Approximations of the GOCE error variance-covariance matrix for least-squares estimation of height datum offsets

One main geodetic objective of the European Space Agency’s satellite mission GOCE (gravity eld and steady-state ocean circulation explorer) is the contribution to global height system uni cation. This can be achieved by applying the Geodetic Boundary Value Problem (GBVP) approach. Thereby one estimates the unknown datum offsets between different height networks (datum zones) by comparing the physical (e.g. orthometric) height values H of benchmarks in different datum zones to the corresponding values derived from the difference between ellipsoidal heights h (e.g. determined by means of global navigation satellite systems) and geoid heights N . In the ideal case, i.e. neglecting data errors, the mis t between H and (h−N) is constant inside one datum zone and represents the datum offset. In practise, the accuracy of the offset estimationdepends on the accuracy of the threequantitiesH, h andN , where the latter canbe computed from the combination of a GOCE-derivedGlobal Potential Model (GPM) for the long tomediumwavelength and terrestrial data for the short wavelength content. Providing an optimumestimation of the datumoffsets alongwith realistic error estimates, theoretically, requires propagation of the full error variance and covariance information of the GOCE spherical harmonic coefficients to geoid heights, respectively geoid height differences. Fromanumerical point of view, this is a very demanding taskwhich cannot simply be run on a single PC. Therefore it is worthwhile to investigate on different levels of approximation of the full variance-covariancematrix (VCM) with the aim of minimizing the numerical effort. In this paper, we compare the estimation error based on three levels of approximation, namely (1) using the full VCM, (2) using only elements of the dominant m-block structure of the VCM and (3) using only the main diagonal of the VCM, i.e. neglecting all error covariances between the spherical harmonic coefficients. We show that the m-block approximation gives almost the same result as provided by the full VCM. The diagonal approximation however overor underestimates the geoid height error, depending on the geographic location and therefore is not regarded to be a suitable approximation.