# Sensitivity analysis of a spatio-temporal hydrological model for water and pesticide transfers

Katarina Radišić, Emilie Rouzies, Claire Lauvernet, Arthur Vidard

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Katarina Radišić, Emilie Rouzies, Claire Lauvernet, Arthur Vidard. Sensitivity analysis of a spatio-temporal hydrological model for water and pesticide transfers. SAMO 2022 - Tenth International Conference on Sensitivity Analysis of Model Output, Mar 2022, Tallahassee, Florida, United States. hal-03800833

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https://hal.inrae.fr/hal-03800833

Submitted on 6 Oct 2022

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# Sensitivity analysis of a spatio-temporal hydrological model for water and pesticide transfers

Katarina Radišić

Emilie Rouzies, Claire Lauvernet, Arthur Vidard

10th International Conference on Sensitivity Analysis of Model Output
(SAMO) March 14-16, 2022
The PESHMELBA model Rouzies et al.<fnref n="1" />,<fnref n="2" />

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

Landscape units and soil horizons, illustration from
Rouzies et al.<fnref n="2" />

* simulates water and pesticide transfers at a catchment scale

* heterogeneous soil types

* 14 landscape units

* heterogeneous layers: soil horizons

* infiltration, surface runoff and lateral subsurface exchanges

* heterogeneous vegetation types

* large number of input parameters ($K = 145$)

* input parameters are independent

* expensive, process based, hydrologic model

<footnote n="1">1. ROUZIES, LAUVERNET, BARACHET et al., "From agricultural catchment to management scenarios", 2019.</footnote>

<footnote n="2">2. ROUZIES, LAUVERNET, SUDRET et al., *How to perform global sensitivity analysis of a catchment-scale, distributed pesticide transfer model?*, 2021.</footnote>
<figure><img src="image_4.png" /></figure>

## 1 Case study

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

## 2 Methodology

* Sobol' indices and PCE metamodels

* Functional principal components

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

## 3 Results

* Interpretation of principal components

* Validation of metamodels

* Sobol' indices
# 1. Case study

Sensitivity analysis on : **surface moisture outputs**.

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

$$ \begin{bmatrix} Y^{(1)}(t) \\ Y^{(2)}(t) \\ \vdots \\ Y^{(14)}(t) \end{bmatrix} = \mathcal{M}(\mathbf{X}, t) $$

* time dependent outputs

* spatialized outputs

* large number of input parameters ($K = 52$ after screening)

Two types of behaviour :

$$ \mathcal{W} \sim \mathcal{G}_{plateau} $$

$$ \mathcal{W} \sim \mathcal{G}_{no\_plateau} $$
<figure><img src="image_8.png" /></figure>

## 1 Case study

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

## 2 Methodology

* Sobol' indices and PCE metamodels

* Functional principal components

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

## 3 Results

* Interpretation of principal components

* Validation of metamodels

* Sobol' indices
## 2. Methodology

### 2.1. Sobol' indices and PCE metamodels

Polynomial Chaos Expansion (PCE), metamodel for analytical Sobol' indices calculation<fnref n="3" />, calculations were done in the UQLab<fnref n="4" /> Matlab environment.

$$ 
\begin{aligned}
Y &= \mathcal{M}(\mathbf{X}) \\
&= \sum_{\alpha \in \mathbb{N}^K} y_{\alpha} \Psi_{\alpha}(\mathbf{X}) \\
D &= \mathrm{Var}(Y) \\
S_i &= \sum_{\substack{\alpha \in \mathbb{N}^K, \\ \alpha_i > 0, \alpha_{j \neq i} = 0}} y_{\alpha}^2 / D
\end{aligned}
$$
$$ 
\begin{aligned}
Y &\approx \mathcal{M}_{\mathrm{PCE}}(\mathbf{X}) \\
&= \sum_{\alpha \in \mathcal{A}_q^{K, p}} y_{\alpha} \Psi_{\alpha}(\mathbf{X}) \\
\hat{D} &= \mathrm{Var}(\mathcal{M}_{\mathrm{PCE}}(\mathbf{X})) \\
\hat{S}_i &= \sum_{\substack{\alpha \in \mathcal{A}_q^{K, p}, \\ \alpha_i > 0, \alpha_{j \neq i} = 0}} y_{\alpha}^2 / \hat{D}
\end{aligned}
$$

* Useful for models with large input spaces, needs little model simulations

* Scalar outputs → application to principal components (PC)

<footnote n="3">3. SUDRET, "Global sensitivity analysis using polynomial chaos expansions", 2008.</footnote>

<footnote n="4">4. MARELLI et SUDRET, "UQLab", 2014.</footnote>
# 2. Methodology

## 2.2. Functional principal components

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

$$Y^{(m)}(t) = \mu^{(m)}(t) + \sum_{j=1}^{\infty} H_j^{(m)} \mathbf{v}_j^{(m)}(t)$$

$$Y^{(m)}(t) \approx \mu^{(m)}(t) + H_1^{(m)} \mathbf{v}_1^{(m)}(t) + H_2^{(m)} \mathbf{v}_2^{(m)}(t)$$

$$H_1^{(m)} = \mathcal{M}_{PC1}^{(m)}(\mathbf{X}) \\ \approx \mathcal{M}_{PC1,PCE}^{(m)}(\mathbf{X})$$

$$H_2^{(m)} = \mathcal{M}_{PC2}^{(m)}(\mathbf{X}) \\ \approx \mathcal{M}_{PC2,PCE}^{(m)}(\mathbf{X})$$
<figure><img src="image_12.png" /></figure>

## 1 Case study

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

## 2 Methodology

* Sobol' indices and PCE metamodels

* Functional principal components

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

## 3 Results

* Interpretation of principal components

* Validation of metamodels

* Sobol' indices
# 3. Results

## 3.1. Interpretation of principal components

<figure><img src="image_15.png" /></figure>
# 3. Results

## 3.2 Validation of metamodels

<table><thead><tr><th>m</th><th>M<sup>m</sup><sub>PC1,PCE</sub></th><th>M<sup>m</sup><sub>PC2,PCE</sub></th></tr></thead><tbody><tr><td>1</td><td>1.00</td><td>NA</td></tr><tr><td>2</td><td>0.99</td><td>0.80</td></tr><tr><td>3</td><td>0.99</td><td>0.77</td></tr><tr><td>4</td><td>0.99</td><td>0.99</td></tr><tr><td>5</td><td>0.99</td><td>NA</td></tr><tr><td>6</td><td>0.99</td><td>0.97</td></tr><tr><td>7</td><td>0.99</td><td>0.82</td></tr><tr><td>8</td><td>1.00</td><td>NA</td></tr><tr><td>9</td><td>0.99</td><td>0.80</td></tr><tr><td>10</td><td>0.99</td><td>0.98</td></tr><tr><td>11</td><td>1.00</td><td>NA</td></tr><tr><td>12</td><td>0.99</td><td>0.95</td></tr><tr><td>13</td><td>0.98</td><td>0.78</td></tr><tr><td>14</td><td>0.99</td><td>0.99</td></tr></tbody></table>

$$Y^{(m)}(t) \approx \mu^{(m)}(t) + H_{1}^{(m)}v_{1}^{(m)}(t) + H_{2}^{(m)}v_{2}^{(m)}(t)$$

$$H_{1}^{(m)} \approx M_{PC1,PCE}^{(m)}(\mathbf{X})$$

$$H_{2}^{(m)} \approx M_{PC2,PCE}^{(m)}(\mathbf{X})$$

$R^2$ of the metamodels built on PC1 and PC2.
# 3. Results

## 3.2 Validation of metamodels

<table><thead><tr><th>m</th><th>M<sup>m</sup><sub>PC1,PCE</sub></th><th>M<sup>m</sup><sub>PC2,PCE</sub></th><th>G</th></tr></thead><tbody><tr><td>1</td><td>1.00</td><td>NA</td><td>G<sub>no-plateau</sub></td></tr><tr><td>2</td><td>0.99</td><td><strong>0.80</strong></td><td><strong>G<sub>plateau</sub></strong></td></tr><tr><td>3</td><td>0.99</td><td><strong>0.77</strong></td><td><strong>G<sub>plateau</sub></strong></td></tr><tr><td>4</td><td>0.99</td><td><strong>0.99</strong></td><td>G<sub>no-plateau</sub></td></tr><tr><td>5</td><td>0.99</td><td>NA</td><td>G<sub>no-plateau</sub></td></tr><tr><td>6</td><td>0.99</td><td><strong>0.97</strong></td><td>G<sub>no-plateau</sub></td></tr><tr><td>7</td><td>0.99</td><td><strong>0.82</strong></td><td><strong>G<sub>plateau</sub></strong></td></tr><tr><td>8</td><td>1.00</td><td>NA</td><td>G<sub>no-plateau</sub></td></tr><tr><td>9</td><td>0.99</td><td><strong>0.80</strong></td><td><strong>G<sub>plateau</sub></strong></td></tr><tr><td>10</td><td>0.99</td><td><strong>0.98</strong></td><td>G<sub>no-plateau</sub></td></tr><tr><td>11</td><td>1.00</td><td>NA</td><td>G<sub>no-plateau</sub></td></tr><tr><td>12</td><td>0.99</td><td><strong>0.95</strong></td><td>G<sub>no-plateau</sub></td></tr><tr><td>13</td><td>0.98</td><td><strong>0.78</strong></td><td><strong>G<sub>plateau</sub></strong></td></tr><tr><td>14</td><td>0.99</td><td><strong>0.99</strong></td><td>G<sub>no-plateau</sub></td></tr></tbody></table>

$$Y^{(m)}(t) \approx \mu^{(m)}(t) + H_{1}^{(m)}v_{1}^{(m)}(t) + H_{2}^{(m)}v_{2}^{(m)}(t)$$

$$H_{1}^{(m)} \approx M_{PC1,PCE}^{(m)}(\mathbf{X})$$

$$H_{2}^{(m)} \approx M_{PC2,PCE}^{(m)}(\mathbf{X})$$

$R^2$ of the metamodels built on PC1 and PC2.
# 3. Results

## 3.3 Sobol' indices

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

Total Sobol' indices on PC1 per landscape unit with bootstrap confidence limits.
# 3. Results

## 3.3 Sobol' indices

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

Total Sobol' indices on PC2 per landscape unit with bootstrap confidence limits.
* Successful application to surface moisture outputs.

• Sobol' indices obtained with high precision.

* Meaningful physical interpretations, increased comprehension of model behaviour.

* information on the surface runoff type (infiltration excess runoff, saturation excess overland flow)

* Further adaptations

* Cluster based SA for landscape units $\in \mathcal{G}_{plateau}$

* Replace polynomial chaos expansion metamodel with more flexible metamodels (random forests or deep GP Gaussian processes)