Semianalytical Equations for Transforming Errors into a Local Geodetic Frame

With the development of space-based systems, the results of various applications are generally provided in an earth-centered and earth-fixed global coordinate frame. However, many times it is necessary to transform those results, together with their errors, into local geodetic frames for physical modeling and geometrical interpretation of results as well as to combine them with terrestrial observations. Although it is common practice to transform errors using the full matrix error propagation law, results of many studies in global Cartesian frames are published in compact form without attaching the necessary variance–covariance parameters, therefore preventing a consistent transformation of covariances into a local geodetic frame. This is especially true for coordinates released as time series of sequential epochs. Examination of coordinate/velocity correlations in a global Cartesian frame (GCF) and a local geodetic frame (LGF) reveals that the transformation matrix from GCF into LGF (which depends on the position and origin of LGF) decorrelates the covariance matrix in the GCF up to 90%, depending on design matrix, a priori weights, and constraints. In this study, utilizing the error decorrelation in the LGF, analytical, transformation equations were inverted for correlation parameters in GCF so as to reconstruct the approximate covariance matrix in GCF from a diagonal matrix of variances. Variances in the LGF were then obtained by a basis change through singular value decomposition. Results show that the new semianalytical expressions could be used successfully to transform variances in GCF into LGF especially in the absence of covariance terms.