A polynomial chaos framework for probabilistic predictions of storm surge events

Abstract

We present a polynomial chaos-based framework to quantify the uncertainties in predicting hurricane-induced storm surges. Perturbation strategies are proposed to characterize poorly known time-dependent input parameters, such as tropical cyclone track and wind as well as space-dependent bottom stresses, using a handful of stochastic variables. The input uncertainties are then propagated through an ensemble calculation and a model surrogate is constructed to represent the changes in model output caused by changes in the model input. The statistical analysis is then performed using the model surrogate once its reliability has been established. The procedure is illustrated by simulating the flooding caused by Hurricane Gustav 2008 using the ADvanced CIRCulation model. The hurricane’s track and intensity are perturbed along with the bottom friction coefficients. A sensitivity analysis suggests that the track of the tropical cyclone is the dominant contributor to the peak water level forecast, while uncertainties in wind speed and in the bottom friction coefficient show minor contributions. Exceedance probability maps with different levels are also estimated to identify the most vulnerable areas.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Berg, E.V.D., Friedlander, M.P.: SPGL1: a solver for large-scale sparse reconstruction. [Available online at http://www.cs.ubc.ca/labs/scl/spgl1] (2007)

  2. 2.

    Berg, E.V.D., Friedlander, M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comp. 31(2), 890–912 (2008). https://doi.org/10.1137/080714488. http://link.aip.org/link/?SCE/31/890

    Article  Google Scholar 

  3. 3.

    Beven, J.L. II, Kimberlain, T.B.: Tropical cyclone report - hurricane Gustav. Tech. rep., National Oceanic and Atmospheric Administration/NHC. [Available online at http://www.nhc.noaa.gov/data/tcr/AL072008_Gustav.pdf] (2009)

  4. 4.

    Brown, J.D., Spencer, T., Moeller, I.: Modeling storm surge flooding of an urban area with particular reference to modeling uncertainties: a case study of Canvey Island, United Kingdom. Water Resour. Res. 43(6). https://doi.org/10.1029/2005WR004597. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2005WR004597 (2007)

  5. 5.

    Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comp. 20, 33–61 (1998)

    Article  Google Scholar 

  6. 6.

    Crestaux, T., Le Maître, O., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. and Syst. Saf. 94(7), 1161–1172 (2009)

    Article  Google Scholar 

  7. 7.

    Cruz-Jiménez, H., Li, G., Mai, P., Hoteit, I., Knio, O.: Bayesian inference of earthquake rupture models using polynomial chaos expansion. Geosci. Model Dev. 11(7), 3071–3088 (2018). https://doi.org/10.5194/gmd-11-3071-2018. https://www.geosci-model-dev.net/11/3071/2018/

    Article  Google Scholar 

  8. 8.

    Dietrich, J.C.: Coauthors: a high-resolution coupled riverine flow, tide, wind, wind wave, and storm surge model for Southern Louisiana and Mississippi. Part II: synoptic description and analysis of hurricanes Katrina and Rita. Mon. Weather Rev. 138(2), 378–404 (2010). https://doi.org/10.1175/2009MWR2907.1

    Article  Google Scholar 

  9. 9.

    Dietrich, J.C.: Coauthors: hurricane Gustav (2008) waves and storm surge: hindcast, synoptic analysis, and validation in Southern Louisiana. Mon. Weather Rev. 139(8), 2488–2522 (2011). https://doi.org/10.1175/2011MWR3611.1

    Article  Google Scholar 

  10. 10.

    Elsheikh, A.H., Hoteit, I., Wheeler, M.F.: Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates. Comput. Methods Appl. Mech. Engrg. 269, 515–537 (2014). https://doi.org/10.1016/j.cma.2013.11.001. http://www.sciencedirect.com/science/article/pii/S004578251300296X

    Article  Google Scholar 

  11. 11.

    Finocchio, P.M., Majumdar, S.J., Nolan, D.S., Iskandarani, M.: Idealized tropical cyclone responses to the height and depth of environmental vertical wind shear. Mon. Weather Rev. 144(6), 2155–2175 (2016). https://doi.org/10.1175/MWR-D-15-0320.1

    Article  Google Scholar 

  12. 12.

    Flowerdew, J., Horsburgh, K., Wilson, C., Mylne, K.: Development and evaluation of an ensemble forecasting system for coastal storm surges. Quart. J. Roy. Meteor. Soc. 136(651), 1444–1456 (2010). https://doi.org/10.1002/qj.648. https://rmets.onlinelibrary.wiley.com/doi/abs/10.1002/qj.648

    Article  Google Scholar 

  13. 13.

    Forbes, C., Luettich, R.A. JR, Mattocks, C.A., Westerink, J.J.: A retrospective evaluation of the storm surge produced by hurricane Gustav. Forecast and Hindcast Results. Weather Forecast. 25(6), 1577–1602 (2010). https://doi.org/10.1175/2010WAF2222416.1 (2008)

    Article  Google Scholar 

  14. 14.

    Formaggia, L., Guadagnini, A., Imperiali, I., Lever, V., Porta, G., Riva, M., Scotti, A., Tamellini, L.: Global sensitivity analysis through polynomial chaos expansion of a basin-scale geochemical compaction model. Comput. Geosci. 17(1), 25–42 (2013). https://doi.org/10.1007/s10596-012-9311-5

    Article  Google Scholar 

  15. 15.

    Ghanem, R.G., Spanos, S.D.: Stochastic Finite Elements: a Spectral Approach. Springer, Berlin (1991)

    Google Scholar 

  16. 16.

    Giraldi, L., Le Maître, O., Mandli, K., Dawson, C., Hoteit, I., Knio, O.: Bayesian inference of earthquake parameters from buoy data using a polynomial chaos-based surrogate. Comput. Geosci. 21(4), 683–699 (2017). https://doi.org/10.1007/s10596-017-9646-z

    Article  Google Scholar 

  17. 17.

    Graham, L., Butler, T., Walsh, S., Dawson, C., Westerink, J.J.: A measure-theoretic algorithm for estimating bottom friction in a coastal inlet: case study of bay St. Louis during hurricane Gustav. Mon. Weather Rev. 145(3), 929–954 (2017). https://doi.org/10.1175/MWR-D-16-0149.1

    Article  Google Scholar 

  18. 18.

    Heaps, N.S.: Storm surges, 1967–1982. Geophys. J. Roy. Astro. Soc. 74(1), 331–376 (1983). https://doi.org/10.1111/j.1365-246X.1983.tb01883.x. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1365-246X.1983.tb01883.x

    Article  Google Scholar 

  19. 19.

    Holland, G.J.: An analytic model of the wind and pressure profiles in hurricanes. Mon. Weather Rev. 108(8), 1212–1218 (1980). https://doi.org/10.1175/1520-0493(1980)108<1212:AAMOTW>2.0.CO;2

    Article  Google Scholar 

  20. 20.

    Homma, T., Saltelli, A.: Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. and Syst. Saf. 52(1), 1–17 (1996). https://doi.org/10.1016/0951-8320(96)00002-6. http://www.sciencedirect.com/science/article/pii/0951832096000026

    Article  Google Scholar 

  21. 21.

    Houze, R.A.: Cloud Dynamics. International Geophysics. Elsevier Science. https://books.google.fr/books?id=5DKWGZwBBEYC (1994)

  22. 22.

    Irish, J.L., Resio, D.T., Cialone, M.A.: A surge response function approach to coastal hazard assessment. Part 2: quantification of spatial attributes of response functions. Nat. Hazards 51(1), 183–205 (2009). https://doi.org/10.1007/s11069-009-9381-4

    Article  Google Scholar 

  23. 23.

    Iskandarani, M., Wang, S., Srinivasan, A., Thacker., W.C., Winokur, J., Knio, O.: An overview of uncertainty quantification techniques with application to oceanic and oil-spill simulations. J. Geophys. Res.: Oceans 121(4), 2789–2808 (2016). https://doi.org/10.1002/2015JC011366

    Article  Google Scholar 

  24. 24.

    Jia, G., Taflanidis, A.A., Nadal-Caraballo, N.C., Melby, J.A., Kennedy, A.B., Smith, J.M.: Surrogate modeling for peak or time-dependent storm surge prediction over an extended coastal region using an existing database of synthetic storms. Nat. Hazards 81(2), 909–938 (2016). https://doi.org/10.1007/s11069-015-2111-1

    Article  Google Scholar 

  25. 25.

    Kennedy, A.B.: Coauthors: origin of the hurricane Ike forerunner surge. Geophys. Res. Lett. 38(L08608). https://doi.org/10.1029/2011GL047090. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2011GL047090 (2011)

    Article  Google Scholar 

  26. 26.

    Köppel, M., Franzelin, F., Kröker, I., Oladyshkin, S., Santin, G., Wittwar, D., Barth, A., Haasdonk, B., Nowak, W., Pflüger, D., Rohde, C.: Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario. Comput. Geosci. 23(2), 339–354 (2019). https://doi.org/10.1007/s10596-018-9785-x

    Article  Google Scholar 

  27. 27.

    Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Scientific Computation, Springer (2010)

    Google Scholar 

  28. 28.

    Li, G., Curcic, M., Iskandarani, M., Chen, S.S., Knio, O.M.: Uncertainty propagation in coupled Atmosphere–Wave–Ocean prediction system: a study of hurricane Earl. Mon. Weather Rev. 147(1), 221–245 (2019). https://doi.org/10.1175/MWR-D-17-0371.1

    Article  Google Scholar 

  29. 29.

    Li, G., Iskandarani, M., Le Hénaff, M., Winokur, J., Le Maître, O.P., Knio, O.M.: Quantifying initial and wind forcing uncertainties in the Gulf of Mexico. Comput. Geosci. 20(5), 1133–1153 (2016). https://doi.org/10.1007/s10596-016-9581-4

    Article  Google Scholar 

  30. 30.

    Lin, N., Chavas, D.: On hurricane parametric wind and applications in storm surge modeling. J. Geophys. Res.: Atmospheres 117(D9), n/a–n/a (2012). https://doi.org/10.1029/2011JD017126. D09120

    Google Scholar 

  31. 31.

    Lorenz, E.N.: Empirical orthogonal functions and statistical weather prediction. Scientific report / MIT. Statistical Forecasting Project. Massachusetts Institute of Technology, Department of Meteorology. https://books.google.com/books?id=2cQIogEACAAJ (1956)

  32. 32.

    Luettich, R.A., Westerink J.J.: Formulation and numerical implementation of the 2D/3D ADCIRC finite element model version 44.XX (2004)

  33. 33.

    Mayo, T., Butler, T., Dawson, C., Hoteit, I.: Data assimilation within the advanced circulation (ADCIRC) modeling framework for the estimation of Manning’s friction coefficient. Ocean Model. 76, 43–58 (2014). https://doi.org/10.1016/j.ocemod.2014.01.001. http://www.sciencedirect.com/science/article/pii/S146350031400002X

    Article  Google Scholar 

  34. 34.

    Mel, R., Viero, D.P., Carniello, L., Defina, A., D’Alpaos, L.: Simplified methods for real-time prediction of storm surge uncertainty: The city of Venice case study. Adv. Water. Resour. 71, 177–185 (2014). https://doi.org/10.1016/j.advwatres.2014.06.014. http://www.sciencedirect.com/science/article/pii/S0309170814001316

    Article  Google Scholar 

  35. 35.

    Molteni, F., Buizza, R., Palmer, T.N., Petroliagis, T.: The ECMWF ensemble prediction system: methodology and validation. Quart. J. Roy. Meteor. Soc. 122(529), 73–119 (1996). https://doi.org/10.1002/qj.49712252905. https://rmets.onlinelibrary.wiley.com/doi/abs/10.1002/qj.49712252905

    Article  Google Scholar 

  36. 36.

    Morokoff, W.J., Caflisch, R.E.: Quasi-Monte Carlo integration. J. Comput. Phys. 122(2), 218–230 (1995). https://doi.org/10.1006/jcph.1995.1209. http://www.sciencedirect.com/science/article/pii/S0021999185712090

    Article  Google Scholar 

  37. 37.

    Myers, V.A.: Storm tide frequencies on the South Carolina Coast. Tech. Rep. NWS. 16 NOAA (1975)

  38. 38.

    Navarro, M., Le Maître, O., Hoteit, I., George, D., Mandli, K., Knio, O.: Surrogate-based parameter inference in debris flow model. Comput. Geosci. 22(6), 1447–1463 (2018). https://doi.org/10.1007/s10596-018-9765-1

    Article  Google Scholar 

  39. 39.

    NOAA/NHC: Tropical cyclone report, hurricane Gustav (AL072008). [Available online at http://ftp.nhc.noaa.gov/atcf/archive/2008/bal072008.dat.gz] (2008)

  40. 40.

    Powell, M.D.: Drag coefficient distribution and wind speed dependence in tropical cyclones. Final report to the NOAA joint hurricane testbed (JHT) program, 26p NOAA/atlantic oceanographic and meteorological laboratory (2006)

  41. 41.

    Resio, D.T., Powell, N.J., Cialone, M.A., Das, H.S., Westerink, J.J.: Quantifying impacts of forecast uncertainties on predicted storm surges. Nat. Hazards 88(3), 1423–1449 (2017).

    Article  Google Scholar 

  42. 42.

    Rohmer, J., Lecacheux, S., Pedreros, R., Quetelard, H., Bonnardot, F., Idier, D.: Dynamic parameter sensitivity in numerical modelling of cyclone-induced waves: a multi-look approach using advanced meta-modelling techniques. Nat. Hazards 84(3), 1765–1792 (2016). https://doi.org/10.1007/s11069-016-2513-8

    Article  Google Scholar 

  43. 43.

    Sobol, I.M.: Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1, 407–414 (1993)

    Google Scholar 

  44. 44.

    Sochala, P., De Martin, F.: Surrogate combining harmonic decomposition and polynomial chaos for seismic shear waves in uncertain media. Comput. Geosci. https://doi.org/10.1007/s10596-017-9677-5 (2017)

    Article  Google Scholar 

  45. 45.

    Sochala, P., Le Maître, O.: Polynomial Chaos expansion for subsurface flows with uncertain soil parameters. Adv. Water. Resour. 62, 139–154 (2013). https://doi.org/10.1016/j.advwatres.2013.10.003. https://hal.archives-ouvertes.fr/hal-00931639

    Article  Google Scholar 

  46. 46.

    Song, Y.K., Irish, J.L., Udoh, I.E.: Regional attributes of hurricane surge response functions for hazard assessment. Nat. Hazards 64(2), 1475–1490 (2012). https://doi.org/10.1007/s11069-012-0309-z

    Article  Google Scholar 

  47. 47.

    Taylor, G.I.: Skin friction of the wind on the earth’s surface. Proc. Roy. Soc. London A92(637), 196–199 (1916). https://doi.org/10.1098/rspa.1916.0005. http://rspa.royalsocietypublishing.org/content/92/637/196

    Google Scholar 

  48. 48.

    Taylor, N.R., Irish, J.L., Udoh, I.E., Bilskie, M.V., Hagen, S.C.: Development and uncertainty quantification of hurricane surge response functions for hazard assessment in coastal bays. Nat. Hazards 77(2), 1103–1123 (2015). https://doi.org/10.1007/s11069-015-1646-5

    Article  Google Scholar 

  49. 49.

    Thompson, E.F., Cardone, V.J.: Practical modeling of hurricane surface wind fields. J. Waterw. Port. Coast. 122(4), 195–205 (1996). https://doi.org/10.1061/(ASCE)0733-950X(1996)122:4(195). http://ascelibrary.org/doi/abs/10.1061/1996

    Article  Google Scholar 

  50. 50.

    Toro, G.R., Resio, D.T., Divoky, D., Niedoroda, A.W., Reed, C.: Efficient joint-probability methods for hurricane surge frequency analysis. Ocean Eng. 37(1), 125–134 (2010). https://doi.org/10.1016/j.oceaneng.2009.09.004. http://www.sciencedirect.com/science/article/pii/S0029801809002236

    Article  Google Scholar 

  51. 51.

    Zheng, F., Westra, S., Leonard, M., Sisson, S.A.: Modeling dependence between extreme rainfall and storm surge to estimate coastal flooding risk. Water Resour. Res. 50(3), 2050–2071 (2014). https://doi.org/10.1002/2013WR014616. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1002/2013WR014616

    Article  Google Scholar 

Download references

Funding

The work of P. Sochala is supported by a funding of BRGM (French Geological Survey) through its Institut Carnot sponsored by the ANR (French National Research Agency). This research was made possible in part by a grant from The Gulf of Mexico Research Initiative to the Consortium for Advanced Research on Transport of Hydrocarbon in the Environment (CARTHE) and by NSF 1639722 and NSF 1818847.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pierre Sochala.

Additional information

Data availability

Data are publicly available through the Gulf of Mexico Research Initiative Information & Data Cooperative (GRIIDC) at https://doi.org/https://data.gulfresearchinitiative.org (https://doi.org/10.7266/N73777BS).

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Holland model

Appendix: Holland model

This parametric model [19] describes a symmetrical vortex and is derived by starting with an empirical analytical pressure field and by using the gradient wind equation to get the wind speed profile. The pressure field p is assumed to have an exponential profile,

$$ p(r,t)=p_{\mathrm{c}}(t)+(p_{\mathrm{a}}(t)-p_{\mathrm{c}}(t))\exp\left( -~\left( R_{\text{mw}}(t)/r\right)^{B}\right), $$
(28)

where r is the radial distance from the eye, t the time variable, pc(t) the central pressure, pa(t) the ambient pressure (at infinite radius), Rmw(t) the maximum wind radius (RMW), and B the Holland parameter. Plugging the pressure profile (28) into the gradient wind equation [21] and neglecting the Coriolis force (assumed to be small in the region of maximum winds) leads to the following tangential wind speed profile,

$$ v(r,t)=v_{\max}(t)\left[\left( R_{\text{mw}}(t)/r\right)^{B} \exp\left( -~\left( R_{\text{mw}}(t)/r\right)^{B}+1\right) \right]^{\frac{1}{2}}, $$
(29)

where the maximum wind speed \(v_{\max \limits }(t)\) has been introduced. This latter quantity is defined as v(r = Rmw,t) = (B(pa(t) − pc(t))/(ρae))1/2 with \(\rho _{\mathrm {a}}=1.18\text {kg/m}^{3}\) the air density, and e = 2.72 the Euler’s number. Assuming that the central pressure and the maximum wind speed are known, the ambient pressure in Eq. 28 is then computed with \(p_{\mathrm {a}}(t)=p_{\mathrm {c}}(t)+\rho _{\mathrm {a}} e v_{\max \limits }^{2}(t)/B\). The shape parameter B of the model determines the steepness of the eyewall and the strength of the winds far from the center. Its value ranges over the interval [0.5, 2.5] as mentioned in [49] and we set B = 1.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sochala, P., Chen, C., Dawson, C. et al. A polynomial chaos framework for probabilistic predictions of storm surge events. Comput Geosci 24, 109–128 (2020). https://doi.org/10.1007/s10596-019-09898-5

Download citation

Keywords

  • Tropical cyclones
  • Uncertainty quantification
  • Empirical orthogonal functions
  • Global sensitivity analysis
  • Exceedance probability
  • Hurricane Gustav