Spatio-temporal Inversion using the Selection Kalman Model

Abstract

Data assimilation in models representing spatio-temporal phenomena poses a challenge, particularly if the spatial histogram of the variable appears with multiple modes. The traditional Kalman model is based on a Gaussian initial distribution and Gauss-linear dynamic and observation models. This model is contained in the class of Gaussian distribution and is therefore analytically tractable. It is however unsuitable for representing multimodality. We define the selection Kalman model that is based on a selection-Gaussian initial distribution and Gauss-linear dynamic and observation models. The selection-Gaussian distribution can be seen as a generalization of the Gaussian distribution and may represent multimodality, skewness and peakedness. This selection Kalman model is contained in the class of selection-Gaussian distributions and therefore it is analytically tractable. An efficient recursive algorithm for assessing the selection Kalman model is specified. The synthetic case study of spatio-temporal inversion of an initial state, inspired by pollution monitoring, containing an extreme event suggests that the use of the selection Kalman model offers significant improvements compared to the traditional Kalman model when reconstructing discontinuous initial states.

Data Assimilation in Spatio-Temporal Models with Non-Gaussian Initial States—The Selection Ensemble Kalman Model

“…The SEnKF and SEnKS algorithms under study provide an ensemble of the augmented posterior models, f (r T+1 , ν|d 0:T ) and f (r 0:T , ν 0:T |d 0:T ), respectively. In order to obtain the posterior models of interest, f (r T+1 |d 0:T ) and f (r 0:T |d 0:T ), the conditioning on ν ∈ A must be made by empirical sampling based inference, see [21,22]. This inference requires the estimation of the expectation vector µr ν and covariance matrix Σr ν .…”

“…This model is denoted the selection ensemble Kalman model. If the forward and likelihood models are Gauss-linear, the posterior model is also selection-Gaussian and analytically tractable, see [22]. When the forward and/or likelihood models are nonlinear, however, approximate or sampling based assessment of the posterior model must be made.…”

“…Both algorithms, SEnKF and SEnKS, contain empirically linearized conditioning and asymptotic results, when the ensemble size goes to infinity, and are consistent only for Gauss-linear forward and likelihood models. Under these assumptions, the model is analytically tractable; however, see [22]. In spite of this lack of asymptotic consistency for general HM models, the ensemble Kalman scheme has proven surprisingly reliable for high-dimensional, weakly nonlinear models even with very modest ensemble sizes [25].…”

“…This forward model is nonlinear due to the product of r t and λ in Equation (20). Consequently, the assumption of Gauss-linearity required for both the traditional Kalman model [1] and the selection Kalman model [22] is violated and necessitates ensemble based algorithms.…”

“…This random field model is conjugate with respect to Gauss-linear forward and observation models, similarly to the Gaussian random field model. The posterior distribution is therefore analytically tractable under these assumptions [22]. For general forward and observation models, ensemble based algorithms along the lines of the EnKF can be designed.…”