Computational Geosciences

, Volume 20, Issue 5, pp 1133–1153 | Cite as

Quantifying initial and wind forcing uncertainties in the Gulf of Mexico

  • Guotu Li
  • Mohamed Iskandarani
  • Matthieu Le Hénaff
  • Justin Winokur
  • Olivier P. Le Maître
  • Omar M. Knio
Original Paper

Abstract

This study aims at analyzing the combined impact of uncertainties in initial conditions and wind forcing fields in ocean general circulation models (OGCM) using polynomial chaos (PC) expansions. Empirical orthogonal functions (EOF) are used to formulate both spatial perturbations to initial conditions and space-time wind forcing perturbations, namely in the form of a superposition of modal components with uniformly distributed random amplitudes. The forward deterministic HYbrid Coordinate Ocean Model (HYCOM) is used to propagate input uncertainties in the Gulf of Mexico (GoM) in spring 2010, during the Deepwater Horizon oil spill, and to generate the ensemble of model realizations based on which PC surrogate models are constructed for both localized and field quantities of interest (QoIs), focusing specifically on sea surface height (SSH) and mixed layer depth (MLD). These PC surrogate models are constructed using basis pursuit denoising methodology, and their performance is assessed through various statistical measures. A global sensitivity analysis is then performed to quantify the impact of individual modes as well as their interactions. It shows that the local SSH at the edge of the GoM main current—the Loop Current—is mostly sensitive to perturbations of the initial conditions affecting the current front, whereas the local MLD in the area of the Deepwater Horizon oil spill is more sensitive to wind forcing perturbations. At the basin scale, the SSH in the deep GoM is mostly sensitive to initial condition perturbations, while over the shelf it is sensitive to wind forcing perturbations. On the other hand, the basin MLD is almost exclusively sensitive to wind perturbations. For both quantities, the two sources of uncertainty have limited interactions. Finally, the computations indicate that whereas local quantities can exhibit complex behavior that necessitates a large number of realizations, the modal analysis of field sensitivities can be suitably achieved with a moderate size ensemble.

Keywords

Polynomial chaos expansion Empirical orthogonal function Sensitivity analysis Basis pursuit denoising 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexanderian, A., Winokur, J., Sraj, I., Srinivasan, A., Iskandarani, M., Thacker, W.C., Knio, O.M.: Global sensitivity analysis in an ocean general circulation model: a sparse spectral projection approach. Comput. Geosci. 16(3), 757–778 (2012)CrossRefGoogle Scholar
  2. 2.
    Alvera-Azcárate, A., Barth, A., Rixen, M., Beckers, J.M.: Reconstruction of incomplete oceanographic data sets using empirical orthogonal functions: application to the adriatic sea surface temperature. Ocean Model. 9 (4), 325–346 (2005)CrossRefGoogle Scholar
  3. 3.
    Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)CrossRefGoogle Scholar
  4. 4.
    Bleck, R.: An oceanic general circulation model framed in hybrid isopycnic cartesian coordinates. Ocean Modell. 4(1), 55–88 (2002). doi:10.1016/S1463-5003(01)00012-9 CrossRefGoogle Scholar
  5. 5.
    Chassignet, E.P., Hurlburt, H.E., Smedstad, O.M., Halliwell, G.R., Hogan, P.J., Wallcraft, A.J., Baraille, R., Bleck, R.: The hycom (hybrid coordinate ocean model) data assimilative system. J. Mar. Syst. 65(1-4), 60–83 (2007). doi:10.1016/j.jmarsys.2005.09.016 CrossRefGoogle Scholar
  6. 6.
    Chen, S.S., Curcic, M.: Ocean surface waves in hurricane ike (2008) and superstorm sandy (2012): Coupled model predictions and observations Ocean Modelling (2015)Google Scholar
  7. 7.
    Conrad, P.R., Marzouk, Y.M.: Adaptive smolyak pseudospectral approximations. SIAM J. Sci. Comput. 35(6), A2643–A2670 (2013)CrossRefGoogle Scholar
  8. 8.
    Constantine, P.G., Eldred, M.S., Phipps, E.T.: Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229, 1–12 (2012)CrossRefGoogle Scholar
  9. 9.
    Crestaux, T., Le Maître, O., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009)CrossRefGoogle Scholar
  10. 10.
    Doostan, A., Owhadi, H.: A non-adapted sparse approximation of pdes with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)CrossRefGoogle Scholar
  11. 11.
    Efron, B., Hastie, T., Johnstone, I., Tibshirani, R., et al.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)CrossRefGoogle Scholar
  12. 12.
    Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res 99(C5), 10,143–10,162 (1994)CrossRefGoogle Scholar
  13. 13.
    Gerstner, T., Griebel, M.: Dimension-adaptive tensor—product quadrature. Computing 71(1), 65–87 (2003)CrossRefGoogle Scholar
  14. 14.
    Ghanem, R.G., Spanos, P.D.: Stochastic finite elements: a spectral approach. Springer-Verlag, New York (1991)CrossRefGoogle Scholar
  15. 15.
    Hodur, R.M.: The naval research laboratorys coupled ocean/atmosphere mesoscale prediction system (coamps). Mon. Weather Rev. 125, 1414–1430 (1997). doi:10.1175/1520-0493(1997)125%3C1414:TNRLSC%3E2.0.CO;2 CrossRefGoogle Scholar
  16. 16.
    Homma, T., Saltelli, A.: Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Saf. 52(1), 1–17 (1996)CrossRefGoogle Scholar
  17. 17.
    Ichiye, T.: Circulation and water mass distribution in the gulf of Mexico. Geofis. Int. 2(3), 47–76 (1962)Google Scholar
  18. 18.
    Kac, M., Siegert, A.: An explicit representation of a stationary gaussian process. Annals of Mathematical Statistics, 438–442 (1947)Google Scholar
  19. 19.
    Karhunen, K.: Über lineare Methoden in der Wahrscheinlichkeitsrechnung, vol. 37. Universitat Helsinki (1947)Google Scholar
  20. 20.
    Keppenne, C., Rienecker, M.: Initial testing of a massively parallel ensemble Kalman filter with the Poseidon isopycnal ocean general circulation model. Mon. Weather. Rev. 130(12), 2951–2965 (2002)CrossRefGoogle Scholar
  21. 21.
    Knio, O.M., Najm, H.N., Ghanem, R.G., et al.: A stochastic projection method for fluid flow: i. basic formulation. J. Comput. Phys. 173(2), 481–511 (2001)CrossRefGoogle Scholar
  22. 22.
    Le Hénaff, M., Kourafalou, V.H., Paris, C., Helgers, J., Aman, Z.M., Hogan, P.J., Srinivasan, A.: Surface evolution of the deepwater horizon oil spill patch: combined effects of circulation and wind-induced drift. Environ. Sci. Tech. 46(13), 7267–7273 (2012)CrossRefGoogle Scholar
  23. 23.
    Le Maître, O.P., Knio, O.M.: Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer Science & Business Media (2010)Google Scholar
  24. 24.
    Le Maître, O.P., Najm, H.N., Pébay, P.P., Ghanem, R.G., Knio, O.M.: Multi-resolution-analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29(2), 864–889 (2007)CrossRefGoogle Scholar
  25. 25.
    Le Maître, O.P., Reagan, M.T., Najm, H.N., Ghanem, R.G., Knio, O.M.: A stochastic projection method for fluid flow: Ii. random process. J. Comput. Phys. 181(1), 9–44 (2002)CrossRefGoogle Scholar
  26. 26.
    Loève, P.: Fonctions Aléatoires Du Second Ordre, a Note in P. Lévy, Processus Stochastiques Et Mouvement Brownien. Gauthier-Villars, Paris (1948)Google Scholar
  27. 27.
    Marzouk, Y.M., Najm, H.N.: Dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems. J. Comput. Phys. 228(6), 1862–1902 (2009)CrossRefGoogle Scholar
  28. 28.
    McKay, M.D., Beckman, R.J., Conover, W.J.: Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)Google Scholar
  29. 29.
    Najm, H.N., Debusschere, B.J., Marzouk, Y.M., Widmer, S., Le Maître, O.: Uncertainty quantification in chemical systems. Int. J. Numer. Methods Eng. 80(6), 789 (2009)CrossRefGoogle Scholar
  30. 30.
    Peng, J., Hampton, J., Doostan, A.: A weighted l1-minimization approach for sparse polynomial chaos expansions. J. Comput. Phys. 267, 92–111 (2014)CrossRefGoogle Scholar
  31. 31.
    Schmitz, W.J.: Cyclones and Westward Propagation in the Shedding of Anticyclonic Rings from the Loop Current. In: Sturges, W., Lugo-Fernandez, A. (eds.) Circulation in the Gulf of Mexico: Observations and Models, Geophysical Monograph Series, vol. 161, pp 241–261. American Geophysical Union, Washington, D.C (2005)Google Scholar
  32. 32.
    Sheather, S.J., Jones, M.C.: A reliable data-based bandwidth selection method for kernel density estimation. J. Royal Stat. Soc. Ser. B (Methodological) 53(3), 683–690 (1991). http://www.jstor.org/stable/2345597 Google Scholar
  33. 33.
    Sobol, I.: Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1, 407–414 (1993)Google Scholar
  34. 34.
    Sraj, I., Iskandarani, M., Srinivasan, A., Thacker, W.C., Winokur, J., Alexanderian, A., Lee, C.Y., Chen, S.S., Knio, O.M.: Bayesian inference of drag parameters using axbt data from typhoon fanapi. Mon. Weather. Rev. 141(7), 2347–2367 (2013)CrossRefGoogle Scholar
  35. 35.
    Sraj, I., Maître, O.P.L., Knio, O.M., Hoteit, I.: Coordinates transformation and polynomial chaos for the bayesian inference of a gaussian process with parametrized prior covariance function. arXiv preprint arXiv:1501.03323 (2015)
  36. 36.
    Thacker, W.C., Srinivasan, A., Iskandarani, M., Knio, O.M., Le Henaff, M.: Propagating boundary uncertainties using polynomial expansions. Ocean Model. 43, 52–63 (2012)CrossRefGoogle Scholar
  37. 37.
    Van Den Berg, E., Friedlander, M.: Spgl1: a solver for large-scale sparse reconstruction (2007)Google Scholar
  38. 38.
    Van Den Berg, E., Friedlander, M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)CrossRefGoogle Scholar
  39. 39.
    Winokur, J., Conrad, P., Sraj, I., Knio, O., Srinivasan, A., Thacker, W.C., Marzouk, Y., Iskandarani, M.: A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database. Comput. Geosci. 17(6), 899–911 (2013)CrossRefGoogle Scholar
  40. 40.
    Winokur, J.G.: Adaptive sparse grid approaches to polynomial chaos expansions for uncertainty quantification. Ph.D. thesis Duke University (2015)Google Scholar
  41. 41.
    Xiu, D., Karniadakis, G.E.: The wiener–askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Guotu Li
    • 1
  • Mohamed Iskandarani
    • 2
  • Matthieu Le Hénaff
    • 3
    • 4
  • Justin Winokur
    • 1
  • Olivier P. Le Maître
    • 5
  • Omar M. Knio
    • 1
  1. 1.Department of Mechanical Engineering and Material ScienceDuke UniversityDurhamUSA
  2. 2.Rosenstiel School of Marine and Atmospheric ScienceUniversity of MiamiMiamiUSA
  3. 3.Cooperative Institute for Marine and Atmospheric StudiesUniversity of MiamiMiamiUSA
  4. 4.NOAA Atlantic Oceanographic and Meteorological LaboratoryMiamiUSA
  5. 5.LIMSI-CNRSOrsayFrance

Personalised recommendations