# Robust calibration of a hydrological model with stochastic surrogates

Katarina Radišić, Claire Lauvernet, Arthur Vidard

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## To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Robust calibration of a hydrological model with stochastic surrogates. MASCOT-NUM 2023 - Workshop Méthodes d'Analyse Stochastique pour les Codes et Traitement NUMériques, Julien Bect (Université Paris-Saclay, CNRS, CentraleSupélec, L2S); Nicolas Bousquet (EDF R&D); Bertrand Iooss (EDF R&D) Anthony Nouy (Centrale Nantes, LMJL); Sidonie Lefebvre (ONERA, DOTA); Anthony Nouy (Centrale Nantes, LMJL); Emmanuel Vazquez (Université Paris-Saclay, CNRS, CentraleSupélec, L2S), Apr 2023, Le Croisic, France. pp.1-2. hal-03983100

HAL Id: hal-03983100

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Submitted on 10 Feb 2023

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# Robust calibration of a hydrological model with stochastic surrogates

Katarina Radišić†,1,2, Claire Lauvernet§,1, Arthur Vidard§,2

† PhD student (presenting author). § PhD supervisor

PhD expected duration: Dec. 2021 – Dec. 2024

¹ INRAE, RiverLy, 69625, Villeurbanne, France
{katarina.radisic,claire.lauvernet}@inrae.fr

² Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
[arthur.vidard@inria.fr](mailto:arthur.vidard@inria.fr)

## Abstract

Misspecifying external forcings (such as rain) on a hydrological model can directly affect subsequent parameter calibrations. Indeed, by using classical calibration and problem inversion methods, the error in the external forcings is propagated to the model output, and then, if not treated correctly, this error is compensated by overcalibrating the model parameters. As a consequence, parameter values that were found optimal for one value of the external forcings, are not granted to be optimal for another one. Ideally however, estimated parameter values (that describe time-invariant soil properties) should be the same no matter the value of the external forcing.

<figure><img src="image_3.png" /><figcaption>Figure 1: The influence of stochastic external forcings $U(\omega)$ on the calibration. The aim is to estimate the parameter $\theta_0$ that minimizes the distance between model simulations $M$ and terrain observations $y_{obs}$. This distance is quantified through a cost function $J(\theta_0, U(\omega))$. However, the cost function depends on the realization of the random variable $U(\omega)$, hence its minimizer $\theta^*$ does too. The aim of robust calibration is to annul this dependence.</figcaption></figure>

Robust calibration was proposed to reduce the dependence of the estimated parameter values on the external forcings [3]. The aim of robust calibration is to propose parameter estimators
which are satisfactory over a large set of values of the external forcings. We model the external forcing (here, the rain) as a random variable, and want the estimated parameter to be "optimal enough" over a set of probable values of the random variable. We will be interested in the following robust estimators: the minimizer of the mean of the cost function, the minimizer of its variance, and the Pareto front of these two objectives.

As robust calibration is difficult to apply on distributed hydrological models, mainly due to their high dimension and computational cost, we use a surrogate model for the cost function. The stochastic surrogate for the cost function is constructed as proposed in [1]. Indeed, in a context where rain forcings are considered stochastic, the cost function to be minimised in the calibration setting is itself stochastic, Figure 1. Furthermore, this approach is non-intrusive as the structure of the stochasticity in the rain is never used, only an external stochastic rain simulator is needed.

We present the robust calibration of the PESHMELBA [2] distributed, process based, hydrological pesticide transfer model used for the simulation of pesticide fate on small agricultural catchments. The case study is a small virtual catchment in the Beaujolais region. Here, we focus on the calibration of two parameters, the soil content at saturation in deep soil layers, and the Van Genuchten infiltration parameter. Our results confirm the interest of the robust approach. Indeed, we show that the estimated parameters satisfy a larger set of rain realizations than when compared with the classic calibration of PESHMELBA for the same rain errors.

## Short biography (PhD student)

I have completed my master studies in applied mathematics - statistical learning in Politecnico di Milano. My PhD focuses on the calibration of a hydrological model for pesticide transfers, developed to be used as a decision making tool in landscape management, as the latter conditions long-term water quality in rivers surrounded by pesticide-treated agricultural areas. The PhD is co-funded by the research centers INRAE and Inria.

## References

[1] Nora Lüthen, Stefano Marelli, and Bruno Sudret. A spectral surrogate model for stochastic simulators computed from trajectory samples, July 2022. arXiv:2207.05653 [cs, math, stat].

[2] Emilie Rouzies, Claire Lauvernet, Christine Barachet, Thierry Morel, Flora Branger, Isabelle Braud, and Nadia Carluer. From agricultural catchment to management scenarios: A modular tool to assess effects of landscape features on water and pesticide behavior. *Science of The Total Environment*, 671:1144–1160, June 2019. doi: 10.1016/j.scitotenv.2019.03.060.

[3] Victor Trappler, Élise Arnaud, Arthur Vidard, and Laurent Debreu. Robust calibration of numerical models based on relative regret. *Journal of Computational Physics*, 426:109952, February 2021. doi: 10.1016/j.jcp.2020.109952.