# Data assimilation to quantify and reduce uncertainty in ecohydrology modelling

Claire Lauvernet, Emilie Rouzies, Arthur Vidard, Alexandre Devers, Jean-Philippe Vidal, Laura Gatel, Katarina Radišić, Claudio Paniconi

► To cite this version:

Claire Lauvernet, Emilie Rouzies, Arthur Vidard, Alexandre Devers, Jean-Philippe Vidal, et al. Data assimilation to quantify and reduce uncertainty in ecohydrology modelling. Séminaire ITES 2023, ITES, Nov 2023, Strasbourg, France. hal-04387935

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Submitted on 11 Jan 2024

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

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# Data assimilation to quantify and reduce uncertainty in ecohydrology modelling
## séminaire ITES, 13 Nov. 2023

Claire Lauvernet¹, Emilie Rouzies¹, Arthur Vidard², Alexandre Devers¹, Jean-Philippe Vidal¹, Laura Gatel², Katarina Radišić¹, Claudio Paniconi³

¹INRAE, UR RiverLy, Lyon

²INRIA, AIRSEA, Grenoble

³INRS/Univ. Laval, Québec, CA
# Data assimilation to quantify and reduce uncertainty in ecohydrology modelling
séminaire ITES, 13 Nov. 2023

Claire Lauvernet¹, Emilie Rouzies¹, Arthur Vidard², Alexandre Devers¹, Jean-Philippe Vidal¹, Laura Gatel², Katarina Radišić¹, Claudio Paniconi³

¹INRAE, UR RiverLy, Lyon

²INRIA, AIRSEA, Grenoble

³INRS/Univ. Laval, Québec, CA

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

RÉPUBLIQUE
FRANÇAISE

Liberté
Égalité
Fronterval

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

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

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

# Plan

## Introduction: some challenges in ecohydrology

## Data assimilation

## An example of data assimilation in ecohydrological model

<figure><img src="image_4.png" /></figure>
<figure><img src="image_3.png" /></figure>

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

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

# Plan

## Introduction: some challenges in ecohydrology

## Data assimilation

## An example of data assimilation in ecohydrological model

<figure><img src="image_6.png" /></figure>
<figure><img src="image_5.png" /></figure>

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

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

# Context: How to improve the water quality ?

⇒ a better understanding of water and pesticide transfer in soil

* Spatial heterogeneity of the soils, at all scales

* Soil and agricultural practices are more and more diverse

* Processes that drive the pesticide fate at the catchment scale are complex :

* Hydrological transfer

* adsorption

* degradation

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

<figure><img src="image_9.png" /></figure>
<figure><img src="image_7.png" /></figure>

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

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

# Context: How to improve the water quality ?

→ a better understanding of water and pesticide transfer in soil

* Spatial heterogeneity of the soils, at all scales

* Soil and agricultural practices are more and more diverse

* Processes that drive the pesticide fate at the catchment scale are complex :

* Hydrological transfer

* adsorption

* degradation

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

<figure><img src="image_11.png" /></figure>
<figure><img src="image_10.png" /></figure>

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

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

Spatially and temporally heterogeneous data...

Availability, quality, quantity of data
are heterogeneous in space and time
:

* remote sensing images

* field data (lysimeters in soil, water table and river measurments)

* geophysical data

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

BUT without heavy experiments, this is very difficult to get the pesticides dynamics

<figure><img src="image_14.png" /></figure>
<figure><img src="image_12.png" /></figure>

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

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

Spatially and temporally heterogeneous data...

Availability, quality, quantity of data
are heterogeneous in space and time
:

* remote sensing images

* field data (lysimeters in soil, water table and river measurments)

* geophysical data

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

BUT without heavy experiments, this is very difficult to get the pesticides dynamics

<figure><img src="image_16.png" /></figure>
<figure><img src="image_15.png" /></figure>

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

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

# Spatially and temporally heterogeneous data... and pesticides modeling at several scales and several complexity degrees

* based on non linear equations and/or conceptual

* unknown boundary and initial conditions

* a large set of spatialized parameters that are difficult to measure/estimate

* many processes affecting pesticide transfer are not (well) represented (e.g., pref. flows)

⇒ a high uncertainty (when we it is considered !)

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

<figure><img src="image_19.png" /></figure>
<figure><img src="image_17.png" /></figure>

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

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

Spatially and temporally heterogeneous data...
...and pesticides modeling at several scales and several complexity degrees

* based on non linear equations and/or conceptual

* unknown boundary and initial conditions

* a large set of spatialized parameters that are
difficult to measure/estimate

* many processes affecting pesticide transfer are not
(well) represented (e.g., pref. flows)

→ a high uncertainty (when we it is considered !)

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

<figure><img src="image_21.png" /></figure>
<figure><img src="image_20.png" /></figure>

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

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

Spatially heterogeneous data...

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

... and spatialized modeling

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

⇒ merging information from the available data and from the model to get as close as possible to the “true” state

Data Assimilation techniques (or model-data fusion)

<figure><img src="image_25.png" /></figure>
<figure><img src="image_22.png" /></figure>

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

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

Spatially heterogeneous data...

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

... and spatialized modeling

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

⇒ merging information from the available data and from the model to get as close as possible to the “true” state

Data Assimilation techniques (or model-data fusion)

<figure><img src="image_27.png" /></figure>
<figure><img src="image_26.png" /></figure>

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

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

# Plan

## Introduction: some challenges in ecohydrology

## Data assimilation

## An example of data assimilation in ecohydrological model

<figure><img src="image_29.png" /></figure>
<figure><img src="image_28.png" /></figure>

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

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

# Plan

## Introduction: some challenges in ecohydrology

## Data assimilation

## An example of data assimilation in ecohydrological model

<figure><img src="image_31.png" /></figure>
Data assimilation: definition

* the systematic use of data to constrain a numerical model

* first used in the 1960s in numerical weather forecasting models for short-term predictions of meteorological conditions

* in the 1970s, development in numerical ocean general circulation models (OGCMs)

* poorly developed in other domains (hydrology)

<figure><img src="image_30.png" /></figure>
<figure><img src="image_32.png" /></figure>

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

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

# Data assimilation: definition

* the systematic use of data to constrain a numerical model

* first used in the 1960s in numerical weather forecasting models for short-term predictions of meteorological conditions

* in the 1970s, development in numerical ocean general circulation models (OGCMs)

* poorly developed in other domains (hydrology)

<figure><img src="image_35.png" /></figure>
<figure><img src="image_31.png" /></figure>

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

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

# Data assimilation: definition

<figure><img src="image_34.png" /><figcaption>Model</figcaption></figure>

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

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

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

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

Data assimilation: definition

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

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

<figure><img src="image_41.png" /></figure>
<figure><img src="image_37.png" /></figure>

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

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

# Data assimilation: definition

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

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

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

<figure><img src="image_43.png" /></figure>
<figure><img src="image_42.png" /></figure>

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

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

Data assimilation: definition

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

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

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

<figure><img src="image_48.png" /></figure>
<figure><img src="image_44.png" /></figure>

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

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

# Data assimilation: definition

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

<figure><img src="image_48.png" /><figcaption>Observations</figcaption></figure>

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

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

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

Data assimilation: definition

<figure><img src="image_52.png" /><figcaption>Model</figcaption></figure>

<figure><img src="image_53.png" /><figcaption>Observations</figcaption></figure>

<figure><img src="image_54.png" /></figure>
<figure><img src="image_50.png" /></figure>

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

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

Data assimilation: definition

<figure><img src="image_53.png" /><figcaption>Model</figcaption></figure>

<figure><img src="image_54.png" /><figcaption>Observations</figcaption></figure>

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

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

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

Data assimilation: definition

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

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

<figure><img src="image_60.png" /></figure>
<figure><img src="image_56.png" /></figure>

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

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

# Data assimilation: definition

> "Approximation of the true state of a physical system at a given time by combining time-distributed observations with a dynamic model in an optimal way" (Asch2016)

⇒ DA has two main goals:

* optimally blend information from observations and model to produce an accurate and physically consistent estimate of the state of the system $x^a$

* quantify the uncertainty of this estimate for future users

<figure><img src="image_59.png" /></figure>
<figure><img src="image_61.png" /></figure>

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

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

# Data assimilation: definition

> "Approximation of the true state of a physical system at a given time by combining time-distributed observations with a dynamic model in an optimal way" (Asch2016)

## ⇒ DA has two main goals:

* optimally blend information from observations and model to produce an accurate and physically consistent estimate of the state of the system $x^a$

* quantify the uncertainty of this estimate for future users

<figure><img src="image_64.png" /></figure>
<figure><img src="image_60.png" /></figure>

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

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

Data assimilation: the ingredients

$x = (x_0, x_1, ..., x_N)^T$ represents the state of system:

streamflow at the outlet, soil moisture, dissolved oxygen concentration in the river, etc. We don't know it, but we do have information from:

* the dynamical model $x_k = M_{k-1 \to k}[x_{k-1}, \text{param}] + \eta_k$
  $\eta_k$ the model error of covariance matrix $P_k$

* the background $x^b$ is the state at $t_{k-1}$ and its associated error $\varepsilon^b = x^b - x_{k-1}^t$

<figure><img src="image_63.png" /></figure>
<figure><img src="image_65.png" /></figure>

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

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

Data assimilation: the ingredients

$x = (x_0, x_1, ..., x_N)^T$ represents the state of system:

streamflow at the outlet, soil moisture, dissolved oxygen concentration in the river, etc. We don't know it, but we do have information from:

* the dynamical model $x_k = M_{k-1 \to k}[x_{k-1}, \text{param}] + \eta_k$
  $\eta_k$ the model error of covariance matrix $P_k$

* the background $x^b$ is the state at $t_{k-1}$ and its associated error $\varepsilon^b = x^b - x_{k-1}^t$

<figure><img src="image_68.png" /></figure>
<figure><img src="image_64.png" /></figure>

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

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

# Data assimilation: the ingredients

$x = (x_0, x_1, ..., x_N)^T$ represents the state of system:

streamflow at the outlet, soil moisture, dissolved oxygen concentration in the river, etc. We don't know it, but we do have information from:

* the dynamical model $x_k = M_{k-1 \to k}[x_{k-1}, param] + \eta_k$

* the background $x^b$ is the state at $t_{k-1}$ and its associated error $\epsilon^b = x^b - x_{k-1}^t$

* the observation model $y_k = H_k[x_k] + \epsilon_k$

$y_k$ is the observation/data at time $k$

$\epsilon_k$ the observation error, of covariance matrix $R_k$, e.g. instrumental error, representativeness

$H: \mathbb{R}^m \to \mathbb{R}^d$ the observation operator that projects from model space to observational space (spatial interpolations, convolutions or spectral-to-physical space transformation in spectral models, etc.)

<figure><img src="image_67.png" /></figure>
<figure><img src="image_69.png" /></figure>

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

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

# Data assimilation: the ingredients

$x = (x_0, x_1, ..., x_N)^T$ represents the state of system:

streamflow at the outlet, soil moisture, dissolved oxygen concentration in the river, etc. We don't know it, but we do have information from:

* the dynamical model $x_k = M_{k-1 \to k}[x_{k-1}, param] + \eta_k$

* the background $x^b$ is the state at $t_{k-1}$ and its associated error $\epsilon^b = x^b - x_{k-1}^t$

* the observation model $y_k = H_k[x_k] + \epsilon_k$

$y_k$ is the observation/data at time $k$

$\epsilon_k$ the observation error, of covariance matrix $R_k$, e.g. instrumental error, representativeness

$H: \mathbb{R}^m \to \mathbb{R}^d$ the observation operator that projects from model space to observational space (spatial interpolations, convolutions or spectral-to-physical space transformation in spectral models, etc.)

<figure><img src="image_72.png" /></figure>
<figure><img src="image_68.png" /></figure>

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

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

# Data assimilation: the ingredients

$x = (x_0, x_1, ..., x_N)^T$ represents the state of system:

streamflow at the outlet, soil moisture, dissolved oxygen concentration in the river, etc. We don't know it, but we do have information from:

* the dynamical model $x_k = M_{k-1} \rightarrow_k [x_{k-1}, param] + \eta_k$

* the background $x^b$ is the state at $t_{k-1}$ and its associated error $\epsilon^b = x^b - x_{k-1}^t$

* the observation model $y_k = H_k[x_k] + \epsilon_k$

* We assume that **model and obs. errors are random variables → described by pdf or by their covariance matrix**

<figure><img src="image_71.png" /></figure>
<figure><img src="image_73.png" /></figure>

<figure><img src="image_74.png" /></figure>

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

Data assimilation: the ingredients

$x = (x_0, x_1, ..., x_N)^T$ represents the state of system:

streamflow at the outlet, soil moisture, dissolved oxygen concentration in the river, etc. We don't know it, but we do have information from:

* the dynamical model $x_k = M_{k-1 \to k}[x_{k-1}, param] + \eta_k$

* the background $x^b$ is the state at $t_{k-1}$ and its associated error $\epsilon^b = x^b - x_{k-1}^t$

* the observation model $y_k = H_k[x_k] + \epsilon_k$

* We assume that **model and obs. errors are random variables → described by pdf or by their covariance matrix**

<figure><img src="image_76.png" /></figure>
<figure><img src="image_72.png" /></figure>

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

<figure><img src="image_74.png" /></figure>

# Data assimilation: approaches

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

(from Asch2016)

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

INRAO
<figure><img src="image_77.png" /></figure>

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

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

# Data assimilation: approaches

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

(from Asch2016)

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

INRAO
<figure><img src="image_77.png" /></figure>

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

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

# Data assimilation: approaches

* **Deterministic methods (used in weather forecasting)**: state variables are assumed to be governed by deterministic laws.
  ⇒ methods based on optimal control, minimization of a cost function

* **Statistical/Stochastic methods (used in ocean forecasting)**: a phenomenon is assumed to be the realization of a random variable: this is justified by the fact that the dynamics of the system under study (weather, ocean) are chaotic and therefore resemble a random system.
  ⇒ methods based on statistical estimation, Bayes theorem and Kalman filter
  ⇒ objective = determine a good approximation of the conditional expectation of the system state (as well as its error covariance matrix) given the observed data
  ⇒ in a perfect context (linear, Gaussian, etc.), the methods are equivalent!

* In hydrology? Chosen method should be suited to heterogeneous structure of the model, highly nonlinear processes but also suited to our high computational cost.

<figure><img src="image_80.png" /></figure>
<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /></figure>

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

Data assimilation: approaches

* **Deterministic methods** (used in weather forecasting): state variables are assumed to be governed by deterministic laws.
  ⇒ methods based on optimal control, minimization of a cost function

* **Statistical/Stochastic methods** (used in ocean forecasting): a phenomenon is assumed to be the realization of a random variable: this is justified by the fact that the dynamics of the system under study (weather, ocean) are chaotic and therefore resemble a random system.
  ⇒ methods based on statistical estimation, Bayes theorem and Kalman filter
  ⇒ objective = determine a good approximation of the conditional expectation of the system state (as well as its error covariance matrix) given the observed data
  ⇒ in a perfect context (linear, Gaussian, etc.), the methods are equivalent!

* In hydrology? Chosen method should be suited to heterogeneous structure of the model, highly nonlinear processes but also suited to our high computational cost.

<figure><img src="image_85.png" /></figure>
<figure><img src="image_81.png" /></figure>

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

<figure><img src="image_83.png" /></figure>

Data assimilation: definition

<figure><img src="image_84.png" /></figure>
<figure><img src="image_86.png" /></figure>

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

<figure><img src="image_88.png" /></figure>

Data assimilation: definition

<figure><img src="image_89.png" /></figure>
<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /></figure>

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

Data assimilation: definition

<figure><img src="image_88.png" /></figure>
<figure><img src="image_90.png" /></figure>

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

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

Data assimilation: definition

<figure><img src="image_93.png" /></figure>
<figure><img src="image_89.png" /></figure>

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

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

Data assimilation: definition

<figure><img src="image_92.png" /></figure>
<figure><img src="image_94.png" /></figure>

<figure><img src="image_95.png" /></figure>

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

Data assimilation: definition

<figure><img src="image_97.png" /></figure>
<figure><img src="image_93.png" /></figure>

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

<figure><img src="image_95.png" /></figure>

Data assimilation: definition

<figure><img src="image_96.png" /></figure>
<figure><img src="image_98.png" /></figure>

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

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Data assimilation: definition

<figure><img src="image_101.png" /></figure>
<figure><img src="image_97.png" /></figure>

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

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

Data assimilation: definition

<figure><img src="image_100.png" /></figure>
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Data assimilation: definition

<figure><img src="image_105.png" /></figure>
<figure><img src="image_101.png" /></figure>

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

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

Data assimilation: definition

<figure><img src="image_104.png" /></figure>
<figure><img src="image_106.png" /></figure>

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

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

Data assimilation: definition

<figure><img src="image_109.png" /></figure>
<figure><img src="image_105.png" /></figure>

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

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Data assimilation: definition

<figure><img src="image_108.png" /></figure>
<figure><img src="image_110.png" /></figure>

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

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

Data assimilation: definition

<figure><img src="image_113.png" /></figure>
<figure><img src="image_109.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

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

0. At time k: $X_k^f$

<figure><img src="image_116.png" /></figure>
<figure><img src="image_114.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

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

0. At time k: $X_k^f$

<figure><img src="image_121.png" /></figure>
<figure><img src="image_117.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

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

1. Forecast step: $X_{k+1}^f = MX_k^f$

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

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

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

1. Forecast step: $X_{k+1}^f = MX_k^f$

<figure><img src="image_127.png" /></figure>
<figure><img src="image_123.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

2. Analysis step: $\mathbf{X}_{k+1}^a = \mathbf{X}_{k+1}^f + \mathbf{K}_{k+1}(\mathbf{Y}_{k+1}-\mathbf{H}_{k+1}\mathbf{X}_{k+1}^f)$
<figure><img src="image_128.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

2. Analysis step: $X_{k+1}^a = X_{k+1}^f + K_{k+1}(Y_{k+1}-H_{k+1}X_{k+1}^f)$
<figure><img src="image_127.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

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

2. Analysis step: $X_{k+1}^f = X_{k+1}^f + K_{k+1}(Y_{k+1}-H_{k+1}X_{k+1}^f)$
with $K_{k+1}=P_{k+1}^f H_{k+1}^T [H_{k+1}P_{k+1}^f H_{k+1}^T + R_{k+1}]^{-1}$

<figure><img src="image_134.png" /></figure>
<figure><img src="image_132.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

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

2. Analysis step: $X_{k+1}^f = X_{k+1}^f + K_{k+1}(Y_{k+1}-H_{k+1}X_{k+1}^f)$
with $K_{k+1}=P_{k+1}^f H_{k+1}^T [H_{k+1}P_{k+1}^f H_{k+1}^T + R_{k+1}]^{-1}$

<figure><img src="image_139.png" /></figure>
<figure><img src="image_135.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

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

2. Analysis step: $X_{k+1}^a = X_{k+1}^f + K_{k+1}(Y_{k+1}-H_{k+1}X_{k+1}^f)$

$$P_{k+1}^a = P_{k+1}^f - K_{k+1}H_{k+1}P_{k+1}^f$$

<figure><img src="image_142.png" /></figure>
<figure><img src="image_140.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

<figure><img src="image_146.png" /></figure>

2. Analysis step: $X_{k+1}^a = X_{k+1}^f + K_{k+1}(Y_{k+1}-H_{k+1}X_{k+1}^f)$

$$P_{k+1}^a = P_{k+1}^f - K_{k+1}H_{k+1}P_{k+1}^f$$

<figure><img src="image_147.png" /></figure>
<figure><img src="image_143.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

<figure><img src="image_146.png" /></figure>

observation

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

forecast/prior for next step

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

analysis

<figure><img src="image_149.png" /></figure>
<figure><img src="image_148.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

<figure><img src="image_154.png" /></figure>
<figure><img src="image_150.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

<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>

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

<figure><img src="image_161.png" /></figure>
<figure><img src="image_155.png" /></figure>

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

X

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

The KF assumes that:

✓ all sources of errors are gaussian

✓ both the observational and dynamical models are linear

→ not realistic in most cases !

k

k+1

k+2

k+3

t

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

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

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

The Kalman Filter (Kalman1960) : estimate the optimal state at each observation time

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

observation

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

forecast/prior for next step

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

analysis

X

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

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

The KF assumes that:

✓ all sources of errors are gaussian

✓ both the observational and dynamical models are linear

→ not realistic in most cases !

k

k+1

k+2

k+3

t

<figure><img src="image_170.png" /></figure>
<figure><img src="image_163.png" /></figure>

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

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

The method for data assimilation should be suited to spatialized models

• models are physically-based but:

• highly nonlinear equations (Richards, ...)

• some are more/less conceptual =>
  discontinuities, thresholds

→ **definitely not gaussian !**

→ **Ensemble filter approaches**

1

$$S_w S_r \frac{\partial \varphi}{\partial r} + \frac{\partial S_w}{\partial r} = \nabla [K, K(\nabla \varphi + n_r)] + \phi_{w}$$

$$\frac{\partial Q}{\partial r} + r_s \frac{\partial Q}{\partial n} = D_s \frac{\partial P Q}{\partial \sigma^2} + c_s q_{w}(h, \epsilon)$$

$$\frac{\partial Q_s}{\partial r} = \nabla \cdot (-\bar{U}e + D\bar{\nabla}\epsilon) + \phi_{s}$$

$$\frac{\partial Q_{w\infty}}{\partial r} + c_s \frac{\partial Q_{w\infty}}{\partial n} = D_s \frac{\partial P Q_{w\infty}}{\partial \sigma^2} + c_s q_{w\infty}$$

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

CATHY-Pesticides

Camporese et al., 2010

Gatel et al., 2019

<figure><img src="image_167.png" /></figure>
<figure><img src="image_171.png" /></figure>

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

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

The method for data assimilation should be suited to spatialized models

* models are physically-based but:

* highly nonlinear equations (Richards, ...)

* some are more/less conceptual => discontinuities, thresholds

→ **definitely not gaussian !**

→ **Ensemble filter approaches**

1

$$S_w S_r \frac{\partial \varphi}{\partial r} + \frac{\partial S_w}{\partial r} = \nabla [K, K(\nabla \varphi + n_r)] + \phi_{w}$$

$$\frac{\partial Q}{\partial r} + r_s \frac{\partial Q}{\partial n} = D_s \frac{\partial P Q}{\partial \varepsilon^2} + c_s q_{w}(h, \varepsilon)$$

$$\frac{\partial Q_s}{\partial r} = \nabla \cdot (-\bar{U}e + D\bar{\nabla}e) + \phi_{s}$$

$$\frac{\partial Q_{w\infty}}{\partial r} + c_s \frac{\partial Q_{w\infty}}{\partial n} = D_s \frac{\partial P Q_{w\infty}}{\partial \varepsilon^2} + c_s q_{w\infty}$$

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

CATHY-Pesticides

Camporese et al., 2010

Gatel et al., 2019

<figure><img src="image_175.png" /></figure>
<figure><img src="image_168.png" /></figure>

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

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

Ensemble-based methods (Evensen_2003)

* a version of the Kalman filter for nonlinear problems at large dimension

* the state variable distribution is represented by an ensemble of state vectors $x_k$

* the error covariance matrices are represented by the ensemble covariance

<figure><img src="image_171.png" /></figure>
<figure><img src="image_176.png" /></figure>

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

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

Ensemble-based methods (Evensen_2003)

* a version of the Kalman filter for nonlinear problems at large dimension

* the state variable distribution is represented by an ensemble of state vectors $x_k$

* the error covariance matrices are represented by the ensemble covariance

<figure><img src="image_179.png" /></figure>
<figure><img src="image_172.png" /></figure>

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

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

On the variationnal side...

* Operational in meteorological centers : can deal with very large problems

* Use of ALL available information by solving a unique system

* Transform the inverse problem into an optimization problem The search of the minimum of the **cost function**

$$J = |sim. - obs.|^2 + apriori info$$

is performed with standard methods (conjugate gradient, Newton-type methods)

* They require the computation of the gradient of J (optimality condition):

$$\nabla J = 0 \Leftrightarrow J \text{ is in an optimum}$$

⇒ analytic approximation: computer time consuming

⇒ **adjoint model method**

<figure><img src="image_175.png" /></figure>
<figure><img src="image_180.png" /></figure>

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

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

On the variationnal side...

* Operational in meteorological centers : can deal with very large problems

* Use of ALL available information by solving a unique system

* Transform the inverse problem into an optimization problem The search of the minimum of the **cost function**

$$J = |sim. - obs.|^2 + apriori info$$

is performed with standard methods (conjugate gradient, Newton-type methods)

* They require the computation of the gradient of J (optimality condition):

$$\nabla J = 0 \Leftrightarrow J \text{ is in an optimum}$$

⇒ analytic approximation: computer time consuming

⇒ **adjoint model method**

<figure><img src="image_183.png" /></figure>
<figure><img src="image_176.png" /></figure>

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

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

... an optimization problem

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

Problem : determine *K*\* that achieves the best LAI simulation closest to the observations, i.e. minimizing *J*

<figure><img src="image_180.png" /></figure>
<figure><img src="image_184.png" /></figure>

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

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

...an optimization problem

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

Problem : determine *K*\* that achieves the best LAI simulation closest to the observations, i.e. minimizing *J*

<figure><img src="image_188.png" /></figure>
<figure><img src="image_181.png" /></figure>

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

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

Variational data assimilation using the adjoint model

$$
\left\{
\begin{array}{l}
\displaystyle \frac{dX}{dt} = F(X, K) \\
X(0) = u \\[1.5em]
\displaystyle \frac{dP}{dt} + \left[ \frac{\partial F}{\partial X} \right]^T P = C^t (C X - X_{obs}) \\
P(T) = 0 \\[1.5em]
\nabla_K J = - \left[ \frac{\partial F}{\partial K} \right]^t P + K - K_0
\end{array}
\right.
\quad
\begin{array}{l}
\text{model} \\
\text{adjoint} \\
\text{model} \\
\text{optimality} \\
\text{condition}
\end{array}
$$

⇒ The optimality system contains all available information: observations, model, statistics ...
In practice, the gradient is computed by running an «adjoint model» derived from the model
(automatic differentiation tool).
<figure><img src="image_189.png" /></figure>

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

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

Variational data assimilation using the adjoint model

$$
\left\{
\begin{array}{l}
\displaystyle \frac{dX}{dt} = F(X, K) \\
X(0) = u \\[1.5em]
\displaystyle \frac{dP}{dt} + \left[ \frac{\partial F}{\partial X} \right]^T P = C^t (C X - X_{obs}) \\
P(T) = 0 \\[1.5em]
\nabla_K J = - \left[ \frac{\partial F}{\partial K} \right]^t P + K - K_0
\end{array}
\right.
\quad
\begin{array}{l}
\text{model} \\
\text{adjoint model} \\
\text{optimality condition}
\end{array}
$$

⇒ The optimality system contains all available information: observations, model, statistics ...
In practice, the gradient is computed by running an «adjoint model» derived from the model
(automatic differentiation tool).
<figure><img src="image_184.png" /></figure>

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

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

# Plan

## Introduction: some challenges in ecohydrology

## Data assimilation

## An example of data assimilation in ecohydrological model

<figure><img src="image_187.png" /></figure>
<figure><img src="image_192.png" /></figure>

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

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

# Plan

## Introduction: some challenges in ecohydrology

## Data assimilation

## An example of data assimilation in ecohydrological model

<figure><img src="image_195.png" /></figure>
<figure><img src="image_188.png" /></figure>

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

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

Context

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

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

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

INRAO

<figure><img src="image_192.png" /></figure>
<figure><img src="image_196.png" /></figure>

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

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

Context

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

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

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

INRAO

<figure><img src="image_200.png" /></figure>
<figure><img src="image_193.png" /></figure>

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

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

Context

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

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

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

INRAO

<figure><img src="image_197.png" /></figure>
<figure><img src="image_201.png" /></figure>

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

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

## Context

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

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

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

INRAO

<figure><img src="image_205.png" /></figure>
<figure><img src="image_198.png" /></figure>

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

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

How to tackle pesticide transfers and fate on small agricultural catchments with modelling tools ?

✓ Integrating landscape elements diversity in a modular model

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

<figure><img src="image_202.png" /></figure>
<figure><img src="image_206.png" /></figure>

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

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

How to tackle pesticide transfers and fate on small agricultural catchments with modelling tools ?

✓ Integrating landscape elements diversity in a modular model

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

<figure><img src="image_210.png" /></figure>
<figure><img src="image_203.png" /></figure>

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

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

# How to tackle pesticide transfers and fate on small agricultural catchments with modelling tools ?

* ✓ Integrating landscape elements diversity in a modular model

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

* ✓ Exploring landscape management scenarios

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

<figure><img src="image_208.png" /></figure>
<figure><img src="image_211.png" /></figure>

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

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

# How to tackle pesticide transfers and fate on small agricultural catchments with modelling tools ?

* ✓ Integrating landscape elements diversity in a modular model

<figure><img src="image_214.png" /></figure>

* ✓ Exploring landscape management scenarios

<figure><img src="image_215.png" /></figure>

<figure><img src="image_216.png" /></figure>
<figure><img src="image_209.png" /></figure>

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

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

# How to tackle pesticide transfers and fate on small agricultural catchments with modelling tools ?

* ✓ Integrating landscape elements diversity in a modular model

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

* ✓ Exploring landscape management scenarios

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

→ Development of the PESHMELBA model (Rouzies2019)

<figure><img src="image_214.png" /></figure>
<figure><img src="image_217.png" /></figure>

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

<figure><img src="image_219.png" /></figure>

# How to tackle pesticide transfers and fate on small agricultural catchments with modelling tools ?

* ✓ Integrating landscape elements diversity in a modular model

<figure><img src="image_220.png" /></figure>

* ✓ Exploring landscape management scenarios

<figure><img src="image_221.png" /></figure>

→ Development of the PESHMELBA model (Rouzies2019)

<figure><img src="image_222.png" /></figure>
<figure><img src="image_215.png" /></figure>

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

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

# The PESHMELBA model (Rouzies2019)

PESticides et Hydrologie: Modélisation à l'EcheLle du BAssin versant

* ✓ Simulation of heterogenous landscapes composed of plots, vegetative filter zones, hedges, ditches and rivers

* ✓ Water transfers on surface and subsurface

* ✓ Solute advection, adsorption and degradation, solid transport

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

<figure><img src="image_219.png" /></figure>
<figure><img src="image_223.png" /></figure>

<figure><img src="image_224.png" /></figure>

<figure><img src="image_225.png" /></figure>

# The PESHMELBA model (Rouzies2019)

PESticides et Hydrologie: Modélisation à l'EcheLle du BAssin versant

* ✓ Simulation of heterogenous landscapes composed of plots, vegetative filter zones, hedges, ditches and rivers

* ✓ Water transfers on surface and subsurface

* ✓ Solute advection, adsorption and degradation, solid transport

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

<figure><img src="image_227.png" /></figure>
<figure><img src="image_220.png" /></figure>

<figure><img src="image_221.png" /></figure>

<figure><img src="image_222.png" /></figure>

# The PESHMELBA model (Rouzies2019)

## PESticides et Hydrologie: Modélisation à l'EcheLle du BAssin versant

* ✓ Simulation of heterogenous landscapes composed of plots, vegetative filter zones, hedges, ditches and rivers

* ✓ Water transfers on surface and subsurface

* ✓ Solute advection, adsorption and degradation, solid transport

* ✓ One module ≡ one process or ensemble of processes on a landscape element

* ✓ Coupling of modules within the OPENFLUID coupler to make it flexible

<figure><img src="image_223.png" /></figure>

Synchronization + Infiltration

<figure><img src="image_224.png" /></figure>

Vertical infiltration
1D Richards'
equation
own time step

<figure><img src="image_225.png" /></figure>

<figure><img src="image_226.png" /></figure>
<figure><img src="image_228.png" /></figure>

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

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

# The PESHMELBA model (Rouzies2019)

## PESticides et Hydrologie: Modélisation à l'EcheLle du BAssin versant

* ✓ Simulation of heterogenous landscapes composed of plots, vegetative filter zones, hedges, ditches and rivers

* ✓ Water transfers on surface and subsurface

* ✓ Solute advection, adsorption and degradation, solid transport

* ✓ One module $≡$ one process or ensemble of processes on a landscape element

* ✓ Coupling of modules within the OPENFLUID coupler to make it flexible

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

<figure><img src="image_232.png" /></figure>
<figure><img src="image_227.png" /></figure>

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

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

Uncertainty in PESHMELBA

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

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation.

<figure><img src="image_231.png" /></figure>
<figure><img src="image_233.png" /></figure>

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

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

# Uncertainty in PESHMELBA

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

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation.

<figure><img src="image_237.png" /></figure>
<figure><img src="image_232.png" /></figure>

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

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

Uncertainty in PESHMELBA

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

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation.

This is our objective...but before, it is necessary to **quantify** and **reduce** the uncertainty associated to PESHMELBA output variables.

<figure><img src="image_236.png" /></figure>
<figure><img src="image_238.png" /></figure>

<figure><img src="image_239.png" /></figure>

<figure><img src="image_240.png" /></figure>

# Uncertainty in PESHMELBA

<figure><img src="image_241.png" /></figure>

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation.

This is our objective...but before, it is necessary to **quantify** and **reduce** the uncertainty associated to PESHMELBA output variables.

<figure><img src="image_242.png" /></figure>
<figure><img src="image_237.png" /></figure>

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

<figure><img src="image_239.png" /></figure>

# Uncertainty in PESHMELBA

<figure><img src="image_240.png" /></figure>

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation.

*This is our objective... but before, it is necessary to **quantify** and **reduce** the uncertainty associated to PESHMELBA output variables.*

## Emilie Rouzies's PhD objectives

1. **Quantify**: performing an **uncertainty analysis** and a **sensitivity analysis** of the model

2. **Reduce**: performing **data assimilation** to integrate different sources of data: soil moisture images, ERT measurements and in-situ data of pesticide concentration

<figure><img src="image_241.png" /></figure>
<figure><img src="image_243.png" /></figure>

<figure><img src="image_244.png" /></figure>

<figure><img src="image_245.png" /></figure>

# Uncertainty in PESHMELBA

<figure><img src="image_246.png" /></figure>

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation.

*This is our objective... but before, it is necessary to **quantify** and **reduce** the uncertainty associated to PESHMELBA output variables.*

## Emilie Rouzies's PhD objectives

1. **Quantify**: performing an **uncertainty analysis** and a **sensitivity analysis** of the model

2. **Reduce**: performing **data assimilation** to integrate different sources of data: soil moisture images, ERT measurements and in-situ data of pesticide concentration

<figure><img src="image_247.png" /></figure>
<figure><img src="image_242.png" /></figure>

<figure><img src="image_243.png" /></figure>

<figure><img src="image_244.png" /></figure>

## Case study

First attempt of DA in the PESHMELBA model: let's keep it simple...but realistic! (types of landscape elements, number of parameters, climate conditions...)

<figure><img src="image_245.png" /></figure>

<figure><img src="image_246.png" /></figure>

Panorama du bassin versant de la Morcille (Crédit photo: INRAE)

<figure><img src="image_247.png" /></figure>

INRAE
<figure><img src="image_248.png" /></figure>

<figure><img src="image_249.png" /></figure>

<figure><img src="image_250.png" /></figure>

## Case study

First attempt of DA in the PESHMELBA model: let's keep it simple...but realistic! (types of landscape elements, number of parameters, climate conditions...)

<figure><img src="image_251.png" /></figure>

<figure><img src="image_252.png" /></figure>

Panorama du bassin versant de la Morcille (Crédit photo: INRAE)

<figure><img src="image_253.png" /></figure>

INRAE
<figure><img src="image_248.png" /></figure>

<figure><img src="image_249.png" /></figure>

<figure><img src="image_250.png" /></figure>

## Case study

First attempt of DA in the PESHMELBA model: let's keep it simple...but realistic! (types of landscape elements, number of parameters, climate conditions...)

<figure><img src="image_251.png" /></figure>

<figure><img src="image_252.png" /></figure>

Panorama du bassin versant de la Moreille (Crédit photo INRAE)

<figure><img src="image_253.png" /></figure>

INRAE
<figure><img src="image_254.png" /></figure>

<figure><img src="image_255.png" /></figure>

<figure><img src="image_256.png" /></figure>

## Case study

First attempt of DA in the PESHMELBA model: let's keep it simple...but realistic! (types of landscape elements, number of parameters, climate conditions...)

<figure><img src="image_257.png" /><figcaption>Panorama du bassin versant de la Moreille (Crédit photo INRAE)</figcaption></figure>

<figure><img src="image_258.png" /></figure>
<figure><img src="image_254.png" /></figure>

<figure><img src="image_255.png" /></figure>

<figure><img src="image_256.png" /></figure>

# Which observations are available ?

## Surface moisture images

* ✓ Surface moisture images got from the synergic use of Sentinel-1 and Sentinel-2 data

<figure><img src="image_257.png" /></figure>

* ✓ One observation of mean moisture in the top 5 cm per landscape element per time step

* ✓ Freq. of observation: 144h, obs. error : assumed Gaussian, std ~ 0.02 cm³cm⁻³ (! may highly differ on vineyard !)

## In-situ moisture profiles

* ✓ Moisture profiles from EMI measurements or probe.

* ✓ Assumption : 2m-moisture profile on some landscape elements, obs. error : assumed Gaussian, std ~ 0.02 cm³cm⁻³

<figure><img src="image_258.png" /></figure>

<figure><img src="image_259.png" /></figure>
<figure><img src="image_259.png" /></figure>

<figure><img src="image_260.png" /></figure>

<figure><img src="image_261.png" /></figure>

# Which observations are available ?

## Surface moisture images

* ✓ Surface moisture images got from the synergic use of Sentinel-1 and Sentinel-2 data

<figure><img src="image_262.png" /></figure>

* ✓ One observation of mean moisture in the top 5 cm per landscape element per time step

* ✓ Freq. of observation: 144h, obs. error : assumed Gaussian, std ~ 0.02 cm³cm⁻³ (! may highly differ on vineyard !)

## In-situ moisture profiles

* ✓ Moisture profiles from EMI measurements or probe.

* ✓ Assumption : 2m-moisture profile on some landscape elements, obs. error : assumed Gaussian, std ~ 0.02 cm³cm⁻³

<figure><img src="image_263.png" /></figure>

<figure><img src="image_264.png" /></figure>
<figure><img src="image_260.png" /></figure>

<figure><img src="image_261.png" /></figure>

<figure><img src="image_262.png" /></figure>

# Data assimilation

## Objectives

* ✓ Improve moisture dynamics modelling both in surface and subsurface

* ✓ Improve estimation of pesticide export at the outlet

* ✓ Estimate input parameters ($\theta_{sat}$) that would be set for the exploration of landscape management scenarios

<figure><img src="image_263.png" /></figure>
<figure><img src="image_265.png" /></figure>

<figure><img src="image_266.png" /></figure>

<figure><img src="image_267.png" /></figure>

# Data assimilation

## Objectives

* ✓ Improve moisture dynamics modelling both in surface and subsurface

* ✓ Improve estimation of pesticide export at the outlet

* ✓ Estimate input parameters ($\theta_{sat}$) that would be set for the exploration of landscape management scenarios

<figure><img src="image_268.png" /></figure>
<figure><img src="image_264.png" /></figure>

<figure><img src="image_265.png" /></figure>

<figure><img src="image_266.png" /></figure>

# DA method

## Ensemble Smoother with Multiple Data Assimilation (Emerick2013)

* Ensemble method that inherits from Kalman Filter

* Iterative smoother well suited to parameter estimation problems in non linear contexts

Ensemble Smoother with Multiple Data Assimilation

<figure><img src="image_267.png" /></figure>

<figure><img src="image_268.png" /></figure>
<figure><img src="image_269.png" /></figure>

<figure><img src="image_270.png" /></figure>

<figure><img src="image_271.png" /></figure>

# DA method

## Ensemble Smoother with Multiple Data Assimilation (Emerick2013)

* Ensemble method that inherits from Kalman Filter

* Iterative smoother well suited to parameter estimation problems in non linear contexts

Ensemble Smoother with Multiple Data Assimilation

<figure><img src="image_272.png" /></figure>

<figure><img src="image_273.png" /></figure>
<figure><img src="image_269.png" /></figure>

<figure><img src="image_270.png" /></figure>

<figure><img src="image_271.png" /></figure>

Twin experiment

<figure><img src="image_272.png" /></figure>
<figure><img src="image_274.png" /></figure>

<figure><img src="image_275.png" /></figure>

<figure><img src="image_276.png" /></figure>

Twin experiment

<figure><img src="image_277.png" /></figure>
<figure><img src="image_273.png" /></figure>

<figure><img src="image_274.png" /></figure>

<figure><img src="image_275.png" /></figure>

# Twin experiment

First step: twin experiments to set and validate the DA framework ( !! Reanalysis context)

<figure><img src="image_276.png" /></figure>
<figure><img src="image_278.png" /></figure>

<figure><img src="image_279.png" /></figure>

<figure><img src="image_280.png" /></figure>

# Twin experiment

First step: twin experiments to set and validate the DA framework ( !! Reanalysis context)

<figure><img src="image_281.png" /></figure>
<figure><img src="image_277.png" /></figure>

<figure><img src="image_278.png" /></figure>

<figure><img src="image_279.png" /></figure>

Twin experiment

First step: twin experiments to set and validate
the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

<figure><img src="image_280.png" /></figure>

<figure><img src="image_281.png" /></figure>
<figure><img src="image_282.png" /></figure>

<figure><img src="image_283.png" /></figure>

<figure><img src="image_284.png" /></figure>

Twin experiment

First step: twin experiments to set and validate
the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

<figure><img src="image_285.png" /></figure>

<figure><img src="image_286.png" /></figure>
<figure><img src="image_282.png" /></figure>

<figure><img src="image_283.png" /></figure>

<figure><img src="image_284.png" /></figure>

Twin experiment

First step: twin experiments to set and validate
the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

<figure><img src="image_285.png" /></figure>

<figure><img src="image_286.png" /></figure>
<figure><img src="image_287.png" /></figure>

<figure><img src="image_288.png" /></figure>

<figure><img src="image_289.png" /></figure>

Twin experiment

First step: twin experiments to set and validate
the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

<figure><img src="image_290.png" /></figure>

<figure><img src="image_291.png" /></figure>
<figure><img src="image_287.png" /></figure>

<figure><img src="image_288.png" /></figure>

<figure><img src="image_289.png" /></figure>

# Twin experiment

First step: twin experiments to set and validate the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

3. Generate the prior/background state (build an ensemble from biased input parameters distrbs)

<figure><img src="image_290.png" /></figure>

<figure><img src="image_291.png" /></figure>
<figure><img src="image_292.png" /></figure>

<figure><img src="image_293.png" /></figure>

<figure><img src="image_294.png" /></figure>

# Twin experiment

First step: twin experiments to set and validate the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

3. Generate the prior/background state (build an ensemble from biased input parameters distrbs)

<figure><img src="image_295.png" /></figure>

<figure><img src="image_296.png" /></figure>
<figure><img src="image_292.png" /></figure>

<figure><img src="image_293.png" /></figure>

Twin experiment

First step: twin experiments to set and validate
the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

3. Generate the prior/background state (build an ensemble from biased input parameters distrbs)

4. Perform ensemble data assimilation to correct input parameters and moisture series towards the reference

<figure><img src="image_294.png" /></figure>
<figure><img src="image_297.png" /></figure>

<figure><img src="image_298.png" /></figure>

<figure><img src="image_299.png" /></figure>

Twin experiment

First step: twin experiments to set and validate
the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

3. Generate the prior/background state (build an ensemble from biased input parameters distrbs)

4. Perform ensemble data assimilation to correct input parameters and moisture series towards the reference

<figure><img src="image_300.png" /></figure>
<figure><img src="image_295.png" /></figure>

<figure><img src="image_296.png" /></figure>

<figure><img src="image_297.png" /></figure>

# Twin experiment

First step: twin experiments to set and validate the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

3. Generate the prior/background state (build an ensemble from biased input parameters distrbs)

4. Perform ensemble data assimilation to correct input parameters and moisture series towards the reference

<figure><img src="image_298.png" /></figure>

<figure><img src="image_299.png" /></figure>
<figure><img src="image_301.png" /></figure>

<figure><img src="image_302.png" /></figure>

<figure><img src="image_303.png" /></figure>

Twin experiment

First step: twin experiments to set and validate
the DA framework ( !! Reanalysis context)

1. Use PESHMELBA to generate a "True" reference simulation

2. Add perturbation to generate virtual observations

3. Generate the prior/background state (build an ensemble from biased input parameters distrbs)

4. Perform ensemble data assimilation to correct input parameters and moisture series towards the reference

<figure><img src="image_304.png" /></figure>

<figure><img src="image_305.png" /></figure>
<figure><img src="image_300.png" /></figure>

<figure><img src="image_301.png" /></figure>

<figure><img src="image_302.png" /></figure>

# Results - Surface moisture images

<figure><img src="image_303.png" /><figcaption>* Moisture estimation</figcaption></figure>

<figure><img src="image_304.png" /></figure>
<figure><img src="image_306.png" /></figure>

<figure><img src="image_307.png" /></figure>

<figure><img src="image_308.png" /></figure>

# Results - Surface moisture images

<figure><img src="image_309.png" /><figcaption>* Moisture estimation</figcaption></figure>

<figure><img src="image_310.png" /></figure>
<figure><img src="image_305.png" /></figure>

<figure><img src="image_306.png" /></figure>

<figure><img src="image_307.png" /></figure>

# Results - Surface moisture images

## Moisture estimation

Surface

<figure><img src="image_308.png" /></figure>

Intermediary (20cm)

<figure><img src="image_309.png" /></figure>

Deep (4m)

<figure><img src="image_310.png" /></figure>

<figure><img src="image_311.png" /></figure>

## Parameter estimation

Surface

<figure><img src="image_312.png" /></figure>

Inter surface

<figure><img src="image_313.png" /></figure>
<figure><img src="image_311.png" /></figure>

<figure><img src="image_312.png" /></figure>

<figure><img src="image_313.png" /></figure>

# Results - Surface moisture images

## Moisture estimation

Surface

<figure><img src="image_314.png" /></figure>

Intermediary (20cm)

<figure><img src="image_315.png" /></figure>

Deep (4m)

<figure><img src="image_316.png" /></figure>

<figure><img src="image_317.png" /></figure>

## Parameter estimation

Surface

<figure><img src="image_318.png" /></figure>

Inter Surface

<figure><img src="image_319.png" /></figure>
<figure><img src="image_314.png" /></figure>

<figure><img src="image_315.png" /></figure>

<figure><img src="image_316.png" /></figure>

# Results - Surface moisture images

## Moisture estimation

<figure><img src="image_317.png" /></figure>

Data assimilation of satellite moisture images

## Parameter estimation

<figure><img src="image_318.png" /></figure>
<figure><img src="image_320.png" /></figure>

<figure><img src="image_321.png" /></figure>

<figure><img src="image_322.png" /></figure>

# Results - Surface moisture images

## Moisture estimation

<figure><img src="image_323.png" /></figure>

Data assimilation of satellite moisture images

## Parameter estimation

<figure><img src="image_324.png" /></figure>
<figure><img src="image_319.png" /></figure>

<figure><img src="image_320.png" /></figure>

<figure><img src="image_321.png" /></figure>

# Results - Surface moisture images

* **Moisture estimation**

<figure><img src="image_322.png" /></figure>

* **Parameter estimation**

<figure><img src="image_323.png" /></figure>

## Data assimilation of satellite moisture images

* Good correction of surface moisture and surface parameters

* Corrections do not propagate to subsurface (lack of correlations between surface and subsurface)

⇒ Idea? Integrate subsurface observations : point *vertical* profiles of moisture

<figure><img src="image_324.png" /></figure>
<figure><img src="image_325.png" /></figure>

<figure><img src="image_326.png" /></figure>

<figure><img src="image_327.png" /></figure>

# Results - Surface moisture images

* **Moisture estimation**

<figure><img src="image_328.png" /></figure>

* **Parameter estimation**

<figure><img src="image_329.png" /></figure>

## Data assimilation of satellite moisture images

* Good correction of surface moisture and surface parameters

* Corrections do not propagate to subsurface (lack of correlations between surface and subsurface)

⇒ Idea? Integrate subsurface observations : point *vertical* profiles of moisture

<figure><img src="image_330.png" /></figure>
<figure><img src="image_325.png" /></figure>

<figure><img src="image_326.png" /></figure>

<figure><img src="image_327.png" /></figure>

# Results - Surface images + vertical profiles

<figure><img src="image_328.png" /></figure>
<figure><img src="image_331.png" /></figure>

<figure><img src="image_332.png" /></figure>

<figure><img src="image_333.png" /></figure>

Results - Surface images + vertical profiles

<figure><img src="image_334.png" /></figure>
<figure><img src="image_329.png" /></figure>

<figure><img src="image_330.png" /></figure>

<figure><img src="image_331.png" /></figure>

# Results - Surface images + vertical profiles

## Moisture estimation

<figure><img src="image_332.png" /></figure>

## Adding subsurface observations

<figure><img src="image_333.png" /></figure>

* improves moisture simulations at all depths

<figure><img src="image_334.png" /></figure>
<figure><img src="image_335.png" /></figure>

<figure><img src="image_336.png" /></figure>

<figure><img src="image_337.png" /></figure>

# Results - Surface images + vertical profiles

## Moisture estimation

<figure><img src="image_338.png" /></figure>

Adding subsurface observations

* improves moisture simulations at all depths

<figure><img src="image_339.png" /></figure>
<figure><img src="image_335.png" /></figure>

<figure><img src="image_336.png" /></figure>

<figure><img src="image_337.png" /></figure>

# Results - Surface images + vertical profiles

## Moisture estimation

<figure><img src="image_338.png" /></figure>

## Parameter estimation

<figure><img src="image_339.png" /></figure>

## Adding subsurface observations

* improves moisture simulations at all depths

* improves $\theta_s$ estimates at all depths but on the plots of the same soil type

<figure><img src="image_340.png" /></figure>
<figure><img src="image_340.png" /></figure>

<figure><img src="image_341.png" /></figure>

<figure><img src="image_342.png" /></figure>

# Results - Surface images + vertical profiles

## Moisture estimation

<figure><img src="image_343.png" /></figure>

## Parameter estimation

<figure><img src="image_344.png" /></figure>

## Adding subsurface observations

* improves moisture simulations at all depths

* improves $\theta_s$ estimates at all depths but on the plots of the same soil type

<figure><img src="image_345.png" /></figure>
<figure><img src="image_341.png" /></figure>

<figure><img src="image_342.png" /></figure>

<figure><img src="image_343.png" /></figure>

# Impact of DA of hydological variables on pesticides variables ?

* Pesticide concentration at outlet

<figure><img src="image_344.png" /></figure>

* (not shown) a relative impact of assimilating integrated concentration of pesticides (only if conc. are measured at high frequency (< 5 days) and accurate)

<figure><img src="image_345.png" /></figure>
<figure><img src="image_346.png" /></figure>

<figure><img src="image_347.png" /></figure>

<figure><img src="image_348.png" /></figure>

# Impact of DA of hydological variables on pesticides variables ?

* Pesticide concentration at outlet

<figure><img src="image_349.png" /></figure>

* (not shown) a relative impact of assimilating integrated concentration of pesticides (only if conc. are measured at high frequency (< 5 days) and accurate)

<figure><img src="image_350.png" /></figure>
<figure><img src="image_346.png" /></figure>

<figure><img src="image_347.png" /></figure>

<figure><img src="image_348.png" /></figure>

# Impact of DA of hydological variables on pesticides variables ?

* Pesticide concentration at outlet

<figure><img src="image_349.png" /></figure>

⇒ Coupled DA assimilation efficiently corrects pest. concentration.

* (not shown) a relative impact of assimilating integrated concentration of pesticides (only if conc. are measured at high frequency (< 5 days) and accurate)

<figure><img src="image_350.png" /></figure>
<figure><img src="image_351.png" /></figure>

<figure><img src="image_352.png" /></figure>

<figure><img src="image_353.png" /></figure>

# Impact of DA of hydological variables on pesticides variables ?

* Pesticide concentration at outlet

<figure><img src="image_354.png" /></figure>

⇒ Coupled DA assimilation efficiently corrects pest. concentration.

* (not shown) a relative impact of assimilating integrated concentration of pesticides (only if conc. are measured at high frequency (< 5 days) and accurate)

<figure><img src="image_355.png" /></figure>
<figure><img src="image_351.png" /></figure>

<figure><img src="image_352.png" /></figure>

<figure><img src="image_353.png" /></figure>

Conclusion

* DA framework set for the first time in PESHMELBA

* Multisource DA of hydrological data is efficient to also improve the pesticide transfer

* Next step : set a DA framework on a real catchment : many challenges ! (get data, characterize real observation errors, handle high computation cost...)

* Include external uncertainties such as forcings (rainfall, ETP, ...)

* Compare with DA in the CATHY model, purely physics-based (⇒ less discontinuities?)

<figure><img src="image_354.png" /></figure>
<figure><img src="image_356.png" /></figure>

<figure><img src="image_357.png" /></figure>

<figure><img src="image_358.png" /></figure>

Conclusion

* DA framework set for the first time in PESHMELBA

* Multisource DA of hydrological data is efficient to also improve the pesticide transfer

* Next step : set a DA framework on a real catchment : many challenges ! (get data, characterize real observation errors, handle high computation cost...)

* Include external uncertainties such as forcings (rainfall, ETP, ...)

* Compare with DA in the CATHY model, purely physics-based (⇒ less discontinuities?)

<figure><img src="image_359.png" /></figure>
<figure><img src="image_355.png" /></figure>

<figure><img src="image_356.png" /></figure>

<figure><img src="image_357.png" /></figure>

# PhD Katarina Radišić : Calibrating a hydrological model robustly to rain perturbations (with stochastic surrogates)

<figure><img src="image_358.png" /></figure>

What about the impact of external uncertainties on the calibration results?

<figure><img src="image_359.png" /></figure>
<figure><img src="image_360.png" /></figure>

<figure><img src="image_361.png" /></figure>

<figure><img src="image_362.png" /></figure>

# PhD Katarina Radišić : Calibrating a hydrological model robustly to rain perturbations (with stochastic surrogates)

<figure><img src="image_363.png" /></figure>

What about the impact of external uncertainties on the calibration results?

<figure><img src="image_364.png" /></figure>
<figure><img src="image_360.png" /></figure>

<figure><img src="image_361.png" /></figure>

<figure><img src="image_362.png" /></figure>

# PhD Katarina Radišić : Calibrating a hydrological model robustly to rain perturbations (with stochastic surrogates)

<figure><img src="image_363.png" /></figure>

What does it mean to find a *robust* minimizer ?

<figure><img src="image_364.png" /></figure>
<figure><img src="image_365.png" /></figure>

<figure><img src="image_366.png" /></figure>

<figure><img src="image_367.png" /></figure>

# PhD Katarina Radišić : Calibrating a hydrological model robustly to rain perturbations (with stochastic surrogates)

<figure><img src="image_368.png" /></figure>

What does it mean to find a *robust* minimizer ?

<figure><img src="image_369.png" /></figure>
References

<figure><img src="image_365.png" /></figure>
References

<figure><img src="image_370.png" /></figure>

INRAO