Low-Rank Kalman Filtering for Efficient State Estimation of Subsurface Advective Contaminant Transport Models
Publication: Journal of Environmental Engineering
Volume 138, Issue 4
Abstract
Accurate knowledge of the movement of contaminants in porous media is essential to track their trajectory and later extract them from the aquifer. A two-dimensional flow model is implemented and then applied on a linear contaminant transport model in the same porous medium. Because of different sources of uncertainties, this coupled model might not be able to accurately track the contaminant state. Incorporating observations through the process of data assimilation can guide the model toward the true trajectory of the system. The Kalman filter (KF), or its nonlinear invariants, can be used to tackle this problem. To overcome the prohibitive computational cost of the KF, the singular evolutive Kalman filter (SEKF) and the singular fixed Kalman filter (SFKF) are used, which are variants of the KF operating with low-rank covariance matrices. Experimental results suggest that under perfect and imperfect model setups, the low-rank filters can provide estimates as accurate as the full KF but at much lower computational effort. Low-rank filters are demonstrated to significantly reduce the computational effort of the KF to almost 3%.
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Acknowledgments
This publication utilized work supported in part by funds from the KAUST GCR Collaborative Fellow program.
References
Ahn, H. (2000). “Ground water drought management by a feedforward control method.” J. Am. Water Resour. Assoc., 36, 501–510.JWRAF5
Astrom, K. J., and Wittenmark, B. (1989). Adaptive control, Addison Wiesley, Reading, MA.
Bidwell, V. J. (1998). “State-space moxing cell model of unsteady solute transport in unsaturated soil.” Environ. Modell. Software, 14, 161–169.EMSOFT
Bowles, D. S., and Grenner, W. J. (1978). “Steady state river quality modeling by sequential extended Kalman filter.” Water Resour. Res., 14, 84–96.WRERAQ
Budhiraja, A., Chen, L., and Lee, C. (2007). “A survey of numerical methods for nonlinear filtering problems.” Physica D (Amsterdam), 230, 27–36.PDNPDT
Cheng, X. (2000). “Kalman filter scheme for three-dimensional subsurface transport simulation with a continuous input.” MS thesis, North Carolina A&T State Univ., Greensboro, NC.
Chang, S. Y., and Jin, A. (2005). “Kalman filtering with regional noise to improve accuracy of contaminant transport models.” J. Environ. Eng.JOEEDU, 131(6), 971–982.
Chang, S. Y., and Latif, S. M. I. (2007). “Use of Kalman filtering and particle filtering in a one dimensional leachate transport model.” Proc., 3rd National Conf. on Environmental Science and Technology, Springer, Greensboro, NC.
Chang, S. Y., and Latif, S. M. I. (2010). “Extended Kalman filtering to improve the accuracy of a subsurface contaminant transport model.” J. Environ. Eng.JOEEDU, 136(5), 466–474.
Cosby, B. J., Hornberger, G. M., and Kelly, M. G. (1984). “Identification of photosynthesis-lightmodels for aquatic systems II: Application to a macrophyte dominated stream.” Ecol. Model.ECMODT, 23, 25–51.
Dawson, C., Sun, S., and Wheeler, M. F. (2004). “Compatible algorithms for coupled flow and transport.” Comput. Methods Appl. Mech. Eng., 193, 2565–2580.CMMECC
Dong, C., Sun, S., and Taylor, G. A. (2011). “Numerical modeling of contaminant transport in fractured porous media using mixed finite-element and finite volume methods.” J. Porous MediaJPMEFB, 14(3), 219–242
El Harrouni, K., Ouazar, D., Wrobel, L. C., and Cheng, A. H. D. (1997). “Aquifer parameter estimation by extended Kalman filtering and boundary elements.” Eng. Anal. Boundary Elem.EABAEL, 19(3), 231–237.
Ferraresi, M., and Marinelli, A. (1996). “An extended formulation of the integrated finite difference method for ground water flow and transport.” J. Hydrol. (Amsterdam)JHYDA7, 175, 453–471.
Ghil, M., and Malanotte-Rizzoli, P. (1991). “Data assimilation in meteorology and oceanography.” Adv. Geophys., 33, 141–266.ADGOAR
Godfrey, J. T., and Foster, G. D. (1996). “Kalman filter method for estimating organic contaminant concentration in major Chesapeake Bay tributaries.” Environ. Sci. Technol., 30, 2312–2317.ESTHAG
Hantush, M. M., and Mariño, M. A. (1997). “Stochastic solution to Inverse problem in ground water.” J. Hydraul. Eng.JHEND8, 123(12), 1139–1146.
Hoteit, I., and Pham, D. (2003). “Evolution of the reduced state space and data assimilation schemes based on the Kalman filter.” J. Metero. Sci. JapanJMSJAU, 81(1), 21–39.
Hoteit, I., Triantafyllou, G., and Korres, G. (2007). “Using low-rank ensemble Kalman filters for data assimilation with high dimensional imperfect models.” J. Numer. Anal., Indust. Appl. Math., 2(1-2), 67–78.
Hoteit, I., Pham, D., and Blum, J. (2001). “A semi-evolutive partially local filter for data assimilation.” Mar. Pollut. Bull., 43, 164–174.MPNBAZ
Hoteit, I., Pham, D., and Blum, J. (2002). “A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific.” J. Marine SystemsJMASE5, 36(1–2), 101–127.
Moradkhan, H., Sorooshia, S., Gupt, H. V., and House, P. R. (2005). “Dual state–parameter estimation of hydrological models using ensemble Kalman filter.” Adv. Water Resour., 28(2), 135–147.AWREDI
Pham, D. T., Verron, J., and Rouband, M. C. (1998). “Singular evolutive Kalman filter with EOF initialization for data assimilation in oceanography.” J. Marine SystemsJMASE5, 16(3–4), 323–340.
Piatyszek, E., Voignier, P., and Graillot, D. (2000). “Fault detection on a sewer network by a combination of a Kalman filter and a binary sequential probability ratio test.” J. Hydrol. (Amsterdam)JHYDA7, 230, 258–268.
Porter, D., Bruce, G., Jones, W., Huyakorn, P., Hamm, L., and Flach, G. (2000). “Data fusion modeling for groundwater systems.” J. Contam. Hydrol., 42, 303–335.JCOHE6
Schreider, S. Y., Young, P. C., and Jakeman, A. J. (2001). “An application of the Kalman filtering technique for streamflow forecasting in the Upper Murray Basin.” Math. Comput. Model.MCMOEG, 33(6-7), 733–743.
Skaggs, T. H., and Mohanty, B. P. (1998). “Water table dynamics in tile-drained fields.” Soil Sci. Soc. Am. J.SSSJD4, 62, 1191–1196.
Sun, S., and Wheeler, M. F. (2005a). “Discontinuous Galerkin methods for coupled flow and reactive transport problems.” Appl. Numer. Math., 52(2-3), 273–298.ANMAEL
Sun, S., and Wheeler, M. F. (2005b). “Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media.” SIAM J. Numer. Anal., 43(1), 195–219.SJNAEQ
Sun, S., and Wheeler, M. F. (2006a). “Anisotropic and dynamic mesh adaptation for discontinuous Galerkin methods applied to reactive transport.” Comput. Methods Appl. Mech. Eng., 195(25-28), 3382–3405.CMMECC
Sun, S., and Wheeler, M. F. (2006b). “A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems.” Comput. Methods Appl. Mech. Eng., 195(7-8), 632–652.CMMECC
Sun, S., and Wheeler, M. F. (2006c). “Projections of velocity data for the compatibility with transport.” Comput. Methods Appl. Mech. Eng., 195, 653–673.CMMECC
van Geer, F. C. (1982). “An equation based theoretical approach to network design for ground water levels using Kalman filters.” Proc., Exeter Symp.: Improvements of Methods of Long Term Prediction of Variations in Groundwater Resources and Regimes Due to Human Activity, Int. Association for Hydrological Science (IAHS), Wallingford, U.K., 136, 241–250.
Whitehead, P. G., and Hornberger, G. M. (1984). “Modeling algal behavior in the river Thames.” Water Resour., 18(8), 945–953.WATRAG
Welch, G., and Bishop, G. (2006). “An introduction to the Kalman filter.” UNC Chapel Hill, Dept. of Computer Science, TR 95-041, Chapel Hill, NC.
Wendroth, O., Rogasik, H., Koszinski, S., Ritsema, C. J., Dekker, L. W., and Nielsen, D. R. (1999). “State-space prediction of field-scale soil water content time series in a sandy loam.” Soil Tillage Res., 50, 85–93.SOTRD5
Yu, Y.-S., Heidari, M., and Guang-Te, W. (1989). “Optimal estimation of contaminant transport in ground water.” J. Am. Water Resour. Assoc.JWRAF5, 25(2), 295–300.
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Information
Published In
Journal of Environmental Engineering
Volume 138 • Issue 4 • April 2012
Pages: 446 - 457
Copyright
© 2012. American Society of Civil Engineers.
History
Received: Feb 15, 2011
Accepted: Sep 2, 2011
Published online: Sep 5, 2011
Published in print: Apr 1, 2012
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