hydrological response units (HRUs) concept (Leavesley et al. 1983), (b) establishment of direct relations between catchment characteristics and model parameters, and (c) assigning a priori values for initial points or narrowing search space. The use of HRUs can be regarded as a regionalization technique as it is often the unit for assigning parameter sets across space. For instance, the HRU-based Soil and Water Assessment Tool (SWAT) model (Arnold et al. 2012) and the topography-driven modelling framework FLEX-Topo (Savenije 2010) calibrate parameters for the spatial units categorized by their characteristics (slope, land use, soil type, etc.). The second subset of methods does not require a classification process but rather transfers catchment characteristics into model parameters with a presumed relationship. Parameters to address this relationship are called super parameters (Tonkin and Doherty 2005, Pokhrel et al. 2008), global parameters (Samaniego et al. 2010), or transfer parameters (Hundecha and Bárdossy 2004). The number of parameters to be calibrated becomes significantly lower, regardless of the increase in the number of sub-catchments. For instance, Samaniego et al. (2010) proposed and applied the multiscale parameter regionalization (MPR) technique in the mesoscale hydrological model (mHM) model: only 62 global parameters needed to be estimated instead of 28 parameters for each modelling unit. In addition, the relationship established by

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>
the MPR technique is valid across different spatial scales. Finally, the *a priori* information provided in calibration allows a more efficient parameter search compared to a plain search with only the boundary conditions. Pokhrel *et al.* (2012) calibrated a distributed model based on the *a priori* estimation of parameter spatial fields derived from soil datasets. In the work of de Lavenne *et al.* (2019), the optimization aims to minimize the difference from the *a priori* parameter value in addition to maximizing the simulation performance.

Rather than constraining parameters, this article will focus more on constraining the model state variables. To be more confident that we are getting the right answer for the right reason, there has been a consensus among the hydrological modelling community about the need for information other than discharge as extra constraints to the model (Beldring 2002, Silvestro *et al.* 2013, Rakovec *et al.* 2016b, Li *et al.* 2018, Nijzink *et al.* 2018, Oliveira *et al.* 2021). Without acquiring additional measurement data, some apply multi-criteria calibration involving hydrological signatures that call attention to different aspects of discharge data (Yadav *et al.* 2007, Bulygina *et al.* 2009, Euser *et al.* 2013). While these constraints still rely on discharge data, they are of limited help in improving the realism of model simulations. To better address the partition of water, He *et al.* (2015) applied a calibration to the four runoff-generation mechanisms partitioned from the discharge hydrograph. Similarly, Shafii *et al.* (2017) proposed a partition-focused multi-criteria calibration approach but considering the interaction between the partitioned flows. Although the comparison with signature-based calibration showed an improvement in model robustness and parameter identifiability, these partition-based calibrations are limited by the strong assumption that the partition theories are correct.

Other researchers chose to rely on measurement or proxy data of the model state variables – for instance, site measurement of piezometry (Beldring 2002, Pelletier and Andréassian 2022), soil moisture (Koren *et al.* 2008, Babaeian *et al.* 2019), actual evaporation (AE; Rakovec *et al.* 2016a), etc. As the aforementioned data are not commonly available in space, a need for remote-sensing data emerged. The type of commonly used remote-sensing data includes total water storage (TWS) anomalies (Tapley *et al.* 2004, Werth *et al.* 2009, Vergnes *et al.* 2012, Mulder *et al.* 2015, Rakovec *et al.* 2016b), soil moisture (Owe *et al.* 2008, Kerr *et al.* 2012, Sriwongsitanon *et al.* 2016, Dorigo *et al.* 2017), AE (Ghilain *et al.* 2011, Kobayashi *et al.* 2015, Martens *et al.* 2017, Nijzink *et al.* 2018, Dembélé *et al.* 2020b), snow state (Hall *et al.* 2006, Riboust *et al.* 2019) and land-surface temperature (Silvestro *et al.* 2013, Corbari *et al.* 2015, Zink *et al.* 2018). These studies apply remote sensing data as extra constraints in calibration or as information for evaluating model realism (e.g. Pimentel *et al.* 2023). Model performance of discharge simulation is not necessarily improved, but the simulation of other fluxes is more realistic (evaluated by remote sensing data) and robust when extra constraints are involved (Rakovec *et al.* 2016b). However, we should keep in mind that most remote sensing data are highly processed data under many assumptions.

The observed data are applied to constrain the corresponding state variables in the model. What about the modelled fluxes that are not easily measured – for instance, the

intercatchment groundwater flow (IGF)? IGF is understood as either the water lost by the catchment to regional aquifers, i.e. part of the catchment rainfall that will feed underground aquifers that do not contribute to the surface flow at the catchment outlet, or the water coming from outside the topographic catchment boundaries, e.g. due to geological structures connecting the catchment with surrounding catchments that will contribute to the surface flow. The experiment in the work of Bouaziz *et al.* (2018) demonstrates the possible compensation between IGF and AE. The simulated AE increases when the groundwater outflow is not allowed in the new model structure. This interaction allows us to assume a better-constrained IGF simulation by constraining AE.

In this work, we apply two sources of AE estimation, which have different temporal resolutions and levels of spatial information, as constraints to a semi-distributed conceptual model. By comparing the typical discharge-only calibration and the simulations with constraints on AE, we attempt to answer the following question: What are the effects of adding an extra constraint on AE, especially on the simulation of other fluxes like IGF?

# 2 Data and methods

## 2.1 General strategy

The importance of accounting for IGF in long-term catchment water balance analysis has been raised in hydrology for many years (Genereux *et al.* 2005, Le Moine *et al.* 2007, Fan 2019). These studies all show that neglecting the IGF can lead to significant errors in the rainfall partition estimation. While some rainfall-runoff models do simulate the IGF as an internal flux, it is weakly constrained and often not examined. Moreover, the nature of IGF makes it impossible to measure. In this work, we assume that the extra constraint on AE would improve the spatial consistency of AE, and that the better simulation of AE should allow for an improved IGF estimation, given the possible compensation between AE and IGF in the long-term water balance (see the graphical explanation in Fig. 1). With better AE and IGF estimates, the hydrological behaviour of the catchment can be described more realistically.

## 2.2 Catchment and data

The study was carried out on the Somme catchment (Fig. 2) in northern France. The drainage area is 5687 km². This catchment was chosen because its surface flows are known to be highly affected by the chalk aquifer, which crosses the topographical boundaries. The catchment is composed of marshes and ponds in the valley surrounded by plateaus. Fractures and

<figure><img src="image_13.png" /><figcaption>Figure 1. Main idea behind the calibration strategy: to constrain actual evaporation (AE) in order to avoid equifinality of fluxes in the water balance equation, and thus indirectly improve the estimation of intercatchment groundwater flow (IGF). ΔS: changes in catchment storage, P: precipitation, Q: discharge.</figcaption></figure>
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 2.** Digital elevation model (DEM) map of the Somme catchment. The 10 red points indicate the location of the internal gauges of the Somme catchment. The Somme catchment is located in the North of France.</figcaption></figure>

fissures exist and increase from the plateau to the dry and wet valleys. The widespread chalk aquifer plays an important role in Somme River discharge, especially during the dry periods. The depth of the unsaturated zone ranges from 1 m in the wet valley to more than 50 m under the plateaus. The groundwater table is consistent with the surface topography in general (Pinault *et al.* 2005). On average, during the last 15 years, the annual precipitation of the catchment is 720 mm, with a potential evaporation of 650 mm. Although precipitation decreases slightly from coast to inland area, the potential evaporation is rather homogeneous over the catchment.

The discharge data was obtained from the French national database HydroPortail (Leleu *et al.* 2014) at the daily time step. The period of available data varies among the gauges. Among the 16 gauges located in the catchment, 10 have data records in a common duration from 1 October 2005 to 30 September 2017 (red points in Fig. 2). Climate forcing data includes the daily precipitation data from SAFRAN reanalysis by Météo-France (Vidal *et al.* 2010), and the daily potential evaporation estimation using the temperature-based formula proposed by Oudin *et al.* (2005). To be used in the semi-distributed model, the 8 × 8 km² gridded data of precipitation and potential evaporation are resampled according to sub-catchment boundaries.

In addition, we applied the remotely sensed AE data from the Terra satellite in Moderate Resolution Imaging Spectroradiometer (MODIS) project (MODIS16A2GF product) of National Aeronautics and Space Administration (NASA) in two of our multi-objective calibration tests. We used the gap-filled 8 day cumulated evaporation time series at 500 m resolution (Running *et al.* 2021). The value is derived

from the Penman-Monteith method. We used the AppEEARS (Application for Extracting and Exploring Analysis Ready Samples) tool (2023) to retrieve the data for the spatial and temporal range needed. Table 1 provides an overview of the dataset applied in this work.

## 2.3 Model

The semi-distributed GR model (GRSD) (Lobliginois *et al.* 2014, de Lavenne *et al.* 2019) is based on a lumped model implemented in each modelling unit. The outflow transfer from the upstream unit to the downstream one is captured by a linear lag-propagation model and its corresponding parameter: average flow velocity C (m/s). In this study, the GR4J (Perrin *et al.* 2003) model is applied on each modelling unit. GR4J is a daily lumped conceptual hydrological model with four free parameters (see the structure in Fig. 3). It consists of a production store (with maximum capacity $X_1$ (mm)) and a routing store (with maximum capacity $X_3$ (mm)), and a unit hydrograph for distributing the water transfer in $2X_4$ (day) time steps. The groundwater exchange process is described by a coefficient $X_2$ (mm/d) and the relative storage in the routing reservoir (water level R over capacity $X_3$), with positive $X_2$ for water gains from external water bodies and negative $X_2$ for water losses (Equation 1).

$$IGF = X_2 \left( \frac{R}{X_3} \right)^{7/2} \quad (1)$$

Like many conceptual hydrological models, GR4J takes the climate forcing data (precipitation (P) and potential

<table><caption>**Table 1.** Overview of the datasets.</caption><thead><tr><th>Variables</th><th>Spatial and temporal resolution</th><th>Sources</th></tr></thead><tbody><tr><td>Climate input</td><td></td><td></td></tr><tr><td>Precipitation</td><td>8 km, daily</td><td>SAFRAN reanalysis by Météo-France (Vidal <i>et al.</i> 2010)</td></tr><tr><td>Potential evaporation</td><td>8 km, daily</td><td>Estimation using the formula proposed by Oudin <i>et al.</i> (2005)</td></tr><tr><td>Model calibration</td><td></td><td></td></tr><tr><td>Discharge</td><td>-, daily</td><td>French national database HydroPortail (Leleu <i>et al.</i> 2014)</td></tr><tr><td>Remotely sensed data of actual evaporation</td><td>500 m, 8 days</td><td>Gap-filled 8-day cumulated MODIS16A2GF product (Running <i>et al.</i> 2021)<sup>a</sup></td></tr></tbody></table>

<footnote>aRetrieved from the tool Application for Extracting and Exploring Analysis Ready Samples (AppEEARS 2023).</footnote>
<figure><img src="image_16.png" /><figcaption>Figure 3. Structure of the lumped GR4J model.</figcaption></figure>

evaporation (PET)) as input, and river discharge is the output flux targeted during calibration. However, GR4J also simulates the AE and the IGF.

In GRSD, catchments are delineated first at (internal) gauge stations, then at ungauged locations in a way that keeps the size of the modelling units as similar as possible. Based on a target size of 280 km², the sub-catchment delineation algorithm yielded a Somme catchment model counting 25 meshes (also referred to as modelling units), among which 10 meshes are constrained by discharge measurement (blue points), as shown in Fig. 4. According to the common period of available data of the 10 river discharge gauge stations, the study period is selected from 01 October 2005 to 30 September 2017. A distinct meteorological input (P and PET) and parameter set are applied for the GR4J model within each modelling unit. A sequential calibration strategy is designed in the current GRSD, to ensure that the number of parameters to be calibrated at one time does not grow with the number of modelling units.

## 2.4 Calibration procedures

### 2.4.1 A sequential calibration

The number of parameters to be calibrated is one of the biggest challenges in applying a semi-distributed model. This

<figure><img src="image_17.png" /><figcaption>Figure 4. The 10 gauges (blue points) and the river network (light blue lines) in the Somme catchment. The colour of meshes indicates the intermediary contributing area (ICA) constrained by the same internal gauge.</figcaption></figure>

challenge is currently addressed in GRSD by applying sequential calibration, following the work of Andersen et al. (2001) and Lerat et al. (2012). The parameter search space is allowed to be the same four-dimensional space as originally in GR4J (plus the celerity parameter) when calibrating sub-catchments from upstream to downstream. A parameter set is estimated at each gauged station and is applied homogeneously over the modelling units in the intermediary contributing area (ICA). An ICA is the additional drainage area between a downstream station and one or several upstream stations (see Fig. 4).

### 2.4.2 Optimization strategies

Four calibration strategies were carried out in this work: a traditional discharge-only calibration case (A) and three multi-objective calibration cases (B_buR, B_moC, and B_moK).

#### A. Mono-objective calibration on discharge (Q)

Model performance on discharge simulation is evaluated by the widely used Kling-Gupta efficiency (KGE) metric (Gupta et al. 2009). KGE consists of three components: correlation coefficient ($r$), bias in average volume ($\beta$), and bias in standard deviation ($\alpha$) between simulated and observed fluxes (Equation 2). The value of KGE ranges from $-\infty$ to 1 as optimum. A typical discharge-only calibration based on KGEQ of daily time series is applied as a benchmark case.

$$ KGE = 1 - \sqrt{(r-1)^2 + (\beta-1)^2 + (\alpha-1)^2} \quad (2) \\ r = \frac{cov(Q_{obs}, Q_{sim})}{\sigma_{obs}^2 \sigma_{sim}^2}, \quad \beta = \frac{\mu_{sim}}{\mu_{obs}}, \quad \alpha = \frac{\sigma_{sim}}{\sigma_{obs}} $$

#### B. Multi-objective calibration with an extra constraint on AE

In this study, we tested three metrics on AE and two reference AE data: Budyko-like estimated long-term average AE and MODIS AE 8-day time series. Unlike discharge measurement, which is only available at the gauge stations, the reference data of extra constraint, here AE, can be obtained for every modelling unit. To be compatible with the sequential calibration procedure, the averaged AE-related criteria value of modelling units in ICA is applied along with KGEQ at the gauged sub-catchment as objectives in the multi-objective calibration.
<figure><img src="image_18.png" /></figure>

* B_buR: extra constraint on bias of AE volume (KGEQ, βAE)

In this calibration approach, we constrain the long-term behaviour of the AE. The ratio between the average annual simulated flux and the Turc-Mezentsev estimation of long-term AE (Andréassian and Sari 2019) is compared to the optimal value of 1. This new criterion, called βAE, is formulated as Equation (3) following the structure of KGE.

$$ \beta_{AE} = \left( 1 - \sqrt{(1 - AE_{sim}/AE_{TM})^2} \right) \quad (3) $$

The Turc-Mezentsev formula is one of the Budyko-like equations for estimating the long-term average of the annual AE from precipitation and potential evaporation (Equation 4). The coefficient n = 2.5 was found suitable for French catchments (Le Moine et al. 2007).

$$ AE_{TM} = (P^{-n} + PET^{-n})^{-1/n}, \quad n = 2.5 \quad (4) $$

* B_moc: extra constraint on correlation between simulated and MOD16 AE data (KGEQ, CORAE)

Satellite data is favoured for its high spatial coverage and regular measurement all over the world, but most of them rely on the assumptions necessary to process the raw data. Therefore, the uncertainty involved in the satellite products implies avoiding using these values as the “answer” or “truth” whereas the spatial pattern or normalized values can be valuable sources of information (Dembélé et al. 2020a, Bouaziz et al. 2021). Here, we constrain the consistency in the temporal pattern between the simulated AE and the remotely sensed AE by the bias-insensitive metric correlation. In the equation of correlation (Equation 5), X and Y refer to the 8-day cumulative time series of simulated AE and MODIS AE data, respectively.

$$ \rho(X, Y) = \frac{\text{COV}(X, Y)}{\sigma_X \sigma_Y} = \frac{E(X, Y) - E(X)E(Y)}{\sqrt{E(X^2) - E(X)^2} \sqrt{E(Y^2) - E(Y)^2}} \quad (5) $$

* B_mok: extra constraint on KGE of AE using MOD16 AE data (KGEQ, KGEAE)

The constraint on the time series of remotely sensed AE is investigated in this calibration approach by KGE calculated on actual evaporation (KGEAE). The daily simulated AE is first transformed into an 8-day time series, by using the same date as the remote-sensing AE data. This approach is carried out to demonstrate the effect of relying on the value of satellite data.

For the multi-objective calibration, we use the caRamel algorithm (Monteil et al. 2020) available as an R package on the Comprehensive R Archive Network (CRAN), originally developed for hydrological model calibration. The resulting Pareto front is derived from the input setting of algorithm parameters, the boundary of model parameters and objective functions. The documentation mentions the effect of randomness embedded in the caRamel algorithm and recommends retrieving the new Pareto front from several independent optimizations. In this work, we used the combined Pareto front from five optimizations.

Since we applied the sequential calibration in this work, an “optimum” parameter set of the upstream sub-catchment is necessary, so we can have the simulated upstream discharge inflow to the downstream sub-catchment. The optimum point is then defined as the point on the Pareto front with the shortest Euclidean distance to the point whose coordinates are defined by the best performance of each objective function. Moreover, to consider the different sensitivity inherent in the objective functions, the distance is calculated after rescaling the metric values by the range of values of the Pareto solutions.

## 2.5 Evaluation procedures

### 2.5.1 Model performance and robustness

The model performance and robustness under extra constraints are tested by a split-sample test and a leave-one-out test. When a second component of an objective function is added, we can expect the first component to be degraded. However, it is also expected that a two-criteria optimization will result in a parameter set with more robust performance, i.e. more stable outside the calibration period. The temporal effect of an extra constraint on model results is evaluated by a split-sample test (Klemeš 1986). The time series is split into two sub-periods, T1 (01 October 2005 to 20 September 2011) and T2 (01 October 2011 to 30 September 2017), which are used for calibration and validation in turn. A 5-year warm-up period is applied prior to periods T1 (i.e. 2000–2005) and T2 (i.e. 2006–2011), respectively, when that period is used for calibration. In the following, we will use the average of the performances over T1 and T2 in calibration and validation, respectively, to talk about the calibration and validation performance of the model in general.

In a semi-distributed model, we can further investigate the spatial effect of the extra constraint on model robustness. Through the leave-one-out test, where the internal gauges are successively treated as ungauged (called “pseudo-ungauged” here) during calibration, the spatial robustness of different calibration strategies is evaluated after calibration based on the discharge measurement that actually exists. ICAs which should apply the parameter set calibrated at this pseudo-ungauged mesh, including the pseudo-ungauged mesh itself, would be assigned the parameter set of the next downstream gauge. The leave-one-out test is done on the nine internal gauges and is not combined with the split-sample test. It was therefore carried out over the entire 2005–2017 period.

### 2.5.2 Spatial efficiency

In addition to discharge simulation performance, we are also interested in the realism of model results in terms of spatial consistency. Modelling the hydrological system in a spatially distributed manner allows investigating spatial consistency. It describes the spatial organization followed by simulated fluxes and states, given by the interaction of hydrological processes and the catchment (Koren et al. 2008, Welch and Allen 2012). Spatial consistency is sometimes accessed visually: for instance, Grayson et al. (2002) described the difference in the pattern
from maps of observed and simulated soil moisture. In this work, we used the SPAtial Efficiency (SPAEF) metric for a quantitative evaluation. Inspired by the multi-component idea of KGE, Demirel et al. (2018) presented the SPAEF metric for distributed model (Equation 6). Aiming to be bias-insensitive, SPAEF consists of three equal-weight components: the correlation coefficient (α), the ratio of the coefficient of variation (β) and the histogram overlap (γ). The optimal value of SPAEF is 1, while the standard for a good performance could vary for the variable it is evaluating.

$$ \text{SPAEF} = 1 - \sqrt{(\alpha - 1)^2 + (\beta - 1)^2 + (\gamma - 1)^2} \\ \alpha = \rho(\text{obs}, \text{sim}), \quad \beta = \left(\frac{\sigma_{\text{sim}}}{\mu_{\text{sim}}}\right) / \left(\frac{\sigma_{\text{obs}}}{\mu_{\text{obs}}}\right), \quad \gamma = \frac{\sum_{j=1}^{m} \min(K_j, L_j)}{\sum_{j=1}^{m} K_j} \quad (6) $$

where K and L are the histograms of observation and simulation, respectively, and m is the number of bins of the histograms.

# 3 Results

## 3.1 Model performance in calibration and validation

The model performance at the 10 gauged sub-catchments under discharge-only calibration (case A, x-axis) and the multi-objective cases (cases B, y-axis) is compared using the metrics applied as objective functions (Fig. 5). As expected, the discharge performances (KGEQ) of multi-objective cases are lower than that of case A in calibration (Fig. 5(a)), because the optimization is not only on KGEQ anymore. In general, the three multi-objective cases perform similarly well. They do not

lose much performance compared to case A. In the validation period, the multi-objective cases' performance is slightly lower than that of case A, but not by more than 5%.

For the metrics related to AE, the average performance over the ICA is computed at each gauged station. The bias of average annual volume towards the Budyko-like estimation (βAE) is one of the objectives for case B_buR, therefore it often scores the highest among the four cases in calibration (Fig. 5(b)). The values of this criterion for all 10 gauges of the four cases are greater than 90% and vary insignificantly between calibration and validation, implying a temporal robustness in terms of this criterion. However, despite this high score, an increase of 0.05 in the criterion value can bring considerable effect on the spatial pattern of simulated AE, as demonstrated later in Fig. 7. More investigation is needed on the definition of good performance for this metric. In terms of the model efficiency of MODIS AE (KGEAE), case B_moK was expected to perform the best in calibration and validation (Fig. 5(c)). To our surprise, however, case B_moC is found to outperform case B_moK for most gauges.

The spatial representation of discharge efficiency during calibration and validation is shown in Fig. 6. We can say that applying multi-objective calibration does not result in a significant decrease in KGE of discharge during calibration and validation. Overall, the performances at these gauges are not as good as that offered by the GRSD model on a national scale. Indeed, the performance of the square root of discharge can reach 0.9 on average over the catchments in France, shown in the work of de Lavenne et al. (2019). As stated in section 2.1, the hydrological behaviour of the Somme catchment is strongly

<figure><img src="image_19.png" /><figcaption>Figure 5. The performance at controlled meshes in the split-sample test: mono-objective case (case A) on the x-axis and multi-objective cases (case B) on the y-axis. First row: calibration period, second row: validation period. The diagonals (y = x) in the sub-figures indicate where the performance of multi-objective cases is equal to that of case A. Three metrics are shown in columns: (a) KGE of discharge (KGEQ) at the internal gauges, (b) mean of ratio of annual AE (βAE) and (c) mean of KGE of AE (KGEAE) for the ICAs of internal gauges.</figcaption></figure>
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 6.** The KGE of discharge in the (a) calibration period and (b) validation period for case A, B_buR, B_moC, and B_moK. (c) The map of mesh ID (small version of Fig. 3) is shown for reference.</figcaption></figure>

influenced by groundwater. GR models often have difficulties in simulating this type of catchment; this will be further discussed in section 4.4.

The reference values in Fig. 8. The compensation between AE and IGF is obvious in the lower-value meshes: when less water is allowed to leave the catchment as AE, the extra water amount is reflected in its larger IGF outflow.

## 3.2 Model flux at annual scale

The maps of the average annual value of simulated discharge, simulated AE and IGF in calibration for the four cases are shown in Fig. 7. The maps of the annual discharge (Fig. 7(d)) agree with Fig. 5(a) that the model produces an almost identical estimation of discharge under different calibration strategies. However, they simulate dissimilar partitions of AE and IGF. In benchmark case A, the simulated AE in the east part of the catchment (mesh ID 6 and 9–13) is significantly lower than the neighbours. Since the land use over the catchment does not vary much in space, we would expect more continuous annual AE amounts among the meshes, as the Budyko-like estimation (Fig. 8(a)) or the MODIS measurement (Fig. 8(b)).

The three multi-objective cases estimate higher AE in these lower-AE meshes in case A. The spatial pattern of simulated AE of multi-objective cases seems much more consistent with

## 3.3 Seasonality of model fluxes

The monthly regime of simulated discharge, 8-day regime of AE, and monthly regime of IGF of the 25 meshes were inspected. As an example, Fig. 9 shows these regimes' two constrained meshes (12 and 13). Among the four calibration strategies, case A often has a distinct pattern. The simulated IGF values of case A are the most negative in most of the meshes and provide stronger seasonal variation than the other three cases. In terms of AE, the difference between calibration cases is not significant. From October to March, the four cases have almost uniform values. Starting from March, the difference in AE simulation between cases grows and stays high from June to August. The compensation behaviour between AE and IGF is demonstrated here. Since case A has lower AE, the model exports water through other fluxes, which are discharge and IGF outflow.
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 7.** Average annual flux of climate input (a,b) and simulated fluxes (d,e,f) in calibration mode. (a) Precipitation and (b) potential evaporation, simulated (c) discharge, (d) actual evaporation, and (e) intercatchment groundwater flow (IGF) for cases A, B_buR, B_moC, B_moK. In addition, (f) a map of mesh ID and the legend of each variable are shown for reference.</figcaption></figure>

## 3.4 Spatial robustness in the leave-one-out test

In the leave-one-out test, we estimate the performance at each internal gauge as if it were ungauged (pseudo-ungauged condition, see section 2.5.1). Figure 10 plots the performance of the discharge-only case (case A) against the multi-objective cases. On average, case B_buR and B_moK outperforms the benchmark case A to a small extent. For case B_moC, the extra constraint on AE improves the performance of some pseudo-

ungauged meshes. But for most meshes, the performance is slightly lower than the discharge-only case.

## 3.5 Spatial consistency to reference data

With the constraint on AE, the multi-objective cases show significant improvement of the SPAEF values compared to the mono-objective case A. When using the Budyko-like
<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /><figcaption>Figure 8. The average annual value of reference actual evaporation: (a) Budyko-like (Turc-Mezentsev) estimation and (b) MODIS data.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure 9. The monthly regime of simulated discharge (Qsim), 8-day regime of actual evaporation (AE), and monthly regime of intercatchment groundwater flow (IGF) of the constrained meshes, (a) mesh 12 and (b) mesh 13. Reference data, discharge measurement (Qobs) and MODIS AE data (ref AE_MODIS) are plotted as dashed lines.</figcaption></figure>

estimation of annual AE as a reference, the three multi-objective cases all have performance values larger than 0.6. On the other hand, if we use the average annual MODIS AE as a reference, the values are not as high as the B_buR case. However, the spatial efficiency still improves a lot from case A to multi-objective cases. Therefore, this criterion should be interpreted in relative rather than in absolute terms when comparing different cases. Case B_moC has the highest score under both reference data, which demonstrates the potential of the spatial pattern information embedded in MODIS AE measurement.

# 4 Discussion

## 4.1 The use of remote-sensing data

Satellite measurement data is usually reanalysis data, which could have systematic discrepancies from the climate input

data. When we extract information from satellite data using an error-based index (e.g. KGE), the systematic difference can result in distinct AE estimation and therefore IGF. Potential solutions include rescaling the satellite AE before using it for calibration or using an error-insensitive metric. In practice, a scaling or correction factor is often applied to the satellite data to be more comparable with the model simulation (e.g. Rajib et al. 2016, Casper et al. 2023). In the work of Casper et al., the correction factor on AE is derived based on the assumption of a watertight catchment, which is not applicable to our case where IGF is considered. The ratio between satellite measurement of PET and the PET data we used can be a correction factor, but the rescaled AE data will then inherit the uncertainty from the two PET data sources. To avoid increasing data uncertainty from rescaling the data, we may consider the use of metrics that are less (or not) sensitive to the exact value of data. For instance, we applied the correlation in case B_moC to optimize the temporal pattern instead of trying
<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /><figcaption>Figure 10. The performance (KGE of discharge) at nine internal gauges under the pseudo-ungauged condition in the leave-one-out test: mono-objective case (case A) on the x-axis and multi-objective cases (case B) on the y-axis. The diagonal (y = x) in the figure indicates the performance of multi-objective cases is equal to that of case A.</figcaption></figure>

to fit the exact value. Unlike other works, the MODIS data we applied has less than 1% bias with the Budyko-like estimation (AE<sub>TM</sub>) on average in the annual scale (Fig. 8). The spatial patterns of AE and IGF of cases B_buR and B_moK are thus very similar to each other.

## 4.2 Effects of constraint on AE

The results of our experiments show the limited trade-off for discharge performance when the constraint on AE is applied in addition to discharge. The evaluation of spatial consistency using the metric SPAEF (Table 2) further suggests the improvement of the spatial pattern of AE of the multi-objective calibration cases. These results corroborate a number of works that applied the SPAEF of remotely sensed AE data as an objective function when calibrating a physically-based gridded model (Demirel et al. 2018, 2024, Koch et al. 2018, Soltani et al. 2021). It is worth noting that although the spatial and temporal scale of the AE constraints applied in this work are different, the three multi-objective calibration cases reach a consensus on the partitioning of the water balance components.

## 4.3 Challenges in sequential calibration

Although sequential calibration can be an answer to the issue of an increasing number of parameters in a semi-distributed model, the parameter identifiability is questionable in downstream meshes (de Lavenne et al. 2016). It is not

easy to identify the optimal parameter set when similar simulated discharge can be the result of a very different parameter set. For downstream meshes, the inflow from upstream plays an important role in their total discharge. Extreme parameter values may result from this lack of sensitivity of downstream ICA, as total discharge is too much controlled by upstream discharge. In a discharge-only calibration case, the calibrated parameter sets may include these extreme values and result in unrealistic estimation of other model fluxes. Unlike the group of calibration strategies that constrain parameter estimation, the strategies tested in this work can still have very different parameter values at neighbouring sub-catchments. The aim of this work was to compare the effects of different constraints on AE. To further reduce the issue of parameter uncertainty, a combination of constraints on both parameter estimation and the model state variables is recommended.

## 4.4 Limit of current model structure

In the current model structure, the groundwater exchange among sub-catchments is still unconstrained. In each mesh, IGF is indirectly estimated as the flux required to close the water balance, as it is done in the lumped GR4J model. As a result, the simulation of IGF can be unrealistic; see, for instance, the larger groundwater outflow (negative IGF) in the east of the catchment in the discharge-only case (Fig. 7 (f)). The four cases produce a similar performance on KGE of discharge and similar annual discharge estimation, but the AE and IGF between the cases can vary up to 30 mm/year. Also, the discharge performance shown in Fig. 6 demonstrates the model was struggling to simulate the meshes having higher elevation (mesh 20 and 21); in other words, the sub-catchments most likely are losing groundwater. Finally, the optimal parameter values derived from the four calibration strategies are not within a reasonable range of the GR4J model. As a reference, GR4J model users often use the 80% confidence interval of parameters mentioned in the work of Perrin et al. (2003), which is derived from the optimal parameter values calibrated over a large set of catchments. The dashed line and shaded area in Fig. 11 correspond to the mean and the 80% confidence intervals of the parameter values of the GR4J model. Compared to the box plot of the optimal parameter values in our experiment, we can tell the model tried to reproduce the low flow period using an overly large unit hydrograph time constant ($X_4$), production store ($X_1$) and routing store ($X_3$).

Nevertheless, the current model is able to provide a simulation comparable to the GR4J model applied in a lumped mode (Appendix A). The simulated average annual water balance components over the catchment of lumped and semi-distributed models differ slightly (<10 mm/year)

<table><caption>Table 2. Spatial efficiency metric (SPAEF) of actual evaporation of each calibration strategy when different references are applied: Budyko-like (Turc-Mezentsev) estimation (AE<sub>TM</sub>) and MODIS data (AE<sub>MODIS</sub>).</caption><thead><tr><th>Case</th><th>A</th><th>B_buR</th><th>B_moC</th><th>B_moK</th></tr></thead><tbody><tr><td>ref = AE<sub>TM</sub></td><td>0.26</td><td>0.65</td><td>0.73</td><td>0.64</td></tr><tr><td>ref = AE<sub>MODIS</sub></td><td>-0.15</td><td>0.17</td><td>0.34</td><td>0.30</td></tr></tbody></table>
<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /><figcaption>Figure 11. Box plot of calibrated parameter set. From the work of Perrin et al. (2003), a reference of the reasonable range of GR4J parameters is shown as a blue shaded area: X₁ = 100 to 1200 (mm), X₂ = -5 to 3 (mm), X₃ = 20 to 300 (mm), and X₄ = 1.1 to 2.9 (days).</figcaption></figure>

regardless of the calibration strategies. In order to improve the physical realism at the sub-catchment scale, we see a need to implement an explicit underground connection between the meshes of the semi-distributed model, i.e. where the IGFs would become interdependent in incoming and outgoing flows between the meshes.

## 4.5 Limit of experiment size

In the work of de Lavenne et al. (2019), the regularization strategy applying extra constraint on the drift to upstream parameter value was found to be beneficial to the performance of pseudo-ungauged sub-catchments in the leave-one-out test. Therefore, in our experiment, the extra constraint during calibration is also expected to bring advantages to model performance compared to the discharge-only case. However, the improvement brought by extra constraints was not significant. The improvement found in de Lavenne et al. (2019)'s work mainly centres on the internal gauges whose ICAs occupy only a small portion of the upstream contributing area in the sequential calibration. In our model setting for the Somme catchment, there is only one mesh (mesh 13 in Fig. 4) in this case. Therefore, the extra constraint on AE might demonstrate its positive effects on model performance if the same experiment were carried out on a larger number of catchments.

## 5 Conclusions and perspective

This paper compares the model performance and estimation of fluxes under four calibration strategies for a semi-distributed conceptual model. In addition to discharge measurement, the Budyko-like estimation and remotely sensed AE are each applied as an extra constraint to the model. The experiments demonstrate how different extra constraints affect the simulation results of AE and IGF. In this case study, optimizing the average annual AE amount (case B_buR) allows the model to simulate discharge almost as well as the discharge-only calibration (case A). In contrast to other works, the use of MODIS AE data produces similar results regardless of the objective function. The use of a correlation coefficient and of the error-based metric KGE both enable constraining the model in terms

of the AE dynamic without compromising discharge performances. Among the three calibration cases that provide comparable results, the correlation of remotely sensed AE data is the most highly recommended metric to apply. The modellers can then benefit from the temporal pattern and avoid the uncertainty on absolute value.

Although the multi-objective cases show limited improvement in model temporal robustness, we considered the significant increase in SPAEF to be valuable. The approaches tested in this paper demonstrate the benefit of constraining AE during calibration and the potential of spatial pattern information hidden in the reference AE data. Furthermore, adding a constraint on simulated fluxes indirectly constrains the estimation of other fluxes, allowing the water balance to be closed more realistically. With the proposed strategies, we hope to reduce the problem of equifinality to a smaller range. To go further in solving the problem of equifinality, one may think of applying constraints on parameter estimation. However, a constraint on fluxes, as we proposed in this paper, is a more model-independent methodology that can provide us a better understanding of the processes in the hydrological system. Nevertheless, knowing the weakness of the model structure mentioned in the discussion, further work will aim at testing new model hypotheses of the groundwater-surface water exchanges, in order to more explicitly describe the underground connections between sub-catchments. If the physical realism of the model structure is improved, even simple regionalization strategies may show a better performance.

Finally, we also see the potential of constraining the spatial pattern of simulated fluxes. The multi-objective calibration cases give a much higher performance on the spatial efficiency metric than the discharge-only case, which is a quantitative assessment of the consistency of the spatial pattern. To be able to implement the constraint on a spatial consistency metric, we need to move from sequential calibration to regional calibration, such as the MPR approach (Samaniego et al. 2010) or the parameter set shuffling (PASS) method (Merz et al. 2020). Indeed, the simulation results of all meshes are required simultaneously to derive the spatial pattern metric. The application of regional calibration is also essential for testing a shared groundwater store in the model structure, as the variation of groundwater level may have an impact on the upstream sub-
<figure><img src="image_31.png" /></figure>

catchments. In addition, the issue of parameter insensitivity found in downstream sub-catchments (de Lavenne et al. 2019) can be avoided by using regional calibration instead of sequential calibration. This work is a relatively parsimonious experiment in terms of input data and model structure. The three constraints on AE are easily applicable to other models. The proposed methodology enables a significant improvement in model realism in the scope of sequential calibration and the conceptual model.

## Acknowledgements

The authors appreciate the financial support of the CIPRHES project funded by the French national research agency ANR. We thank Météo-France and HydroPortail for making the climatic and hydrological data available. The authors also thank Nicolas Le Moine for sharing his experience and suggestions on the use of caRamel package. The review comments by Reza Barati, two anonymous reviewers and the associate editor Maria Grodzka-Łukaszewska helped to improve the quality of the manuscript and are thankfully acknowledged.

## Disclosure statement

No potential conflict of interest was reported by the author(s).

## Funding

This work was supported by the Agence Nationale de la Recherche for the CIPRHES project [grant number ANR-20-CE04-0009-02] and the European Union's Horizon Europe Research and Innovation Actions Programme under Grant Agreement No. [101059372] (STARS4Water project).

## ORCID

Shu-Chen Hsu http://orcid.org/0009-0002-7831-8150
Alban de Lavenne http://orcid.org/0000-0002-9448-3490
Charles Perrin http://orcid.org/0000-0001-8552-1881
Vazken Andréassian http://orcid.org/0000-0001-7124-9303

## Data availability statement

Climate data are available upon request from Météo-France for research use. River discharge data are available on the website of the French national database HydroPortail (https://hydro.eaufrance.fr/). The MODIS data was retrieved from the online tool AppEEARS (https://appears.earthdatacloud.nasa.gov/).

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# Appendix A

## Water balance of the Somme catchment

Table A1 compares the water balance flux results from the GR4J model applied in lumped and semi-distributed modes. For each calibration cases, the difference in the average annual value of simulated discharge between the two models is negligible. The same applies for the simulated actual evaporation and total water balance.

<table><caption>**Table A1.** Water balance components of Somme catchment (mm/year) in lumped and semi-distributed (semi-D) mode. The potential evaporation (PET) is given for reference. Water balance (WB) = precipitation (P) + intercatchment groundwater flow (IGF) - simulated discharge (Qsim) - actual evaporation (AE).</caption><thead><tr><th>Case</th><th>Model</th><th>WB</th><th>P</th><th>PET</th><th>IGF</th><th>Qsim</th><th>AE</th></tr></thead><tbody><tr><td rowspan="2">A</td><td>Lumped</td><td>2</td><td>719</td><td>648</td><td>-11</td><td>197</td><td>509</td></tr><tr><td>Semi-D</td><td>1</td><td>718</td><td>651</td><td>-27</td><td>194</td><td>494</td></tr><tr><td rowspan="2">B_buR</td><td>Lumped</td><td>1</td><td>719</td><td>648</td><td>-8</td><td>197</td><td>513</td></tr><tr><td>Semi-D</td><td>-1</td><td>718</td><td>651</td><td>-18</td><td>194</td><td>507</td></tr><tr><td rowspan="2">B_moC</td><td>Lumped</td><td>1</td><td>719</td><td>648</td><td>-6</td><td>196</td><td>516</td></tr><tr><td>Semi-D</td><td>-1</td><td>718</td><td>651</td><td>-18</td><td>194</td><td>506</td></tr><tr><td rowspan="2">B_moK</td><td>Lumped</td><td>1</td><td>719</td><td>648</td><td>-7</td><td>196</td><td>515</td></tr><tr><td>Semi-D</td><td>0</td><td>718</td><td>651</td><td>-19</td><td>194</td><td>505</td></tr></tbody></table>