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# Global Sensitivity Analysis applied to the Telemac2D numerical forecast model of high water levels in the Gironde estuary

Vanessya Laborie, Nicole Goutal, Sophie Ricci, Matthias de Lozzo, Philippe Sergent

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## To cite this version:

Vanessya Laborie, Nicole Goutal, Sophie Ricci, Matthias de Lozzo, Philippe Sergent. Global Sensitivity Analysis applied to the Telemac2D numerical forecast model of high water levels in the Gironde estuary. XXIVth TELEMAC-MASCARET, Oct 2017, Graz, Austria. hal-02535631

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Laborie, Vanessya; Goutal, Nicole; Ricci, Sophie; De Lozzo, Matthias; Sergent, Philippe

# Global Sensitivity Analysis applied to the Telemac2D numerical forecast model of high water levels in the Gironde estuary

Zur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:
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Vorgeschlagene Zitierweise/Suggested citation:
Laborie, Vanessya; Goutal, Nicole; Ricci, Sophie; De Lozzo, Matthias; Sergent, Philippe (2017): Global Sensitivity Analysis applied to the Telemac2D numerical forecast model of high water levels in the Gironde estuary. In: Dorfmann, Clemens; Zenz, Gerald (Hg.): Proceedings of the XXIVth TELEMAC-MASCARET User Conference, 17 to 20 October 2017, Graz University of Technology, Austria. Graz: Graz University of Technology. S. 81-89.

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# Global Sensitivity Analysis applied to the Telemac2D numerical forecast model of high water levels in the Gironde estuary

Vanessya Laborie¹,², Nicole Goutal²,³, Sophie Ricci⁴, Matthias De Lozzo⁴, Philippe Sergent¹

¹ French Center For Studies and Expertise on Risks, Environment, Mobility, and Urban and Country planning (Cerema)

² Saint-Venant Laboratory for Hydraulics

³ EDF R&D National Laboratory for Hydraulics and Environment (LNHE)

⁴CECI, CERFACS/CNRS

¹Compiègne, France, ²,³Chatou, France, ⁴Toulouse, France

e-mail: vanessya.laborie@cerema.fr

***Abstract***—In the context of the development and the implementation of data assimilation techniques in Gironde estuary for flood forecasting, a Telemac 2D model is used to calculate water depths and velocity fields at each node of an unstructured mesh. Upstream, the model boundaries are respectively La Réole and Pessac on the Garonne and Dordogne river. The maritime boundary is 32 km off the mouth of Gironde estuary, located in Verdon.

This model, which contains 7351 nodes and 12838 finite elements, does not take into account overflows. It has been calibrated on 4 non-overflowing events and then validated on 6 overflowing events.

In a first step, a propagation and quantification of uncertainties by an unidirectional analysis method (creation of a set of 2000 members perturbed for each parameter and input forcings and analysis of output water depths) was carried out on the numerical parameters (wind influence coefficient, Strickler friction coefficients for 4 zones) and forcings of the model (rivers discharges and maritime boundary conditions, meteorological forcings). The objective is to determine the variation coefficient of water depths for 13 major events between 1981 and 2016. The exploitation of 1981 event results shows a predominance of the influence of the maritime boundary conditions and the Strickler coefficient for the estuarine part and the confluence, to which must be added the Garonne discharge as a predominant parameter for the latter. Unsurprisingly, river zones are influenced primarily by the coefficient of friction and the respective river flows of Garonne and Dordogne rivers.

On the second hand, a Global Sensitivity Analysis (GSA) by variance analysis (ANOVA) was carried out, by calculating the total and partial Sobol' indices, integrating the time- and/or space dependent forcing variables. It has led to the identification of parameters and forcings to which the model is most sensitive, as well as their inter-dependencies, in order to choose the variables to assimilate. The GSA shows that the tidal

signal imposed at the maritime boundary condition and provided by a more extended surge levels model is the key input variable. Moving from the mouth to the upstream part of the Garonne and Dordogne rivers, the influence of the friction coefficient increases and hydrological forcings have a very local influence upstream the rivers.

## I. INTRODUCTION

Hydrodynamic numerical softwares based on shallow-water equations are commonly used for management and protection of urban infrastructures located near rivers or coasts. They are also used for operational flood forecasting with strong computational constraints. Yet, these numerical codes remain imperfect as uncertainties in the model (numerical schemes, time and space resolution, etc.) and its inputs (model parameters, boundary conditions, geometry, etc.) translate into uncertainties in the outputs. Quantifying uncertainties goes beyond the limits of deterministic forecast and represent a great challenge for Decision Support Systems for risk assessment of crisis management. This study presents a Sensitivity Analysis (SA) that aims at identifying the major sources of uncertainties for water level simulation. Once identified and quantified, these uncertainties can be reduced with data assimilation methods in further works, thus improving water level forecast in the context of flood forecasting on the Gironde estuary.

A wide range of SA methods are proposed in the literature [2] to explain the contribution of the uncertain model parameters to the uncertainty in the model Quantities of Interest (QoI). On the one hand, local SA approaches provide the sensitivity of the model outputs with respect to the model inputs around a reference value using the tangent linear of the model when available or finite differences techniques. A univariate global uncertainty propagation technique is first proposed here. It consists in perturbing uncertain inputs one at a time within physical ranges, computing the corresponding QoI and comparing the
associated relative variations. On the other hand, Global SA approaches (GSA) provide the contribution to the QoI's uncertainty from the uncertain input parameter when varying over the whole input parameter space. A multivariate GSA method (ANOVA) is then presented. It consists in the QoI variance decomposition in terms of elementary variances associated to the different parameters and their interactions.

This paper presents a SA study in the context of flood forecasting in the Gironde estuary. It aims at identifying which input variables should be better described for water levels to be better simulated and forecasted in the estuary. The structure of the paper is as follows: Sect. 2 presents the Gironde estuary hydrodynamic model implemented with TELEMAC2D. The experimental settings for the GSA (univariate and ANOVA) study is presented in Sect. 3. Results are given in Sect. 4. Conclusions and perspectives for the study are finally given.

## II. HYDRODYNAMIC MODEL FOR THE GIRONDE ESTUARY

### A. Shallow water equations in TELEMAC2D.

The non-conservative form of shallow water equations is derived from two principles: mass conservation and momentum conservation, after expansion of the derivatives. The equations are written in terms of water depth (h) and horizontal components of velocity (u and v).

$$ 
\begin{aligned}
& \frac{\partial h}{\partial t} + \frac{\partial}{\partial x}(hu) + \frac{\partial}{\partial y}(hv) = 0 \\
& \frac{\partial hu}{\partial t} + u \frac{\partial}{\partial x}(hu) + v \frac{\partial}{\partial y}(hu) = -gH \frac{\partial h}{\partial x} - \frac{1}{K^2} \frac{g|q|uH}{H^3} \\
& \qquad - \frac{h}{\rho} \frac{\partial P_{atm}}{\partial x} + \frac{\rho_{air}}{\rho} C_{Dz} U_{w,x} \sqrt{U_{w,x}^2 + U_{w,y}^2} \\
& \qquad + hF_x + \nabla \cdot (hv_e \vec{\nabla}(u))
\end{aligned}
\quad (1) $$

$$ 
\begin{aligned}
& \frac{\partial hv}{\partial t} + u \frac{\partial}{\partial x}(hv) + v \frac{\partial}{\partial y}(hv) = -gH \frac{\partial h}{\partial y} - \frac{1}{K^2} \frac{g|q|vH}{H^3} \\
& \qquad - \frac{h}{\rho} \frac{\partial P_{atm}}{\partial y} + \frac{\rho_{air}}{\rho} C_{Dz} U_{w,y} \sqrt{U_{w,x}^2 + U_{w,y}^2} \\
& \qquad + hF_y + \nabla \cdot (hv_e \vec{\nabla}(v))
\end{aligned}
$$

where $\rho / \rho_{air}$ [kg / m³] are the water/air density, $P_{atm}$ [Pa] is the atmospheric pressure, $U_{w,x}$ and $U_{w,y}$ [m/s] are the horizontal wind velocity components, $C_{Dz}$ [-] is the wind influence coefficient, $K$ [m¹³/s] is the river bed and flood plain friction coefficient, using Strickler formulation

This decomposition is obtained from an orthogonal decomposition of the uncertain QoI over the probabilized parameter space [1]. A set of sensitivity indices, called Sobol' indices, is estimated. They represent the contribution of each parameter and their interactions to the model output variance. This approach is efficient even for non linear and non monotonic models ([6], [13]).

[3]. $F_x$ and $F_y$ [m/s²] are the horizontal components of external forces (Coriolis force, for example), h [m] is the water depth, H [m NGF69] is the water level (h=H-zf if zf [m NGF69] is the bottom level), u and v [m/s] are the horizontal components of velocity, w [m/s] is the vertical component of velocity and $v_e$ [m²/s] is the water diffusion coefficient. g is the gravity (9,814 m/s²). t [s] is time, x [m] and y [m] are the coordinates respectively along longitude and latitude axis.

To solve the set of equations (1), initial conditions $h(x,y,t=0)=h_0(x,y)$; $u(x,y,t=0)=u_0(x,y)$; $v(x,y,t=0)=v_0(x,y)$ are provided along with wall boundary conditions at the coastline (slip: $u=0$ and no normal flow:

$$ u \frac{\partial h}{\partial x} + v \frac{\partial h}{\partial y} + w=0 ) \quad \text{and at the upstream and downstream frontiers} (h(x_{BC}, y_{BC}, t)=h_{BC}(t)). $$

### B. Gironde estuary numerical Model.

A hydrodynamics numerical model for the Gironde estuary (presented in Fig. 1) implemented with TELEMAC2D [4] is used to compute water depths and velocities in the estuary and on Garonne and Dordogne rivers. This model is used operationally by the Flood Forecast Service for Gironde Adour Dordogne watersheds (SPC GAD) and SCHAPI (Service Central d'Hydrométéorologie et d'Appui pour la Prévision des Inondations). The maritime boundary is located in the Gascogne Gulf, 35 km away from le Verdon. The upstream boundaries are located on the Garonne River (at La Réole) and on the Dordogne River (at Pessac). It should be noted that inflows from the Isle River and the Dronne are artificially injected at Pessac [5] and that flood plains are not taken into account. The model that covers about 125 km long from east to west features 12838 finite elements and is composed of 7351 nodes.

Surface forcing wind velocity and pressure fields from the regional meteorological model ALADIN are provided by Meteo-France at a 3-hour time step. Hydrological upstream forcing for the Dordogne and Garonne rivers are provided by DREAL (Direction Régionale de l'Environnement, de l'Aménagement des Territoires et du Logement) Nouvelle Aquitaine. Water levels at the maritime boundary, which are the sum of the predicted astronomical tide and surge levels, are also provided by Meteo-France at a 10 to 15 min time step.

The friction coefficient is described over 4 homogeneous areas as described in Fig. 1. The model calibration was achieved over 11 storm events using water level Root Mean Square Error (RMSE) and Nash criteria at high tides
(NASH HT) computed between simulated and observed water levels where observations are available at the 26 observing stations shown in Fig. 1 (red stars). Two sets of friction coefficients were obtained from calibration and are presented in Tab. 1. The wind influence coefficient $C_{Dz}$ formulates the wind shear stress at the free surface from the wind velocity [8]. A uniform and constant value was chosen here ($C_{Dz} = 2,1410^{-3}$) in coherence with the calibration of the surge levels numerical model for Atlantic Ocean, English Channel and North Sea [9].

<table><caption>TABLE 1. CALIBRATED STRICKLER (K) PARAMETERS COMPUTED FROM THE NASH-HT AND RMSE CRITERIA.</caption><thead><tr><th>Input variable</th><th>Updated Strickler coefficients with NASH-HT criterion</th><th>Updated Strickler coefficients with RMSE criterion</th><th>Uniform distribution over</th></tr></thead><tbody><tr><td>Ks<sub>1</sub></td><td>55</td><td>70</td><td>[50; 70]</td></tr><tr><td>Ks<sub>2</sub></td><td>70</td><td>70</td><td>[45; 75]</td></tr><tr><td>Ks<sub>3</sub></td><td>75</td><td>65</td><td>[25; 75]</td></tr><tr><td>Ks<sub>4</sub></td><td>50</td><td>55</td><td>[40; 80]</td></tr><tr><td>C<sub>Dz</sub></td><td>2.57.10<sup>-6</sup></td><td>2.57.10<sup>-6</sup></td><td>[0.678.10<sup>-6</sup>; 3.016.10<sup>-6</sup>]</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Figure 1. extension and location of the numerical model of the Gironde estuary and delimitation of the Strickler coefficient areas 1 to 4. Circles represent the nodes of the numerical model based on a mesh built with finite elements (in black). Red stars show the main measurement stations of interest for the water levels forecast.</figcaption></figure>

# I. METHODOLOGY FOR GSA

## A. Uncertain variables for SA

Eight uncertain input variables are considered for the univariate GSA and ANOVA in the following: the 4-zone distributed friction coefficients (K1, K2, K3, K4), the wind influence coefficient $C_{Dz}$ and time-dependent boundary conditions at the hydrological limits (QDOR and QGAR for the Dordogne and Garonne rivers respectively) and at the

maritime limit (CLMAR). The uncertain input vector is denoted by $X$, of components $X_i$ with $i$ in [1, 8]. The QoI is the water level at a specific node and a specific time, it is a scalar and it is denoted by $Y$ in the following. The GSA is applied for water level at the 26 observing stations in Fig. 1, over time.

For each $X_i$, a Sobol' sequence is used for space filling in a normalized space and the sample is then mapped onto the physical space. Scalar uncertain variables are described by their Probability Density Function (PDF). Here, wind influence coefficient and friction coefficient PDFs are supposed to be uniform with ranges described in Table 1. The critical point is the description of the ensemble from which sensitivity indices are stochastically estimated. For ANOVA, all physical variables should be mapped onto a unit hypercube.

When uncertainty relates to time dependent variables (hydrological time vectors for this study), the dimension of the input space should be reduced for computational reasons. This applies to the time-dependent upstream discharge forcings (Dordogne and Garonne), downstream water level forcing and wind and pressure surface forcing (not treated in this paper). The sampling procedure for these quantities aims at preserving temporal correlation of errors. The discharge chronicles $q_T$ are supposed to be Gaussian Processes with Gaussian correlation function $f_T$ with a correlation length scale $L_T$, estimated from observed chronicles over the 1981-2016 period. $L_T$ is assumed to depend on the event for the hydrological boundary conditions. It is set to 1.25 days and 3.6 days on the Dordogne and Garonne reaches for 1981. It is set to 1.21 days and 2.91 days for the 2003 event. For both events, $L_T$ is set to 6 hours (half-tide) at the maritime boundary. $q_T$ is written by a Karhunen-Loève decomposition, as the truncated sum of $n_p$ orthogonal functions where $\epsilon_i$ coefficients (modes) are independent standard normal variables:

$$q_T(t) = \sum \sqrt{\lambda_i} \, {}_i(t) \quad (2)$$

where $\sqrt{\lambda_i}$, ${}_i(t)$ are the eigen values and eigen functions for the Fredholm equation with $C$ the Gaussian correlation function:

$$\forall i \in \int C(t_1, t_2) \, {}_i(t_2) dt_2 = \lambda_i \, {}_i(t_1) t_1 \in T \quad (3)$$

Eq. (3) is discretized:

$$C\psi_i = \alpha_i \psi_i \quad (4)$$

with $C_{ij} = f(t_i, t_j)$, $\alpha_i$ and $\psi_i$ (vector of size $t_N$) are the $i^{th}$ eigen values and vectors of $C$, computed with a Singular Value decomposition (SVD). This leads to

$$q_T(t_1, \dots, t_N) = \sum \sqrt{\alpha_i} \psi_i \quad (5)$$

From Eq. (5), a set of $N_e$ boundary condition time dependent vectors are generated with a sampling of $\epsilon_i$. The
amplitude of the perturbation is set proportional to the observed chronicle (1.04 multiplying factor for Dordogne and Garonne, 1.012 for the maritime boundary). This leads to a maximum discharge variance of 20 % and a maximum water level variance of 50 cm for the maritime boundary. A set of perturbed forcing for the Garonne boundary conditions are shown in Fig. 2.

The GSA is thus carried out in an uncertain space described by 20 variables: 4 Strickler coefficients, the wind influence coefficient Cdz, the 4 principal modes for each river discharge q(t) QDOR and QGAR and the 7 principal modes for the maritime boundary tide signal. However, for each time-dependent variables (river discharges and tide signal at the maritime boundary), perturbations for each mode are then aggregated by to the physical input space.

<figure><img src="image_5.png" /><figcaption>Figure 2. Observed (in black) and Perturbed (other colors) Garonne discharge during 20030202 event.</figcaption></figure>

## B. Coefficients of variation for univariate analysis

The coefficient of variation is defined with respect to each uncertain input variable $X_i$, at a given time step and at a given mesh node, as $CV = \frac{\sigma}{\mu}$, with $\sigma$ and $\mu$ the standard deviation/ensemble-mean of the uncertain output Y (resp. uncertain input $X_i$) variable computed over the sample

$E_u(X_i)$. The ratio $\frac{CV(Y)}{CV(X_i)}$ is then formulated.

## C. ANOVA method and Sobol' indices

We consider a numerical model of the form $Y=f(X_1, X_2, \dots, X_k)$ with uncertain input vector X and uncertain output scalar variable Y. It is assumed in the following that the input parameters are scaled with respect to physical parameters. $V_i$ is referred to as the “variance of conditional expectation” of Y given $X_i$ and the subscript $X_{i-}$ denotes the vector X deprived from of all factors other than $X_i$.

The variance of the output Y is the sum of the contribution from the different input parameters:

$$V_{i,j} + \dots + V_{1,2,\dots,k} \quad (6)$$

where $V_i = V_{X_i}(E_{X_{i-}}(Y | X_i))$ and

$$V_{i,j} = V_{X_i, X_j} \left[ E_{X_{i-}, (Y | X_i, X_j)} \right] - V_{X_i} \left( E_{X_{i-}, (Y | X_i)} \right) + V_{X_j} \left( E_{X_{i-}, (Y | X_j)} \right)$$

Dividing Eq. (6) by $V(Y)$ leads to:

$$S_{i,j} + \dots + S_{1,2,\dots,k} = 1 \quad (7)$$

In Eq. (7), the $S_i$ indices are the sensitivity coefficients (also called first order Sobol' indices). The second order indices $S_{i,j}$ (second-order Sobol' indices) measure the variance in Y for higher orders. The $S_{Ti}$ shows all contribution related to $X_i$:

$$S_{Ti} = 1 - \frac{V_{X_i}(E_{X_i}(Y | X_i))}{V(Y)} = \frac{E_{X_i}(V_{X_i}(Y | X_i))}{V(Y)} \quad (8)$$

It should be noted that $S_i = 1$ for numerical models without interaction between the input parameters.

The main steps for the stochastic estimation of the Sobol' indices are described in the following. For more details, the reader can refer to [12] and [10].

1. Generation of an ensemble of size $N_e$ (here $N_e = 1000$) for the normalized input parameters of size k. The ($N_e$, k) matrix is denoted by **A**. The space filling strategy is carried out with a Sobol' sequence rather than a classical Monte-Carlo strategy. The conditional probability computation requires the formulation of another sampling denoted by **B**:

$$A = \begin{bmatrix} x_1^{(1)} & x_2^{(1)} & \dots & x_i^{(1)} & \dots & x_k^{(1)} \\ x_1^{(2)} & x_2^{(2)} & \dots & x_i^{(2)} & \dots & x_k^{(2)} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ x_1^{(N-1)} & x_2^{(N-1)} & \dots & x_i^{(N-1)} & \dots & x_k^{(N-1)} \\ x_1^{(N)} & x_2^{(N)} & \dots & x_i^{(N)} & \dots & x_k^{(N)} \end{bmatrix}$$

$$B = \begin{bmatrix} x_{k+1}^{(1)} & x_{k+2}^{(1)} & \dots & x_{k+i}^{(1)} & \dots & x_{2k}^{(1)} \\ x_{k+1}^{(2)} & x_{k+2}^{(2)} & \dots & x_{k+i}^{(2)} & \dots & x_{2k}^{(2)} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ x_{k+1}^{(N-1)} & x_{k+2}^{(N-1)} & \dots & x_{k+i}^{(N-1)} & \dots & x_{2k}^{(N-1)} \\ x_{k+1}^{(N)} & x_{k+2}^{(N)} & \dots & x_{k+i}^{(N)} & \dots & x_{2k}^{(N)} \end{bmatrix}$$

2. Definition of k matrices $C_i$ formed by all columns of **A** except the i-th column taken from **B** [12].

3. Computation of the model output for all the input values in the sample matrices **A**, **B** and the k matrices $C_i$, obtaining $(k+2)$ vector outputs of
$$
\begin{array}{ll}
\text{dimension N*1:} & y_A = f(A), \quad y_B = f(B), \\
& y_{C_i} = f(C_i) \text{ with } i=1, \dots, k.
\end{array}
$$

In this study, according to [12], [7] and [11] which
describes the best practices for the simultaneous computation
of $S_i$ and $S_{Ti}$ for each input variable, the following
estimators have been chosen for the computation of:

$$
\bullet S_i, \qquad V_{X_i}(E_{X_{\sim i}}(Y | X_i)) = \frac{1}{N} \sum_{j=1}^{N} y_B^{(j)} (y_{C_i}^{(j)} - y_A^{(j)})
$$

$$
\bullet S_{Ti}, \qquad E_{X_i}(V_{X_i}(Y | X_i)) = \frac{1}{2N} \sum_{j=1}^{N} (y_A^{(j)} - y_{C_i}^{(j)})^2
$$

$$
\bullet V(Y), \quad V(Y) = \frac{1}{N} \sum_{j=1}^{N} (y_A^j)^2 - f_0^2 \quad \text{with} \\
f_0^2 = \frac{1}{N} \sum_{j=1}^{N} y_A^j y_B^j
$$

II. GSA RESULTS

1. *Uni-variate GSA results*

The univariate GSA was carried for 12th-17th December 1981 event, characterized by a tide coefficient of [57 ; 106] and Dordogne (resp. Garonne) discharge of [1000 ; 2350] m³/s (resp. [1700 ; 7050] m³/s). In this study, for each input variable, a sample of 2000 perturbed realisations is generated.

The normalized variation coefficient is displayed over
time in Fig. 3 for each uncertain variable Xᵢ along the
curvilinear abscissa from Royan to the confluence at bec
d'Ambès (1st black vertical line), from the confluence to La
Réole (Garonne river area located between both black
vertical lines) and from the confluence to Pessac on the
Dordogne river (Dordogne river area on the right of the 2nd
black vertical line). The red vertical lines represent the limits
between the 4 friction coefficients areas. The dashed lines
represent the time-aggregated values for standardized
coefficients of variation along the Gironde estuary.

Fig. 3a highlights the predominant influence of the
maritime boundary condition.

This influence is strongly related to the tide periodicity over
the estuary and beyond the confluence. When the influence
of the maritime boundary condition decreases, the
coefficient of variation for the friction coefficient increases
especially in areas 1 to 3 (Fig. 3c, d, e).

This influence is strongly related to the tide periodicity over
the estuary and beyond the confluence. When the influence
of the maritime boundary condition decreases, the
coefficient of variation for the friction coefficient increases
especially in areas 1 to 3 (Fig. 3c, d, e).

The friction coefficient in area 4 has barely no impact on
water level (with very low coefficient of variation in Fig. 3f)
except upstream of the Dordogne river.

The hydrological boundary conditions have an impact
that is limited to the extreme upstream part of the domain
(Fig. 3g, h) for both Garonne and Dordogne rivers. The wind
influence coefficient in Fig. 3c has a negligible influence on
water level over the entire domain. It could thus remain
fixed to a nominal value for further analysis. Overall, there
is complementary behaviour between maritime boundary
conditions and friction coefficients chosen for areas 1, 2 and
3 where Ks₁, Ks₂ and Ks₃ are applied.

This univariate SA only provides a qualitative feedback
on the impact of each uncertain input variables on the water
level uncertainty. A quantitative feedback is given by the
ANOVA; it also describes how uncertainties in the inputs
variables interacts.
<figure><img src="image_6.png" /><figcaption>Figure 3. Normalized coefficient of variation along the Gironde estuary during decembre 1981 storm for a) the maritime boundary conditions b) the wind influence coefficient c) friction coefficient in area 1, Ks1 d) friction coefficient in area 2, Ks2, e) friction coefficient in area 3, Ks3, f) friction coefficient in area 4, Ks4, g) the discharge of the Dordogne and Isle rivers at Pessac and h) the discharge of the Garonne river at La Réole.</figcaption></figure>

## 2. ANOVA results

The ANOVA GSA study was carried for 2^nd^-9^th^ February 2003 event, characterized by a tide coefficient of [43 ; 90] and Dordogne (resp. Garonne) discharge of [600 ; 2200] m³/s (resp. [1200 ; 5900] m³/s). For each input variable, a sample of 1000 perturbed realisations is generated.

Sobol' indices are displayed in Fig. 4 with a blue-red color bar for the 8 uncertain inputs $X_i$, over time and along the curvilinear abscissa. The x-axis is similar to that of Fig. 2, with the 26 observing stations classified from left to right for downstream to upstream. The red vertical lines represent the limits between the 4 friction coefficients areas. The black vertical lines represent the limits between the estuary, the confluence and the Garonne and Dordogne rivers. For each input variable, the total Sobol' indices are plotted in panels -a and the difference between total and first order indices are plotted in panels -b. For panels -a, blue/red means small/large Sobol' indices; for panels -b, blue/red means small/large interactions of $X_i$ with other uncertain variables. The thick dashed line represents time-averaged Sobol' indices. Fig. 5 provides a ranked interpretation of Fig. 4 where input variables are classified by order of predominance over time and over the curvilinear abscissa (1 for most significant to 8 for non-significant). Total Sobol' indices and rank are time-averaged and plotted in Fig. 6 for all $X_i$ along the y-axis. along the Gironde estuary.

Fig. 4 clearly shows the predominance of the maritime boundary condition and the dependency of all variable impact on the tidal signal. As previously mentioned, the wind influence coefficient, the hydrological boundary conditions and the friction coefficient in area 4 have no influence on the water level variability except at the upstream location of Garonne and Dordogne.

Over area 1, the water level variance is explained by the variance in the boundary condition with Sobol' indices closed to 1 from Port-Royan (station 1) to Richard (station 5) near the mouth of the estuary. It should be noted the part of water level variance that is not explained by CLMAR is explained by Ks1.

Over area 2, CLMAR and $Ks_2$ are the most significant sources of uncertainty with a predominance of CLMAR at high tide. At low tide, the Sobol' index for $Ks_2$ reaches 0.8. The influence of the maritime boundary condition decreases from Fort-Medoc to the confluence between Garonne and Dordogne and the influence of $Ks_2$ and $Ks_3$ increases. At La Réole and Pessac, Sobol' index for $Ks_4$ reaches 0.8.

It should be noted that only the maritime boundary condition and $Ks_3$ have interactions with the other input variations, these cross-effects reach 30% for CLMAR.
Fig. 6 represents the time-averaged total Sobol' indices and corresponding ranks for each input variable. As expected, the maritime boundary conditions is predominant for the entire domain with significant influence of friction coefficients, while hydrological boundary conditions have a limited impact to the upstream locations of Garonne and Dordogne.

### III. Conclusions

The numerical T2D model operationally used by SPC GAD to forecast water levels along the Gironde estuary was studied through a uni-variate uncertainty analysis and a global sensitivity analysis based of variance decomposition (ANOVA) to provide Sobol' indices. The uni-variate uncertainty analysis over December 1981 event leads to similar conclusions as the ANOVA over the 2003 event. However, it also enables to quantify the contribution of each variable during the storm and the part of the variance linked to interactions between variables. SA show that the tidal signal imposed at the maritime boundary condition and provided by a more extended surge levels model is the key input variable. Moving from the mouth to the upstream part of the Garonne and Dordogne rivers, the influence of the friction coefficient increases and the hydrological forcing have a very local influence upstream the rivers.

A perspective for this study is to use a bootstrap method to compute the confidence interval for Sobol' indices. Moreover, time and spatial dependent uncertain input variables such as the meteorological forcings associated to a tide signal should also be included in the SA.

Finally, this SA approach allows to identify the significant sources of uncertainty that should be reduced with data assimilation, for instance with an ensemble Kalman Filter, in order to improve the water level at key location on the estuary in simulation and forecast mode.

### ACKNOWLEDGEMENT

We would like to thank the service in charge of flood forecast on Garonne, Adour and Dordogne watersheds (SPC GAD), METEO-FRANCE and Greater Maritime Port Councils of Bordeaux (GPMB) for bathymetric and observation data they provided for the needs of this study. The sea level observations along Gironde estuary are the property of GPMB and of the French Ministry in charge of sustainable

development (MEEM). A lot of thanks, too, to Yoann AUDOUIN (EDF/LNHE) for his support in the implementation and use of TELEMAC2D.

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<figure><img src="image_7.png" /><figcaption>Figure 4. Total Sobol' indices (STi) in panels a- (resp. interactions STi - Si in panels b-) along Gironde estuary during february 2003 storm for 8 uncertain input variables (maritime boundary conditions (CLMAR), friction coefficients (Ks1, Ks2, Ks3, Ks4), the wind influence coefficient CdZ and hydrological boundary conditions (Dordogne river discharge: QDOR; Garonne river discharge: QGAR)).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 5. Predominance rank of the 8 input variables over time along Gironde estuary during february 2003 storm for panel a- to h- (CLMAR, Ks1, Ks2, Ks3, Ks4, CDz, QDOR and QGAR) and time aggregated ranks (panel i-).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6. representation of time aggregated Sti (on the left) and of the rank of the input variable with respect to Sti (on the right) along Gironde estuary during February 2003 storm.</figcaption></figure>