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

Katarina Radišić, Claire Lauvernet, Arthur Vidard

► 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

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

Submitted on 24 Jun 2024

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# 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
# 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_2.png" /></figure>
Introduction: Model calibration = minimization of a cost function

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

<figure><img src="image_4.png" /></figure>
Introduction: Model calibration = minimization of a cost function

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

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Introduction: Model calibration = minimization of a cost function

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

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Introduction: Model calibration = minimization of a cost function

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

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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>

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# 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>

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

What criterion for the minimization of
a stochastic cost function ?

<figure><img src="image_19.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>

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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_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_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_28.png" /></figure>

What criterion for the minimization of
a stochastic cost function ?

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

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

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

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

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Methods: Robust minimizers

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

What criterion for the minimization of
a stochastic cost function ?

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

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

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

...

Very expensive simulations

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

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# 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>
# 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>
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>
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_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_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: Moisture profiles

A simplified case study: moisture profile of one plot

<figure><img src="image_44.png" /></figure>

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# 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
# 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>
Results: Problem formulation

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

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

Cost function

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

<figure><img src="image_51.png" /></figure>
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_53.png" /></figure>
Results: Problem formulation

<figure><img src="image_54.png" /></figure>
Results: Problem formulation

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

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

<figure><img src="image_56.png" /></figure>
Results: Problem formulation

<figure><img src="image_57.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_58.png" /></figure>
Results: Problem formulation

<figure><img src="image_59.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_60.png" /></figure>
Results: Minimizers

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

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

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minimizer

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minimizer

<figure><img src="image_67.png" /></figure>
Results: Minimizers

<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>

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

minimizer

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

minimizer

<figure><img src="image_74.png" /></figure>
Results: Minimizers

<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>

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

minimizer

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

minimizer

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

Is the calibration better with robust minimizers ?

<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_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_84.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_85.png" /></figure>

<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_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_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_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_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_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_92.png" /></figure>
Conclusion:

<figure><img src="image_93.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_94.png" /></figure>

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