ALBERT

Description: To avoid computational burden, diagonal variance covariance matrices (VCM) are preferred to describe the stochasticity of terrestrial laser scanner (TLS) measurements. This simplification neglects correlations and affects least-squares (LS) estimates that are trustworthy with minimal variance, if the correct stochastic model is used. When a linearization of the LS functional model is performed, a bias of the parameters to be estimated and their dispersions occur, which can be investigated using a second-order Taylor expansion. Both the computation of the second-order solution and the account for correlations are linked to computational burden. In this contribution, we study the impact of an enhanced stochastic model on that bias to weight the corresponding benefits against the improvements. To that aim, we model the temporal correlations of TLS measurements using the Matérn covariance function, combined with an intensity model for the variance. We study further how the scanning configuration influences the solution. Because neglecting correlations may be tempting to avoid VCM inversions and multiplications, we quantify the impact of such a reduction and propose an innovative yet simple way to account for correlations with a “diagonal VCM.” Originally developed for GPS measurements and linear LS, this model is extended and validated for TLS range and called the diagonal correlation model (DCM).

Description: The study intercompares three stochastic interpolation methods originating from the same geostatistical family: least-squares collocation (LSC) known from geodesy, as well as ordinary kriging (OKR) and universal kriging (UKR) known from geology and other geosciences. The objective of this work is to assess advantages and drawbacks of fundamental differences in modeling between these methods in imperfect data conditions. These differences primarily refer to the treatment of the reference field, commonly called ‘mean value’ or ‘trend’ in geostatistical language. The trend in LSC is determined globally before the interpolation, whereas OKR and UKR detrend the observations during the modeling process. The approach to detrending leads to the evident differences between LSC, OKR and UKR, especially in severe conditions such as far from the optimal data distribution. The theoretical comparisons of LSC, OKR and UKR often miss the numerical proof, while numerical prediction examples do not apply cross-validation of the estimates, which is proven to be a reliable measure of the prediction precision and a validation of empirical covariances. Our study completes the investigations with precise parametrization of all these methods by leave-one-out validation. It finds the key importance of the detrending schemes and shows the advantage of LSC prior global detrending scheme in unfavorable conditions of sparse data, data gaps and outlier occurrence. The test case is the modeling of vertical total electron content (VTEC) derived from GNSS station data. This kind of data is a challenge for precise covariance modeling due to weak signal at higher frequencies and existing outliers. The computation of daily set of VTEC maps using the three techniques reveals the weakness of UKR solutions with a local detrending type in imperfect data conditions.

Description: A new approach to recover time-variable gravity fields from satellite laser ranging (SLR) is presented. It takes up the concept of lumped coefficients by representing the temporal changes of the Earth’s gravity field by spatial patterns via combinations of spherical harmonics. These patterns are derived from the GRACE mission by decomposing the series of monthly gravity field solutions into empirical orthogonal functions (EOFs). The basic idea of the approach is then to use the leading EOFs as base functions in the gravity field modelling and to adjust the respective scaling factors straightforward within the dynamic orbit computation; only for the lowest degrees, the spherical harmonic coefficients are estimated separately. As a result, the estimated gravity fields have formally the same spatial resolution as GRACE. It is shown that, within the GRACE time frame, both the secular and the seasonal signals in the GRACE time series are reproduced with high accuracy. In the period prior to GRACE, the SLR solutions are in good agreement with other techniques and models and confirm, for instance, that the Greenland ice sheet was stable until the late 1990s. Further validation is done with the first monthly fields from GRACE Follow-On, showing a similar agreement as with GRACE itself. Significant differences to the reference data only emerge occasionally when zooming into smaller river basins with strong interannual mass variations. In such cases, the approach reaches its limits which are set by the low spectral sensitivity of the SLR satellites and the strong constraints exerted by the EOFs. The benefit achieved by the enhanced spatial resolution has to be seen, therefore, primarily in the proper capturing of the mass signal in medium or large areas rather than in the opportunity to focus on isolated spatial details.

Description: We derive an empirical model of the sub-daily polar motion (PM) based on the multi-GNSS processing incorporating GPS, GLONASS, and Galileo observations. The sub-daily PM model is based on 3-year multi-GNSS solutions with a 2 h temporal resolution. Firstly, we discuss differences in sub-daily PM estimates delivered from individual GNSS constellations, including GPS, GLONASS, Galileo, and the combined multi-GNSS solutions. Secondly, we evaluate the consistency between the GNSS-based estimates of the sub-daily PM with three independent models, i.e., the model recommended in the International Earth Rotation and Reference Systems Service (IERS) 2010 Conventions, the Desai–Sibois model, and the Gipson model. The sub-daily PM estimates, which are derived from system-specific solutions, are inherently affected by artificial non-tidal signals. These signals arise mainly from the resonance between the Earth rotation period and the satellite revolution period. We found strong spurious signals in GLONASS-based and Galileo-based results with amplitudes up to 30 µas. The combined multi-GNSS solution delivers the best estimates and the best consistency of the sub-daily PM with external geophysical and empirical models. Moreover, the impact of the non-tidal spurious signals in the frequency domain diminishes in the multi-GNSS combination. After the recovery of the tidal coefficients for 38 tides, we infer better consistency of the GNSS-based empirical models with the new Desai–Sibois model than the model recommended in the IERS 2010 Conventions. The consistency with the Desai–Sibois model, in terms of the inter-quartile ranges of tidal amplitude differences, reaches the level of 1.6, 5.7, 6.3, 2.2 µas for the prograde diurnal tidal terms and 1.2/2.1, 2.3/6.0, 2.6/5.5, 2.1/5.1 µas for prograde/retrograde semi-diurnal tidal terms, for the combined multi-GNSS, GPS, GLONASS, and Galileo solutions, respectively.

Description: We use global data from the Lunar Orbiter Laser Altimeter (LOLA) to retrieve the lunar tidal Love number $$h_2$$ h 2 and find $$h_2 = 0.0387pm 0.0025$$ h 2 = 0.0387 ± 0.0025 . This result is in agreement with previous estimates from laser altimetry using crossover points of LOLA profiles. The Love numbers $$k_2$$ k 2 and $$h_2$$ h 2 are key constraints on planetary interior models. We further develop and apply a retrieval method based on a simultaneous inversion for the topography and the tidal signal benefiting from the large volume of LOLA data. By the application to the lunar tides, we also demonstrate the potential of the method for future altimetry experiments at other planetary bodies. The results of this study are very promising with respect to the determination of Mercury’s and Ganymede’s $$h_2$$ h 2 from future altimeter measurements.

Description: The article “Gravity reference at the Argentinean–German Geodetic Observatory (AGGO) by co-location of superconducting and absolute gravity measurements”, written by Ezequiel D. Antokoletz, Hartmut Wziontek, Claudia N. Tocho, Reinhard Falk, was originally published Online First without Open Access.

Description: Current International Association of Geodesy efforts within regional geoid determination include the comparison of different computation methods in the quest for the “1-cm geoid.” Internal (formal) and external (empirical) approaches to evaluate geoid errors exist, and ideally they should agree. Spherical radial base functions using the spline kernel (SK), least-squares collocation (LSC), and Stokes’s formula are three commonly used methods for regional geoid computation. The three methods have been shown to be theoretically equivalent, as well as to numerically agree on the millimeter level in a closed-loop environment using synthetic noise-free data (Ophaug and Gerlach in J Geod 91:1367–1382, 2017. 10.1007/s00190-017-1030-1). This companion paper extends the closed-loop method comparison using synthetic data, in that we investigate and compare the formal error propagation using the three methods. We use synthetic uncorrelated and correlated noise regimes, both on the 1-mGal ($$=10^{-5}~{mathrm {m s}}^{-2}$$ = 10 - 5 m s - 2 ) level, applied to the input data. The estimated formal errors are validated by comparison with empirical errors, as determined from differences of the noisy geoid solutions to the noise-free solutions. We find that the error propagations of the methods are realistic in both uncorrelated and correlated noise regimes, albeit only when subjected to careful tuning, such as spectral band limitation and signal covariance adaptation. For the SKs, different implementations of the L-curve and generalized cross-validation methods did not provide an optimal regularization parameter. Although the obtained values led to a stabilized numerical system, this was not necessarily equivalent to obtaining the best solution. Using a regularization parameter governed by the agreement between formal and empirical error fields provided a solution of similar quality to the other methods. The errors in the uncorrelated regime are on the level of $$sim $$ ∼ 5 mm and the method agreement within 1 mm, while the errors in the correlated regime are on the level of $$sim $$ ∼ 10 mm, and the method agreement within 5 mm. Stokes’s formula generally gives the smallest error, closely followed by LSC and the SKs. To this effect, we note that error estimates from integration and estimation techniques must be interpreted differently, because the latter also take the signal characteristics into account. The high level of agreement gives us confidence in the applicability and comparability of formal errors resulting from the three methods. Finally, we present the error characteristics of geoid height differences derived from the three methods and discuss them qualitatively in relation to GNSS leveling. If applied to real data, this would permit identification of spatial scales for which height information is preferably derived by spirit leveling or GNSS leveling.

Description: Absolute gravimeters are used in geodesy, geophysics and physics for a wide spectrum of applications. Stable gravimetric measurements over timescales from several days to decades are required to provide relevant insight into geophysical processes. Users of absolute gravimeters participate in comparisons with a metrological reference in order to monitor the temporal stability of the instruments and determine the bias to that reference. However, since no measurement standard of higher-order accuracy currently exists, users of absolute gravimeters participate in key comparisons led by the International Committee for Weights and Measures. These comparisons provide the reference values of highest accuracy compared to the calibration against a single gravimeter operated at a metrological institute. The construction of stationary, large-scale atom interferometers paves the way for a new measurement standard in absolute gravimetry used as a reference with a potential stability up to $$1,hbox {nm}{/}{hbox {s}^{2}}$$ 1 nm / s 2 at 1 s integration time. At the Leibniz University Hannover, we are currently building such a very long baseline atom interferometer with a 10-m-long interaction zone. The knowledge of local gravity and its gradient along and around the baseline is required to establish the instrument’s uncertainty budget and enable transfers of gravimetric measurements to nearby devices for comparison and calibration purposes. We therefore established a control network for relative gravimeters and repeatedly measured its connections during the construction of the atom interferometer. We additionally developed a 3D model of the host building to investigate the self-attraction effect and studied the impact of mass changes due to groundwater hydrology on the gravity field around the reference instrument. The gravitational effect from the building 3D model is in excellent agreement with the latest gravimetric measurement campaign which opens the possibility to transfer gravity values with an uncertainty below the $${10},hbox {nm}{/}{hbox {s}^{2}}$$ 10 nm / s 2 level.