Comparison of polynomial chaos and Gaussian process surrogates for uncertainty quantification and correlation estimation of spatially distributed open-channel steady flows

  • Pamphile T. RoyEmail author
  • Nabil El Moçayd
  • Sophie Ricci
  • Jean-Christophe Jouhaud
  • Nicole Goutal
  • Matthias De Lozzo
  • Mélanie C. Rochoux
Original Paper


Data assimilation is widely used to improve flood forecasting capability, especially through parameter inference requiring statistical information on the uncertain input parameters (upstream discharge, friction coefficient) as well as on the variability of the water level and its sensitivity with respect to the inputs. For particle filter or ensemble Kalman filter, stochastically estimating probability density function and covariance matrices from a Monte Carlo random sampling requires a large ensemble of model evaluations, limiting their use in real-time application. To tackle this issue, fast surrogate models based on polynomial chaos and Gaussian process can be used to represent the spatially distributed water level in place of solving the shallow water equations. This study investigates the use of these surrogates to estimate probability density functions and covariance matrices at a reduced computational cost and without the loss of accuracy, in the perspective of ensemble-based data assimilation. This study focuses on 1-D steady state flow simulated with MASCARET over the Garonne River (South-West France). Results show that both surrogates feature similar performance to the Monte-Carlo random sampling, but for a much smaller computational budget; a few MASCARET simulations (on the order of 10–100) are sufficient to accurately retrieve covariance matrices and probability density functions all along the river, even where the flow dynamic is more complex due to heterogeneous bathymetry. This paves the way for the design of surrogate strategies suitable for representing unsteady open-channel flows in data assimilation.


Hydraulic modelling Uncertainty quantification Surrogate model Polynomial chaos Gaussian process Covariance matrix 

List of symbols

\(\varvec{\zeta }\)

Standardized random variable


Kronecker delta function


Correlation length scale


Basis coefficient


Singular value

\({\mathbb {E}}\)

Expectation operator

\({\mathbb {V}}\)


\(\mathbf {\Lambda }\)

Rectangular singular value matrix

\(\mathbf {\Pi }\)

Correlation matrix evaluated for the database \({\mathcal {D}}_N\)

\({\mathbf {C}}\)

Snapshot covariance matrix

\({\mathbf {h}}\)

Output random vector of size M

\({\mathbf {U}}\)

Orthogonal square left singular matrix

\(\mathbf {V}\)

Orthogonal square right singular matrix

\({\mathbf {x}}\)

Input random vector of size d

\({\mathbf {x}}^*\)

Input random vector not included in the training set \({\mathcal {X}}\)

\({\mathbf {Y}}\)

Centred snapshot matrix

\({\mathcal {D}}_N\)

MASCARET simulation database of size N

\({\mathcal {M}}\)

MASCARET forward model operator

\({\mathcal {N}}\)

Normal distribution characterized by mean and STD

\({\mathcal {U}}\)

Uniform distribution characterized by minimum and maximum

\({\mathcal {X}}\)

Input training set of size \(N \times d\)

\({\mathcal {Y}}\)

Output training set of size \(N \times M\)


Mean value


Gaussian quadrature weight


Correlation function


Basis function


Joint probability density function




Nugget effect


Estimated water level

\(a \in [a_{\text {in}}; a_{\text {out}}]\)

Curvilinear abscissa (km)


Hydraulic section (\(\hbox {m}^2\))

\(c(\alpha )\)

\(\alpha\)-level tabulated value associated with D


Kolmogorov–Smirnov statistic


Uncertain space size


Water level part of the hydraulic state (hQ) (m)


Surrogate decomposition index


Snapshot index


Strickler friction coefficient (\(\hbox {m}^{1/3}\,\hbox {s}^{-1}\))

\(M = {14}\)

Number of observation stations


Training set size

\(N_{\text {ref}} = {100,000}\)

Validation set size


Wet perimeter (m)


Discharge part of the hydraulic state (hQ) (\(\hbox {m}^{3}\,\hbox {s}^{-1}\))


Predictive squared correlation coefficient


Hydraulic radius (m)


Number of terms in surrogate model


Upstream water level-discharge local rating curve


Channel slope (\(\hbox {m}\,\hbox {km}^{-1}\))


Friction slope


First order Sobol’ index


Total order Sobol’ index


River width (m)


Cumulative distribution function


Data assimilation


Ensemble Kalman filter


Gaussian process


Monte Carlo


Polynomial chaos


Probability density function


Particle filter


POD-based Gaussian process


Proper orthogonal decomposition


Root mean square error


Sensitivity analysis


Standard deviation


Singular value decomposition


Shallow water equation


Uncertainty quantification



The financial support provided by CNES and EDF R&D is greatly appreciated. The authors acknowledge Michael Baudin, Géraud Blatman, Anne Dutfoy, Bertrand Iooss and Anne-Laure Popelin from MRI (EDF R&D) as well as Cédric Goeury from LNHE (EDF R&D) for helpful discussions on uncertainty quantification and support on OpenTURNS.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.CFD Team, CERFACSToulouse Cedex 1France
  2. 2.CECI, CERFACS – CNRSToulouse Cedex 1France
  3. 3.Laboratory for Hydraulics Saint-Venant (LHSV)ChatouFrance

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