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

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# Sensitivity analysis with external stochastic forcings : application to a water and pesticide transfer model

Katarina Radišić, Claire Lauvernet, Arthur Vidard

► To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Sensitivity analysis with external stochastic forcings: application to a water and pesticide transfer model. MEXICO 2023 - Rencontres annuelles du réseau Mexico, AgroParisTech, Dec 2023, Palaiseau, France. pp.1-36. hal-04383540

HAL Id: hal-04383540

https://hal.inrae.fr/hal-04383540

Submitted on 9 Jan 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
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émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Sensitivity analysis with external stochastic forcings: application to a water and pesticide transfer model

Katarina Radišić, Claire Lauvernet, Arthur Vidard

► To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Sensitivity analysis with external stochastic forcings: application to a water and pesticide transfer model. MEXICO 2023 - Rencontres annuelles du réseau Mexico, AgroParisTech, Dec 2023, Palaiseau, France. pp.1-36. hal-04383540

HAL Id: hal-04383540

https://hal.inrae.fr/hal-04383540v1

Submitted on 9 Jan 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_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Sensitivity analysis with external stochastic forcings : application to a water and pesticide transfer model

Katarina Radišić¹²

Claire Lauvernet¹, Arthur Vidard²

¹INRAE, RiverLy, Lyon-Villeurbanne

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

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

RÉPUBLIQUE
FRANÇAISE

École
Égalité
Femmes

INRAE

Inria
# Sensitivity analysis with external stochastic forcings : application to a water and pesticide transfer model

Katarina Radišić¹²

Claire Lauvernet¹, Arthur Vidard²

¹INRAE, RiverLy, Lyon-Villeurbanne

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

<figure><img src="image_2.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

École
Égalité
Femmes

INRAE

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

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

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

Context: PESHMELBA

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

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

⇒ The configuration of the catchment can influence the water quality.

<figure><img src="image_11.png" /></figure>
<figure><img src="image_3.png" /></figure>

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

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

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

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

Context: PESHMELBA

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

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

© Lucien Bélanger

⇒ The configuration of the catchment can influence the water quality.

<figure><img src="image_10.png" /></figure>
<figure><img src="image_12.png" /></figure>

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

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

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

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

## Context: PESHMELBA

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

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

⇒ The configuration of the catchment can influence the water quality.

<figure><img src="image_19.png" /></figure>
<figure><img src="image_11.png" /></figure>

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

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

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

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

## Context: PESHMELBA

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

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

⇒ The configuration of the catchment can influence the water quality.

<figure><img src="image_17.png" /></figure>
Context: PESHMELBA

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

Figure: PESHMELBAª, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

<footnote>ªEmilie Rouzies 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.</footnote>
Context: PESHMELBA

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

**Figure:** PESHMELBAª, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

<footnote>ªEmilie Rouzies 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.</footnote>
Context: PESHMELBA

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

Figure: PESHMELBAª, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

• Not all model parameters can be measured directly

<footnote>ªEmilie Rouzies 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.</footnote>
Context: PESHMELBA

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

Figure: PESHMELBAª, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

* Not all model parameters can be measured directly

<footnote>ªEmilie Rouzies 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.</footnote>
Context: PESHMELBA

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

Figure: PESHMELBAª, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

* Not all model parameters can be measured directly

* → calibrate these model parameters with terrain observations

<footnote>ªEmilie Rouzies 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.</footnote>
Context: PESHMELBA

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

Figure: PESHMELBAª, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

* Not all model parameters can be measured directly

* → calibrate these model parameters with terrain observations

<footnote>ªEmilie Rouzies 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.</footnote>
Context: PESHMELBA

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

Figure: PESHMELBAª, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

* Not all model parameters can be measured directly

* → calibrate these model parameters with terrain observations

* Impact of external uncertainties on the calibration results

<footnote>ªEmilie Rouzies 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.</footnote>
Context: PESHMELBA

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

Figure: PESHMELBA<sup>a</sup>, a process-based, physical, spatially distributed water and pesticide transfer model, representing pesticide fate in agricultural catchments, highly non-linear.

* Not all model parameters can be measured directly

* → calibrate these model parameters with terrain observations

* Impact of external uncertainties on the calibration results

<footnote><sup>a</sup>Emilie Rouzies 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.</footnote>
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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Context: model calibration

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Before passing to calibration: a sensitivity analysis on the cost function can provide the information on the identifiability of the parameters, (Mai 2023).
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Context: model calibration

<figure><img src="image_102.png" /></figure>

Before passing to calibration: a sensitivity analysis on the cost function can provide the information on the identifiability of the parameters, (Mai 2023).
<figure><img src="image_105.png" /></figure>

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# 1. Introduction: Sensitivity analysis

The variability in $J$ comes from two types of inputs :

<figure><img src="image_111.png" /></figure>

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

<figure><img src="image_107.png" /></figure>

<figure><img src="image_108.png" /></figure>

# 1. Introduction: Sensitivity analysis

The variability in $J$ comes from two types of inputs :

<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>
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<figure><img src="image_118.png" /></figure>

# 1. Introduction: Sensitivity analysis

<figure><img src="image_119.png" /></figure>

The variability in $J$ comes from two types of inputs :

1. parameters ($\theta$) : hydrodynamic soil properties.

<figure><img src="image_120.png" /></figure>
<figure><img src="image_111.png" /></figure>

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

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

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

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

<figure><img src="image_116.png" /></figure>

# 1. Introduction: Sensitivity analysis

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

The variability in $J$ comes from two types of inputs :

1. parameters ($\theta$) : hydrodynamic soil properties.

<figure><img src="image_118.png" /></figure>
<figure><img src="image_121.png" /></figure>

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# 1. Introduction: Sensitivity analysis

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

The variability in $J$ comes from two types of inputs:

1. parameters ($\theta$) : hydrodynamic soil properties.

2. stochastic ($U(\omega)$) : external forcing (rain).

<figure><img src="image_128.png" /></figure>
<figure><img src="image_119.png" /></figure>

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

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

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

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

# 1. Introduction: Sensitivity analysis

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

The variability in $J$ comes from two types of inputs:

1. parameters ($\theta$) : hydrodynamic soil properties.

2. stochastic ($U(\omega)$) : external forcing (rain).

<figure><img src="image_127.png" /></figure>
<figure><img src="image_129.png" /></figure>

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

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# 1. Introduction: Sensitivity analysis

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

The variability in $J$ comes from two types of inputs:

1. parameters ($\theta$) : hydrodynamic soil properties.

2. stochastic ($U(\omega)$) : external forcing (rain)

Only the parameters can be controlled by the modeler.

<figure><img src="image_136.png" /></figure>
<figure><img src="image_128.png" /></figure>

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

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# 1. Introduction: Sensitivity analysis

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

The variability in $J$ comes from two types of inputs:

1. parameters ($\theta$) : hydrodynamic soil properties.

2. stochastic ($U(\omega)$) : external forcing (rain)

Only the parameters can be controlled by the modeler.

<figure><img src="image_135.png" /></figure>
<figure><img src="image_137.png" /></figure>

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

<figure><img src="image_139.png" /></figure>

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<figure><img src="image_142.png" /></figure>

# 1. Introduction: Sensitivity analysis

<figure><img src="image_143.png" /></figure>

The variability in $J$ comes from two types of inputs:

1. parameters ($\theta$) : hydrodynamic soil properties.

2. stochastic ($U(\omega)$) : external forcing (rain)

Only the parameters can be controlled by the modeler.

*What is the sensitivity of the output to the parameters, given the uncertainty about the stochastic inputs? (Dell'Oca 2023)*

<figure><img src="image_144.png" /></figure>
<figure><img src="image_136.png" /></figure>

<figure><img src="image_137.png" /></figure>

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

<figure><img src="image_139.png" /></figure>

<figure><img src="image_140.png" /></figure>

<figure><img src="image_141.png" /></figure>

# 1. Introduction: Sensitivity analysis

<figure><img src="image_142.png" /></figure>

The variability in $J$ comes from two types of inputs:

1. parameters ($\theta$) : hydrodynamic soil properties.

2. stochastic ($U(\omega)$) : external forcing (rain)

Only the parameters can be controlled by the modeler.

*What is the sensitivity of the output to the parameters, given the uncertainty about the stochastic inputs? (Dell'Oca 2023)*

<figure><img src="image_143.png" /></figure>
## 2. Methodology: Sobol' indices as random variables

<figure><img src="image_145.png" /></figure>

$$J : \mathcal{D} \times \Omega \to \mathbb{R},$$

$$ (\theta, \omega) \mapsto J(\theta, U(\omega)), $$

<figure><img src="image_146.png" /></figure>
## 2. Methodology: Sobol' indices as random variables

<figure><img src="image_144.png" /></figure>

$$J : \mathcal{D} \times \Omega \to \mathbb{R},$$

$$(\theta, \omega) \mapsto J(\theta, U(\omega)),$$

<figure><img src="image_145.png" /></figure>
<figure><img src="image_147.png" /></figure>

<figure><img src="image_148.png" /></figure>

<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>

<figure><img src="image_151.png" /></figure>

<figure><img src="image_152.png" /></figure>

## 2. Methodology: Sobol' indices as random variables

<figure><img src="image_153.png" /></figure>

$$J: \mathcal{D} \times \Omega \to \mathbb{R},$$

$$ (\theta, \omega) \mapsto J(\theta, U(\omega)), $$

$$ S_i(\omega) = \frac{\mathrm{Var}_{\theta_i} [\mathbb{E}[Y | \theta_i, \omega]]}{\mathrm{Var}[Y | \omega]} $$

Sobol' indices as random variables,ª.

Estimation of Sobol' indices with polynomial chaos expansion,ª:

$$ J(\theta) = \sum_{\alpha \in \mathbb{N}^K} c_{\alpha} \psi_{\alpha}(\theta) \approx \sum_{\alpha \in A} c_{\alpha} \psi_{\alpha}(\theta) $$

$$ \hat{S}_i = \sum_{\substack{\alpha \in A: \\ \alpha_i > 0, \\ \alpha_{j \neq i} = 0}} c_{\alpha}^2 / D, $$

$$ D = \mathrm{Var} \left[ \sum_{\alpha \in A} c_{\alpha} \psi_{\alpha}(\theta) \right] = \sum_{\substack{\alpha \in A \\ \alpha \neq 0}} c_{\alpha}^2 $$

<footnote>ªSudret 2008; Marelli and Sudret 2014.</footnote>

<footnote>ªHart, Alexanderian, and Gremaud 2017; Jimenez, Le Maître, and Knio 2017; Zhu and Sudret 2021.</footnote>
<figure><img src="image_146.png" /></figure>

<figure><img src="image_147.png" /></figure>

<figure><img src="image_148.png" /></figure>

<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>

<figure><img src="image_151.png" /></figure>

## 2. Methodology: Sobol' indices as random variables

<figure><img src="image_152.png" /></figure>

$$J: \mathcal{D} \times \Omega \to \mathbb{R},$$

$$ (\theta, \omega) \mapsto J(\theta, U(\omega)), $$

$$ S_i(\omega) = \frac{\mathrm{Var}_{\theta_i} [\mathbb{E}[Y | \theta_i, \omega]]}{\mathrm{Var}[Y | \omega]} $$

Sobol' indices as random variables,ª.

Estimation of Sobol' indices with polynomial chaos expansion,ª:

$$ J(\theta) = \sum_{\alpha \in \mathbb{N}^K} c_{\alpha} \psi_{\alpha}(\theta) \approx \sum_{\alpha \in A} c_{\alpha} \psi_{\alpha}(\theta) $$

$$ \hat{S}_i = \sum_{\substack{\alpha \in A: \\ \alpha_i > 0, \\ \alpha_{j \neq i} = 0}} c_{\alpha}^2 / D, $$

$$ D = \mathrm{Var} \left[ \sum_{\alpha \in A} c_{\alpha} \psi_{\alpha}(\theta) \right] = \sum_{\substack{\alpha \in A \\ \alpha \neq 0}} c_{\alpha}^2 $$

<footnote>ªSudret 2008; Marelli and Sudret 2014.</footnote>

<footnote>ªHart, Alexanderian, and Gremaud 2017; Jimenez, Le Maître, and Knio 2017; Zhu and Sudret 2021.</footnote>
<figure><img src="image_154.png" /></figure>

<figure><img src="image_155.png" /></figure>

<figure><img src="image_156.png" /></figure>

<figure><img src="image_157.png" /></figure>

<figure><img src="image_158.png" /></figure>

## 2. Methodology: metamodel validation

The $Q^2$ of the PCE metamodels are calculated on an independent test set for each rain:

<figure><img src="image_159.png" /></figure>

We deem the metamodels a correct approximation of the original and proceed to Sobol' indices calculation.

INRAQ

<figure><img src="image_160.png" /></figure>
<figure><img src="image_153.png" /></figure>

<figure><img src="image_154.png" /></figure>

<figure><img src="image_155.png" /></figure>

<figure><img src="image_156.png" /></figure>

<figure><img src="image_157.png" /></figure>

## 2. Methodology: metamodel validation

The $Q^2$ of the PCE metamodels are calculated on an independent test set for each rain:

<figure><img src="image_158.png" /></figure>

We deem the metamodels a correct approximation of the original and proceed to Sobol' indices calculation.

INRAQ

<figure><img src="image_159.png" /></figure>
<figure><img src="image_161.png" /></figure>

<figure><img src="image_162.png" /></figure>

<figure><img src="image_163.png" /></figure>

<figure><img src="image_164.png" /></figure>

<figure><img src="image_165.png" /></figure>

<figure><img src="image_166.png" /></figure>

## 3. Case study: moisture profiles

<figure><img src="image_167.png" /></figure>

<figure><img src="image_168.png" /></figure>
<figure><img src="image_160.png" /></figure>

<figure><img src="image_161.png" /></figure>

<figure><img src="image_162.png" /></figure>

<figure><img src="image_163.png" /></figure>

<figure><img src="image_164.png" /></figure>

<figure><img src="image_165.png" /></figure>

## 3. Case study: moisture profiles

<figure><img src="image_166.png" /></figure>
<figure><img src="image_169.png" /></figure>

<figure><img src="image_170.png" /></figure>

<figure><img src="image_171.png" /></figure>

<figure><img src="image_172.png" /></figure>

<figure><img src="image_173.png" /></figure>

<figure><img src="image_174.png" /></figure>

3. Case study: moisture profiles

<figure><img src="image_175.png" /></figure>

Rain - $u_A$ ... $u_B$ — other

<figure><img src="image_176.png" /></figure>

Different rains on soil moisture

<figure><img src="image_177.png" /></figure>
<figure><img src="image_167.png" /></figure>

<figure><img src="image_168.png" /></figure>

<figure><img src="image_169.png" /></figure>

<figure><img src="image_170.png" /></figure>

<figure><img src="image_171.png" /></figure>

<figure><img src="image_172.png" /></figure>

## 3. Case study: moisture profiles

<figure><img src="image_173.png" /></figure>

<figure><img src="image_174.png" /></figure>

<figure><img src="image_175.png" /></figure>
<figure><img src="image_178.png" /></figure>

<figure><img src="image_179.png" /></figure>

<figure><img src="image_180.png" /></figure>

<figure><img src="image_181.png" /></figure>

<figure><img src="image_182.png" /></figure>

<figure><img src="image_183.png" /></figure>

## 3. Case study: Sobol' indices under one rain realization

<figure><img src="image_184.png" /></figure>

<table><thead><tr><th>Name</th><th>Definition</th><th>Unit</th><th>Distribution</th></tr></thead><tbody><tr><td>θ<sub>s,surf.</sub></td><td>water content at saturation (surface)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.3375, 0.0338<sup>2</sup>)</td></tr><tr><td>θ<sub>s,inter.</sub></td><td>water content at saturation (intermediary)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.3322, 0.0332<sup>2</sup>)</td></tr><tr><td>θ<sub>s,deep</sub></td><td>water content at saturation (deep)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.316, 0.0316<sup>2</sup>)</td></tr><tr><td>θ<sub>r,deep</sub></td><td>residual water content (deep)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.0612, 0.0153<sup>2</sup>)</td></tr><tr><td>mn,deep</td><td>Van Genuchten retention curve parameter (deep)</td><td>[-]</td><td>N (0.1791, 0.0179<sup>2</sup>)</td></tr><tr><td>hg,deep</td><td>Van Genuchten retention curve parameter (deep)</td><td>[-]</td><td>N (-9.69, 0.969<sup>2</sup>)</td></tr></tbody></table>
<figure><img src="image_176.png" /></figure>

<figure><img src="image_177.png" /></figure>

<figure><img src="image_178.png" /></figure>

<figure><img src="image_179.png" /></figure>

<figure><img src="image_180.png" /></figure>

<figure><img src="image_181.png" /></figure>

## 3. Case study: Sobol' indices under one rain realization

<figure><img src="image_182.png" /></figure>

<table><thead><tr><th>Name</th><th>Definition</th><th>Unit</th><th>Distribution</th></tr></thead><tbody><tr><td>θ<sub>s,surf.</sub></td><td>water content at saturation (surface)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.3375, 0.0338<sup>2</sup>)</td></tr><tr><td>θ<sub>s,inter.</sub></td><td>water content at saturation (intermediary)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.3322, 0.0332<sup>2</sup>)</td></tr><tr><td>θ<sub>s,deep</sub></td><td>water content at saturation (deep)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.316, 0.0316<sup>2</sup>)</td></tr><tr><td>θ<sub>r,deep</sub></td><td>residual water content (deep)</td><td>[L<sup>3</sup>L<sup>-3</sup>]</td><td>N (0.0612, 0.0153<sup>2</sup>)</td></tr><tr><td>mn,deep</td><td>Van Genuchten retention curve parameter (deep)</td><td>[-]</td><td>N (0.1791, 0.0179<sup>2</sup>)</td></tr><tr><td>hg,deep</td><td>Van Genuchten retention curve parameter (deep)</td><td>[-]</td><td>N (-9.69, 0.969<sup>2</sup>)</td></tr></tbody></table>
<figure><img src="image_185.png" /></figure>

<figure><img src="image_186.png" /></figure>

<figure><img src="image_187.png" /></figure>

<figure><img src="image_188.png" /></figure>

<figure><img src="image_189.png" /></figure>

<figure><img src="image_190.png" /></figure>

## 4. Results: Sobol' indices depending on rain

<figure><img src="image_191.png" /></figure>

<table><thead><tr><th>Parameter</th><th>θ<sub>s.surf</sub></th><th>θ<sub>s.inter</sub></th><th>θ<sub>s.deep</sub></th><th>mn<sub>deep</sub></th><th>θ<sub>r.deep</sub></th><th>hg<sub>deep</sub></th></tr></thead><tbody><tr><td>μ(S<sub>i</sub><sup>T</sup>)</td><td>0.11</td><td>0.17</td><td>0.58</td><td>0.09</td><td>0.10</td><td>&lt; 0.01</td></tr><tr><td>σ(S<sub>i</sub><sup>T</sup>)</td><td>0.04</td><td>0.07</td><td>0.19</td><td>0.11</td><td>0.11</td><td>&lt; 0.01</td></tr></tbody></table>
<figure><img src="image_183.png" /></figure>

<figure><img src="image_184.png" /></figure>

<figure><img src="image_185.png" /></figure>

<figure><img src="image_186.png" /></figure>

<figure><img src="image_187.png" /></figure>

<figure><img src="image_188.png" /></figure>

## 4. Results: Sobol' indices depending on rain

<figure><img src="image_189.png" /></figure>

<table><thead><tr><th>Parameter</th><th>θ<sub>s.surf</sub></th><th>θ<sub>s.inter</sub></th><th>θ<sub>s.deep</sub></th><th>mn<sub>deep</sub></th><th>θ<sub>r.deep</sub></th><th>hg<sub>deep</sub></th></tr></thead><tbody><tr><td>μ(S<sub>i</sub><sup>T</sup>)</td><td>0.11</td><td>0.17</td><td>0.58</td><td>0.09</td><td>0.10</td><td>&lt; 0.01</td></tr><tr><td>σ(S<sub>i</sub><sup>T</sup>)</td><td>0.04</td><td>0.07</td><td>0.19</td><td>0.11</td><td>0.11</td><td>&lt; 0.01</td></tr></tbody></table>
<figure><img src="image_192.png" /></figure>

<figure><img src="image_193.png" /></figure>

<figure><img src="image_194.png" /></figure>

<figure><img src="image_195.png" /></figure>

<figure><img src="image_196.png" /></figure>

<figure><img src="image_197.png" /></figure>

## 4. Results: Sobol' indices depending on rain

<figure><img src="image_198.png" /></figure>

variable

<figure><img src="image_199.png" /></figure>

count

INRAO
<figure><img src="image_190.png" /></figure>

<figure><img src="image_191.png" /></figure>

<figure><img src="image_192.png" /></figure>

<figure><img src="image_193.png" /></figure>

<figure><img src="image_194.png" /></figure>

<figure><img src="image_195.png" /></figure>

## 4. Results: Sobol' indices depending on rain

<figure><img src="image_196.png" /></figure>

variable

<figure><img src="image_197.png" /></figure>

count

INRAO
<figure><img src="image_200.png" /></figure>

<figure><img src="image_201.png" /></figure>

<figure><img src="image_202.png" /></figure>

<figure><img src="image_203.png" /></figure>

<figure><img src="image_204.png" /></figure>

<figure><img src="image_205.png" /></figure>

## 5. Conclusion:

1. Sobol' indices were obtained for 6 input parameters in 200 different rain realizations.

2. the sensitivity analysis disentangles the variability in the parameters $\theta$ from the one in the stochastic forcing $\omega$.

3. the parameter $hg$ was found non-identifiable in all rain realizations.

4. the ranking of the input parameters varies depending on $\omega$.

What's next? :

1. other ways of synthesizing information of $S(\omega)$ ?

2. based on the sensitivity analysis results, implement a robust calibration method

3. study different model outputs, such as transferred pesticide mass.

4. study different stochastic inputs, such as pesticide application dates.

<figure><img src="image_206.png" /></figure>
<figure><img src="image_198.png" /></figure>

<figure><img src="image_199.png" /></figure>

<figure><img src="image_200.png" /></figure>

<figure><img src="image_201.png" /></figure>

<figure><img src="image_202.png" /></figure>

## 5. Conclusion:

1. Sobol' indices were obtained for 6 input parameters in 200 different rain realizations.

2. the sensitivity analysis disentangles the variability in the parameters $θ$ from the one in the stochastic forcing $ω$.

3. the parameter $hg$ was found non-identifiable in all rain realizations.

4. the ranking of the input parameters varies depending on $ω$.

What's next? :

1. other ways of synthesizing information of $S(ω)$ ?

2. based on the sensitivity analysis results, implement a robust calibration method

3. study different model outputs, such as transferred pesticide mass.

4. study different stochastic inputs, such as pesticide application dates.

<figure><img src="image_203.png" /></figure>
<figure><img src="image_207.png" /></figure>

<figure><img src="image_208.png" /></figure>

<figure><img src="image_209.png" /></figure>

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

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

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

Bibliography I

Dell'Oca, A. (June 2023). "Sensitivity Analysis: An Operational Picture". en. In: *Water Resources Research* 59.6, e2022WR033780. DOI: 10.1029/2022WR033780.

Hart, J. L., A. Alexanderian, and P. A. Gremaud (Jan. 2017). "Efficient Computation of Sobol' Indices for Stochastic Models". en. In: *SIAM Journal on Scientific Computing* 39.4, A1514–A1530. DOI: 10.1137/16M106193X.

Jimenez, M. Navarro, O. P. Le Maître, and O. M. Knio (Jan. 2017). "Nonintrusive Polynomial Chaos Expansions for Sensitivity Analysis in Stochastic Differential Equations". en. In: *SIAM/ASA Journal on Uncertainty Quantification* 5.1, pp. 378–402. DOI: 10.1137/16M1061989.

Mai, Juliane (May 2023). "Ten strategies towards successful calibration of environmental models". en. In: *Journal of Hydrology* 620, p. 129414. DOI: 10.1016/j.jhydrol.2023.129414.

<figure><img src="image_213.png" /></figure>
<figure><img src="image_204.png" /></figure>

<figure><img src="image_205.png" /></figure>

<figure><img src="image_206.png" /></figure>

<figure><img src="image_207.png" /></figure>

<figure><img src="image_208.png" /></figure>

<figure><img src="image_209.png" /></figure>

Bibliography I

Dell'Oca, A. (June 2023). "Sensitivity Analysis: An Operational Picture". en. In: *Water Resources Research* 59.6, e2022WR033780. DOI: 10.1029/2022WR033780.

Hart, J. L., A. Alexanderian, and P. A. Gremaud (Jan. 2017). "Efficient Computation of Sobol' Indices for Stochastic Models". en. In: *SIAM Journal on Scientific Computing* 39.4, A1514–A1530. DOI: 10.1137/16M106193X.

Jimenez, M. Navarro, O. P. Le Maître, and O. M. Knio (Jan. 2017). "Nonintrusive Polynomial Chaos Expansions for Sensitivity Analysis in Stochastic Differential Equations". en. In: *SIAM/ASA Journal on Uncertainty Quantification* 5.1, pp. 378–402. DOI: 10.1137/16M1061989.

Mai, Juliane (May 2023). "Ten strategies towards successful calibration of environmental models". en. In: *Journal of Hydrology* 620, p. 129414. DOI: 10.1016/j.jhydrol.2023.129414.

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

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.

Sudret, Bruno (July 2008). "Global sensitivity analysis using polynomial chaos expansions". en. In: *Reliability Engineering & System Safety* 93.7, pp. 964-979. DOI: 10.1016/j.ress.2007.04.002.

Zhu, Xujia and Bruno Sudret (Oct. 2021). "Global sensitivity analysis for stochastic simulators based on generalized lambda surrogate models". en. In: *Reliability Engineering & System Safety* 214, p. 107815. DOI: 10.1016/j.ress.2021.107815.
Bibliography II

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.

Sudret, Bruno (July 2008). "Global sensitivity analysis using polynomial chaos expansions". en. In: *Reliability Engineering & System Safety* 93.7, pp. 964-979. DOI: 10.1016/j.ress.2007.04.002.

Zhu, Xujia and Bruno Sudret (Oct. 2021). "Global sensitivity analysis for stochastic simulators based on generalized lambda surrogate models". en. In: *Reliability Engineering & System Safety* 214, p. 107815. DOI: 10.1016/j.ress.2021.107815.