Given a measurement value that has been corrupted with additive multivariate Gaussian noise with nonzero correlation in its covariance matrix, this paper derives two new Taylor series approximations to estimating an unbiased mean and a consistent covariance matrix. The conversion produces mean and covariance estimates that are more consistent than other expansions in the literature when the covariance matrix is not a diagonal matrix.

The paper first derives expressions for the MSE and bias of a mismatched linear filter applied to an arbitrary trajectory and describes how such expressions can be used for process noise selection. The performance of such algorithms when applied to the selection of process noise terms for a 1D sinusoidal dynamic model and for a 2D weaving dynamic model is then considered. Previously, the only algorithm specifically designed for process noise selection of 1D sinusoidal models is a 1995 paper by Sudano. Optimal gains are compared with suboptimal techniques in 1D and 2D, including with Sudano’s algorithm. It is shown that the algorithm optimizing over the explicit MSE outperforms the other techniques both in minimizing the MSE and also in guaranteeing that the peak error is never worse than “connecting the dots” between measurements.

When using long-range sensors to track targets located on the Earth's surface, such as ships at sea or cars on land, the literature often discusses the use of a local 2D tangent-plane coordinate system. This paper derives and compares a number of methods of propagating target motion over short and long distances on an ellipsoidal Earth under particle filter or tangent-plane Gaussian approximations.

The first two moments of a converted bistatic range (delay)-direction-cosine-range-rate measurement are derived taking into account the Doppler information and maximum bounds on the target velocity in orthogonal unobservable directions. Cubature integration is used to very efficiently evaluate the necessary multivariate integrals allowing correlations between measurement components to be easily taken into account. Such single-point track initialization is useful in many tracking algorithms, such as some variants of the joint integrated probabilistic data association filter (JIPDAF).