The classical Kalman filter provides optimal least-squares estimates of all of the states of a linear time-varying system under process and measurement noise. This technical is essential for both terrestrial and space weather data assimilation. In many applications, however, optimal estimates are desired for a specified subset of the system states, rather than all of the system states. For example, for systems arising from discretized partial differential equations for terrestrial and space weather prediction, the chosen subset of states can represent the desire to estimate state variables associated with a subregion of the spatial domain. However, it is well known that the optimal state estimator for a subset of system states coincides with the classical Kalman filter.

For applications involving high-order systems, it is often difficult to implement the classical Kalman filter, and thus it is of interest to consider computationally simpler filters that yield suboptimal estimates of a specified subset of states. One approach to this problem is to consider reduced-order Kalman filters. Such reduced-complexity controllers provide estimates of the desired states that are suboptimal relative to the classical Kalman filter. Alternative variants of the classical Kalman filter have been developed for computationally demanding applications such as weather forecasting, where the classical Kalman filter gain and covariance are directly modified so as to reduce the computational requirements.

The present paper is motivated by computationally demanding applications for which a high-order simulation model is assumed to be available, and the derivation of a reduced-order filter is not feasible due to the lack of a tractable analytic model. Instead, we consider the use of a full-order state estimator based directly on the simulation model. However, rather than implementing the classical Kalman filter, we derive a suboptimal spatially localized Kalman filter in which the structure of the filter gain is constrained a priori to reflect the desire to estimate a specified subset of states. Our development is also more general than the classical treatment since the state dimension can be time varying. This extension is useful for variable-resolution discretizations of partial differential equations, which arise in finite-volume CFD techniques for hydrodynamics and magnetohydrodynamics.

The use of a spatially localized Kalman filter in place of the classical Kalman filter is motivated by computational architecture constraints arising from a multi-processor implementation of the Kalman filter in which the Kalman filter operations can be confined to the subset of processors associated with the states whose estimates are desired. An additional motivation is the use of the extended Kalman filter for nonlinear systems. For systems with sparse measurements, observability may not hold for the entire system. In this case, the spatially localized Kalman filter can be used to obtain state estimates for the observable portion of the system.

The spatially localized Kalman filter is applied to a 2-dimensional MHD simulation based on a finite-volume, upwind differencing method. Comparisons are made with the classical Kalman filter in terms of data assimilation accuracy and computational requirements. In addition, for highly nonlinear behavior involving shocks, the spatially localized Kalman filter is used to account for regions of observability. The technique should be of equal interest to practitioners of data assimilation for both terrestrial and space weather applications.