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Quantifying uncertainties in fault slip distribution during the Tōhoku tsunami using polynomial chaos

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Abstract

An efficient method for inferring Manning’s n coefficients using water surface elevation data was presented in Sraj et al. (Ocean Modell 83:82–97 2014a) focusing on a test case based on data collected during the Tōhoku earthquake and tsunami. Polynomial chaos (PC) expansions were used to build an inexpensive surrogate for the numerical model GeoClaw, which were then used to perform a sensitivity analysis in addition to the inversion. In this paper, a new analysis is performed with the goal of inferring the fault slip distribution of the Tōhoku earthquake using a similar problem setup. The same approach to constructing the PC surrogate did not lead to a converging expansion; however, an alternative approach based on basis pursuit denoising was found to be suitable. Our result shows that the fault slip distribution can be inferred using water surface elevation data whereas the inferred values minimize the error between observations and the numerical model. The numerical approach and the resulting inversion are presented in this work.

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References

  • Alexanderian A, Le Maître O, Najm H, Iskandarani M, Knio O (2011a) Multiscale stochastic preconditioners in non-intrusive spectral projection. J Sci Comput 50:306–340. https://doi.org/10.1007/s10915-011-9486-2

    Article  Google Scholar 

  • Alexanderian A, Le Maître O, Najm H, Iskandarani M, Knio O (2011b) Multiscale stochastic preconditioners in non-intrusive spectral projection. J Sci Comput. https://doi.org/10.1007/s10915-011-9486-2

    Article  Google Scholar 

  • Alexanderian A, Winokur J, Sraj I, Srinivasan A, Iskandarani M, Thacker W, Knio O (2012) Global sensitivity analysis in an ocean general circulation model: a sparse spectral projection approach. Comput Geosci 16:757–778

    Article  Google Scholar 

  • Amante C, Eakins BW (2009) ETOPO1 1 Arc-Minute global relief model: procedures, data sources and analysis, Technical Report. NGDC-24 NOAA

  • Ammon CJ, Lay T, Kanamori H, Cleveland M (2011) A rupture model of the 2011 off the Pacific coast of Tohoku Earthquake. Earth Planet Space 63(7):693–696

    Article  Google Scholar 

  • Anderson JL (2001) An ensemble adjustment Kalman filter for data assimilation. Mon Weather Rev 129:2884–2903

    Article  Google Scholar 

  • Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite element : a non intrusive approach by regression. Eur J Comput Mech 15:81–92

    Article  Google Scholar 

  • Berg Ev, Friedlander MP (2007) SPGL1: a solver for large-scale sparse reconstruction

  • Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230(6):2345–2367

    Article  Google Scholar 

  • Constantine P, Eldred M, Phipps E (2012) Sparse pseudospectral approximation method. Comput Methods Appl Mech Eng 229-232:1–12

    Article  Google Scholar 

  • Conrad PR, Marzouk Y (2013) Adaptive smolyak speudospectral approximations. SIAM J Sci Comp 35 (6):2643–2670

    Article  Google Scholar 

  • Crestaux T, Maitre OL, Martinez J-M (2009) Polynomial chaos expansion for sensitivity analysis. Reliab Eng Syst Saf 94(7):1161–1172. special Issue on Sensitivity Analysis

    Article  Google Scholar 

  • Das SK, Lardner RW (1992) Variational parameter estimation for a two-dimensional numerical tidal model. Int J Numer Methods Fluids 15(3):313–327

    Article  Google Scholar 

  • Davis G, Mallat S, Avellaneda M (1997) Adaptive greedy approximations. Constr Approx 13(1):57–98

    Article  Google Scholar 

  • Donoho D (2006) Compressed sensing. IEEE Trans Inform Theory 52:1289–1306

    Article  Google Scholar 

  • Fukuda J, Johnson KM (2008) A fully Bayesian inversion for spatial distribution of fault slip with objective smoothing. Bullet Seismol Soc Amer 98(3):1128–1146

    Article  Google Scholar 

  • Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New York

    Book  Google Scholar 

  • Gerstner T, Griebel M (2003) Dimension-adaptive tensor-product quadrature. Computing 71(1):65–87

    Article  Google Scholar 

  • Gica E, Spillane M, Titov VV, Chamberlin C, Newman JC (2008) Development of the forecast propagation database for NOAA’s short-term inundation forecast for tsunamis (SIFT). NOAA Technical Memo. OAR PMEL-139, 89 pp

  • González FI, LeVeque RJ, Varkovitzky J, Chamberlain P, Hirai B, George DL (2011) GeoClaw results for the NTHMP tsunami benchmark problems, http://depts.washington.edu/clawpack/links/nthmp-benchmarks/

  • Haario H, Saksman E, Tamminen J (2001) An adaptive metropolis algorithm. Bernoulli 7(2):223–242. http://www.jstor.org/stable/3318737

    Article  Google Scholar 

  • Heemink AW, Mouthaan E, Roest M, Vollebregt E, Robaczewska KB, Verlaan M (2002) Inverse 3D shallow water flow modelling of the continental shelf. Cont Shelf Res 22(3):465–484

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Lardner RW, Song Y (1995) Optimal estimation of Eddy viscosity and friction coefficients for a quasi-three-dimensional numerical tidal model. Atmosphere-Ocean 33(3):581–611

    Article  Google Scholar 

  • Le Maître OP, Knio O (2010) Spectral methods for uncertainty quantification. Springer, New York

    Book  Google Scholar 

  • Li G, Iskandarani M, Le Henaff M, Winokur J, Le Maitre O, Knio O (2016) Quantifying initial and wind forcing uncertainties in the Gulf of Mexico. Comput Geosci 20(5):1133–1153

    Article  Google Scholar 

  • Luo X, Hoteit I (2014) Ensemble Kalman filtering with a divided state-space strategy for coupled data assimilation problems. Mon Weather Rev 142(12):4542–4558

    Article  Google Scholar 

  • MacInnes BT, Gusman AR, LeVeque RJ, Tanioka Y (2013) Comparison of earthquake source models for the 2011 Tohoku event using tsunami simulations and near-field observations. Bullet Seismol Soc Amer 103 (2B):1256–1274

    Article  Google Scholar 

  • Malinverno A (2002) Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem. Geophys J Int 151(3):675–688. https://doi.org/10.1046/j.1365-246X.2002.01847.x

    Article  Google Scholar 

  • Marzouk Y, Najm H, Rahn LA (2007) Stochastic spectral methods for efficient bayesian solution of inverse problems. J Comput Phys 224(2):560–586. https://doi.org/10.1016/j.jcp.2006.10.010

    Article  Google Scholar 

  • Marzouk Y, Najm H (2009) Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J Comput Phys 228(6):1862–1902. https://doi.org/10.1016/j.jcp.2008.11.024

    Article  Google Scholar 

  • Mattern P, Fennel K, Dowd M (2012) Estimating time-dependent parameters for a biological ocean model using an emulator app- roach. J Mar Syst 96–97:32–47. https://doi.org/10.1016/j.jmarsys.2012.01.015

    Article  Google Scholar 

  • Mayo T, Butler T, Dawson C, Hoteit I (2014) Data assimilation within the Advanced Circulation (ADCIRC) modeling framework for the estimation of Manning’s friction coefficient. Ocean Modell 76:43–58

    Article  Google Scholar 

  • Milburn HB, Nakamura AI, González FI (1996) Real-time tsunami reporting from the deep ocean. In: Oceans 96 MTS/IEEE Conference. Fort Lauderdale, FL, pp 390–394

  • Mungov G, Eblé M, Bouchard R (2013) DART tsunameter retrospective and real-time data: a reflection on 10 years of processing in support of tsunami research and operations. Pure Appl Geophys 170(9-10):1369–1384. https://doi.org/10.1007/s00024-012-0477-5

    Article  Google Scholar 

  • Okada Y (1985) Surface deformation due to shear and tensile faults in a half- space. Bullet Seismol Soc Amer 75:1135–1154

    Google Scholar 

  • Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33(3):1065–1076

    Article  Google Scholar 

  • Peng J, Hampton J, Doostan A (2014) A weighted l 1 minimization approach for sparse polynomial chaos expansions. J Comput Phys 267:92–111

    Article  Google Scholar 

  • Percival DB, Arcas D, Denbo DW, Eble MC, Gica E, Mofjeld HO, Spillane M, Tang L, Titov VV (2009) Extracting tsunami source parameters via inversion of DART buoy data, Techncal Memo. OAR PMEL–144. Government Print Office, Seattle, p 22

    Google Scholar 

  • Petras K (2000) On the Smolyak cubature error for analytic functions. Adv Comput Math 12:71–93

    Article  Google Scholar 

  • Reagan M, Najm H, Ghanem R, Knio O (2003) Uncertainty quantification in reacting flow simulations through non-intrusive spectral projection. Combust Flame 132:545–555

    Article  Google Scholar 

  • Roberts GO, Rosenthal JS (2009) Examples of adaptive MCMC. J Comput Graphical Stat 18(2):349–367. https://doi.org/10.1198/jcgs.2009.06134

    Article  Google Scholar 

  • Sarri A, Guillas S, Dias F (2012) Statistical emulation of a tsunami model for sensitivity analysis and uncertainty quantification. Nat Hazards Earth Syst Sci 12(6):2003–2018. https://doi.org/10.5194/nhess-12-2003-2012. http://www.nat-hazards-earth-syst-sci.net/12/2003/2012/

    Article  Google Scholar 

  • Shao G, Li X, Ji C, Maeda T (2011) Focal mechanism and slip history of 2011 Mw 9.1 off the Pacific coast of Tohoku earthquake, constrained with teleseismic body and surface waves. Earth Planets Space 63:559–564

    Article  Google Scholar 

  • Sivia DS (2006) Data analysis—a bayesian tutorial. Oxford Science Publications, Oxford

    Google Scholar 

  • Silverman BW (1986) Density estimation: for statistics and data analysis. Chapman and Hall, London

    Book  Google Scholar 

  • Smolyak S (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl Akad Nauk SSSR 4:240–243

    Google Scholar 

  • Sraj I, Iskandarani M, Srinivasan A, Thacker W, Winokur J, Alexanderian A, Lee C-Y, Chen SS, Knio O (2013) Bayesian inference of drag parameters using Fanapi AXBT data. Mont Weather Rev 141 (7):2347–2367

    Article  Google Scholar 

  • Sraj I, Mandli K, Knio O, Hoteit I (2014a) Uncertainty quantification and inference of Manning’s friction coefficients using DART buoy data during the Tohoku tsunami. Ocean Modell 83:82–97. https://doi.org/10.1016/j.ocemod.2014.09.001

    Article  Google Scholar 

  • Sraj I, Iskandarani M, Srinivasan A, Thacker W, Knio O (2014b) Drag parameter estimation using gradients and hessian from a polynomial chaos model surrogate. Mont Weather Rev 142(2):933–941

    Article  Google Scholar 

  • Sraj I, Zedler S, Jackson C, Knio O, Hoteit I (2016a) Polynomial chaos-based Bayesian inference of K-profile parameterization in a general circulation model of the tropical pacific. Mon Weather Rev 144(12):4621–4640

    Article  Google Scholar 

  • Sraj I, Le Maître OP, Knio O, Hoteit I (2016b) Coordinate transformation and polynomial chaos for the Bayesian inference of a Gaussian process with parametrized prior covariance function. Comput Methods Appl Mech Eng 298:205–228. https://doi.org/10.1016/j.cma.2015.10.002

    Article  Google Scholar 

  • Tang L et al (2012) Direct energy estimation of the 2011 Japan tsunami using deep-ocean pressure measurements. J Geophys Res 117:C08008. https://doi.org/10.1029/2011JC007635

    Article  Google Scholar 

  • Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM, Philadelphia

    Book  Google Scholar 

  • Tibshirani R (1996) Regression shrinkage and selection via the lasso. J Royal Stat Soc Ser B (Methodol) 58 (1):267–288

    Google Scholar 

  • van den Berg E, Friedlander MP (2008) Probing the Pareto frontier for basis pursuit solutions. SIAM J Sci Comput 31(2):890–912

    Article  Google Scholar 

  • Verlaan M, Heemink AW (1997) Tidal flow forecasting using reduced rank square root filters. Stoch Hydrol Hydraul 11(5):349–368

    Article  Google Scholar 

  • Wang S, Iskandarani M, Srinivasan A, Thacker C, Winokur J, Knio O (2016) Propagation of uncertainty and sensitivity analysis in an integral oil-gas plume model. J Geophys Res: Oceans 121(5):3488–3501

    Article  Google Scholar 

  • Winokur J, Conrad P, Sraj I, Knio O, Srinivasan A, Thacker W, Marzouk Y, Iskandarani M (2013) A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database. Comput Geosci 17(6):899–911

    Article  Google Scholar 

  • Xiu D, Tartakovsky D (2004) Uncertainty quantification for flow in highly heterogeneous porous media. In: Cass WGG, Miller T, Farthing MW, Pinder GF (eds) Computational methods in water resources: volume 1, Vol. 55, Part 1 of Developments in Water Science. Elsevier, pp 695–703

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Acknowledgments

The authors would like to thank Dr. Olivier Le Maitre for the helpful discussions of the results.

Funding

Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia grant number CRG3-2156.

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Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, Duke University, 144 Hudson Hall, Durham, NC, 27708, USA

    Ihab Sraj & Omar M. Knio

  2. Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    Ihab Sraj & Ibrahim Hoteit

  3. Department of Applied Physics and Applied Mathematics, Columbia University, 500 W. 120th St., New York, NY, 10027, USA

    Kyle T. Mandli

  4. Division of Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    Omar M. Knio & Ibrahim Hoteit

  5. Institute for Computational Engineering and Science, University of Texas at Austin, 201 E 24th ST. Stop C0200, Austin, TX, 78712-1229, USA

    Clint N. Dawson

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  1. Ihab Sraj
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Correspondence to Ihab Sraj.

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Responsible Editor: Pierre F.J. Lermusiaux

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Sraj, I., Mandli, K.T., Knio, O.M. et al. Quantifying uncertainties in fault slip distribution during the Tōhoku tsunami using polynomial chaos. Ocean Dynamics 67, 1535–1551 (2017). https://doi.org/10.1007/s10236-017-1105-9

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  • DOI: https://doi.org/10.1007/s10236-017-1105-9

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