# Robust calibration of a water and pesticide transfer model at the catchment scale

Katarina Radišić, Claire Lauvernet, Arthur Vidard

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To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Robust calibration of a water and pesticide transfer model at the catchment scale. mexico 2024 - Rencontres annuelles du réseau Mexico, INRAE Lyon-Villeurbanne, Dec 2024, Villeurbanne, France. hal-04828019

HAL Id: hal-04828019

https://hal.inrae.fr/hal-04828019v1

Submitted on 9 Dec 2024

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Distributed under a Creative Commons Attribution 4.0 International License
# Robust calibration of a water and pesticide transfer model at the catchment scale

Katarina Radišić¹²

Claire Lauvernet¹, Arthur Vidard²

¹INRAE, RiverLy, Lyon-Villeurbanne

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

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RÉPUBLIQUE
FRANÇAISE

Liberté
Égalité
Fraternité

INRAE

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Context: Pesticide transfer dynamics

Landscape features speed up or slow down pesticide transfer from the plots to the river.

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

→ The configuration of the catchment influences the water quality.

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

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Context: Pesticide transfer dynamics

Landscape features speed up or slow down pesticide transfer from the plots to the river.

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

→ The configuration of the catchment influences the water quality.

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

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Context: PESHMELBA model

<figure><img src="image_9.png" /><figcaption>Figure: PESHMELBA model, [7].</figcaption></figure>

* not all **parameters** can be measured

* → **calibration** of parameters through field observations

* **calibration** sensitive to **forcing uncertainties**

* 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_10.png" /></figure>

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# Table of contents

## Introduction

- Classic calibration

- Robust calibration

## Methodology

- Polynomial chaos expansion (PCE)

- Stochastic emulator $\hat{f}_s$

- Fitting

- Generate new trajectories

- Validation vs $f_s$

- Robust calibration with $\hat{f}_s$

## Case study

- Moisture profiles

## Results

- Fit and validate $\hat{f}_s$

- Robust calibration with different thresholds c

- Compare robust calibration with classic approach

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Introduction: Classic calibration

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Introduction: Classic calibration

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Introduction: Classic calibration

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Introduction: Classic calibration

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Introduction: Classic calibration

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Introduction: Classic calibration

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

**Calibration robuste** : satisfait des conditions d'optimalité sous un ensemble de forçages

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

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Introduction: Classic calibration

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

**Calibration robuste** : satisfait des conditions d'optimalité sous un ensemble de forçages

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Introduction: Classic calibration

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**Robust calibration:** satisfies optimality conditions under a set of forcings

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# Introduction: Robust calibration

1. Find $\mathbf{x}_{\text{robust}}^*$ minimizing a Qol: the mean $\mathbb{E}$, the variance $\text{Var}$, or Pareto of the two [2, 6].

2. other definitions of robustness, excursion sets, relative regret [1, 8].

→ a thing in common : computationally expensive

* metamodel on the Qol of interest, or

* metamodel on the entire $\mathcal{D}_X \times \Omega$,

→ however the parametrization of $\Omega$ is highly model specific [5].

## Our approach:

* take a non-intrusive approach in the space $\Omega$ [9, 4]

* estimate a stochastic emulator $\hat{f}_s(\mathbf{x}, \omega) \approx f_s(\mathbf{x}, \omega)$ over the whole space $\mathcal{D}_X \times \Omega$ [3]

→ use $\hat{f}_s(\mathbf{x}, \omega)$ to estimate different $\mathbf{x}_{\text{robust}}^*$

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# Table of contents

## Introduction

- Classic calibration

- Robust calibration

## Methodology

- Polynomial chaos expansion (PCE)

- Stochastic emulator $\hat{f}_s$

- Fitting

- Generate new trajectories

- Validation vs $f_s$

- Robust calibration with $\hat{f}_s$

## Case study

- Moisture profiles

## Results

- Fit and validate $\hat{f}_s$

- Robust calibration with different thresholds c

- Compare robust calibration with classic approach

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

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Methodology: Polynomial chaos expansion (PCE)

$f_s(x, \omega)$

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$f_s(x, \omega_1) \approx \dots$

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$f_s(x, \omega)$

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$$f_s(x, \omega_1) \approx f_{PCE}^{(1)}(x)$$

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$$f_s(\mathbf{x}, \omega_1) \approx f_{PCE}^{(1)}(\mathbf{x}) = \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(\mathbf{x})$$

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Methodology: Polynomial chaos expansion (PCE)

$f_s(x, \omega)$

<figure><img src="image_37.png" /></figure>

$$
\begin{align*}
f_s(\mathbf{x}, \omega_1) &\approx f_{PCE}^{(1)}(\mathbf{x}) = \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(\mathbf{x}) \\
&= c_{\alpha_1} \psi_{\alpha_1}(\mathbf{x}) + c_{\alpha_2} \psi_{\alpha_2}(\mathbf{x}) + c_{\alpha_3} \psi_{\alpha_3}(\mathbf{x})
\end{align*}
$$

<figure><img src="image_38.png" /></figure>
Methodology: Polynomial chaos expansion (PCE)

$f_s(x, \omega)$

<figure><img src="image_39.png" /></figure>

$$
\begin{align*}
f_s(\mathbf{x}, \omega_1) &\approx f_{PCE}^{(1)}(\mathbf{x}) = \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(\mathbf{x}) \\
&= c_{\alpha_1} \psi_{\alpha_1}(\mathbf{x}) + c_{\alpha_2} \psi_{\alpha_2}(\mathbf{x}) + c_{\alpha_3} \psi_{\alpha_3}(\mathbf{x})
\end{align*}
$$

<figure><img src="image_40.png" /></figure>
Methodology: Polynomial chaos expansion (PCE)

$f_s(x, \omega)$

<figure><img src="image_41.png" /></figure>

$$ 
\begin{align*}
f_s(x, \omega_1) &\approx f_{PCE}^{(1)}(x) = \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(x) \\
&= c_{\alpha_1} \psi_{\alpha_1}(x) + c_{\alpha_2} \psi_{\alpha_2}(x) + c_{\alpha_3} \psi_{\alpha_3}(x)
\end{align*}
$$

$$ 
\begin{align*}
f_s(x, \omega_1) &\approx f_{PCE}^{(1)}(x) = \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(x) \\
&= c_{\alpha_1} \psi_{\alpha_1}(x) + c_{\alpha_2} \psi_{\alpha_2}(x) + c_{\alpha_3} \psi_{\alpha_3}(x)
\end{align*}
$$

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Methodology: Polynomial chaos expansion (PCE)

<figure><img src="image_43.png" /></figure>

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Methodology: Polynomial chaos expansion (PCE)

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Methodology: Stochastic emulator ſ: Fitting

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Methodology: Stochastic emulator $\hat{f}_s$: Fitting

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Methodology: Stochastic emulator ſ<sub>s</sub>: Fitting

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Methodology: Stochastic emulator ſ<sub>s</sub>: Fitting

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Methodology: Stochastic emulator ſ<sub>s</sub>: Fitting

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(c) inference of distributions
on the principal component scores

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Methodology: Stochastic emulator $\hat{f}_s$: Generate new trajectories

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Methodology: Stochastic emulator $\hat{f}_s$: Generate new trajectories

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Methodology: Stochastic emulator $\hat{f}_s$: Generate new trajectories

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(c) inference of distributions
on the principal component scores

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

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# Methodology: Stochastic emulator $\hat{f}_s$: Validation vs $f_s$
## Averaged normalized Wasserstein distance

Cost function: metamodel and test

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Histograms
in one test point

Katarina Radišić

$$d_{\text{WS}}(Y_1, Y_2) = \sqrt{\int_0^1 (Q_1(u) - Q_2(u))^2 du}$$

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Cumulative

Rencontres dans le

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Quantile

5-6 déc. 2024, Villeurbanne

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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Methodology: Robust calibration with $\hat{f}_s$

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INRAO
# Table of contents

## Introduction

- Classic calibration

- Robust calibration

## Methodology

- Polynomial chaos expansion (PCE)

- Stochastic emulator $\hat{f}_s$

- Fitting

- Generate new trajectories

- Validation vs $f_s$

- Robust calibration with $\hat{f}_s$

## Case study

- Moisture profiles

## Results

- Fit and validate $\hat{f}_s$

- Robust calibration with different thresholds c

- Compare robust calibration with classic approach

<figure><img src="image_88.png" /></figure>
# Case study: Moisture profiles

## (c) PESHMELBA configuration

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

* parameters x (prior GSA), 5 params.

* forcing uncertainty Ω, rain error

* observation $y_{obs}$, moisture profile

* cost function f
# Case study: Moisture profiles

## (c) PESHMELBA configuration

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

* parameters x (prior GSA), 5 params.

* forcing uncertainty Ω, rain error

* observation $y_{obs}$, moisture profile

* cost function f
# Case study: Moisture profiles

## (c) PESHMELBA configuration

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

## (b) Parameters to calibrate

<table><thead><tr><td>Name</td><td>Definition</td></tr></thead><tbody><tr><td>θ<sub>k</sub><sub>surf.</sub></td><td>water content at saturation (surface)</td></tr><tr><td>θ<sub>k</sub><sub>inter.</sub></td><td>water content at saturation (intermediary)</td></tr><tr><td>θ<sub>k</sub><sub>deep</sub></td><td>water content at saturation (deep)</td></tr><tr><td>θ<sub>t</sub><sub>deep</sub></td><td>residual water content (deep)</td></tr><tr><td>rm<sub>n</sub><sub>deep</sub></td><td>Van Genuchten retention curve parameter (deep)</td></tr></tbody></table>

* parameters x (prior GSA), 5 params.

* forcing uncertainty Ω, rain error

* observation $y_{obs}$, moisture profile

* cost function $f$

INRAO
# Case study: Moisture profiles

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

(d) Observation: moisture profile of plot 4

* parameters x (prior GSA), 5 params.

* forcing uncertainty Ω, rain error

* observation $y_{obs}$, moisture profile

* cost function $f$

INRAO
Case study: Moisture profiles

(a) Forcing input

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

(b) Parameters to calibrate

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

(c) PESHMELBA configuration

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

(d) Observation: moisture profile of plot 4

* parameters x (prior GSA), 5 params.

* forcing uncertainty Ω, rain error

* observation $y_{obs}$, moisture profile

* cost function $f$

INRAO
# Table of contents

## Introduction

- Classic calibration

- Robust calibration

## Methodology

- Polynomial chaos expansion (PCE)

- Stochastic emulator $\hat{f}_s$

- Fitting

- Generate new trajectories

- Validation vs $f_s$

- Robust calibration with $\hat{f}_s$

## Case study

- Moisture profiles

## Results

- Fit and validate $\hat{f}_s$

- Robust calibration with different

- thresholds c

- Compare robust calibration with classic

- approach

<figure><img src="image_95.png" /></figure>
$\omega_1, \omega_2, \dots, \omega_n$

Model

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Model

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Observation

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$\mathcal{Y}_{\text{obs}}$

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Results: Overview

$\omega_1$

Model

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Cost function

$\rightarrow f_s^{(1)}(x)$

$\rightarrow f_s^{(2)}(x)$

$\rightarrow f_s^{(n)}(x)$

Observation

<figure><img src="image_103.png" /></figure>

$y_{obs}$
Results: Overview

$\omega_1$

Model

<figure><img src="image_104.png" /></figure>

Cost function

Metamodel

$\rightarrow f_s^{(1)}(x) \rightarrow f_{s\_PCE}^{(1)}(x)$

$\rightarrow f_s^{(2)}(x) \rightarrow f_{s\_PCE}^{(2)}(x)$

$\rightarrow f_s^{(n)}(x) \rightarrow f_{s\_PCE}^{(n)}(x)$

Observation

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$y_{obs}$
Results: Overview

$\omega_1$

Model

<figure><img src="image_106.png" /></figure>

Cost function

$\rightarrow f_s^{(1)}(x)$

$\rightarrow f_{s\_PCE}^{(1)}(x)$

$\rightarrow \hat{x}_{classic}^{*(1)}$

Minimization

$\rightarrow f_s^{(2)}(x)$

$\rightarrow f_{s\_PCE}^{(2)}(x)$

$\rightarrow \hat{x}_{classic}^{*(2)}$

$\rightarrow \dots \rightarrow f_s^{(n)}(x)$

$\rightarrow \dots \rightarrow f_{s\_PCE}^{(n)}(x)$

$\rightarrow \dots \rightarrow \hat{x}_{classic}^{*(n)}$

Observation

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$y_{obs}$
Results: Overview

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Results: Overview

<figure><img src="image_109.png" /></figure>
Results: Overview

$\omega_1$ $\xrightarrow{\omega_2}$ $\vdots$ $\omega_n$

Observation

$y_{\text{obs}}$

Model

<figure><img src="image_110.png" /></figure>

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Cost function

$$ \rightarrow f_s^{(1)}(x) \rightarrow $$

$$ \rightarrow f_s^{(2)}(x) \rightarrow $$

$$ \rightarrow \dots \rightarrow f_s^{(n)}(x) \rightarrow $$

Metamodel

$$ \rightarrow f_{s\_PCE}^{(1)}(x) \rightarrow $$

$$ \rightarrow f_{s\_PCE}^{(2)}(x) \rightarrow $$

$$ \rightarrow \dots \rightarrow f_{s\_PCE}^{(n)}(x) \rightarrow $$

Minimization

$$ \rightarrow \hat{x}_{\text{classic}}^{*(1)} \rightarrow $$

$$ \rightarrow \hat{x}_{\text{classic}}^{*(2)} \rightarrow $$

$$ \rightarrow \dots \rightarrow \hat{x}_{\text{classic}}^{*(n)} \rightarrow $$

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

Stochastic
emulator

$\hat{f}_s(x)$

<figure><img src="image_115.png" /></figure>

Robust
minimization

$\hat{x}_{\text{robust}}^{*}_{f_s}$

<figure><img src="image_116.png" /></figure>
Results: Overview

<figure><img src="image_117.png" /></figure>

1. Fit and validate $\hat{f}_s$.

2. Get robust calibration for different thresholds c.

3. Compare robust calibration to classic approach.
Results: Overview

<figure><img src="image_118.png" /></figure>

1. Fit and validate $\hat{f}_s$.

2. Get robust calibration for different thresholds c.

3. Compare robust calibration to classic approach.
# Table of contents

## Introduction

- Classic calibration

- Robust calibration

## Methodology

- Polynomial chaos expansion (PCE)

- Stochastic emulator $\hat{f}_s$

- Fitting

- Generate new trajectories

- Validation vs $f_s$

- Robust calibration with $\hat{f}_s$

## Case study

- Moisture profiles

## Results

- Fit and validate $\hat{f}_s$

- Robust calibration with different

- thresholds c

- Compare robust calibration with classic

- approach

<figure><img src="image_119.png" /></figure>
Results: Fit and validate ſ

<figure><img src="image_120.png" /></figure>

<figure><img src="image_121.png" /></figure>

<figure><img src="image_122.png" /></figure>

latent space original
(test set) vs
metamodel
Results: Fit and validate ſ̂₅

<figure><img src="image_123.png" /></figure>

<figure><img src="image_124.png" /></figure>

latent space original
(test set) vs
metamodel
Results: Fit and validate ſ

<figure><img src="image_125.png" /></figure>

<figure><img src="image_126.png" /></figure>

latent space original
(test set) vs
metamodel
Results: Fit and validate ſ̂

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /></figure>

coefficients space
original (test set) vs
metamodel
Results: Fit and validate ſ

<figure><img src="image_129.png" /></figure>

<figure><img src="image_130.png" /></figure>

coefficients space
original (test set) vs
metamodel
Results: Fit and validate ſ̂

<figure><img src="image_131.png" /></figure>

<figure><img src="image_132.png" /></figure>

coefficients space
original (test set) vs
metamodel
Results: Fit and validate $\hat{f}_S$

<figure><img src="image_133.png" /></figure>

$\hat{f}_S(x_{test,i}^{(i)})$

<figure><img src="image_134.png" /></figure>

physical space (cost function values)
original (test set) vs metamodel
Results: Fit and validate ſ̂ₕ

<figure><img src="image_135.png" /></figure>

<figure><img src="image_136.png" /></figure>

physical space (cost
function values)
original (test set) vs
metamodel
# Table of contents

## Introduction

- Classic calibration

- Robust calibration

## Methodology

- Polynomial chaos expansion (PCE)

- Stochastic emulator $\hat{f}_s$

- Fitting

- Generate new trajectories

- Validation vs $f_s$

- Robust calibration with $\hat{f}_s$

## Case study

- Moisture profiles

## Results

- Fit and validate $\hat{f}_s$

- Robust calibration with different thresholds c

- Compare robust calibration with classic approach

<figure><img src="image_137.png" /></figure>
Results: Robust calibration with different thresholds c

<figure><img src="image_138.png" /></figure>

1. Fit and validate $\hat{f}_s$.

2. Get robust calibration for different thresholds c.

3. Compare robust calibration to classic approach.

<figure><img src="image_139.png" /></figure>
Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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Results: Robust calibration with different thresholds c

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# Table of contents

## Introduction

- Classic calibration

- Robust calibration

## Methodology

- Polynomial chaos expansion (PCE)

- Stochastic emulator $\hat{f}_s$

- Fitting

- Generate new trajectories

- Validation vs $f_s$

- Robust calibration with $\hat{f}_s$

## Case study

- Moisture profiles

## Results

- Fit and validate $\hat{f}_s$

- Robust calibration with different

- thresholds c

- Compare robust calibration with classic approach

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Results: Compare robust calibration with classic approach

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1. Fit and validate $\hat{f}_s$.

2. Get robust calibration for different thresholds c.

3. Compare robust calibration to classic approach.

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Results: Compare robust calibration with classic approach

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$$| x_{prior}^* |$$

Classic approach

$$\hat{x}_{classic}^{*(r)}$$

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Results: Compare robust calibration with classic approach

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Results: Compare robust calibration with classic approach

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$x_{\text{robust}}^*(I_{s<0.01})$

$\hat{x}_{\text{classic}}^{*(r)}$

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Results: Compare robust calibration with classic approach

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Katarina Radišić

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Classic approach

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$x^*_{\text{robust}}$

$x^*_{\text{robust}}$

$x^*_{\text{prior}}$

Rencontres Mexico 5-6 déc. 2024, Villeurbanne
Results: Compare robust calibration with classic approach

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Soil moisture [cm3/cm3]

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Cost function $f_s(x^*, \cdot)$

Calibrated values

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$x_{prior}^*$

$x_{classic}^*$

$x_{robust}_{f_s < 0.02}^*$

$x_{robust}_{f_s < 0.01}^*$

<table><thead><tr><th></th><th>f<sub>s</sub>(x*,·)</th><th>1&#773;{f<sub>s</sub>(x*,·)>0.01}</th><th>1&#773;{f<sub>s</sub>(x*,·)>0.02}</th><th>max(f<sub>s</sub>(x*,·))</th><th>Var(f<sub>s</sub>(x*,·))</th></tr></thead><tbody><tr><td>x<sup>*</sup><sub>prior</sub></td><td>0.0209</td><td>0.99</td><td>0.39</td><td>0.057</td><td>4.57e-05</td></tr></tbody></table>
Results: Compare robust calibration with classic approach

<figure><img src="image_210.png" /></figure>

Soil moisture [cm3/cm3]

<figure><img src="image_211.png" /></figure>

Cost function $f_s(x^*, \cdot)$

Calibrated values

<figure><img src="image_212.png" /></figure>

<table><thead><tr><th></th><th>f<sub>s</sub>(x<sup>*</sup>, ·)</th><th>1̅<sub>f<sub>s</sub>(x<sup>*</sup>, ·)>0.01</sub></th><th>1̅<sub>f<sub>s</sub>(x<sup>*</sup>, ·)>0.02</sub></th><th>max(f<sub>s</sub>(x<sup>*</sup>, ·))</th><th>Var(f<sub>s</sub>(x<sup>*</sup>, ·))</th></tr></thead><tbody><tr><th>x<sup>*</sup><sub>prior</sub></th><td>0.0209</td><td>0.99</td><td>0.39</td><td>0.057</td><td>4.57e-05</td></tr><tr><th>x̂<sup>*</sup><sub>classic</sub></th><td>0.0173</td><td>0.99</td><td>0.25</td><td>0.042</td><td><strong>2.91e-05</strong></td></tr></tbody></table>
Results: Compare robust calibration with classic approach

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<table><thead><tr><th></th><th>f&#773;<sub>s</sub>(x*, ·)</th><th>1&#773;{f<sub>s</sub>(x*, ·)>0.01}</th><th>1&#773;{f<sub>s</sub>(x*, ·)>0.02}</th><th>max(f<sub>s</sub>(x*, ·))</th><th>Var(f<sub>s</sub>(x*, ·))</th></tr></thead><tbody><tr><td>x*<sub>prior</sub></td><td>0.0209</td><td>0.99</td><td>0.39</td><td>0.057</td><td>4.57e-05</td></tr><tr><td>x&#771;*<sub>classic</sub></td><td>0.0173</td><td>0.99</td><td>0.25</td><td>0.042</td><td><strong>2.91e-05</strong></td></tr><tr><td>x&#771;*<sub>robust<sub>f<sub>s</sub><sub>&lt;0.02</sub></sub></sub></td><td>0.0105</td><td>0.41</td><td><strong>0.09</strong></td><td><strong>0.038</strong></td><td>3.36e-05</td></tr></tbody></table>
Results: Compare robust calibration with classic approach

<figure><img src="image_216.png" /></figure>

Soil moisture [cm3/cm3]

<figure><img src="image_217.png" /></figure>

Cost function $f_s(x^*, \cdot)$

Calibrated values

<figure><img src="image_218.png" /></figure>

<table><thead><tr><th></th><th>f&#773;<sub>s</sub>(x*, ·)</th><th>1&#773;{f<sub>s</sub>(x*, ·)>0.01}</th><th>1&#773;{f<sub>s</sub>(x*, ·)>0.02}</th><th>max(f<sub>s</sub>(x*, ·))</th><th>Var(f<sub>s</sub>(x*, ·))</th></tr></thead><tbody><tr><td>x<sup>*</sup><sub>prior</sub></td><td>0.0209</td><td>0.99</td><td>0.39</td><td>0.057</td><td>4.57e-05</td></tr><tr><td>x<sup>*</sup><sub>classic</sub></td><td>0.0173</td><td>0.99</td><td>0.25</td><td>0.042</td><td><strong>2.91e-05</strong></td></tr><tr><td>x<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> &lt; 0.02</sub></td><td>0.0105</td><td>0.41</td><td><strong>0.09</strong></td><td><strong>0.038</strong></td><td>3.36e-05</td></tr><tr><td>x<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> &lt; 0.01</sub></td><td><strong>0.0091</strong></td><td><strong>0.32</strong></td><td>0.13</td><td>0.043</td><td>6.64e-05</td></tr></tbody></table>
Results: Pesticide concentration at outlet

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• forcing uncertainty Ω, pesticide application date

• observation yobs, pesticide concentration at outlet
Results: Pesticide concentration at outlet

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Rain

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$x_{\text{true}}$

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$\hat{x}_{\text{classic}}^*$

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$\hat{x}_{\text{robust}}^*$

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$\mu_{\text{prior}}^*$

Calibrated
parameter values

* forcing uncertainty Ω, pesticide application date

* observation $y_{\text{obs}}$, pesticide concentration at outlet
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

<figure><img src="image_226.png" /></figure>
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

* Improvement in robustness in two PESHMELBA case studies.

<figure><img src="image_227.png" /></figure>
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

* Improvement in robustness in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

<figure><img src="image_228.png" /></figure>
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

• Improvement in robustness in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of **two robustness criteria** for humidity profiles

<figure><img src="image_229.png" /></figure>

INRAO
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

* Improvement in robustness in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of two robustness criteria for humidity profiles

Limitations

<figure><img src="image_230.png" /></figure>

INRAO
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

* Improvement in robustness in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of two robustness criteria for humidity profiles
Limitations

* For real (complex) models, the sample size required to fit $\hat{f}_s$ can be very large.

<figure><img src="image_231.png" /></figure>
Conclusion:

* **Generic** method, non-intrusive in Ω (only needs a representative sample)

* **Improvement in robustness** in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of **two robustness criteria** for humidity profiles
Limitations

* For real (complex) models, the sample size required to fit $\hat{f}_s$ can be very large.

* Especially in the presence of **interactions between $\omega$ and X** in the model/cost function.

<figure><img src="image_232.png" /></figure>
Conclusion:

* **Generic** method, non-intrusive in Ω (only needs a representative sample)

* **Improvement in robustness** in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of **two robustness criteria** for humidity profiles

Limitations

* For real (complex) models, the sample size required to fit $\hat{f}_s$ can be very large.

* Especially in the presence of **interactions between $\omega$ and X** in the model/cost function.

Possibilities for the future...

<figure><img src="image_233.png" /></figure>
Conclusion:

* **Generic** method, non-intrusive in Ω (only needs a representative sample)

* **Improvement in robustness** in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of **two robustness criteria** for humidity profiles

Limitations

* For real (complex) models, the sample size required to fit $\hat{f}_s$ can be very large.

* Especially in the presence of **interactions between $\omega$ and X** in the model/cost function.

Possibilities for the future...

* Development and comparison with adaptive methods [2].

<figure><img src="image_234.png" /></figure>

INRAO
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

* Improvement in robustness in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of two robustness criteria for humidity profiles

Limitations

* For real (complex) models, the sample size required to fit $\hat{f}_s$ can be very large.

* Especially in the presence of interactions between ω and X in the model/cost function.

Possibilities for the future...

* Development and comparison with adaptive methods [2].

* Study and definition of robustness criteria for pesticide concentrations.

<figure><img src="image_235.png" /></figure>

INRAO
Conclusion:

* Generic method, non-intrusive in Ω (only needs a representative sample)

* Improvement in robustness in two PESHMELBA case studies.

* Possibility to estimate several $x_{robust}^*$ from $\hat{f}_s$, without additional cost

* Comparison of two robustness criteria for humidity profiles

Limitations

* For real (complex) models, the sample size required to fit $\hat{f}_s$ can be very large.

* Especially in the presence of interactions between ω and X in the model/cost function.

Possibilities for the future...

* Development and comparison with adaptive methods [2].

* Study and definition of robustness criteria for pesticide concentrations.

* Another definition of Ω: natural variability, interannual variability, or uncertainty in future projections.

<figure><img src="image_236.png" /></figure>
Bibliography I

[1] Clément Duhamel et al. "A SUR version of the Bichon criterion for excursion set estimation". en. In: *Statistics and Computing* 33.2 (Apr. 2023), p. 41. DOI: 10.1007/s11222-023-10208-4.

[2] Mohamed Reda El Amri et al. "Data-driven stochastic inversion via functional quantization". en. In: *Statistics and Computing* 30.3 (May 2020), pp. 525-541. DOI: 10.1007/s11222-019-09888-8.

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

[4] Nora Lüthen, Stefano Marelli, and Bruno Sudret. "Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark". en. In: *SIAM/ASA Journal on Uncertainty Quantification* 9.2 (Jan. 2021), pp. 593-649. DOI: 10.1137/20M1315774.

<figure><img src="image_237.png" /></figure>

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Bibliography II

[5] Victor Picheny, Ronan Trépos, and Pierre Casadebaig. "Optimization of black-box models with uncertain climatic inputs—Application to sunflower ideotype design". en. In: *PLOS ONE* 12.5 (May 2017). Ed. by Yongtang Shi, e0176815. DOI: 10.1371/journal.pone.0176815.

[6] Mélina Ribaud et al. "Robust optimization: A kriging-based multi-objective optimization approach". en. In: *Reliability Engineering & System Safety* 200 (Aug. 2020), p. 106913. DOI: 10.1016/j.ress.2020.106913.

[7] Emilie Rouzies et al. "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 (June 2019), pp. 1144-1160. DOI: 10.1016/j.scitotenv.2019.03.060.

[8] Victor Trappler et al. "Robust calibration of numerical models based on relative regret". en. In: *Journal of Computational Physics* 426 (Feb. 2021), p. 109952. DOI: 10.1016/j.jcp.2020.109952.

<figure><img src="image_238.png" /></figure>
Bibliography III

[9] Dongbin Xiu and George Em Karniadakis. "The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations". en. In: *SIAM Journal on Scientific Computing* 24.2 (Jan. 2002), pp. 619-644. DOI: 10.1137/S1064827501387826.

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Average norm. wass. distance w.r.t. number of training trajectories and
size of the latent space I

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