# Calibration robuste d'un modèle hydrologique de transfert de pesticides par métamodélisation

Katarina Radišić, Claire Lauvernet, Arthur Vidard

<figure><img src="image_1.png" /></figure>

## To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Calibration robuste d'un modèle hydrologique de transfert de pesticides par métamodélisation. JMSC 2024 - 5ème édition des Journées de Modélisation des Surfaces Continentales, Jun 2024, Strasbourg, France. pp.1-49. hal-04622409

HAL Id: hal-04622409

https://hal.inrae.fr/hal-04622409

Submitted on 24 Jun 2024

HAL is a multi-disciplinary open access
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entific research documents, whether they are pub-
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<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Calibration robuste d'un modèle hydrologique de transfert de pesticides par métamodélisation

Katarina Radišić, Claire Lauvernet, Arthur Vidard

<figure><img src="image_1.png" /></figure>

## To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Calibration robuste d'un modèle hydrologique de transfert de pesticides par métamodélisation. JMSC 2024 - 5ème édition des Journées de Modélisation des Surfaces Continentales, Jun 2024, Strasbourg, France. pp.1-49. hal-04622409

HAL Id: hal-04622409

https://hal.inrae.fr/hal-04622409v1

Submitted on 24 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>
# Calibration robuste d'un modèle hydrologique de transfert de pesticides par métamodélisation

Katarina Radišić¹²³

Claire Lauvernet¹, Arthur Vidard²

¹INRAE, RiverLy, Lyon-Villeurbanne

²Univ. Grenoble-Alpes, Inria, CNRS, Grenoble-INP, LJK

³ED MSTII
# Calibration robuste d'un modèle hydrologique de transfert de pesticides par métamodélisation

Katarina Radišić¹²³

Claire Lauvernet¹, Arthur Vidard²

¹INRAE, RiverLy, Lyon-Villeurbanne

²Univ. Grenoble-Alpes, Inria, CNRS, Grenoble-INP, LJK

³ED MSTII

<figure><img src="image_3.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Légale
Égalité
Fondamentale
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_3.png" /></figure>
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_4.png" /></figure>

INRAO
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>
Introduction: Model calibration = minimization of a cost function

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

<figure><img src="image_12.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_15.png" /></figure>

How to minimize a stochastic function? → **robust calibration**

<figure><img src="image_16.png" /></figure>

INRAO
Introduction: Model calibration = minimization of a cost function

<figure><img src="image_16.png" /></figure>

How to minimize a stochastic function? → **robust calibration**

<figure><img src="image_17.png" /></figure>

INRAO
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_17.png" /></figure>
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_18.png" /></figure>

INRAO
Methods: Robust minimizers

<figure><img src="image_18.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

<figure><img src="image_19.png" /></figure>
<figure><img src="image_19.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

<figure><img src="image_20.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_20.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

<figure><img src="image_21.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_21.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

<figure><img src="image_22.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_22.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

<figure><img src="image_23.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_23.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

<figure><img src="image_24.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_24.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

<figure><img src="image_25.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_25.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

<figure><img src="image_26.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_26.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

$$
x_{\text{mean}}^{*} = \underset{x}{\operatorname{argmin}} \mathbb{E}_{\omega}[f(x, \omega)],
$$

<figure><img src="image_27.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_27.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

$$
x_{\text{mean}}^{*} = \underset{x}{\operatorname{argmin}} \mathbb{E}_{\omega}[f(x, \omega)],
$$

<figure><img src="image_28.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_28.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{x} [\arg\min_{x} f(x, \omega)],
$$

$$
x_{mean}^* = \operatorname*{argmin}_{x} \mathbb{E}_\omega [f(x, \omega)],
$$

$$
x_{\mathbf{Var}}^{*} = \operatorname*{argmin}_{x} \mathcal{Var}_{\omega}[f(x, \omega)],
$$

<figure><img src="image_29.png" /></figure>

INRAO
Methods: Robust minimizers

<figure><img src="image_29.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

$$
x_{mean}^* = \underset{x}{\operatorname{argmin}} \mathbb{E}_\omega [f(x, \omega)],
$$

$$
x_{\mathbb{V}ar}^{*} = \underset{x}{\operatorname{argmin}} \mathbb{V}ar_{\omega}[f(x, \omega)],
$$

<figure><img src="image_30.png" /></figure>

INRAO
Methods: Robust minimizers

<figure><img src="image_30.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

$$
x_{mean}^* = \underset{x}{\operatorname{argmin}} \mathbb{E}_\omega [f(x, \omega)],
$$

$$
x_{\mathbb{V}ar}^{*} = \underset{x}{\operatorname{argmin}} \mathbb{V}ar_{\omega}[f(x, \omega)],
$$

...

<figure><img src="image_31.png" /></figure>
Methods: Robust minimizers

<figure><img src="image_31.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

$$
x_{mean}^* = \underset{x}{\operatorname{argmin}} \mathbb{E}_\omega [f(x, \omega)],
$$

$$
x_{\mathbb{V}ar}^* = \underset{x}{\operatorname{argmin}} \mathbb{V}ar_{\omega}[f(x, \omega)],
$$

...

<figure><img src="image_32.png" /></figure>

INRAO
Methods: Robust minimizers

<figure><img src="image_32.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

$$
x_{mean}^* = \underset{x}{\operatorname{argmin}} \mathbb{E}_\omega [f(x, \omega)],
$$

$$
x_{\mathbb{V}ar}^{*} = \underset{x}{\operatorname{argmin}} \mathbb{V}ar_{\omega}[f(x, \omega)],
$$

...

Very expensive simulations

<figure><img src="image_33.png" /></figure>

INRAO
Methods: Robust minimizers

<figure><img src="image_33.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

$$
\bar{x}^* = \mathbb{E}_{\omega}[\underset{x}{\operatorname{argmin}} f(x, \omega)],
$$

$$
x_{mean}^* = \underset{x}{\operatorname{argmin}} \mathbb{E}_\omega [f(x, \omega)],
$$

$$
x_{\bar{V}ar}^{*} = \underset{x}{\operatorname{argmin}} \mathbb{V}ar_{\omega}[f(x, \omega)],
$$

...

Very expensive simulations

<figure><img src="image_34.png" /></figure>

INRAO
# Methods: Robust minimizers

<figure><img src="image_34.png" /></figure>

What criterion for the minimization of a stochastic cost function ?

$$ 
\begin{align*}
\bar{x}^* &= \mathbb{E}_\omega[\underset{x}{\operatorname{argmin}} f(x, \omega)], \\
x_{\text{mean}}^* &= \underset{x}{\operatorname{argmin}} \mathbb{E}_\omega[f(x, \omega)], \\
x_{\text{Var}}^* &= \underset{x}{\operatorname{argmin}} \text{Var}_\omega[f(x, \omega)], \\
\dots
\end{align*}
$$

Very expensive simulations

How to limit number of model evaluations ?

→ *Metamodels* = approximation of the original model (*or of the cost function*) : linear regression, neural networks ...

* **Polynomial Chaos Expansion**, Marelli and Sudret 2014

* can be used for a large number of input parameters (+)

* uses one-shot experimental designs (can be parallelized) (+)

* approximate the cost function globally (-)

* have to be validated (-)

<figure><img src="image_35.png" /></figure>
# Methods: Robust minimizers

<figure><img src="image_35.png" /></figure>

What criterion for the minimization of a stochastic cost function ?

$$ 
\begin{align*}
\bar{x}^* &= \mathbb{E}_\omega[\underset{x}{\operatorname{argmin}} f(x, \omega)], \\
x_{\text{mean}}^* &= \underset{x}{\operatorname{argmin}} \mathbb{E}_\omega[f(x, \omega)], \\
x_{\text{Var}}^* &= \underset{x}{\operatorname{argmin}} \text{Var}_\omega[f(x, \omega)], \\
\dots
\end{align*}
$$

**Very expensive simulations**

How to limit number of model evaluations ?

→ *Metamodels* = approximation of the original model (*or of the cost function*) : linear regression, neural networks ...

* **Polynomial Chaos Expansion**, Marelli and Sudret 2014

* can be used for a large number of input parameters (+)

* uses one-shot experimental designs (can be parallelized) (+)

* approximate the cost function globally (-)

* have to be validated (-)

<figure><img src="image_36.png" /></figure>
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_36.png" /></figure>
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_37.png" /></figure>
Case study: PESHMELBA

<figure><img src="image_37.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

<figure><img src="image_38.png" /></figure>

INRAO
Case study: PESHMELBA

<figure><img src="image_38.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

<figure><img src="image_39.png" /></figure>
Case study: PESHMELBA

<figure><img src="image_39.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

* process-oriented, physically-based, coupling with landscape features

<figure><img src="image_40.png" /></figure>
Case study: PESHMELBA

<figure><img src="image_40.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

* process-oriented, physically-based, coupling with landscape features

<figure><img src="image_41.png" /></figure>
Case study: PESHMELBA

<figure><img src="image_41.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

* process-oriented, physically-based, coupling with landscape features

* simulates water and pesticide transfers on an agricultural catchment

<figure><img src="image_42.png" /></figure>
Case study: PESHMELBA

<figure><img src="image_42.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

* process-oriented, physically-based, coupling with landscape features

* simulates water and pesticide transfers on an agricultural catchment

<figure><img src="image_43.png" /></figure>
Case study: PESHMELBA

<figure><img src="image_43.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

* process-oriented, physically-based, coupling with landscape features

* simulates water and pesticide transfers on an agricultural catchment

* distributed model, numerous parameters to calibrate
Case study: PESHMELBA

<figure><img src="image_44.png" /><figcaption>Figure: PESHMELBA model, Rouzies et al. 2019</figcaption></figure>

* process-oriented, physically-based, coupling with landscape features

* simulates water and pesticide transfers on an agricultural catchment

* distributed model, numerous parameters to calibrate

<figure><img src="image_45.png" /></figure>
# Case study: Moisture profiles

A simplified case study: moisture profile of one plot

<figure><img src="image_44.png" /></figure>
# Case study: Moisture profiles

A simplified case study: moisture profile of one plot

<figure><img src="image_46.png" /></figure>
# Case study: Moisture profiles

A simplified case study: moisture profile of one plot

## Forcing input

<figure><img src="image_45.png" /></figure>

## Parameters to calibrate

<table><thead><tr><td>Name</td><td>Definition</td></tr></thead><tbody><tr><td>θ<sub>s, surf.</sub></td><td>water content at saturation (surface)</td></tr><tr><td>θ<sub>s, inter.</sub></td><td>water content at saturation (intermediary)</td></tr><tr><td>θ<sub>r, deep</sub></td><td>water content at saturation (deep)</td></tr><tr><td>θ<sub>r, deep</sub></td><td>residual water content (deep)</td></tr><tr><td>mn<sub>deep</sub></td><td>Van Genuchten retention curve parameter (deep)</td></tr></tbody></table>

## PESHMELBA configuration

<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>

Observation = moisture profile of parcel 4
# Case study: Moisture profiles

A simplified case study: moisture profile of one plot

## Forcing input

<figure><img src="image_47.png" /></figure>

## Parameters to calibrate

<table><thead><tr><td>Name</td><td>Definition</td></tr></thead><tbody><tr><td>θ<sub>s, surf.</sub></td><td>water content at saturation (surface)</td></tr><tr><td>θ<sub>s, inter.</sub></td><td>water content at saturation (intermediary)</td></tr><tr><td>θ<sub>r, deep</sub></td><td>water content at saturation (deep)</td></tr><tr><td>θ<sub>r, deep</sub></td><td>residual water content (deep)</td></tr><tr><td>mn<sub>deep</sub></td><td>Van Genuchten retention curve parameter (deep)</td></tr></tbody></table>

## PESHMELBA configuration

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

Observation = moisture profile of parcel 4
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_48.png" /></figure>
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_50.png" /></figure>
# Results: Problem formulation

<figure><img src="image_49.png" /></figure>
# Results: Problem formulation

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

INRAO
Results: Problem formulation

<figure><img src="image_50.png" /></figure>

Cost function

$$ \begin{align*} \omega_1 &\rightarrow f(x, \omega_1) \\ \omega_2 &\rightarrow f(x, \omega_2) \\ \vdots &\rightarrow \vdots \\ \omega_n &\rightarrow f(x, \omega_n) \end{align*} $$

<figure><img src="image_51.png" /></figure>
# Results: Problem formulation

<figure><img src="image_53.png" /></figure>

Cost function

<figure><img src="image_54.png" /></figure>

$$ \omega_1 \rightarrow f(x, \omega_1) $$

<figure><img src="image_55.png" /></figure>

$$ \omega_2 \rightarrow f(x, \omega_2) $$

<figure><img src="image_56.png" /></figure>

$$ \omega_n \rightarrow f(x, \omega_n) $$

<figure><img src="image_57.png" /></figure>

INRAO
Results: Problem formulation

<figure><img src="image_52.png" /></figure>

Cost function

$$ \omega_1 \rightarrow f(x, \omega_1) $$

$$ \omega_2 \rightarrow f(x, \omega_2) $$

$$ \vdots \rightarrow f(x, \omega_n) $$

Metamodel

$$ \omega_1 \rightarrow PCE_{\omega_1}(x) $$

$$ \omega_2 \rightarrow PCE_{\omega_2}(x) $$

$$ \vdots \rightarrow PCE_{\omega_n}(x) $$
Results: Problem formulation

<figure><img src="image_58.png" /></figure>

Cost function

$$ \omega_1 \rightarrow f(x, \omega_1) $$

$$ \omega_2 \rightarrow f(x, \omega_2) $$

$$ \vdots \rightarrow f(x, \omega_n) $$

Metamodel

$$ \omega_1 \rightarrow PCE_{\omega_1}(x) $$

$$ \omega_2 \rightarrow PCE_{\omega_2}(x) $$

$$ \vdots \rightarrow PCE_{\omega_n}(x) $$
# Results: Problem formulation

<figure><img src="image_53.png" /></figure>

$$ \omega_1 \rightarrow f(x, \omega_1) $$

$$ \omega_2 \rightarrow f(x, \omega_2) $$

$$ \omega_n \rightarrow f(x, \omega_n) $$

$$ \omega_1 \rightarrow PCE_{\omega_1}(x) $$

$$ \omega_2 \rightarrow PCE_{\omega_2}(x) $$

$$ \omega_n \rightarrow PCE_{\omega_n}(x) $$

$$ \omega_1 \rightarrow x_{\omega_1}^* $$

$$ \omega_2 \rightarrow x_{\omega_2}^* $$

$$ \omega_n \rightarrow x_{\omega_n}^* $$
# Results: Problem formulation

<figure><img src="image_59.png" /></figure>

$$ \omega_1 \to f(x, \omega_1) $$

$$ \omega_2 \to f(x, \omega_2) $$

$$ \omega_n \to f(x, \omega_n) $$

$$ \omega_1 \to PCE_{\omega_1}(x) $$

$$ \omega_2 \to PCE_{\omega_2}(x) $$

$$ \omega_n \to PCE_{\omega_n}(x) $$

$$ \omega_1^* \to x_1^* $$

$$ \omega_2^* \to x_2^* $$

$$ \omega_n^* \to x_n^* $$
Results: Problem formulation

<figure><img src="image_54.png" /></figure>

<figure><img src="image_55.png" /></figure>
Results: Problem formulation

<figure><img src="image_60.png" /></figure>
Results: Problem formulation

<figure><img src="image_56.png" /></figure>

* the spread of minimizer histogram = the uncertainty propagation from forcings to calibrated values

<figure><img src="image_57.png" /></figure>
Results: Problem formulation

<figure><img src="image_61.png" /></figure>

* the spread of minimizer histogram = the uncertainty propagation from forcings to calibrated values

<figure><img src="image_62.png" /></figure>
# Results: Problem formulation

<figure><img src="image_58.png" /></figure>

* the spread of minimizer histogram = the uncertainty propagation from forcings to calibrated values

* all metamodels show very good R2 values > 0.95 (validation).

<figure><img src="image_59.png" /></figure>
Results: Problem formulation

<figure><img src="image_63.png" /></figure>

* the spread of minimizer histogram = the uncertainty propagation from forcings to calibrated values

* all metamodels show very good R2 values > 0.95 (validation).

<figure><img src="image_64.png" /></figure>
Results: Problem formulation

<figure><img src="image_60.png" /></figure>

* the spread of minimizer histogram = the uncertainty propagation from forcings to calibrated values

* all metamodels showed very good R2 values > 0.95.

Next: Compare classic minimizers with robust minimizers : $x_{\text{mean}}^*$ and $\bar{x}^*$.

<figure><img src="image_61.png" /></figure>
Results: Problem formulation

<figure><img src="image_65.png" /></figure>

* the spread of minimizer histogram = the uncertainty propagation from forcings to calibrated values

* all metamodels showed very good R2 values > 0.95.

Next: Compare classic minimizers with robust minimizers : $x_{\text{mean}}^*$ and $\bar{x}^*$.

<figure><img src="image_66.png" /></figure>
Results: Minimizers

<figure><img src="image_62.png" /></figure>

<figure><img src="image_63.png" /></figure>

<figure><img src="image_64.png" /></figure>

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

minimizer
$x^*(\omega)$

minimizer
$x_{\text{true}}$
Results: Minimizers

<figure><img src="image_67.png" /></figure>

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.png" /></figure>

minimizer
$x^*(\omega)$

minimizer
$x_{\text{true}}$
Results: Minimizers

<figure><img src="image_67.png" /></figure>

cond.min

<figure><img src="image_68.png" /></figure>

cond.min

<figure><img src="image_69.png" /></figure>

cond.min

<figure><img src="image_70.png" /></figure>

cond.min

<figure><img src="image_71.png" /></figure>

cond.min

minimizer

<figure><img src="image_72.png" /></figure>

minimizer

<figure><img src="image_73.png" /></figure>
Results: Minimizers

<figure><img src="image_72.png" /></figure>

cond.min

<figure><img src="image_73.png" /></figure>

cond.min

<figure><img src="image_74.png" /></figure>

cond.min

<figure><img src="image_75.png" /></figure>

cond.min

<figure><img src="image_76.png" /></figure>

cond.min

minimizer

<figure><img src="image_77.png" /></figure>

minimizer

<figure><img src="image_78.png" /></figure>
Results: Minimizers

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>

minimizer

<figure><img src="image_79.png" /></figure>

minimizer

<figure><img src="image_80.png" /></figure>
Results: Minimizers

<figure><img src="image_79.png" /></figure>

<figure><img src="image_80.png" /></figure>

<figure><img src="image_81.png" /></figure>

<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /></figure>

minimizer

<figure><img src="image_84.png" /></figure>

minimizer

<figure><img src="image_85.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

<figure><img src="image_81.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

<figure><img src="image_86.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

→ compare cost function for new realisations of forcing uncertainty.

<figure><img src="image_82.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

→ compare cost function for new realisations of forcing uncertainty.

<figure><img src="image_87.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

→ compare cost function for new realisations of forcing uncertainty.

<figure><img src="image_83.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

→ compare cost function for new realisations of forcing uncertainty.

<figure><img src="image_88.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

→ compare cost function for new realisations of forcing uncertainty.

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /></figure>
# Results: New simulations cost function

Is the calibration better with robust minimizers ?

→ compare cost function for new realisations of forcing uncertainty.

<figure><img src="image_89.png" /></figure>

<figure><img src="image_90.png" /></figure>
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_86.png" /></figure>
# Table of contents

## Introduction

Model calibration = minimization of a cost function

## Methods

Robust minimizers

## Case study

PESHMELBA

Moisture profiles

## Results

Problem formulation

Minimizers

New simulations cost function

## Conclusion

<figure><img src="image_91.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

<figure><img src="image_87.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

<figure><img src="image_92.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

<figure><img src="image_88.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

<figure><img src="image_93.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

What's next :

* complexify case study, passage to catchment scale, study propagation of uncertainty on pesticide application dates to the model calibration

<figure><img src="image_89.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

What's next :

* complexify case study, passage to catchment scale, study propagation of uncertainty on pesticide application dates to the model calibration

<figure><img src="image_94.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

What's next :

* complexify case study, passage to catchment scale, study propagation of uncertainty on pesticide application dates to the model calibration

* compare Polynomial Chaos Expansion metamodelization with other metamodelization methods

<figure><img src="image_90.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

What's next :

* complexify case study, passage to catchment scale, study propagation of uncertainty on pesticide application dates to the model calibration

* compare Polynomial Chaos Expansion metamodelization with other metamodelization methods

<figure><img src="image_95.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

What's next :

* complexify case study, passage to catchment scale, study propagation of uncertainty on pesticide application dates to the model calibration

* compare Polynomial Chaos Expansion metamodelization with other metamodelization methods

* reduce the number of rain simulations used, interpolation in the rains → stochastic metamodel, Lüthen, Marelli, and Sudret 2023

<figure><img src="image_91.png" /></figure>
Conclusion:

1. a method is proposed to calibrate models by taking external factors into account (here, rainfall, but the method is valid for any forcing: discretization, evapotranspiration, given a representative sample).

2. the interest of the method and the chosen robust minimizers is illustrated on a simple case study

What's next :

* complexify case study, passage to catchment scale, study propagation of uncertainty on pesticide application dates to the model calibration

* compare Polynomial Chaos Expansion metamodelization with other metamodelization methods

* reduce the number of rain simulations used, interpolation in the rains → stochastic metamodel, Lüthen, Marelli, and Sudret 2023

<figure><img src="image_96.png" /></figure>
Conclusion:

<figure><img src="image_92.png" /></figure>
Conclusion:

<figure><img src="image_97.png" /></figure>
Bibliography I

Lüthen, Nora, Stefano Marelli, and Bruno Sudret (Mar. 2023). "A spectral surrogate model for stochastic simulators computed from trajectory samples". en. In: *Computer Methods in Applied Mechanics and Engineering* 406, p. 115875. DOI: 10.1016/j.cma.2022.115875.

Marelli, Stefano and Bruno Sudret (July 2014). "UQLab: A Framework for Uncertainty Quantification in Matlab". en. In: *Vulnerability, Uncertainty, and Risk*. University of Liverpool, United Kingdom: American Society of Civil Engineers, pp. 2554-2563. DOI: 10.1061/9780784413609.257.

Rouzies, Emilie et al. (June 2019). "From agricultural catchment to management scenarios: A modular tool to assess effects of landscape features on water and pesticide behavior". en. In: *Science of The Total Environment* 671, pp. 1144-1160. DOI: 10.1016/j.scitotenv.2019.03.060.

<figure><img src="image_93.png" /></figure>

INRAO
Bibliography I

Lüthen, Nora, Stefano Marelli, and Bruno Sudret (Mar. 2023). "A spectral surrogate model for stochastic simulators computed from trajectory samples". en. In: *Computer Methods in Applied Mechanics and Engineering* 406, p. 115875. DOI: 10.1016/j.cma.2022.115875.

Marelli, Stefano and Bruno Sudret (July 2014). "UQLab: A Framework for Uncertainty Quantification in Matlab". en. In: *Vulnerability, Uncertainty, and Risk*. University of Liverpool, United Kingdom: American Society of Civil Engineers, pp. 2554–2563. DOI: 10.1061/9780784413609.257.

Rouzies, Emilie et al. (June 2019). "From agricultural catchment to management scenarios: A modular tool to assess effects of landscape features on water and pesticide behavior". en. In: *Science of The Total Environment* 671, pp. 1144–1160. DOI: 10.1016/j.scitotenv.2019.03.060.

<figure><img src="image_98.png" /></figure>

INRAO