The Meissl scheme for the geodetic ellipsoid

We present a variant of the Meissl scheme to relate surface spherical harmonic coefficients of the disturbing potential of the Earth's gravity field on the surface of the geodetic ellipsoid to surface spherical harmonic coefficients of its first- and second-order normal derivatives on the same or any other ellipsoid. It extends the original (spherical) Meissl scheme, which only holds for harmonic coefficients computed from geodetic data on a sphere. In our scheme, a vector of solid spherical harmonic coefficients of one quantity is transformed into spherical harmonic coefficients of another quantity by pre-multiplication with a transformation matrix. This matrix is diagonal for transformations between spheres, but block-diagonal for transformations involving the ellipsoid. The computation of the transformation matrix involves an inversion if the original coefficients are defined on the ellipsoid. This inversion can be performed accurately and efficiently (i.e., without regularisation) for transformation among different gravity field quantities on the same ellipsoid, due to diagonal dominance of the matrices. However, transformations from the ellipsoid to another surface can only be performed accurately and efficiently for coefficients up to degree and order 520 due to numerical instabilities in the inversion

Towards an International Height Reference System: insights from the Colorado geoid experiment using AUSGeoid computation methods

We apply the AUSGeoid data processing and computation methodologies to data provided for the International Height Reference System (IHRS) Colorado experiment as part of the International Association of Geodesy Joint Working Groups 0.1.2 and 2.2.2. This experiment is undertaken to test a range of different geoid computation methods from international research groups with a view to standardising these methods to form a set of conventions that can be established as an IHRS. The IHRS can realise an International Height Reference Frame to be used to study physical changes on and within the Earth. The Colorado experiment study site is much more mountainous (maximum height 4401 m) than the mostly flat Australian continent (maximum height 2228 m), and the available data over Colorado are different from Australian data (e.g. much more extensive airborne gravity coverage). Hence, we have tested and applied several modifications to the AUSGeoid approach, which had been tailored to the Australian situation. This includes different methods for the computation of terrain corrections, the gridding of terrestrial gravity data, the treatment of long-wavelength errors in the gravity anomaly grid and the combination of terrestrial and airborne data. A new method that has not previously been tested is the application of a spherical harmonic high-pass filter to residual anomalies. The results indicate that the AUSGeoid methods can successfully be used to compute a high accuracy geoid in challenging mountainous conditions. Modifications to the AUSGeoid approach lead to root-mean-square differences between geoid models up to ~ 0.028 m and agreement with GNSS-levelling data to ~ 0.044 m, but the benefits of these modifications cannot be rigorously assessed due to the limitation of the GNSS-levelling accuracy over the computation area

Variance component estimation uncertainty for unbalanced data: Application to a continent-wide vertical datum

Variance component estimation (VCE) is used to update the stochastic model in least-squares adjustments, but the uncertainty associated with the VCE-derived weights is rarely considered. Unbalanced data is where there is an unequal number of observations in each heterogeneous dataset comprising the variance component groups. As a case study using highly unbalanced data, we redefine a continent-wide vertical datum from a combined least-squares adjustment using iterative VCE and its uncertainties to update weights for each data set. These are: (1) a continent-wide levelling network, (2) a model of the ocean’s mean dynamic topography and mean sea level observations, and (3) GPS-derived ellipsoidal heights minus a gravimetric quasigeoid model. VCE uncertainty differs for each observation group in the highly unbalanced data, being dependent on the number of observations in each group. It also changes within each group after each VCE iteration, depending on the magnitude of change for each observation group’s variances. It is recommended that VCE uncertainty is computed for VCE updates to the weight matrix for unbalanced data so that the quality of the updates for each group can be properly assessed. This is particularly important if some groups contain relatively small numbers of observations. VCE uncertainty can also be used as a threshold for ceasing iterations, as it is shown—for this data set at least—that it is not necessary to continue time-consuming iterations to fully converge to unity