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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
To cite this version:
Vanessya Laborie, Nicole Goutal, Sophie Ricci, Matthias de Lozzo, Philippe Sergent. Global Sensitiv- ity 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�
Conference Paper, Published Version
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:
TELEMAC-MASCARET Core Group
Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/104488 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)
4 CECI, CERFACS/CNRS
1 Compiègne, France, 2,3 Chatou, France, 4 Toulouse, 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
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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 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]).
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).
hν
u
+ hF +
U + U U ρ C + ρ x P ρ h
H uH q g K x gH h
= y hu v + x hu u t + hu
= y hv + x hu t + h
e x
2 y w, x w, x w, Dz air atm
.
3 1 1 0
2
2
(1)
hν
v
+ hF +
U + U U ρ C +ρ y P ρ h
H vH q g K y gH h
= y hv v + x hv u t + hv
e y
2 y w, x w, y w, Dz air atm
.
3 1 1
2 2
where ρ / ρair [kg / m3] are the water/air density , Patm [Pa] is the atmospheric pressure, Uw,x and Uw,y [m/s]
are the horizontal wind velocity components, CDz [-] is the wind influence coefficient, K [m1/3/s] is the river bed and flood plain friction coefficient, using Strickler formulation
[3]. Fx and Fy [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 e [m²/s] is the water diffusion coefficient. g is the gravity (9,814 m/s-²). 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)=h0(x,y) ; u(x,y,t=0)=u0(x,y) ; v(x,y,t=0)=v0(x,y) are provided along with wall boundary conditions at the coastline (slip: u=0 and no normal flow:
u∂h
∂x+v∂h
∂y+w=0 ) and at the upstream and
downstream frontiers (h(xBC, yBC,t)=hBC(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
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24th Telemac-Mascaret User Conference Graz, Austria, 17-20 October, 2017
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 2. Observed (in black) and Perturbed (other colors) Garonne discharge during 20030202 event.
B. Coefficients of variation for univariate analysis
The coefficient of variation is defined with respect to each uncertain input variable Xi, at a given time step and at a given mesh node, as CV=σ
, with σ and the standard deviation/ensemble-mean of the uncertain output Y (resp. uncertain input Xi) variable computed over the sample Eu(Xi). The ratio
CV
(
Y)
CV
(
Xi)
is then formulated.C. ANOVA method and Sobol’ indices
We consider a numerical model of the form Y=f(X1, X2,
… Xk) 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. Vi
is referred to as the “variance of conditional expectation” of Y given Xi and the subscript X~i denotes the vector X deprived from of all factors other than Xi.
The variance of the output Y is the sum of the contribution from the different input parameters:
Vi,j+. . .+V1,2, .. . ,k (6) where Vi=VXi
EX i
Y|Xi
∼ and
Vi,j=VX
i,Xj
[
EX∼i,j(
Y∣Xi,Xj) ]
−VX
i
(
EX∼i(
Y∣Xi) )
−VXj(
EX∼ij(
Y∣X j) )
Dividing Eq. (6) by V(Y) leads to:
Si,j+. . .+S1,2,. . . ,k=1 (7)
In Eq. (7), the Si indices are the sensitivity coefficients (also called first order Sobol’ indices) .The second order indices Si,j (second-order Sobol’ indices) measure the variance in Y for higher orders. The STi shows all contribution related to Xi: .
Y VX
| Y V E Y =
V X
| Y E V
=
STi 1 Xi Xi i Xi Xi i (8)
It should be noted that Si=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 Ne (here Ne = 1000) for the normalized input parameters of size k. The (Ne, 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=
[
−− − − −− − −xx1(1(12)) −− −xx2(2(12)) − − −−⋯⋯ − − −xxi(i(12)) − − − − −− − − −⋯⋯ xx((kk12)) x1(N−1) x2
(N−1)
⋯ xi (N−1)
⋯ xk (N−1) x1
(N)
x2 (N)
⋯ x i (N)
⋯ x
k (N)
]
B=
[
− − − −− − −−xx((k+1k+112)) xx− − −k+2(k+2(12)) −− − −⋯⋯ −− −xx((k+ik+i12)) − − −− − − − −−⋯⋯ xx2k(2k(12)) xk+1(N−1)
xk+2 (N−1)
⋯ xk+i (N−1)
⋯ x2k (N−1) xk+1
(N)
xk+2 (N)
⋯ x k+i (N)
⋯ x
2k (N)
]
2. Definition of k matrices Ci formed by all columns of A except the ith 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 Ci, obtaining (k+2) vector outputs of
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24th Telemac-Mascaret User Conference Graz, Austria, 17-20 October, 2017
dimension N*1: yA=f
(
A)
, yB=f(
B)
, yCi=f
(
Ci)
with i=1, …, k.In this study, according to [12], [7] and [11] which describes the best practices for the simultaneous computation of Si and STi for each input variable, the following estimators have been chosen for the computation of:
Si ,
Aj
Ci j B N
= j i i
i X
X j y
y Σ y
=N X
| Y E
V 1
∼
1
STi ,
1
22N
1
j
y Σ y
= X
| Y V
E Ci
j A N
= j i i
i X X
V
(
Y)
, V(
Y)
= 1N ΣNj=1
(
yA j)
2− f02 with
f02= 1
N ΣNj=1yAj yBj
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.
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24th Telemac-Mascaret User Conference Graz, Austria, 17-20 October, 2017
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.
REFERENCES
[1] [Efron & Stein, 1981] Efron, B.; Stein, C., (1981), The Jackknife Estimate of Variance. Ann. Statist. 9 ,no. 3, 586--596.
doi:10.1214/aos/1176345462-
http://projecteuclid.org/euclid.aos/1176345462.
[2] [Iooss & Lemaître, 2014] Iooss, B, Lemaître, P, (2014), A Review on Global Sensitivity Analysis Methods, Chapter Uncertainty Management in Simulation-Optimization of Complex Systems, Volume 59 of the series Operations Research/Computer Science Interfaces Series, pp 101-122
[3] [Gauckler, 1867] Gauckler, Ph., (1867), Etudes Théoriques et Pratiques sur l'Ecoulement et le Mouvement des Eaux, Tome 64, Paris, France: Comptes Rendues de l'Académie des Sciences, pp. 818–822
[4] [HERVOUET, 2006] Hervouet J.M. (2006) - Hydrodynamics of free surface flows. Ed. Wiley. 390 p.
[5] [Hissel, 2010] Hissel, F, (2010), Projet Gironde – rapport final d’évaluation du modèle Gironde, 70 p.
[6] [Homma & Saltelli, 1996] Homma, T, Saltelli. A, (1996), Importance Measures in Global Sensitivity Analysis of Nonlinear Models, Reliability Engineering & System Safety 52 (1), pp 1-17
[7] [Janssen, 1999] Janssen, M.J.W., (1999), Analysis of variance designs for model output, Computer Physics Communications 117, pp 35–43.
[8] [Levy, 2013] Levy, F., (2013), Construction d'un modèle de surcotes sur la façade atlantique, rapport provisoire
[9] [Levy & Joly, 2013] Levy, F., Joly, A., (2013), Modélisation des surcotes avec TELEMAC2D, rapport EDF à accessibilité restreinte, 79 p.
[10] [Nossent & al., 2011] Nossent, J., Elsen, P., Bauwens W., (2011,), Sobol’ sensitivity analysis of a complex environmental model, Environmental Modelling & Software Volume 26, Issue 12, pp. 1515–
1525
[11] [Saltelli, 2002] Saltelli, A., (2002), Making best use of model evaluations to compute sensitivity indices, Computer Physics Communications 145, p. 280–297
[12] [Saltelli, 2010] Saltelli, A., Annoni, P., (2010) How to avoid a perfunctory sensitivity analysis, Environmental Modelling & Software 25 (12), p. 1508-1517
[13] [Saltelli 2015] Becker, W, Saltelli, A, (2015), Design for Sensitivity Analysis, Handbook of Design and Analysis of Experiments, p. 627- 673
[14] [Sobol’, 1991] Sobol', I. M. 1993. Sensitivity Analysis for nonlinear mathematical models. Mathematical Modeling & Computational
Experiment 1, p 407-414.
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