# Effect of forcing uncertainty in the sensitivity and calibration of a pesticide transfer model

Katarina Radišić, Claire Lauvernet, Arthur Vidard

► To cite this version:

Katarina Radišić, Claire Lauvernet, Arthur Vidard. Effect of forcing uncertainty in the sensitivity and calibration of a pesticide transfer model. SAMO 2025 - 11th International Conference on Sensitivity Analysis of Model Output, Apr 2025, Grenoble, France. hal-05076842

HAL Id: hal-05076842

https://hal.inrae.fr/hal-05076842v1

Submitted on 21 May 2025

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

Distributed under a Creative Commons Attribution 4.0 International License
Effect of forcing uncertainty in the sensitivity and
calibration of a pesticide transfer model

Katarina Radišić¹²

Claire Lauvernet¹, Arthur Vidard²

¹INRAE, RiverLy, Lyon-Villeurbanne

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

11th International Conference on Sensitivity Analysis of Model Output
April 23–25 2025, Grenoble, France

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

RÉPUBLIQUE
FRANÇAISE

Liberté
Égalité
Fraternité

INRAE

Inria
Context

Pesticide transfer modeling

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

Pesticide transfer modeling

* Only 37% of Europe's surface waters have good or high ecological status; 29% a good chemical status. a

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

Pesticide transfer modeling

* Only 37% of Europe's surface waters have good or high ecological status; 29% a good chemical status.ª

* Main stressor: agriculture, with the intensive use of nutrients and pesticides.

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

Pesticide transfer modeling

* Only 37% of Europe's surface waters have good or high ecological status; 29% a good chemical status. a

* Main stressor: agriculture, with the intensive use of nutrients and pesticides.

* Landscape management for prevention and mitigation.

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

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

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

<footnote>aEuropean Environment Agency. 2024.</footnote>
Context

Pesticide transfer modeling

* Only 37% of Europe's surface waters have good or high ecological status; 29% a good chemical status. a

* Main stressor: agriculture, with the intensive use of nutrients and pesticides.

* Landscape management for prevention and mitigation.

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

→ We need adapted modeling to evaluate
landscape management scenarios across river
catchments

Europe's state of water 2024

The need for improved water resilience

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

<footnote>ªEuropean Environment Agency. 2024.</footnote>
Context

PESHMELBA model

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

PESHMELBAª simulates water and pesticide transfers on an **agricultural catchment**.

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

PESHMELBA model

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

PESHMELBAª simulates water and pesticide transfers on an **agricultural catchment**.

* Consideration of **landscape elements**

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

PESHMELBA model

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

PESHMELBAª simulates water and pesticide transfers on an **agricultural catchment**.

* Consideration of **landscape elements**

* Process oriented, physically-based model.

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

## PESHMELBA model

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

PESHMELBAª simulates water and pesticide transfers on an **agricultural catchment**.

* Consideration of **landscape elements**

* Process oriented, physically-based model.

* Operational use: **comparison of landscape management scenarios.**

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

<footnote>ªRouzies et al. 2019.</footnote>
Context

PESHMELBA model

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

Distributed model, presenting **numerous parameters**.

PESHMELBA<sup>a</sup> simulates water and pesticide transfers on an **agricultural catchment**.

* Consideration of **landscape elements**

* Process oriented, physically-based model.

* Operational use: **comparison of landscape management scenarios**.

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

<footnote><sup>a</sup>Rouzies et al. 2019.</footnote>
Context

PESHMELBA model

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

Distributed model, presenting numerous parameters.

* Soil properties: Ks, hydraulic conductivity at saturation, θs, water content at saturation...

* Vegetation, river, topography, pesticide molecule...

* for each soil and vegetation type, soil horizon...

PESHMELBAª simulates water and pesticide transfers on an agricultural catchment.

* Consideration of landscape elements

* Process oriented, physically-based model.

* Operational use: comparison of landscape management scenarios.

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

<footnote>ªRouzies et al. 2019.</footnote>
Context

PESHMELBA model

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

Distributed model, presenting **numerous parameters**.

* Soil properties: $K_s$, hydraulic conductivity at saturation, $\theta_s$, water content at saturation...

* Vegetation, river, topography, pesticide molecule...

* for each soil and vegetation type, soil horizon...

→ = **145 model parameters !**

PESHMELBAª simulates water and pesticide transfers on an **agricultural catchment**.

* Consideration of **landscape elements**

* Process oriented, physically-based model.

* Operational use: **comparison of landscape management scenarios.**

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

<footnote>ªRouzies et al. 2019.</footnote>
Context

PESHMELBA model

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

Distributed model, presenting **numerous parameters**.

* Soil properties: $K_s$, hydraulic conductivity at saturation, $\theta_s$, water content at saturation...

* Vegetation, river, topography, pesticide molecule...

* for each soil and vegetation type, soil horizon...

→ = **145 model parameters!**

PESHMELBAª simulates water and pesticide transfers on an **agricultural catchment**.

* Consideration of **landscape elements**

* Process oriented, physically-based model.

* Operational use: **comparison of landscape management scenarios.**

How to set the values of model parameters?

<footnote>ªRouzies et al. 2019.</footnote>
Introduction

Classical calibration approach

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

• x: Ks, θs, θr, LAI, Koc...

* z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

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

Classical calibration approach

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

• x: Ks, θs, θr, LAI, Koc...

• z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

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

Classical calibration approach

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

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

x: Ks, θs, θr, LAI, Koc...

z: rainfall, pesticide
applications dates and
quantities, ETP, mesh,
irrigation

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

Classical calibration approach

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

streamflow
soil moisture
pesticide concentration

• x: Ks, θs, θr, LAI, Koc...

* z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

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

## Classical calibration approach

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

* $x: K_s, \theta_s, \theta_r, LAI, Koc...$

* $z$: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

1. What parameters ($x$) are the most important?

<figure><img src="image_36.png" /></figure>
Introduction

Classical calibration approach

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

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

f(x, z₁) • x: Ks, θs, θr, LAI, Koc...

z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

1. What parameters (x) are the most important?

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

Classical calibration approach

<figure><img src="image_40.png" /></figure>

* $x: K_s, \theta_s, \theta_r, LAI, Koc...$

* $z$: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

→ **Uncertainty on forcing conditions**
$\omega \in \Omega$

1. What parameters ($x$) are the most important?

→ Study the output's sensitivity to parameters over the domain of forcing uncertainty.

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

Classical calibration approach

<figure><img src="image_42.png" /></figure>

streamflow
soil moisture
pesticide concentration

• x: $K_s, \theta_s, \theta_r, LAI, Koc...$

• z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

→ **Uncertainty on forcing conditions**
$\omega \in \Omega$

1. What parameters (x) are the most important?

→ Study the output's **sensitivity** to parameters **over the domain of forcing uncertainty**.

2. How to calibrate the model parameters robustly to forcing uncertainty?

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

## Classical calibration approach

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

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

* x: $K_s, \theta_s, \theta_r, LAI, Koc...$

* z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

→ Uncertainty on forcing conditions
$\omega \in \Omega$

1. What parameters (x) are the most important?

→ Study the output's sensitivity to parameters over the domain of forcing uncertainty.

2. How to calibrate the model parameters robustly to forcing uncertainty?
Introduction

Classical calibration approach

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

• x: $K_s, \theta_s, \theta_r, LAI, KOC...$

• z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

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

1. What parameters (x) are the most important?

→ Study the output's **sensitivity** to parameters **over the domain of forcing uncertainty**.

2. How to calibrate the model parameters robustly to forcing uncertainty?

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

Classical calibration approach

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

• x: K_s, θ_s, θ_r, LAI, Koc...

• z: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

→ Uncertainty on forcing conditions

ω ∈ Ω

1. What parameters (x) are the most important?

→ Study the output's sensitivity to parameters over the domain of forcing uncertainty.

2. How to calibrate the model parameters robustly to forcing uncertainty?

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

Classical calibration approach

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

* $x: K_s, \theta_s, \theta_r, LAI, Koc...$

* $z$: rainfall, pesticide applications dates and quantities, ETP, mesh, irrigation

→ **Uncertainty on forcing conditions**
$\omega \in \Omega$

1. What parameters ($x$) are the most important?

→ Study the output's **sensitivity** to parameters **over the domain of forcing uncertainty**.

2. How to calibrate the model parameters robustly to forcing uncertainty?

→ Find $x_{\text{robust}}^*$ with highest probability to be below threshold c, **computationally expensive**.
Methodology

Construction of the stochastic emulator $\hat{f}_s$:

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

$f_s(x, \omega)$

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

Construction of the stochastic emulator $\hat{f}_s$:

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

$f_s(x, \omega)$

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

Construction of the stochastic emulator $\hat{f}_s$:

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

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

Construction of the stochastic emulator $\hat{f}_s$:

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

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

Construction of the stochastic emulator $\hat{f}_s$:

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

$f_s(x, \omega)$

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

Construction of the stochastic emulator $\hat{f}_s$:

$f_s(x, \omega)$

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

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

$$ 
\begin{aligned}
f_s(x, \omega_2) &\approx f_{PCE}^{(2)}(x) = \sum_{\alpha \in A} c_{\alpha}^{(2)} \psi_{\alpha}(x) \\
&= c_{\alpha_1}^{(2)} \psi_{\alpha_1}(x) + c_{\alpha_2}^{(2)} \psi_{\alpha_2}(x) + c_{\alpha_3}^{(2)} \psi_{\alpha_3}(x)
\end{aligned}
$$
Methodology

Construction of the stochastic emulator $\hat{f}_s$:

$f_s(x, \omega)$

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

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

$$ 
\begin{align*} 
f_s(x, \omega_2) &\approx f_{PCE}^{(2)}(x) = \sum_{\alpha \in A} c_{\alpha}^{(2)} \psi_{\alpha}(x) \\ 
&= c_{\alpha_1}^{(2)} \psi_{\alpha_1}(x) + c_{\alpha_2}^{(2)} \psi_{\alpha_2}(x) + c_{\alpha_3}^{(2)} \psi_{\alpha_3}(x) 
\end{align*} 
$$

* Get Sobol' indices from PCE coefficients $c_{\alpha}^{(i)}$, conditionally to each forcing condition $\omega^{(i)}$.

<footnote>ªSudret 2008.</footnote>
Methodology

Construction of the stochastic emulator $\hat{f}_s$:

$f_s(x, \omega)$

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

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

$$ 
\begin{aligned}
f_s(x, \omega_2) &\approx f_{PCE}^{(2)}(x) = \sum_{\alpha \in A} c_{\alpha}^{(2)} \psi_{\alpha}(x) \\
&= c_{\alpha_1}^{(2)} \psi_{\alpha_1}(x) + c_{\alpha_2}^{(2)} \psi_{\alpha_2}(x) + c_{\alpha_3}^{(2)} \psi_{\alpha_3}(x)
\end{aligned}
$$

* Get Sobol' indices from PCE coefficients $c_{\alpha}^{(i)}$ conditionally to each forcing condition $\omega^{(i)}$.$^a$

* Deduce regularity from coefficients.$^b$

<footnote>$^a$Sudret 2008.</footnote>

<footnote>$^b$Lüthen et al. 2023.</footnote>
Methodology

Construction of the stochastic emulator $\hat{f}_s$:

$f_s(x, \omega)$

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

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

$$ 
\begin{aligned}
f_s(x, \omega_2) & \approx f_{PCE}^{(2)}(x) = \sum_{\alpha \in A} c_{\alpha}^{(2)} \psi_{\alpha}(x) \\
& = c_{\alpha_1}^{(2)} \psi_{\alpha_1}(x) + c_{\alpha_2}^{(2)} \psi_{\alpha_2}(x) + c_{\alpha_3}^{(2)} \psi_{\alpha_3}(x)
\end{aligned}
$$

* Get Sobol' indices from PCE coefficients $c_{\alpha}^{(i)}$ conditionally to each forcing condition $\omega^{(i)}$.$^a$

* Deduce regularity from coefficients.$^b$

<footnote>$^a$Sudret 2008.</footnote>

<footnote>$^b$Lüthen et al. 2023.</footnote>
Methodology

Construction of the stochastic emulator $\hat{f}_s$:

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

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

Construction of the stochastic emulator $\hat{f}_s$:

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

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

Construction of the stochastic emulator $\hat{f}_s$:

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

Construction of the stochastic emulator $\hat{f}_s$:

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

Construction of the stochastic emulator $\hat{f}_s$:

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

Construction of the stochastic emulator $\hat{f}_s$:

<figure><img src="image_68.png" /><figcaption>(c) histogram of principal component scores</figcaption></figure>

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

Construction of the stochastic emulator $\hat{f}_s$:

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

(c) inference of distributions
on the principal component scores

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

Generate new simulations from $\hat{f}_s$:

<figure><img src="image_72.png" /><figcaption>(c) inference of distributions
on the principal component scores</figcaption></figure>

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

Generate new simulations from $\hat{f}_s$:

<figure><img src="image_74.png" /><figcaption>(c) inference of distributions
on the principal component scores</figcaption></figure>

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

Generate new simulations from $\hat{f}_s$:

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

(c) inference of distributions
on the principal component scores

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

Generate new simulations from $\hat{f}_s$:

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

(c) inference of distributions
on the principal component scores

→ Generate simulations at a low computational cost.

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

Generate new simulations from $\hat{f}_s$:

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

(c) inference of distributions
on the principal component scores

→ Generate simulations at a low computational cost.

→ Find $x_{robust}^*$ at a lower computational cost.
# Uncertain pesticide application date

## Case study: Pesticide concentration

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

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

<figure><img src="image_83.png" /></figure>
# Uncertain pesticide application date

## Case study: Pesticide concentration

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

<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /></figure>
# Uncertain pesticide application date

## Case study: Pesticide concentration

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

<figure><img src="image_88.png" /></figure>
# Uncertain pesticide application date

## Case study: Pesticide concentration

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

## Dominant transfer processes:

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

Subsurface lateral flow

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

Surface runoff

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

<figure><img src="image_93.png" /></figure>
# Uncertain pesticide application date

Sobol' indices as random variables:

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

$$S_{X_i}(\omega) = \frac{\mathrm{Var}_{x_i}(\mathbb{E}[Y | x_i, \omega])}{\mathrm{Var}(Y|\omega)}$$

1. **parameters x**: catchment properties (hydrodynamic soil properties, vegetation, topology).

2. **forcing uncertainty Z(ω)**: pesticide application dates

<figure><img src="image_95.png" /></figure>
# Uncertain pesticide application date

## Results: Sensitivity index distributions

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

Total GSI indices

Pesticide application after before

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

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

## Generalized Sensitivity Indices, GSIª

<figure><img src="image_99.png" /></figure>

<footnote>ªLamboni et al. 2011.</footnote>
# Uncertain pesticide application date

## Results: Sensitivity index distributions

<figure><img src="image_100.png" /></figure>

<figure><img src="image_101.png" /></figure>

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

Generalized Sensitivity
Indices, GSIª

<footnote>ªLamboni et al. 2011.</footnote>
# Uncertain pesticide application date

## Results: Sensitivity index distributions

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

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

<figure><img src="image_105.png" /></figure>

Generalized Sensitivity
Indices, GSIª

<footnote>ªLamboni et al. 2011.</footnote>
# Uncertain pesticide application date

## Results: Sensitivity index distributions

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

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

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

Generalized Sensitivity
Indices, GSIª

<footnote>ªLamboni et al. 2011.</footnote>
# Uncertain rainfall forcing

## Case study: Moisture profiles

### (c) PESHMELBA configuration

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

* Forcing uncertainty Ω: rainfall error.

* Observation $y_{obs}$: moisture profile.

<figure><img src="image_110.png" /></figure>
# Uncertain rainfall forcing

## Case study: Moisture profiles

### (c) PESHMELBA configuration

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

#### Synthetic observation

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

### (d) Observation: moisture profile of plot 4

### (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>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>

* Forcing uncertainty Ω: rainfall error.

* Observation $y_{obs}$: moisture profile.

<figure><img src="image_113.png" /></figure>
# Uncertain rainfall forcing

## Case study: Moisture profiles

### (a) Forcing input

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

### (c) PESHMELBA configuration

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

### (d) Observation: moisture profile of plot 4

* Forcing uncertainty $\Omega$: rainfall error.

* Observation $y_{obs}$: moisture profile.
Uncertain rainfall forcing

Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

<figure><img src="image_116.png" /></figure>
Uncertain rainfall forcing

Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

<figure><img src="image_118.png" /></figure>
# Uncertain rainfall forcing

## Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

<figure><img src="image_120.png" /></figure>
Uncertain rainfall forcing

Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

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

<figure><img src="image_123.png" /></figure>
Uncertain rainfall forcing

Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

<figure><img src="image_125.png" /></figure>
# Uncertain rainfall forcing

## Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

Soil moisture [cm3/cm3]

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

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

Calibrated values

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

<table><thead><tr><th></th><th>f&#773;{f<sub>s</sub>(x<sup>*</sup>, .) > 0.01}</th><th>f&#773;{f<sub>s</sub>(x<sup>*</sup>, .) > 0.02}</th><th>f<sub>s</sub>(x<sup>*</sup>, .)</th><th>max(f<sub>s</sub>(x<sup>*</sup>, .))</th></tr></thead><tbody><tr><td>x<sup>*</sup><sub>prior</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>x&#773;<sup>*</sup><sub>classic</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>x&#773;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> < 0.02</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>x&#773;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> < 0.01</sub></td><td></td><td></td><td></td><td></td></tr></tbody></table>
# Uncertain rainfall forcing

## Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

Soil moisture [cm3/cm3]

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

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

Calibrated values

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

<table><thead><tr><th></th><th>1/f<sub>s</sub>{f<sub>s</sub>(x*, .) > 0.01}</th><th>1/f<sub>s</sub>{f<sub>s</sub>(x*, .) > 0.02}</th><th>f<sub>s</sub>(x*, .)</th><th>max(f<sub>s</sub>(x*, .))</th></tr></thead><tbody><tr><td>x<sup>*</sup><sub>prior</sub></td><td>0.99</td><td></td><td></td><td></td></tr><tr><td>x&#770;<sup>*</sup><sub>classic</sub></td><td>0.99</td><td></td><td></td><td></td></tr><tr><td>x&#770;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> < 0.02</sub></td><td>0.41</td><td></td><td></td><td></td></tr><tr><td>x&#770;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> < 0.01</sub></td><td><strong>0.32</strong></td><td></td><td></td><td></td></tr></tbody></table>
Uncertain rainfall forcing

Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

Soil moisture [cm3/cm3]

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

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

Calibrated values

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

<table><thead><tr><th></th><th>1/f<sub>s</sub>{f<sub>s</sub>(x*, .) > 0.01}</th><th>1/f<sub>s</sub>{f<sub>s</sub>(x*, .) > 0.02}</th><th>f<sub>s</sub>(x*, .)</th><th>max(f<sub>s</sub>(x*, .))</th></tr></thead><tbody><tr><td>x<sup>*</sup><sub>prior</sub></td><td>0.99</td><td>0.39</td><td></td><td></td></tr><tr><td>x&#x0302;<sup>*</sup><sub>classic</sub></td><td>0.99</td><td>0.25</td><td></td><td></td></tr><tr><td>x&#x0302;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> < 0.02</sub></td><td>0.41</td><td><strong>0.09</strong></td><td></td><td></td></tr><tr><td>x&#x0302;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> < 0.01</sub></td><td><strong>0.32</strong></td><td>0.13</td><td></td><td></td></tr></tbody></table>
# Uncertain rainfall forcing

## Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

Soil moisture [cm3/cm3]

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

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

Calibrated values

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

<table><thead><tr><th></th><th>∫{f<sub>s</sub>(x*,·)>0.01}</th><th>∫{f<sub>s</sub>(x*,·)>0.02}</th><th>f<sub>s</sub>(x*,·)</th><th>max(f<sub>s</sub>(x*,·))</th></tr></thead><tbody><tr><td>x<sup>*</sup><sub>prior</sub></td><td>0.99</td><td>0.39</td><td>0.021</td><td></td></tr><tr><td>x&#x0302;<sup>*</sup><sub>classic</sub></td><td>0.99</td><td>0.25</td><td>0.017</td><td></td></tr><tr><td>x&#x0302;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> &lt; 0.02</sub></td><td>0.41</td><td><strong>0.09</strong></td><td>0.011</td><td></td></tr><tr><td>x&#x0302;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> &lt; 0.01</sub></td><td><strong>0.32</strong></td><td>0.13</td><td><strong>0.009</strong></td><td></td></tr></tbody></table>
Uncertain rainfall forcing

Results: Comparison of PESHMELBA robust vs classical calibrations under rainfall uncertainty

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

Soil moisture [cm3/cm3]

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

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

Calibrated values

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

<table><thead><tr><th></th><th>∫{f<sub>s</sub>(x*, ·)>0.01}</th><th>∫{f<sub>s</sub>(x*, ·)>0.02}</th><th>f<sub>s</sub>(x*, ·)</th><th>max(f<sub>s</sub>(x*, ·))</th></tr></thead><tbody><tr><td>x<sup>*</sup><sub>prior</sub></td><td>0.99</td><td>0.39</td><td>0.021</td><td>0.057</td></tr><tr><td>x&#x0302;<sup>*</sup><sub>classic</sub></td><td>0.99</td><td>0.25</td><td>0.017</td><td>0.042</td></tr><tr><td>x&#x0302;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> &lt; 0.02</sub></td><td>0.41</td><td><strong>0.09</strong></td><td>0.011</td><td><strong>0.038</strong></td></tr><tr><td>x&#x0302;<sup>*</sup><sub>robust<sub>f<sub>s</sub></sub> &lt; 0.01</sub></td><td><strong>0.32</strong></td><td>0.13</td><td><strong>0.009</strong></td><td>0.043</td></tr></tbody></table>
Conclusion

Considered two forcing uncertainties of **pesticide application date** and **rainfall** in water and pesticide transfers (**PESHMELBA**), on:

1. Global sensitivity analysis

* Separated parameter and forcing variability.

* Showed output sensitivity to model inputs depends on application date:

* Application before rainfall: primary contributions from the left riverbank.

* Application after rainfall: primary contributions from river-water table exchanges.

2. Robust model calibration

* Proposed a generic **stochastic emulator** methodology.

* Built a **non-intrusive** emulator in Ω (requires only a representative sample).

* Estimated **multiple robust solutions** $x_{\text{robust}}^*$ from $\hat{f}_s$, without additional simulations.

Future possibilities...

1. Reduce sample size over the forcing uncertainty space Ω.

2. Define guidelines for thresholds $c, x_{\text{robust}}^*$ with field experts.

3. Study other types of forcing uncertainty Ω: natural variability, interannual variability, or future projections.

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

Thank you for your attention!

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

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

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

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.

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