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Since 2006, several different groups have computed geoid and/or quasigeoid (quasi/geoid) models for the Auvergne test area in central France using various approaches. In this contribution, we compu...

Least-squares collocation (LSC) is a widely used method applied in physical geodesy to separate observations into a signal and noise part but has received only little attention when interpolating velocity fields. The advantage of the LSC is the possibility to filter and interpolate as well as extrapolate the observations. Here, we will present several extensions to the traditional LSC technique, which allows the combined interpolation of both horizontal velocity components (horizontal velocity (HV)-LSC), the separation of velocity observations on different tectonic plates, and the removal of stationarity by moving variance (the latter as HV-LSC-ex(tended)$$^2$$ 2 ). Furthermore, the covariance analysis, which is required to find necessary input parameters for the LSC, is extended by finding a suitable variance and correlation length using both horizontal velocity components at the same time. The traditional LSC and all extensions are tested on a synthetic dataset to find the signal at known as well as newly defined points, with stations separated on four different plates with distinct plate velocities. The methodologies are evaluated by calculation of a misfit to the input data, and implementation of a leave-one-out cross-validation and a Jackknife resampling. The largest improvement in terms of reduced misfit and stability of the interpolation can be obtained when plate boundaries are considered. In addition, any small-scale changes can be filtered out using the moving-variance approach and a smoother velocity field is obtained. In comparison with interpolation using the Kriging method, the fit is better using the new HV-LSC-ex$$^2$$ 2 technique.

We present the official land uplift model NKG2016LU of the Nordic Commission of Geodesy (NKG) for northern Europe. The model was released in 2016 and covers an area from 49° to 75° latitude and 0° to 50° longitude. It shows a maximum absolute uplift of 10.3 mm/a near the city of Umea in northern Sweden and a zero-line that follows the shores of Germany and Poland. The model replaces the NKG2005LU model from 2005. Since then, we have collected more data in the core areas of NKG2005LU, specifically in Norway, Sweden, Denmark and Finland, and included observations from the Baltic countries as well. Additionally, we have derived an underlying geophysical glacial isostatic adjustment (GIA) model within NKG as an integrated part of the NKG2016LU project. A major challenge is to estimate a realistic uncertainty grid for the model. We show how the errors in the observations and the underlying GIA model propagate through the calculations to the final uplift model. We find a standard error better than 0.25 mm/a for most of the area covered by precise levelling or uplift rates from Continuously Operating Reference Stations and up to 0.7 mm/a outside this area. As a check, we show that two different methods give approximately the same uncertainty estimates. We also estimate changes in the geoid and derive an alternative uplift model referring to this rising geoid. Using this latter model, the maximum uplift in Umea reduces from 10.3 to 9.6 mm/a and with a similar reduction ratio elsewhere. When we compare this new NKG2016LU with the former NKG2005LU, we find the largest differences where the GIA model has the strongest influence, i.e. outside the area of geodetic observation. Here, the new model gives from − 3 to 4 mm/a larger values. Within the observation area, similar differences reach − 1.5 mm/a at the northernmost part of Norway and − 1.0 mm/a at the north-western coast of Denmark, but generally within the range of − 0.5 to 0.5 mm/a.

The primary objective of the 1-cm geoid experiment in Colorado (USA) is to compare the numerous geoid computation methods used by different groups around the world. This is intended to lay the foundations for tuning computation methods to achieve the sought after 1-cm accuracy, and also evaluate how this accuracy may be robustly assessed. In this experiment, (quasi)geoid models were computed using the same input data provided by the US National Geodetic Survey (NGS), but using different methodologies. The rugged mountainous study area (730 km $$\times $$ 560 km) in Colorado was chosen so as to accentuate any differences between the methodologies, and to take advantage of newly collected GPS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17) which are now available to be used as an accurate and independent test dataset. Fourteen groups from fourteen countries submitted a gravimetric geoid and a quasigeoid model in a 1′ $$\times $$ 1′ grid for the study area, as well as geoid heights, height anomalies, and geopotential values at the 223 GSVS17 marks. This paper concentrates on the quasigeoid model comparison and evaluation, while the geopotential value investigations are presented as a separate paper (Sanchez et al. in J Geodesy 95(3):1. https://doi.org/10.1007/s00190-021-01481-0 , 2021). Three comparisons are performed: the area comparison to show the model precision, the comparison with the GSVS17 data to estimate the relative accuracy of the models, and the differential quasigeoid (slope) comparison with GSVS17 to assess the relative accuracy of the height anomalies at different baseline lengths. The results show that the precision of the 1′ × 1′ models over the complete area is about 2 cm, while the accuracy estimates along the GSVS17 profile range from 1.2 cm to 3.4 cm. Considering that the GSVS17 does not pass the roughest terrain, we estimate that the quasigeoid can be computed with an accuracy of ~ 2 cm in Colorado. The slope comparisons show that RMS values of the differences vary from 2 to 8 cm in all baseline lengths. Although the 2-cm precision and 2-cm relative accuracy have been estimated in such a rugged region, the experiment has not reached the 1-cm accuracy goal. At this point, the different accuracy estimates are not a proof of the superiority of one methodology over another because the model precision and accuracy of the GSVS17-derived height anomalies are at a similar level. It appears that the differences are not primarily caused by differences in theory, but that they originate mostly from numerical computations and/or data processing techniques. Consequently, recommendations to improve the model precision toward the 1-cm accuracy are also given in this paper.

Traditionally, sea level is observed at tide gauge stations, which usually also serve as height reference stations for national levelling networks and therefore define a height system of a country. Thus, sea level research across countries is closely linked to height system unification and needs to be regarded jointly. The project aims to make use of a new observation technique, namely SAR positioning, which can help to connect the GNSS basic network of a country to tide gauge stations and as such to link the sea level records of tide gauge stations to the geometric network. By knowing the geoid heights at the tide gauge stations in a global height reference frame with high precision, one can finally obtain absolute sea level heights of the tide gauge stations in a common reference system and can link them together. By this method, on the one hand national height systems can be connected and on the other hand the absolute sea level at the tide gauge stations can be determined. By analyzing time series of absolute sea level heights their changes can be determined in an absolute sense in a global reference frame and the impact of climate change on sea level can be quantified (e.g. by ice sheet and glacier melting, water inflow, global warming). The paper presents the main scientific questions to be addressed by the project, introduces the idea of using SAR transponders for this application and describes the observation network implemented for this feasibility study.

Geoid and quasigeoid modelling from gravity anomalies by the method of least squares modification of Stokes's formula with additive corrections is adapted for the usage with gravity disturbances an ...

The rate of change of surface gravity, dg/dt, and vertical deformation rate of the solid surface, du/dt, are two observables of glacial isostatic adjustment (GIA). They contribute with different information on the same phenomenon. Their relation contains information of the underlying physics and a trustworthy relation allows to combine these observations to strengthen the overall observational accuracy of the phenomenon. In this paper we investigate the predicted relation between dg/dt and du/dt in previously glaciated areas. We use the normal mode approach for one dimensional earth models and solutions of the sea level equation with time-dependent coastline geometry. Numerical predictions of dg/dt and du/dt are computed for Laurentia, Fennoscandia and the British Isles respectively, using six different earth models. Within each region a linear trend is then fitted using the relation dg/dt = C du/dt + dg_0/dt. The estimated C and dg_0/dt differ more between the regions than between different earth models within each region. For Fennoscandia C ≈ −0.163 μGal/mm and for Laurentia C ≈ −0.152 μGal/mm. Maximum residuals between the linear trend and spatially varying model predictions of dg/dt are 0.04 μGal/yr in Fennoscandia and 0.17 μGal/yr in Laurentia. For the British Isles the results are harder to interpret, mainly since this region is located on the zero uplift isoline of Fennoscandia. In addition, we show temporal variation of the relation since the last glacial maximum till present-day. The temporal and spatial variation of the relation between dg/dt and du/dt can be explained by (i) the elastic respectively viscous proportion of the total signal and (ii) the spectral composition of the regional signal. Additional local effects, such as the Newtonian attraction and elastic deformation from local sea level changes, are examined in a case study for six stations in the Nordic absolute gravity network. The influence of these local effects on the relation between View the dg/dt and du/dt is negligible except for extreme locations close to the sea.

We present the results of the third European Comparison of Absolute Gravimeters held in Walferdange, Grand Duchy of Luxembourg, in November 2011. Twenty-two gravimeters from both metrological and non-metrological institutes are compared. For the first time, corrections for the laser beam diffraction and the self-attraction of the gravimeters are implemented. The gravity observations are also corrected for geophysical gravity changes that occurred during the comparison using the observations of a superconducting gravimeter. We show that these corrections improve the degree of equivalence between the gravimeters. We present the results for two different combinations of data. In the first one, we use only the observations from the metrological institutes. In the second solution, we include all the data from both metrological and non-metrological institutes. Those solutions are then compared with the official result of the comparison published previously and based on the observations of the metrological institutes and the gravity differences at the different sites as measured by non-metrological institutes. Overall, the absolute gravity meters agree with one another with a standard deviation of 3.1 µGal. Finally, the results of this comparison are linked to previous ones. We conclude with some important recommendations for future comparisons.

The deduction of a regularly spaced gravity anomaly grid from scattered survey data is studied, addressing mainly two aspects: reduction of gravity to anomalies and subsequent interpolation by various methods. The problem is illustrated in a heterogeneous study area and contrasting test areas including mountains, low terrains, and a marine area. Provided with realistic error estimates, Least Squares Collocation interpolation of Residual Terrain Model anomalies yields the highest quality gravity grid. In most cases, the Bouguer reduction and other interpolation methods tested are equally viable. However, spline-based interpolation should be avoided in marine areas with trackwise survey data.

This paper deals with the modelling of surface gravity change in Fennoscandia, induced by postglacial rebound or Glacial Isostatic Adjustment (GIA). The theoretical foundation is based on the theory introduced by [Peltier, 1974] and [Peltier, 1976] for a spherical, non-rotating, laterally homogenous, viscoelastic, Maxwell Earth and the solution of the Sea Level Equation, originally introduced by Farrel and Clark (1976), with time-dependent coastline geometry. The ice history is defined by the ice model ICE-5G. Rotational feedback is not included. The sensitivity of predictions of present day gravity rates View the MathML source, with respect to a selection of assumptions and approximations, is investigated numerically. Six model variants are defined: (i) linear relation between View the MathML source and the vertical deformation rate View the MathML source, (ii) direct attraction expressed in terms of internal and (iii) external harmonic series expansions as well as by (iv) analytical integration over rectangular prisms. For the most rigorous treatment of the direct attraction, the effect of simplified modelling of the sea level is also investigated. These modelling approximations of the sea level change include (v) fixed shorelines and (vi) eustatic sea level change. Predictions of View the MathML source for the model variants are plotted, compared and discussed. The most rigorous model (iv) and the linear model (i) differ less than 0.03 μGal yr−1 over land and close to, or over, the ocean the difference reaches maximally ∼0.5 μGal yr−1. Due to truncation at 180°, the high frequent nature of the direct attraction is not properly described by models (ii) and (iii). The two simplified sea level modelling approximations (v) and (vi) induce differences, compared to the most rigorous model exceeding 0.2 μGal yr−1 over land, i.e. about one order of magnitude worse than the linear model.

Comparison of remove-compute-restore and least squares modification of Stokes' formula techniques to quasi-geoid determination over the Auvergne test areaThe remove-compute-restore (RCR) technique for regional geoid determination implies that both topography and low-degree global geopotential model signals are removed before computation and restored after Stokes' integration or Least Squares Collocation (LSC) solution. The Least Squares Modification of Stokes' Formula (LSMS) technique not requiring gravity reductions is implemented here with a Residual Terrain Modelling based interpolation of gravity data. The 2-D Spherical Fast Fourier Transform (FFT) and the LSC methods applying the RCR technique and the LSMS method are tested over the Auvergne test area. All methods showed a reasonable agreement with GPS-levelling data, in the order of a 3-3.5 cm in the central region having relatively smooth topography, which is consistent with the accuracies of GPS and levelling. When a 1-parameter fit is used, the FFT method using kernel modification performs best with 3.0 cm r.m.s difference with GPS-levelling while the LSMS method gives the best agreement with GPS-levelling with 2.4 cm r.m.s after a 4-parameter fit is used. However, the quasi-geoid models derived using two techniques differed from each other up to 33 cm in the high mountains near the Alps. Comparison of quasi-geoid models with EGM2008 showed that the LSMS method agreed best in term of r.m.s.

The main purpose of this paper is to investigate numerically the effects of non-tidal sea level variations in the Baltic Sea on gravity with special emphasis on the Swedish stations in the Nordic Absolute Gravity Project. To calculate the ocean loading effect on gravity the method described by Farrell (1972) is widely used. This method is based on convolution of a Green's function for gravity with the ocean load, but does not include the direct attraction depending on the height of the observation point. It is described how this effect is included in the Green's functions and how numerical integration is performed over a dense grid bounded by a very high resolution coastline. The importance of this high resolution is shown. The major part of the direct attraction for stations close to the coast comes from relatively small water masses close to the station. The total effect from the Baltic Sea, crustal loading and direct attraction, is calculated for 12 Swedish and one Finnish absolute gravity stations. The distance from the coast for these stations varies from 10 m to 150 km. It is shown that the total non-tidal gravity effect is significant, easily reaching values of 2–3 μgal for stations with high elevation close to the coast. In modelling the Glacial Isostatic Adjustment (GIA), the relation between the change of gravity and the absolute land uplift ( g ˙ / h ˙ ) contains information about the viscoelastic properties of the upper mantle. The Baltic Sea is located in the Fennoscandian postglacial land uplift area and experiences therefore a long-term sea level decrease. It is also shown that the magnitude of this long-term effect is not negligible for determination of the unknown part of g ˙ / h ˙ .

Since 2006 the gravity change at 17 points in Sweden has been observed with Lantmateriet’s absolute gravimeter FG5-233. The main purpose of the observations is to study the postglacial rebound in Fennoscandia. In 2010, a suspected jump of a few μGal can be seen in the gravity time series that significantly affects the estimated gravity rate of changes. It is shown that if the jump is not considered, then the absolute value of the gravity rate of change is systematically underestimated compared to the land uplift model NKG2014LU_test. In this paper two different ways to estimate and apply corrections of the jump are demonstrated. The first is to estimate the jump from the observations themselves within a least squares adjustment, while the second is to assume the instrument bias obtained in international comparisons of absolute gravimeters. The best agreement between land uplift model and estimated rates of change of gravity is achieved by correcting the data with the official biases reported from the international comparisons.

One important application of an Earth Gravity Model (EGM) is to determine the geoid. Since an EGM is represented by an external-type series of spherical harmonics, a biased geoid model is obtained when the EGM is applied inside the masses in continental regions. In order to convert the downward-continued height anomaly to the corresponding geoid undulation, a correction has to be applied for the analytical continuation bias of the geoid height. This technique is here called the geoid bias method. A correction for the geoid bias can also be utilised when an EGM is combined with terrestrial gravity data, using the combined approach to topographic corrections. The geoid bias can be computed either by a strict integral formula, or by means of one or more terms in a binomial expansion. The accuracy of the lowest binomial terms is studied numerically. It is concluded that the first term (of power H2) can be used with high accuracy up to degree 360 everywhere on Earth. If very high mountains are disregarded, then the use of the H2 term can be extended up to maximum degrees as high as 1800. It is also shown that the geoid bias method is practically equal to the technique applied by Rapp, which utilises the quasigeoid-to-geoid separation. Another objective is to carefully consider how the combined approach to topographic corrections should be interpreted. This includes investigations of how the above-mentioned H2 term should be computed, as well as how it can be improved by a correction for the residual geoid bias. It is concluded that the computation of the combined topographic effect is efficient in the case that the residual geoid bias can be neglected, since the computation of the latter is very time consuming. It is nevertheless important to be able to compute the residual bias for individual stations. For reasonable maximum degrees, this can be used to check the quality of the H2 approximation in different situations.

In November 2013 an International Key Comparison, CCM.G-K2, was organized in the Underground Laboratory for Geodynamics in Walferdange. The comparison has assembled 25 participants coming from 19 countries and four different continents. The comparison was divided into two parts: the key comparison that included 10 NMIs or DIs, and the pilot study including all participants. The global result given by the pilot study confirms that all instruments are absolutely coherent to each other. The results obtained for the key comparison confirm a good agreement between the NMI instruments. Main text. To reach the main text of this paper, click on Final Report [http://www.bipm.org/utils/common/pdf/final_reports/M/G-K2/CCM.G-K2.pdf] . Note that this text is that which appears in Appendix B of the BIPM key comparison database kcdb.bipm.org/ [http://kcdb.bipm.org/] . The final report has been peer-reviewed and approved for publication by CCM, according to the provisions of the CIPM Mutual Recognition Arrangement (CIPM MRA).

One of the key activities in Geodesy 2010, the Swedish strategic plan for geodetic activities during the period 2011–2020, is the restoration of the gravity network and data in order to improve the accuracy of the national quasigeoid model. One weak point has been that very few gravity observations have been available over Lake Vanern, Sweden’s largest lake. During the extremely cold winters 2010 and 2011, the ice became sufficiently thick to make ice observation of gravity. The main purpose of this paper is to present the 2011 ice gravity campaign, summarise the experiences made and investigate how much the new ice observations improve the computed quasigeoid model in the area. This is investigated under the assumption that a modern Earth Gravitational Model based on GRACE and GOCE is used. It is found that new ice measurements improve the quasigeoid with a RMS of about 2–3 cm in and around the lake with a maximum improvement of 7 cm.

When GNSS height determination improves in the future, users will ask for increasingly better geoid models. It is not unlikely that a standard error of 5 mm will more or less be required in a couple of years. The main purpose of this paper is to investigate the gravity data requirements to compute a Swedish gravimetric quasigeoid model to that order. The propagation of errors in the terrestrial gravity observations and the Earth Gravitational Model (EGM) are studied using both variance-covariance analysis in the spectral domain and least squares collocation. These errors are also checked by computing a new gravimetric quasigeoid model and comparing it with GNSS/levelling height anomalies. It is concluded that it will be possible to compute a 5 mm model over Sweden in the case that the gravity data set is updated to fulfil the following requirements: the resolution should be at least 5 km and there should be no data gaps nearby. Finally, the standard errors of the uncorrelated and correlated gravity anomaly noises should be below 0.5 and 0.1 mGal, respectively.

The 8th International Comparison of Absolute Gravimeters (ICAG2009) took place at the headquarters of the International Bureau of Weights and Measures (BIPM) from September to October 2009. It was the first ICAG organized as a key comparison in the framework of the CIPM Mutual Recognition Arrangement of the International Committee for Weights and Measures (CIPM MRA) (CIPM 1999). ICAG2009 was composed of a Key Comparison (KC) as defined by the CIPM MRA, organized by the Consultative Committee for Mass and Related Quantities (CCM) and designated as CCM.G-K1. Participating gravimeters and their operators came from national metrology institutes (NMIs) or their designated institutes (DIs) as defined by the CIPM MRA. A Pilot Study (PS) was run in parallel in order to include gravimeters and their operators from other institutes which, while not signatories of the CIPM MRA, nevertheless play important roles in international gravimetry measurements. The aim of the CIPM MRA is to have international acceptance of the measurement capabilities of the participating institutes in various fields of metrology. The results of CCM.G-K1 thus constitute an accurate and consistent gravity reference traceable to the SI (International System of Units), which can be used as the global basis for geodetic, geophysical and metrological observations of gravity. The measurements performed afterwards by the KC participants can be referred to the international metrological reference, i.e. they are SI-traceable. The ICAG2009 was complemented by a number of associated measurements: the Relative Gravity Campaign (RGC2009), high-precision levelling and an accurate gravity survey in support of the BIPM watt balance project. The major measurements took place at the BIPM between July and October 2009. Altogether 24 institutes with 22 absolute gravimeters (one of the 22 AGs was ultimately withdrawn) and nine relative gravimeters participated in the ICAG/RGC campaign. This paper is focused on the absolute gravity campaign. We review the history of the ICAGs and present the organization, data processing and the final results of the ICAG2009. After almost thirty years of hosting eight successive ICAGs, the CIPM decided to transfer the responsibility for piloting the future ICAGs to NMIs, although maintaining a supervisory role through its Consultative Committee for Mass and Related Quantities. © 2012 BIPM & IOP Publishing Ltd.

The least squares modification of Stokes formula has been developed in a series of papers published in Journal of Geodesy between 1984 and 2008. It consists of a least squares (stochastic) Stokes k ...

The increased need for improved geoid models for Global Navigation Satellite Systems (GNSS) height determination calls for additional gravity observations and quality assurance of existing data. In this perspective, a modern gravity system and the renovation of an already existing high order gravity network is considered as a moderate strategic investment which provides a firm foundation for further activities. Here the new gravity reference frame RG 2000 for Sweden is presented. RG 2000 is realized by absolute gravity observations at 109 stations. The absolute points are connected via old and new relative gravity observations, including another 216 points. Points and observations have been chosen so that good overlap with the older Swedish reference frames, RG 62 and RG 82, is achieved, allowing to evaluate the older frames and transformations between them. RG 2000 is based on a zero permanent tide system with epoch 2000.