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# Consideration of external uncertainties in the estimation of parameters of a water and pesticide transfer model at the watershed scale

Katarina Radišić, Claire Lauvernet, Arthur Vidard

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Katarina Radišić, Claire Lauvernet, Arthur Vidard. Consideration of external uncertainties in the estimation of parameters of a water and pesticide transfer model at the watershed scale. Séminaire du réseau Eaux et systèmes agricoles, Feb 2023, Montpellier, France. hal-04008833

HAL Id: hal-04008833

https://hal.inrae.fr/hal-04008833

Submitted on 28 Feb 2023

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# Prise en compte d'incertitudes externes dans l'estimation de paramètres d'un modèle de transfert d'eau et de pesticides à l'échelle du bassin versant

Katarina Radišić¹²³

Arthur Vidard<fnref n="2" />, Claire Lauvernet<fnref n="3" />

<footnote n="1">¹École doctorale Mathématiques, Sciences et technologies de l'information, Informatique (ED MSTII)</footnote>

<footnote n="2">²Univ. Grenoble-Alpes, Inria, CNRS, Grenoble-INP, LJK</footnote>

<footnote n="3">³INRAE, RiverLy, Lyon-Villeurbanne</footnote>
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Context : Landscape management in the service of water quality

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Figure: Landscape elements influence pesticide transfer dynamics, image credit Rouzies et al. (2018) (hal-02608211)

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Context : The PESHMELBA model

<figure><img src="image_12.png" /><figcaption>**Figure:** The catchment of assembled parcels, their vertical decomposition, and the main processes of water and pesticide transfer in PESHMELBA.
(Rouzies et al., 2019) doi: 10.1016/j.scitotenv.2019.03.060, image credit E. Rouzies</figcaption></figure>

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Context : The PESHMELBA model

<figure><img src="image_14.png" /><figcaption>**Figure:** The catchment of assembled parcels, their vertical decomposition, and the main processes of water and pesticide transfer in PESHMELBA,</figcaption></figure>

(Rouzies et al., 2019) doi: 10.1016/j.scitotenv.2019.03.060, image credit E. Rouzies

Complex model with many parameters -> needs to be calibrated.

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# Table of contents

## Introduction

## Methods

## Case study

## Results

## Conclusion

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# Table of contents

## Introduction

## Methods

## Case study

## Results

## Conclusion

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# Introduction: From classical to robust calibration

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# Introduction: From classical to robust calibration

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# Introduction: From classical to robust calibration

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# Introduction: From classical to robust calibration

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# Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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# Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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Introduction: From classical to robust calibration

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All hydrological model suffer from this.
Here, only rain uncertainties will be considered.

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# Introduction: From classical to robust calibration

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All hydrological model suffer from this.
Here, only rain uncertainties will be considered.

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Introduction: From classical to robust calibration

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All hydrological model suffer from this.
Here, only rain uncertainties will be considered.

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# Table of contents

Introduction

Methods

Case study

Results

Conclusion

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# Methods: Definition of robust estimators

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# Methods: Definition of robust estimators

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# Methods: Definition of robust estimators

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# Methods: Definition of robust estimators

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$$1. \theta_{mean}^* = \underset{\theta}{\operatorname{argmin}}(\mathbb{E}_U[J(\theta, U)]),$$

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# Methods: Definition of robust estimators

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1. $\theta_{mean}^* = \operatorname{argmin}_{\theta}(\mathbb{E}_U[J(\theta, U)])$,

2. $\theta_{variance}^* = \operatorname{argmin}_{\theta}(\operatorname{Var}_U[J(\theta, U)])$,

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# Methods: Definition of robust estimators

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1. $\theta_{mean}^* = \underset{\theta}{\operatorname{argmin}}(\mathbb{E}_U[J(\theta, U)])$,

2. $\theta_{variance}^* = \underset{\theta}{\operatorname{argmin}}(\mathrm{Var}_U[J(\theta, U)])$,

Other definition of robustness exist.

The one we choose depends on our goal.

No matter the robustness we choose, they will all be computationally expensive.

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# Methods: Definition of robust estimators

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1. $\theta_{mean}^* = \operatorname{argmin}_{\theta}(\mathbb{E}_U[J(\theta, U)])$,

2. $\theta_{variance}^* = \operatorname{argmin}_{\theta}(\operatorname{Var}_U[J(\theta, U)])$,

Other definition of robustness exist.

The one we choose depends on our goal.

No matter the robustness we choose, they will all be computationally expensive.

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# Table of contents

## Introduction

## Methods

## Case study

## Results

## Conclusion

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Case study: PESHMELBA virtual catchment

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Figure: Rouzies et al. 2023

https://doi.org/10.5194/gmd-2021-425

INRAO

* 3 soil types

* 2 vegetation types

* 4 soil horizons for each soil type

* 9 soil parameters for each horizon

- We focus on parcel 4.

- Calibrate parameters of parcel 4.

- Observations on parcel 4.

- Our observations are moisture profiles of the soil column after 100 hours of rain.
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# Case study: Observations, cost function and rain perturbations

We calibrate only two parameters :

1. the water content in the soil at saturation (thetas10)

2. the parameter governing Van Genuchten infiltration equation (mn10)

$$\theta_{\text{as10}} \sim \mathcal{N}(0.3160, 0.0316^2),$$

$$mn_{10} \sim \mathcal{N}(0.1791, 0.0179^2),$$

$$\theta_{\text{true}} = (0.164, 0.292).$$

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Case study: Observations, cost function and rain perturbations

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$$J(\theta, u) = (\mathcal{M}(\theta, u) - y_{\text{true}})^2 = \sum_{cell_i=1}^{25} (\mathcal{M}(\theta, u)|_{cell_i} - y_{\text{true}}|_{cell_i})^2$$

$$y_{\text{true}} = \mathcal{M}(\theta_{\text{true}}, u_{\text{true}}) \in \mathbb{R}^{25}$$

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# Table of contents

Introduction

Methods

Case study

Results

Conclusion

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# Results:

We compare the following minimization problems :

$$1. \theta^*(u_{true}) = \underset{\theta}{\operatorname{argmin}} J(\theta, u_{true})$$

$$2. \theta^*(u_{false}) = \underset{\theta}{\operatorname{argmin}} J(\theta, u_{false})$$

3. Robust estimators

$$3.1 \theta_{mean}^* = \underset{\theta}{\operatorname{argmin}} (\mathbb{E}_U[J(\theta, U)]),$$

$$3.2 \theta_{variance}^* = \underset{\theta}{\operatorname{argmin}} (\operatorname{Var}_U[J(\theta, U)]),$$

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# Results: Calibration with $u_{true}$ vs $u_{false}$

We compare the following minimization problems :

$$1. \quad \theta^*(u_{true}) = \arg\min_{\theta} J(\theta, u_{true})$$

$$2. \quad \theta^*(u_{false}) = \arg\min_{\theta} J(\theta, u_{false})$$

3. Robust estimators

$$3.1 \quad \theta_{mean}^* = \arg\min_{\theta} (E_U[J(\theta, U)]),$$

$$3.2 \quad \theta_{variance}^* = \arg\min_{\theta} (\mathrm{Var}_U[J(\theta, U)]),$$

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# Results: Calibration with $u_{true}$ vs $u_{false}$

We compare the following minimization problems :

$$1. \quad \theta^*(u_{true}) = \underset{\theta}{\operatorname{argmin}} J(\theta, u_{true})$$

$$2. \quad \theta^*(u_{false}) = \underset{\theta}{\operatorname{argmin}} J(\theta, u_{false})$$

3. Robust estimators

$$3.1 \quad \theta_{mean}^* = \underset{\theta}{\operatorname{argmin}} (\mathbb{E}_U[J(\theta, U)]$$

$$3.2 \quad \theta_{variance}^* = \underset{\theta}{\operatorname{argmin}} (\operatorname{Var}_U[J(\theta, U)]$$

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Results: Calibration with $u_{true}$ vs $u_{false}$

<figure><img src="image_147.png" /><figcaption>Figure: Minimizers when $u = u_{true}$ in red and when $u = u_{false}$ in blue.</figcaption></figure>

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# Results: Robust estimators

We compare the following minimization problems :

$$1. \quad \theta^*(u_{true}) = \underset{\theta}{\operatorname{argmin}} J(\theta, u_{true})$$

$$2. \quad \theta^*(u_{false}) = \underset{\theta}{\operatorname{argmin}} J(\theta, u_{false})$$

3. Robust estimators

$$3.1 \quad \theta_{mean}^* = \underset{\theta}{\operatorname{argmin}} (E_u[J(\theta, U)]),$$

$$3.2 \quad \theta_{variance}^* = \underset{\theta}{\operatorname{argmin}} (\operatorname{Var}_u[J(\theta, U)]),$$

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# Results: Robust estimators

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Robust estimators : the minimizer of the mean and variance for the two estimated parameters.

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# Table of contents

## Introduction

## Methods

## Case study

## Results

## Conclusion

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Suite :

* Différentes définitions de robustesse

* Différentes perturbations de pluie

* Le coût numérique de PESHMELBA -> metamodèles

* passage à plusieurs paramètres, plusieurs parcelles

* passage aux pluies spatialisées

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Pluies spatialisées

<figure><img src="image_169.png" /><figcaption>Figure: Simulations de champs de pluie provenant de SAMPO, (Leblois and Creutin, 2013) doi : 10.1002/wrcr.20190.</figcaption></figure>

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