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# Consideration of uncertain forcings in the global sensitivity analysis and metamodeling of a pesticide transfer model

Katarina Radišić, Claire Lauvernet, Arthur Vidard

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Katarina Radišić, Claire Lauvernet, Arthur Vidard. Consideration of uncertain forcings in the global sensitivity analysis and metamodeling of a pesticide transfer model. EGU General Assembly, Apr 2024, Vienna, Austria, Austria. 10.5194/egusphere-egu24-19838 . hal-04626611

HAL Id: hal-04626611

https://hal.inrae.fr/hal-04626611

Submitted on 27 Jun 2024

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HAL
open science

# Consideration of uncertain forcings in the global sensitivity analysis and metamodeling of a pesticide transfer model

Katarina Radišić, Claire Lauvernet, Arthur Vidard

► To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Consideration of uncertain forcings in the global sensitivity analysis and metamodeling of a pesticide transfer model. EGU General Assembly 2024, Apr 2024, Vienna, Austria, Austria. pp.1-1, 2024, 10.5194/egusphere-egu24-19838. hal-04626611

HAL Id: hal-04626611

https://hal.inrae.fr/hal-04626611v1

Submitted on 27 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.

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Distributed under a Creative Commons Attribution 4.0 International License
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# Consideration of uncertain forcings in the global sensitivity analysis and metamodeling of a pesticide transfer model

K. Radišić¹,², C. Lauvernet¹, A. Vidard², B. Denise³, F. De Freitas³

¹ RIVERLY, INRAE Lyon, France

² AIRSEA, Inria, France

³ Ecole Centrale de Lyon, France

## Uncertainty quantification in decision support for water quality

Pesticide transfer modeling at the catchment scale (e.g, [1]) simulates surface/subsurface hydrology and reactive solute transport:

* based on non linear equations

* need for a large set of spatialized parameters

* many interactive processes not (well) represented

controllable model inputs: the model parameters

uncontrollable model inputs: the forcings : rainfall / pesticide app. dates (typ. known within a 2/3 day range)

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

Uncertainty in forcings is propagated to:

* the estimation of the model input parameters

* the **global sensitivity analysis (GSA)** of model outputs: this is generally not taken into account due to the complexity of the models [2,3] ⇒ this study objectives

## Case study: uncertainty on the pesticide application date

<figure><img src="image_4.png" /><figcaption>Application of pesticides on the vineyard parcels of the catchment</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>What if this is uncertain whether pesticide applications occurred before or after a heavy rainfall?</figcaption></figure>

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

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

* ✓ pesticides are applied over a small catchment

* ✓ output var. studied: concentration at the outlet

* ✓ simulated pesticide concentrations differ w.r.t. the application date and model parameters

* ⇒ need for a 2-steps global S.A. to investigate [5-6]

## 2-steps GSA

### ① Screening with HSIC independence test

$H_0: X_i \perp Y$ vs. $H_1: X_i$ and $Y$ are not independent

$$p_{\text{val}_B} = \frac{1}{B} \sum_{b=1}^{B} \mathbb{1}_{\widehat{\text{HSIC}}}[b](X_i, Y) > \widehat{\text{HSIC}}(X_i, Y) \quad (1)$$

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

HSIC indices, ©Guerlain Lambert

### ② Sobol' indices with Polynomial Chaos Exp.

$$ \begin{gathered} Y = f_d(\mathbf{X}) \approx f_{\text{PCE}}(\mathbf{X}) = \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(\mathbf{X}), \\ \begin{aligned} \hat{S}_i &= \sum_{\alpha \in \mathcal{A}_i} c_{\alpha}^2 / \hat{D}, & \mathcal{A}_i &= \{\alpha \in \mathcal{A} : \alpha_i > 0, \alpha_{j \neq i} = 0\}, \\ \hat{S}_{T_i} &= \sum_{\alpha \in \mathcal{A}_{T_i}} c_{\alpha}^2 / \hat{D}, & \mathcal{A}_{T_i} &= \{\alpha \in \mathcal{A} : \alpha_i > 0\}, \\ \hat{D} &= \text{Var} \left[ \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(\mathbf{X}) \right] = \sum_{\substack{\alpha \in \mathcal{A} \\ \alpha \neq \{0\}}} c_{\alpha}^2 \end{aligned} \end{gathered} \quad (2) $$

## Sensitivity in two contrasting scenarios

<figure><img src="image_9.png" /><figcaption>Total Sobol' indices</figcaption></figure>

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

* The HSIC identifies 28 influential parameters on 150

* The Sobol' indices for the 28 parameters are calculated

* GSA results are different in the two cases

* what about the application dates between $\omega_A$ and $\omega_B$?

## References

[1] Rouzies et al., 2019. 10.1016/j.scitotenv.2019.03.060 [2] Gatel, L. et al., 2019. 10.3390/w12010121 [3] Rouzies et al., 2022. 10.5194/egusphere-egu22-10384 [4] Sudret et al., 2008 10.1016/j.ress.2007.04.002 [5] Gretton et al., 2005 10.5555/1046920.1194914

[6] Lüthen et al., 2023 10.1016/j.cma.2022.115875

## A more global approach: Sobol' indices as random variables

<figure><img src="image_11.png" /><figcaption>Distribution of Total Sobol' indices</figcaption></figure>

<caption>Application Before and during rain After rain</caption>

* ✓ Sobol' indices are seen as random variables,

* ✓ Sobol' indices represented as histograms (depending on the application date)

* ✓ the sensitivity of the output on the pest.Koc.1 (pesticide mobility) and others par. differs visibly w.r.t. the application moment

## Conclusion / Next steps

* The GSA results lead to contrasting conclusions depending on the uncertain pesticide application date

* Considering Sobol' indices as random variables reveals a difference in the influence of some input factors (pest.mobility Koc, $\theta_s$, $K_s$ of some horizons, hpond, etc.)) on the concentration at the outlet.

* Next step : building a **stochastic metamodel** of PESHMELBA by inferring the distributions of the PCE coefficients [6]:

<figure><img src="image_12.png" /><figcaption>inference of marginal distributions on the principal component scores</figcaption></figure>

BUT: in our case complex nonlinear interactions between determ. and stochastic inputs ⇒ other methods are tested for inferring distrib. of the coeff. such as KDE, GMM, Principal Curve Analysis in 3D on the PCs

1. Approximate each trajectory with a PCE

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

3. Use GMM for new simulations $c_{\alpha}^{\text{new}}$

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

2. Infer the distribution of $c_\alpha$ (out of $R_{\text{train}}$ available realizations)

a) Dimensionality reduction of $c_\alpha$

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

b) Infer a GMM on the PCs
# Consideration of uncertain forcings in the global sensitivity analysis and metamodeling of a pesticide transfer model

K. Radišić¹,², C. Lauvernet¹, A. Vidard², B. Denise³, F. De Freitas³

¹ RIVERLY, INRAE Lyon, France

² AIRSEA, Inria, France

³ Ecole Centrale de Lyon, France

## Uncertainty quantification in decision support for water quality

Pesticide transfer modeling at the catchment scale (e.g, [1]) simulates surface/subsurface hydrology and reactive solute transport:

* based on non linear equations

* need for a large set of spatialized parameters

* many interactive processes not (well) represented

controllable model inputs: the model parameters

uncontrollable model inputs: the forcings : rainfall / pesticide app. dates (typ. known within a 2/3 day range)

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

Uncertainty in forcings is propagated to:

* the estimation of the model input parameters

* the **global sensitivity analysis (GSA)** of model outputs: this is generally not taken into account due to the complexity of the models [2,3] ⇒ this study objectives

## Case study: uncertainty on the pesticide application date

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

Application of pesticides on the vineyard parcels of the catchment

What if this is uncertain whether pesticide applications occurred before or after a heavy rainfall?

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

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

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

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

* ✓ pesticides are applied over a small catchment

* ✓ output var. studied: concentration at the outlet

* ✓ simulated pesticide concentrations differ w.r.t. the application date and model parameters

* ⇒ need for a 2-steps global S.A. to investigate [5-6]

## 2-steps GSA

### ① Screening with HSIC independence test

$H_0: X_i \perp Y$ vs. $H_1: X_i$ and $Y$ are not independent

$$p_{\text{val}_B} = \frac{1}{B} \sum_{b=1}^{B} \left| \widehat{\text{HSIC}}^{[b]} (X_i, Y) - \widehat{\text{HSIC}} (X_i, Y) \right| \quad (1)$$

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

HSIC indices, ©Guerlain Lambert

### ② Sobol' indices with Polynomial Chaos Exp.

$$ 
\begin{align} 
Y &= f_d(\mathbf{X}) \approx f_{\text{PCE}}(\mathbf{X}) = \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(\mathbf{X}), \nonumber \\
\hat{S}_i &= \sum_{\alpha \in \mathcal{A}_i} c_{\alpha}^2 / \hat{D}, \quad \mathcal{A}_i = \{\alpha \in \mathcal{A} : \alpha_i > 0, \alpha_{j \neq i} = 0\}, \nonumber \\
\hat{S}_{T_i} &= \sum_{\alpha \in \mathcal{A}_{T_i}} c_{\alpha}^2 / \hat{D}, \quad \mathcal{A}_{T_i} = \{\alpha \in \mathcal{A} : \alpha_i > 0\}, \nonumber \\
\hat{D} &= \text{Var} \left[ \sum_{\alpha \in \mathcal{A}} c_{\alpha} \psi_{\alpha}(\mathbf{X}) \right] = \sum_{\substack{\alpha \in \mathcal{A} \\ \alpha \neq \{0\}}} c_{\alpha}^2 \tag{2}
\end{align} 
$$

## Sensitivity in two contrasting scenarios

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

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

* The HSIC identifies 28 influential parameters on 150

* The Sobol' indices for the 28 parameters are calculated

* **GSA results are different in the two cases**

what about the application dates between $\omega_A$ and $\omega_B$?

## References

[1] Rouzies et al., 2019. 10.1016/j.scitotenv.2019.03.060 [2] Gatel, L. et al., 2019. 10.3390/w12010121 [3] Rouzies et al., 2022. 10.5194/egusphere-egu22-10384 [4] Sudret et al., 2008 10.1016/j.ress.2007.04.002 [5] Gretton et al., 2005 10.5555/1046920.1194914 [6] Lüthen et al., 2023 10.1016/j.cma.2022.115875

## A more global approach: Sobol' indices as random variables

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

* ✓ Sobol' indices are seen as random variables,

* ✓ Sobol' indices represented as histograms (depending on the application date)

* ✓ the sensitivity of the output on the pest.Koc.1 (pesticide mobility) and others par. differs visibly w.r.t. the application moment

## Conclusion / Next steps

* The GSA results lead to contrasting conclusions depending on the uncertain pesticide application date

* Considering Sobol' indices as random variables reveals a difference in the influence of some input factors (pest.mobility Koc, $\theta_s$, $K_s$ of some horizons, hpond, etc.)) on the concentration at the outlet.

* Next step : building a **stochastic metamodel** of PESHMELBA by inferring the distributions of the PCE coefficients [6]:

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

inference of marginal distributions on the principal component scores

**BUT**: in our case complex nonlinear interactions between determ. and stochastic inputs ⇒ other methods are tested for inferring distrib. of the coeff. such as KDE, GMM, Principal Curve Analysis in 3D on the PCs

1. Approximate each trajectory with a PCE

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

3. Use GMM for new simulations $c_{\alpha}^{\text{new}}$

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

2. Infer the distribution of $c_{\alpha}$ (out of $R_{\text{train}}$ available realizations)

a) Dimensionality reduction of $c_{\alpha}$

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

b) Infer a GMM on the PCs