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

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

Abstract

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.

Keywords

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

List of symbols

\(\varvec{\zeta }\)

Standardized random variable

\(\delta\)

Kronecker delta function

\(\ell\)

Correlation length scale

\(\gamma\)

Basis coefficient

\(\lambda\)

Singular value

\({\mathbb {E}}\)

Expectation operator

\({\mathbb {V}}\)

Variance

\(\mathbf {\Lambda }\)

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 \(N \times M\)

\(\mu\)

Mean value

\(\omega\)

Gaussian quadrature weight

\(\pi\)

Correlation function

\(\varPsi\)

Basis function

\(\rho\)

Joint probability density function

\(\sigma\)

STD

\(\tau\)

Nugget effect

\(\widehat{h}\)

Estimated water level

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

Curvilinear abscissa (km)

A

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

\(c(\alpha )\)

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

D

Kolmogorov–Smirnov statistic

d

Uncertain space size

h

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

i

Surrogate decomposition index

k

Snapshot index

\(K_s\)

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

\(M = {14}\)

Number of observation stations

N

Training set size

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

Validation set size

P

Wet perimeter (m)

Q

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

\(Q_2\)

Predictive squared correlation coefficient

R

Hydraulic radius (m)

r

Number of terms in surrogate model

RC

Upstream water level-discharge local rating curve

\(S_0\)

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

\(S_f\)

Friction slope

\(S_i\)

First order Sobol’ index

\(S_T\)

Total order Sobol’ index

W

River width (m)

CDF

Cumulative distribution function

DA

Data assimilation

EnKF

Ensemble Kalman filter

GP

Gaussian process

MC

Monte Carlo

PC

Polynomial chaos

PDF

Probability density function

PF

Particle filter

PGP

POD-based Gaussian process

POD

Proper orthogonal decomposition

RMSE

Root mean square error

SA

Sensitivity analysis

STD

Standard deviation

SVD

Singular value decomposition

SWE

Shallow water equation

UQ

Uncertainty quantification

Notes

Acknowledgements

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.

References

  1. Barthélémy S (2015) Assimilation de donnée ensembliste et couplage de modèles hydrauliques 1D-2D pour la prévision des crues en temps réel: application au réseau hydraulique Adour Maritime. PhD thesis, Institut National Polytechnique de ToulouseGoogle Scholar
  2. Baudin M, Dutfoy A, Iooss B, Popelin A-L (2015) OpenTURNS: an industrial software for uncertainty quantification in simulationGoogle Scholar
  3. Baudin M, Boumhaout K, Delage T, Iooss B, Martinez J-M (2016) Numerical stability of Sobol’ indices estimation formula. In: 8th International conference on sensitivity analysis of model output, Réunion IslandGoogle Scholar
  4. Berveiller M (2005) Eléments finis stochastiques: approches intrusive et non intrusive pour des analyses de fiabilité. PhD thesis, Université Blaise Pascal, Clermont-FerrandGoogle Scholar
  5. Besnard A, Goutal N (2011) Comparaison de modèles 1D à casiers et 2D pour la modélisation hydraulique d’une plaine d’inondation-Cas de la Garonne entre Tonneins et La Réole. La Houille Blanche 3:42–47CrossRefGoogle Scholar
  6. Birolleau A, Poette G, Lucor D (2014) Adaptive Bayesian inference for discontinuous inverse problems, application to hyperbolic conservation laws. Commun Comput Phys 16:1–34CrossRefGoogle Scholar
  7. Blatman G (2009) Adaptative sparse Polynomial Chaos expansions for uncertainty propagation and sensitivity analysis. PhD thesis, Université Blaise Pascal, Clermont-FerrandGoogle Scholar
  8. Bozzi S, Passoni G, Bernardara P, Goutal N, Arnaud A (2014) Roughness and discharge uncertainty in 1D water level calculations. Environ Model Assess. doi: 10.1007/s10666-014-9430-6
  9. Braconnier T, Ferrier M, Jouhaud J-C, Montagnac M, Sagaut P (2011) Towards an adaptive POD/SVD surrogate model for aeronautic design. Comput Fluids 40(1):195–209. doi: 10.1016/j.compfluid.2010.09.002 CrossRefGoogle Scholar
  10. Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817Google Scholar
  11. Ciriello V, Di Federico V, Riva M, Cadini F, De Sanctis J, Zio E, Guadagnini A (2013) Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer. Stoch Env Res Risk Assess 27(4):945–954CrossRefGoogle Scholar
  12. Clarke B, Fokoue E, Zhang HH (1992) Breakthroughs in statistics, Springer series in statistics. Springer, New York. doi: 10.1007/978-1-4612-4380-9 ISBN 978-0-387-94039-7Google Scholar
  13. Cloke HL, Pappenberger F (2009) Ensemble flood forecasting: a review. J Hydrol 375:613–626. doi: 10.1016/j.jhydrol.2009.06.005 ISSN 00221694CrossRefGoogle Scholar
  14. Damblin G, Couplet M, Iooss B (2013) Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties. J Simul 7(4):276–289CrossRefGoogle Scholar
  15. De Lozzo M, Marrel A (2017) Sensitivity analysis with dependence and variance-based measures for spatio-temporal numerical simulators. Stoch Env Res Risk Assess 31(6):1437–1453CrossRefGoogle Scholar
  16. Dechant CM, Moradkhani H (2011) Improving the characterization of initial condition for ensemble streamflow prediction using data assimilation. Hydrol Earth Syst Sci 15:3399–3410. doi: 10.5194/hess-15-3399-2011 CrossRefGoogle Scholar
  17. Deman G, Konakli K, Sudret B, Kerrou J, Perrochet P, Benabderrahmane H (2015) Using sparse polynomial chaos expansions for the global sensitivity analysis of groundwater lifetime expectancy in a multi-layered hydrogeological model. Reliab Eng Syst Saf 147:156–169CrossRefGoogle Scholar
  18. Després B, Poette G, Lucor G (2013) Robust uncertainty propagation in systems of conservation laws with the entropy closure method. Springer, Cham, pp 105–149. doi: 10.1007/978-3-319-00885-1_3
  19. Dubreuil S, Berveiller M, Petitjean F, Salaün M (2014) Construction of bootstrap confidence intervals on sensitivity indices computed by polynomial chaos expansion. Reliab Eng Syst Saf 121:263–275. doi: 10.1016/j.ress.2013.09.011 CrossRefGoogle Scholar
  20. Durand M, Andreadis KM, Alsdorf DE, Lettenmaier DP, Moller D, Wilson M (2008) Estimation of bathymetric depth and slope from data assimilation of swath altimetry into a hydrodynamic model. Geophys Res Lett 35:1–5. doi: 10.1029/2008GL034150 CrossRefGoogle Scholar
  21. Dutka-Malen I, Lebrun I, Saassouh B, Sudret B (2009) Implementation of a polynomial chaos toolbox in openturns with test-case application. In: Conference: Proceedings of the 10th international conference structures safety and reliability (ICOSSAR’2009), Osaka, JapanGoogle Scholar
  22. El Moçayd N (2017) La décomposition en polynômes du chaos pour lámélioration de lássimilation de données ensembliste en hydraulique fluviale. PhD thesis, Institut National Polytechnique de ToulouseGoogle Scholar
  23. El Moçayd N, Ricci S, Goutal N, Rochoux MC, Boyaval S, Goeury C, Lucor D, Thual O (2017) Polynomial surrogate model for open-channel flows in steady stateGoogle Scholar
  24. ELSheikh AH, Pain CC, Fang F, Gomes JLMA, Navon IM (2013) Parameter estimation of subsurface flow models using iterative regularized ensemble Kalman filter. Stoch Env Res Risk Assess 27(4):877–897CrossRefGoogle Scholar
  25. Goutal N, Maurel F (2002) A finite volume solver for 1D shallow-water equations applied to an actual river. Int J Numer Meth Fluids 38(1):1–19CrossRefGoogle Scholar
  26. Goutal N, Lacombe J-M, Zaoui F, El-Kadi-Adberrezzak K (2012) MASCARET: a 1-D open-souce software for flow hydrodynamic and water quality in open channel networks. In: Murillo (ed) River Flow. Taylor & Francis group, London, pp 1169–1174Google Scholar
  27. Habert J, Ricci S, Le Pape E, Thual O, Piacentini A, Goutal N, Jonville G, Rochoux M (2016) Reduction of the uncertainties in the water level-discharge relation of a 1D hydraulic model in the context of operational flood forecasting. J Hydrol 532:52–64. doi: 10.1016/j.jhydrol.2015.11.023 CrossRefGoogle Scholar
  28. Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning, volume 2 of Springer series in statistics. Springer, New York. ISBN 978-0-387-84857-0. doi: 10.1007/978-0-387-84858-7
  29. Horritt MS, Bates PD (2002) Evaluation of 1D and 2D numerical models for predicting river flood inundation. J Hydrol 268:87–99CrossRefGoogle Scholar
  30. Hosder R, Perez S, Walters RW (2006) A non-intrusive polynomial chaos method for uncertainty propagation in cfd simulations. In: 48th AIAA aerospace sciences meeting and exhibit, number AIAA-2010-0129. The American Institute of Aeronautics and Astronautics, IncGoogle Scholar
  31. Iooss B, Saltelli A (2016) Introduction to sensitivity analysis. In: Handbook of uncertainty quantification, 1–20. Springer, Berlin. doi: 10.1007/978-3-319-11259-6_31-1
  32. Iooss B, Boussouf L, Feuillard V, Marrel A (2010) Numerical studies of the metamodel fitting and validation processes. Int J Adv Syst Meas 3(1):11–21Google Scholar
  33. Ishigami T, Homma T (1990) An importance quantification technique in uncertainty analysis for computer models. IEEE, pp 398–403. doi: 10.1109/ISUMA.1990.151285
  34. Krause P, Boyle DP, Bäse F (2005) Comparison of different efficiency criteria for hydrological model assessment. Adv Geosci 5(89):89–97. doi: 10.5194/adgeo-5-89-2005 CrossRefGoogle Scholar
  35. Lamboni M, Monod H, Makowski D (2011) Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models. Reliab Eng Syst Saf 96(4):450–459. doi: 10.1016/j.ress.2010.12.002 CrossRefGoogle Scholar
  36. Le Gratiet L, Cannamela C, Iooss B (2014) A Bayesian approach for global sensitivity analysis of (multifidelity) computer codes. SIAM/ASA J Uncertain Quantif 2(1):336–363. doi: 10.1137/130926869 CrossRefGoogle Scholar
  37. Le Gratiet L, Marelli S, Sudret B (2017) Metamodel-based sensitivity analysis: polynomial chaos expansions and Gaussian processes. In: Handbook of uncertainty quantification. Springer, Berlin, pp 1–37. doi: 10.1007/978-3-319-11259-6_38-1
  38. Le Maitre O, Knio O (2010) Spectral methods for uncertainty quantification. Springer, BerlinCrossRefGoogle Scholar
  39. Li J, Xiu D (2008) On numerical properties of the ensemble Kalman filter for data assimilation. Comput Methods Appl Mech Eng 197:3574–3583CrossRefGoogle Scholar
  40. Li J, Xiu D (2009) A generalized polynomial chaos based ensemble Kalman filter with high accuracy. J Comput Phys 228(15):5454–5469. doi: 10.1016/j.jcp.2009.04.029 CrossRefGoogle Scholar
  41. Liang G, Fai C Kwok, Kobayashi MH (2008) Stochastic solution for uncertainty propagation in nonlinear shallow-water equations. J Hydraul Eng 134(12):1732–1743. doi: 10.1061/(ASCE)0733-9429 (2008) 134:12(1732)CrossRefGoogle Scholar
  42. Lockwood BA, Anitescu M (2012) Gradient-enhanced universal kriging for uncertainty propagation. Nucl Sci Eng 170(2):168–195CrossRefGoogle Scholar
  43. Lucor D, Meyers J, Sagaut P (2007) Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos. J Fluid Mech 585:255–279. doi: 10.1017/S0022112007006751 CrossRefGoogle Scholar
  44. Marrel A, Iooss B, Laurent B, Roustant O (2009) Calculations of sobol indices for the Gaussian process metamodel. Reliab Eng Syst Saf 94(3):742–751. doi: 10.1016/j.ress.2008.07.008 CrossRefGoogle Scholar
  45. Marrel A, Perot G, Mottet C (2015) Development of a surrogate model and sensitivity analysis for spatio-temporal numerical simulators. Stoch Env Res Risk Assess 29(3):959–974CrossRefGoogle Scholar
  46. Matgen P, Montanari M, Hostache R, Pfister L, Hoffmann L, Plaza D, Pauwels VRN, De Lannoy GJM, De Keyser R, Savenije HHG (2010) Towards the sequential assimilation of SAR-derived water stages into hydraulic models using the particle filter: proof of concept. Hydrol Earth Syst Sci 14:1773–1785. doi: 10.5194/hess-14-1773-2010 CrossRefGoogle Scholar
  47. Migliorati G, Nobile F, Von Schwerin E, Tempone R (2013) Approximation of quantities of interest in stochastic PDEs by the random Discret L2 Projection on polynomial spaces. SIAM J Sci Comput 35(3):A1440–A1460CrossRefGoogle Scholar
  48. Molga M, Smutnicki C (2005) Test functions for optimization needs (c), pp 1–43Google Scholar
  49. Moradkhani H, Sorooshian S, Gupta HV, Houser PR (2005) Dual state-parameter estimation of hydrological models using ensemble Kalman filter. Adv Water Resour 28:135–147. doi: 10.1016/j.advwatres.2004.09.002 CrossRefGoogle Scholar
  50. Oakley JE, O’Hagan A (2004) Probabilistic sensitivity analysis of complex models: a Bayesian approach. J R Stat Soc B (Stat Methodol) 66(3):751–769. doi: 10.1111/j.1467-9868.2004.05304.x CrossRefGoogle Scholar
  51. Owen NE, Challenor P, Menon PP, Bennani S (2017) Comparison of surrogate-based uncertainty quantification methods for computationally expensive simulators. SIAM/ASA J Uncertain Quantif 5(1):403–435. doi: 10.1137/15M1046812 CrossRefGoogle Scholar
  52. Parrish MA, Moradkhani H, Dechant CM (2012) Toward reduction of model uncertainty: integration of Bayesian model averaging and data assimilation. Water Resour Res 48(W03519):1–18. doi: 10.1029/2011WR011116 Google Scholar
  53. Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, Duchesnay É (2012) Scikit-learn: machine learning in python. J Mach Learn Res 12:2825–2830Google Scholar
  54. Rasmussen CE, Williams C (2006) Gaussian processes for machine learning. MIT Press, CambridgeGoogle Scholar
  55. Rochoux MC, Ricci S, Lucor B, Cuenot D, Trouvé A (2014) Towards predictive data-driven simulations of wildfire spread—Part 1: reduced-cost ensemble Kalman filter based on a polynomial chaos surrogate model for parameter estimation. Nat Hazards Earth Syst Sci 14(11):2951–2973CrossRefGoogle Scholar
  56. Roy PT (2016) Uncertainty Quantification applied to Turbine Design. Technical report, CERFACS, ToulouseGoogle Scholar
  57. Saad GA (2007) Stochastic data assimilation with application to multi-phase flow and health monitoring problems. PhD thesis, Faculty of the Graduate School, University of Southern CaliforniaGoogle Scholar
  58. Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2007) Global sensitivity analysis. The primer. Wiley, Chichester. doi: 10.1002/9780470725184 CrossRefGoogle Scholar
  59. Schoebi R, Sudret B, Wiart J (2015) Polynomial-chaos-based kriging. Int J Uncertain Quantif 5(2):171–193CrossRefGoogle Scholar
  60. Sirovich L (1987) Turbulence and the dynamics of coherent structures part I: coherent structures. Q Appl Math XLV(3):561–571CrossRefGoogle Scholar
  61. Smirnov NV (1939) Estimate of difference between empirical distribution curves in two independent samples. Byull Mosk Gos Univ 2(2)Google Scholar
  62. Sobol IM (1993) Sensitivity analysis for nonlinear mathematical models. Math Model Comput Exp 1(4):407–414Google Scholar
  63. Storlie CB, Swiler LP, Helton JC, Sallaberry CJ (2009) Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models. Reliab Eng Syst Saf 94(11):1735–1763. doi: 10.1016/j.ress.2009.05.007 CrossRefGoogle Scholar
  64. Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979. doi: 10.1016/j.ress.2007.04.002 CrossRefGoogle Scholar
  65. Thual O (2010) Hydrodynamique de l’environnement. Ecole polytechniqueGoogle Scholar
  66. Wales DJ, Doye JPK (1997) Global optimization by basin-hopping and the lowest energy structures of Lennard–Jones clusters containing up to 110 atoms. J Phys Chem A 101(28):5111–5116. doi: 10.1021/jp970984n CrossRefGoogle Scholar
  67. Wand MP, Jones MC (1995) Kernel smoothing. Springer, Boston. doi: 10.1007/978-1-4899-4493-1 ISBN 978-0-412-55270-0CrossRefGoogle Scholar
  68. Weerts AH, Winsemius HC, Verkade JS (2011) Estimation of predictive hydrological uncertainty using quantile regression: examples from the National Flood Forecasting System (England and Wales). Hydrol Earth Syst Sci 15:255–265. doi: 10.5194/hess-15-255-2011 CrossRefGoogle Scholar
  69. Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, PrincetonGoogle Scholar
  70. Xiu D, Karniadakis GE (2002) The wiener-askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(2):619–644. doi: 10.1137/S1064827501387826 CrossRefGoogle Scholar

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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