High-precision square-root filter/smoother for near-singular system in estimation of terrestrial reference frame

Abstract:

In most space and ground-based Earth observations, the accuracy with which objects can be positioned depends ultimately on the accuracy of the underlying terrestrial reference frame (TRF). For example, errors in the currently available TRFs could account for as much as 15% of the observed sea level rise signal. Determination of a TRF typically involves precise estimation of the positions and velocities of surface stations using decades of space geodetic observations. The model of station motions needs to address heterogeneous time scales including secular drifts of plate tectonics and seasonal periodicities due to atmospheric and ground water loadings. Seismic events and post-seismic relaxation tend to lead to discontinuities in the observation time series, which become sources of nonlinearity in the estimation problem. The state dimension of the smoothing problem is usually on the order of ten thousands, which is two to four orders of magnitude smaller than that of a typical atmospheric and ocean data assimilation problem. However, TRF requires a precision of 1mm in position out of a mean dynamic range of about 1 meter, as well as a full posterior error covariance matrix. These requirements challenge the available computation capabilities. Moreover, the use of prior models are kept minimal to prevent introduction of biases in the observations, leading to a smoothing problem that involves matrices with relatively high condition numbers. A filter/smoother solution based on the information matrix is desirable here since the prior model tends to be only partial or implicit. A "square-root" solution is also desirable since a major source of numerical instability is round-off error affecting positive definiteness of covariance and information matrices. The well-known Square-Root Information Filter followed by the Dyer-McReynolds Covariance Smoother seem suitable although this approach requires a large array for Householder transformations. The Modified Bryson-Frazier smoother and a new information filter/smoother require only a single matrix inversion per time step for the entire procedure, while not offering numerical stability of a square-root approach. These methods are applied to emulate the ITRF2008 solution, a TRF standard, in order to evaluate computational advantages of each method.