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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 × 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′ × 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 (Sánchez 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.

Numerical methods, like the finite element method (FEM) or finite volume method (FVM), are widely used to provide solutions in many boundary value problems. In previous studies, these numerical methods have also been applied in geodesy but demanded extensive computations because the upper boundary condition was usually set up at the satellite orbit level, hundreds of kilometers above the Earth. The relatively large distances between the lower boundary of the Earth's surface and the upper boundary exacerbate the computation loads because of the required discretization in between. Considering that many areas, such as the US, have uniformly distributed airborne gravity data just a few kilometers above the topography, we adapt the upper boundary from the satellite orbit level to the mean flight level of the airborne gravimetry. The significant decrease in the domain of solution dramatically reduces the large computation demand for FEM or FVM. This paper demonstrates the advantages of using FVM in the decreased domain in simulated and actual field cases in study areas of interest. In the simulated case, the FVM numerical results show that precision improvement of about an order of magnitude can be obtained when moving the upper boundary from 250 to 10 km, the upper altitude of the GRAV-D flights. A 2–3 cm level of accurate quasi-geoid model can be obtained for the actual datasets depending on different schemes used to model the topographic mass. In flat areas, the FVM solution can reach to about 1 cm precision, which is comparable with the counterparts from classical methods. The paper also demonstrates how to find the upper boundary if no airborne data are available. Finally, the numerical method provides a 3D discrete representation of the entire local gravity field instead of a surface solution, a (quasi) geoid model.

This article investigates the role of different approaches and interpolation methods in gridding terrestrial gravity anomalies. In this regard, first of all, simple and complete Bouguer anomalies are considered in gravity data gridding. In the comparison results of gridding these two Bouguer anomaly datasets, the effect of the high-frequency contribution of topographic gravitation (by means of the terrain correction) is clarified. After that, the role of the used interpolation algorithm on the resulting grid of mean gravity anomalies and hence on the geoid modeling accuracy is inspected. For this purpose, four different interpolation methods including geostatistical Kriging, nearest neighbor, inverse distance to a power (IDP), and artificial neural networks (ANNs) are applied. Here, the IDP and nearest neighbor methods represent simple-structured algorithms among the interpolation methods tested in this study. The ANN method, on the other hand, is preferred as a complex, optimization-based soft computing method that has been applied in recent years. In addition, the geostatistical Kriging method is one of the conventional methods that is mostly applied for gridding gravity data in geodesy and geophysics. The calculated gravity anomalies in grids are employed in high-resolution geoid model computations using the least squares modifications of Stokes formula with additive corrections (LSMSA) technique. The investigations are carried out using the test datasets of Auvergne, France that are provided by the International Service for the Geoid for scientific research. It is concluded that the interpolation algorithms affect the gravity gridding results and hence the geoid model determination. The ANN method does not provide superior results compared to the conventional algorithms in gravity gridding. The geoid model with 4.1 cm accuracy is computed in the test area.

This work is an investigation of three methods for regional geoid computation: Stokes’s formula, least-squares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223–232, 1995 . doi: 10.1007/BF00806734 ) that the equivalence of Stokes’s formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes’s formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level.

The determination of physical heights by transformation of the ellipsoidal GNSS heights to the Geoid (or quasi-Geoid) height reference surface (HRS) - the 2nd component of the Geodetic Infrastructure for GNSS positioning services (GIPS) - requires the computation of Geoid/QGeoid models as RTCM-capable databases for sustainable geodetic task in the establishment of GIPS. In the work, overview is given about different GIPS components as well as about realized concepts and relevant software. The authors present in detail mathematical models of the general world-wide valid concept for the HRS computation for the mentioned models as essential part of GIPS. The concept is illustrated by the Baltic QGeoid meant for the territories of Estonia, Latvia and Lithuania. The DFHRS-software for the computation was developed at the Karlsruhe University of Applied Sciences. Fizikālo augstumu virs jūras līmeņa noteikšana, pārrēķinot elipsiodālos GNSS augstumus, kad izmantojam geoīda (jeb kvazigeoīda) augstuma atskaites virsmu (HRS) kā otro svarīgāko geodēziskās infrastruktūras komponenti GNSS pozicionēšanas pakalpojumos (GIPS), prasa, ka aprēķinātajiem geoīda - kvazigeoīda modeļiem jābūt kā RTCM datu formāta savietojamai datu bāzei, kas derētu dažādu praktisku geodēzisku uzdevumu veikšanai. Šajā darbā tiek sniegts detalizēts pārskats par dažādām GIPS komponentēm, kā arī izpildītiem aprēķiniem un atbilstošo programmatūru. Autori līdz atsevišķām niansēm izklāsta iegūtos matemātiskos modeļus atbilstoši vispārējām pasaules koncepcijas nostādnēm augstuma atskaites virsmu (HRS) aprēķināšanā, kas nodrošina būtisku GIPS komponenti, t.i. augstumu virs jūras līmeni. Izklāstītā koncepcija ir sekmīgi realizēta Baltijas kvazigeoīda aprēķināšanā, ietverot Igaunijas, Latvijas un Lietuvas valstu teritorijas. Speciālā programmatūra atskaites augstuma līmeņvirsmas DFHRS aprēķināšanai tikusi izstrādāta Pielietojamo zinātņu Karlsrūes Augstskolā (HsKA).

Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.

The mean dynamic topography (MDT) is a key reference surface for altimetry. It is needed for the calculation of the ocean absolute dynamic topography, and under the geostrophic approximation, the estimation of surface currents. CNES-CLS mean dynamic topography (MDT) solutions are calculated by merging information from altimeter data, GRACE, and GOCE gravity field and oceanographic in situ measurements (drifting buoy velocities, hydrological profiles). The objective of this paper is to present the newly updated CNES-CLS18 MDT. The main improvement compared to the previous CNES-CLS13 solution is the use of updated input datasets: the GOCO05S geoid model is used based on the complete GOCE mission (November 2009–October 2013) and 10.5 years of GRACE data, together with all drifting buoy velocities (SVP-type and Argo floats) and hydrological profiles (CORA database) available from 1993 to 2017 (instead of 1993–2012). The new solution also benefits from improved data processing (in particular a new wind-driven current model has been developed to extract the geostrophic component from the buoy velocities) and methodology (in particular the computation of the medium-scale GOCE-based MDT first guess has been revised). An evaluation of the new solution compared to the previous version and to other existing MDT solutions show significant improvements in both strong currents and coastal areas.

This paper studies the contribution of airborne gravity data to improvement of gravimetric geoid modelling across the mountainous area in Colorado, USA. First, airborne gravity data was processed, filtered, and downward-continued. Then, three gravity anomaly grids were prepared; the first grid only from the terrestrial gravity data, the second grid only from the downward-continued airborne gravity data, and the third grid from combined downward-continued airborne and terrestrial gravity data. Gravimetric geoid models with the three gravity anomaly grids were determined using the least-squares modification of Stokes’ formula with additive corrections (LSMSA) method. The absolute and relative accuracy of the computed gravimetric geoid models was estimated on GNSS/levelling points. Results exhibit the accuracy improved by 1.1 cm or 20% in terms of standard deviation when airborne and terrestrial gravity data was used for geoid computation, compared to the geoid model computed only from terrestrial gravity data. Finally, the spectral analysis of surface gravity anomaly grids and geoid models was performed, which provided insights into specific wavelength bands in which airborne gravity data contributed and improved the power spectrum.

An accurate knowledge of the ocean mean dynamic topography (MDT) is mandatory for the optimal use of altimetric data, including their assimilation into operational ocean forecasting systems. A new global 1/4° resolution MDT was computed for the 1993–1999 time period with improved data and methodology compared to the previous RIO05 MDT field. First, a large‐scale MDT is obtained from the CLS01 altimetric Mean Sea Surface and a recent geoid model computed from 4.5 years of GRACE (Gravity Recovery and Climate Experiment) data. Altimetric sea level anomalies and in situ measurements are then combined to compute synthetic estimates of the MDT and the corresponding mean currents. While the RIO05 MDT was based on 10 years of in situ dynamic heights and drifting buoy velocities, the new field benefits from an enlarged data set of in situ measurements ranging from 1993 to 2008 and includes all hydrological profiles from the Argo array. Moreover, the processing of the in situ data has been updated. A new Ekman model was developed to extract the geostrophic velocity component from the drifting buoy measurements. The handling of hydrologic measurements has also been revisited. Compared to the previous RIO05 solution, the new global MDT resolves much stronger gradients in western boundary currents, with mean velocities being doubled in some places. Moreover, in comparison to several other recent MDT estimates, we find that the new CNES‐CLS09 MDT is in better agreement with independent in situ observations. Key Points Computation of a new mean dynamic topography Development of a new Ekman model Optimal filtering of MSS‐Geoid

The International Service for the Geoid (ISG, https://www.isgeoid.polimi.it/, last access: 31 March 2021) provides free access to a dedicated and comprehensive repository of geoid models through its website. In the archive, both the latest releases of the most important and well-known geoid models, as well as less recent or less known ones, are freely available, giving to the users a wide range of possible applications to perform analyses on the evolution of the geoid computation research field. The ISG is an official service of the International Association of Geodesy (IAG), under the umbrella of the International Gravity Field Service (IGFS). Its main tasks are collecting, analysing, and redistributing local, regional, and continental geoid models and providing technical support to people involved in geoid-related topics for both educational and research purposes. In the framework of its activities, the ISG performs research taking advantage of its archive and organizes seminars and specific training courses on geoid determination, supporting students and researchers in geodesy as well as distributing training material on the use of the most common algorithms for geoid estimation. This paper aims at describing the data and services, including the newly implemented DOI Service for geoid models (https://dataservices.gfz-potsdam.de/portal/?fq=subject:isg, last access: 31 March 2021), and showing the added value of the ISG archive of geoid models for the scientific community and technicians, like engineers and surveyors (https://www.isgeoid.polimi.it/Geoid/reg_list.html, last access: 31 March 2021).