Using Fractional Gaussian Noise Models in Orbit Determination

Abstract

Motivated by a recent requirement to provide accurate covariance matrices as well as orbit estimates for cata-loging space objects, a Bayesian estimator to handle autocorrelated process and measurement noises is presented. Although the application is orbit determination, the estimation method itself is general. Techniques are presented to model autocorrelated noises in terms of fractional Gaussian noise and increments of fractional Brownian motion. The Bayesian estimator is derived in both batch and sequential forms, along with explicit formulas for calculating the estimation error covariance matrix. The sequential form generalizes the Kalman filter to the case of autocorrelated noise processes. Also discussed is a batch least-squares estimator formulated with a whitening transformation that accounts for both measurement and process noise. Basic properties of the fractional Gaussian noise model are presented, showing how it contains the special case of white noise in a one-parameter family of self-similar random processes. The parameter characterizes the extent of the autocorrelation and is known as the Hurst parameter. When the measurement noise can be isolated from the process noise by appropriate sensor calibrations, statistical test-of-hypothesis techniques can be used to estimate the Hurst parameter by adjusting it to optimize thep value of the test. For a measurement noise sample that is sufficiently dense in time, the Hurst parameter can be calculated directly from the sample's fractal dimension. In contrast, the process noise usually cannot be isolated from the total noise because the measurements are the only information available. However, assuming the Hurst parameter of the measurement noise is either estimated first or known a priori, test-of-hypothesis techniques can be used to estimate the Hurst parameters of the process noise.