A stochastic collocation based Kalman filter for data assimilation |
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Authors: | Lingzao Zeng Dongxiao Zhang |
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Institution: | (1) Cold and Arid Regions Environmental and Engineering Research Institute, CAS, 730000 Lanzhou, China;(2) Department of Geological Sciences, Florida State University, Tallahassee, FL 32306, USA;(3) Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA; |
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Abstract: | In this paper, a stochastic collocation-based Kalman filter (SCKF) is developed to estimate the hydraulic conductivity from
direct and indirect measurements. It combines the advantages of the ensemble Kalman filter (EnKF) for dynamic data assimilation
and the polynomial chaos expansion (PCE) for efficient uncertainty quantification. In this approach, the random log hydraulic
conductivity field is first parameterized by the Karhunen–Loeve (KL) expansion and the hydraulic pressure is expressed by
the PCE. The coefficients of PCE are solved with a collocation technique. Realizations are constructed by choosing collocation
point sets in the random space. The stochastic collocation method is non-intrusive in that such realizations are solved forward
in time via an existing deterministic solver independently as in the Monte Carlo method. The needed entries of the state covariance
matrix are approximated with the coefficients of PCE, which can be recovered from the collocation results. The system states
are updated by updating the PCE coefficients. A 2D heterogeneous flow example is used to demonstrate the applicability of
the SCKF with respect to different factors, such as initial guess, variance, correlation length, and the number of observations.
The results are compared with those from the EnKF method. It is shown that the SCKF is computationally more efficient than
the EnKF under certain conditions. Each approach has its own advantages and limitations. The performance of the SCKF decreases
with larger variance, smaller correlation ratio, and fewer observations. Hence, the choice between the two methods is problem
dependent. As a non-intrusive method, the SCKF can be easily extended to multiphase flow problems. |
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