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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.

Modern height systems are based on the combination of satellite positioning and gravity field models of high resolution. However, in many regions, especially developing or newly industrializing countries, there is no (reliable) regional gravity model at all, due to challenges such as limited data availability, unknown/low data quality, and missing metadata. This paper addresses this issue in a case study of Colombia, where eight decades of historical terrestrial and airborne gravity measurements are available but widely contain systematic errors, outliers, and biases. Correspondingly, processing strategies and structures are proposed and applied to validate and improve the quality of old gravity datasets. A novel method is developed based on spherical radial basis functions (SRBFs) for estimating biases, which are found in different airborne surveys with values exceeding 40 mGal. The validity of this bias estimation method is demonstrated both by a simulation test and by the evaluation of the airborne data in comparison to the SATOP (SAtellite-TOPography) model, which merges the satellite-only global gravity model GOCO06s with the Earth2014 topography model. The terrestrial and airborne data are then combined with a global gravity model (GGM), ultra-high-resolution topography models, as well as altimetry-derived gravity anomalies from DTU21GRA for the offshore areas. The results are presented in terms of height anomalies (QGeoidCOL2023), and they are thoroughly validated using GPS/leveling data both in the absolute and relative manner. The standard deviation in comparison to the GPS/leveling data after applying a correction surface to account for the datum inconsistencies amounts to 15.76 cm, which is 27% smaller compared to the mean standard deviation value given by five recent high-resolution GGMs, and 36% smaller than the one delivered by the latest South American quasi-geoid model QGEOID2021. The relative validation results show that QGeoidCOL2023 performs better, i.e., delivers lower RMS errors than the GGMs and QGEOID2021 in all the baseline length groups. These results indicate the validity and benefits of the developed methods and procedures, which can be used for other data-challenging areas to facilitate the realization of geopotential-based height systems. Graphical Abstract

The launch of dedicated satellite missions at the beginning of the 2000s led to significant improvement in the determination of Earth gravity field models. As a consequence of this progress, both the accuracies and the spatial resolutions of the global geopotential models increased. However, the spectral behaviors and the accuracies of the released models vary mainly depending on their computation strategies. These strategies are briefly explained in this article. Comprehensive quality assessment of the gravity field models by means of spectral and statistical analyses provides a comparison of the gravity field mapping accuracies of these models, as well as providing an understanding of their progress. The practical benefit of these assessments by means of choosing an optimal model with the highest accuracy and best resolution for a specific application is obvious for a broad range of geoscience applications, including geodesy and geophysics, that employ Earth gravity field parameters in their studies. From this perspective, this study aims to evaluate the GOCE High-Level Processing Facility geopotential models including recently published sixth releases using different validation methods recommended in the literature, and investigate their performances comparatively and in addition to some other models, such as GOCO05S, GOGRA04S and EGM2008. In addition to the validation statistics from various countries, the study specifically emphasizes the numerical test results in Turkey. It is concluded that the performance improves from the first generation RL01 models toward the final RL05 models, which were based on the entire mission data. This outcome was confirmed when the releases of different computation approaches were considered. The accuracies of the RL05 models were found to be similar to GOCO05S, GOGRA04S and even to RL06 versions but better than EGM2008, in their maximum expansion degrees. Regarding the results obtained from these tests using the GPS/leveling observations in Turkey, the contribution of the GOCE data to the models was significant, especially between the expansion degrees of 100 and 250. In the study, the tested geopotential models were also considered for detailed geoid modeling using the remove-compute-restore method. It was found that the best-fitting geopotential model with its optimal expansion degree (please see the definition of optimal degree in the article) improved the high-frequency regional geoid model accuracy by almost 15%.

Three Geoid Slope Validation Surveys were planned by the National Geodetic Survey for validating geoid improvement gained by incorporating airborne gravity data collected by the “Gravity for the Redefinition of the American Vertical Datum” (GRAV-D) project in flat, medium and rough topographic areas, respectively. The first survey GSVS11 over a flat topographic area in Texas confirmed that a 1-cm differential accuracy geoid over baseline lengths between 0.4 and 320 km is achievable with GRAV-D data included (Smith et al. in J Geod 87:885–907, 2013 ). The second survey, Geoid Slope Validation Survey 2014 (GSVS14) took place in Iowa in an area with moderate topography but significant gravity variation. Two sets of geoidal heights were computed from GPS/leveling data and observed astrogeodetic deflections of the vertical at 204 GSVS14 official marks. They agree with each other at a ± 1.2 cm level, which attests to the high quality of the GSVS14 data. In total, four geoid models were computed. Three models combined the GOCO03/5S satellite gravity model with terrestrial and GRAV-D gravity with different strategies. The fourth model, called xGEOID15A, had no airborne gravity data and served as the benchmark to quantify the contribution of GRAV-D to the geoid improvement. The comparisons show that each model agrees with the GPS/leveling geoid height by 1.5 cm in mark-by-mark comparisons. In differential comparisons, all geoid models have a predicted accuracy of 1–2 cm at baseline lengths from 1.6 to 247 km. The contribution of GRAV-D is not apparent due to a 9-cm slope in the western 50-km section of the traverse for all gravimetric geoid models, and it was determined that the slopes have been caused by a 5 mGal bias in the terrestrial gravity data. If that western 50-km section of the testing line is excluded in the comparisons, then the improvement with GRAV-D is clearly evident. In that case, 1-cm differential accuracy on baselines of any length is achieved with the GRAV-D-enhanced geoid models and exhibits a clear improvement over the geoid models without GRAV-D data. GSVS14 confirmed that the geoid differential accuracies are in the 1–2 cm range at various baseline lengths. The accuracy increases to 1 cm with GRAV-D gravity when the west 50 km line is not included. The data collected by the surveys have high accuracy and have the potential to be used for validation of other geodetic techniques, e.g., the chronometric leveling. To reach the 1-cm height differences of the GSVS data, a clock with frequency accuracy of 10 - 18 is required. Using the GSVS data, the accuracy of ellipsoidal height differences can also be estimated.

Nowadays, Global Geopotential Models (GGMs) can be used as a reference to develop more detailed regional/local geoids, or they can be used to provide geoid heights on their own. Since 2000, several GGMs have been released, and they are mainly derived from satellite gravity measurements, satellite-only models, terrestrial gravimetry, altimeter-derived gravity data in marine areas, and airborne gravity data. With a precise geoid model, ellipsoidal heights obtained from GPS can be converted to orthometric heights, which is reasonably quite needed in Geodesy, Civil Engineering, etc. These heights reflect changes in topography as well as local variations in gravity. This paper evaluates some of the latest releases of high degree reference models and the satellite-only global gravity field model over Sudan using 19 GPS/Leveling stations. We have been selected 6 GGMs based on Gravity field Goce and Grace, and they released in 2020, 2019, 2014, 2008, and 1996 as shown in the International Centre for Global Earth Models website (ICGEM). The accuracy evaluation of the GGM models have been discussed, the accurate GGMs over Sudan are XGM2019e_2159 and GOCO05s, which have indicated -0.019 and 0.046 meters, respectively. The evaluation results produce valuable information to academia and geoid modeling research topics in Sudan, which shows the precise model from the selected GGMs in Sudan by using the available GPS/Leveling data.

Global geopotential models are widely used in the remove-compute-restore technique for local gravity field modeling. In this paper, a method for regional improvement of global geopotential models using GPS/Leveling data is presented. The part of the spherical harmonic expansion degrees that can be subject to the regional improvement is determined depending on the spatial resolution of the GPS/Leveling data and the size of the study region. In this method, a global geopotential model is required as the original model. Using the GPS/Leveling data corrected for the systematic errors, the geoid surface is obtained at the GPS/Leveling points. By expanding the gravity potential of the geoid surface into the spherical harmonics, a mathematical model is made to estimate the spherical harmonic coefficients of the regionally improved geopotential model. To stabilize the mathematical model, pseudo data of the gravitational potential type produced by the original model on the entire Earth’s surface are added to the GPS/Leveling data. The relative weight of the two types of the data, i.e., the GPS/Leveling data and the pseudo data, is selected based on fitting the original model to the GPS/Leveling data. As numerical tests, the regionally improved geopotential model of the USA from degree 8 to 779 and the regionally improved geopotential model of Iran from degree 12 to 339 are developed. To develop both regionally improved geopotential models, the EGM2008 model up to degree 2160 is selected as the original model. The assessments at the GPS/Leveling checkpoints show that the regionally improved geopotential model of the USA has a 23% improvement and the regionally improved geopotential model of Iran has an 8% improvement with respect to the original model. The numerical tests confirm the efficiency of the proposed method for the regional improvement of global geopotential models using the GPS/Leveling data.

This work focuses on the development of a combined quasigeoid model for Tierra del Fuego province. The Equivalent Source Technique (EST) is applied together with the remove-compute-restore technique in order to combine gravity and GPS/levelling observations and to obtain a quasigeoid model. This model features an improved accuracy in relation to previous models. A discussion about the geodetic reference system is also presented. Geodetic coordinates of all stations used were transformed to TDF08 to be in accordance with the new geodetic reference frame of Argentina. After a cross validation procedure it is determined that a 5cm (r.m.s.) quasigeoid model has been achieved for the major part of the province, fulfilling the requirements for its practical use. New Global Geopotential Models (GGM) are introduced in the discussion, particularly the EGM2008 which is used for evaluation purposes. It shows a 9cm agreement after its evaluation on the levelling lines.

The purpose of this study is evaluation of geostatistical surface model for transformation of GPS derived ellipsoidal heights to orthometric heights. The model was handled as its accuracy for surveying applications. 1148 reference points were used covering an area of 115×112 km with GPS and leveling data from the \"Izmir geodetic infrastructure for the production of 1/5000 scaled digital photogrammetric maps and orthophotos\" project. As a basic data, the differences between ellipsoidal and orthometric heights for each benchmarks were modeled by geostatistical interpolation method namely kriging. ArcGIS 10.0 Geostatistical Analyst was used with optimized parameters for modeling. The quality of the model was analysed by Cross Validation, splitted data and external data validation. The model provide about ± 5 cm absolute, 1 ppm relative accuracy. Also the consistency of the model with several geoid models namely TG03 (Turkish Geoid 2003) and EGM08 (Earth Gravitational Model 2008) geoids was approximately ± 7-10 cm. / V članku je obravnavano vrednotenje geostatističnega modela ploskve za pretvorbo elipsoidnih višin, pridobljenih z meritvami GPS, v ortometrične višine. V raziskavi je bilo uporabljenih 1148 referenčnih točk, katerih višine so bile določene z geometričnim nivelmanom in meritvami GPS. Zajeto je območje velikosti 115 x 112 kilometrov v okviru projekta »Geodetska infrastruktura območja Izmir za izdelavo digitalnih fotogrametričnih zemljevidov in ortofotov v merilu 1 : 5000«. Osnovne podatke pomenijo razlike med elipsoidnimi in ortometričnimi višinami, ki so modelirane z metodo geostatistične interpolacije, in sicer krigingom. Uporabili smo programsko orodje ArcGIS 10.0 ’Geostatistical Analyst’ z optimiziranimi parametri modeliranja. Kakovost modela je bila analizirana z navzkrižnim preverjanjem, delitvijo podatkov in njihovim zunanjim preverjanjem. Ocenjena natančnost modela je približno 5 cm absolutno in približno 1 ppm relativno. Tudi skladnost modela z drugimi modeli geoida, in sicer z modeloma TG03 (turški geoid 2003) in EGM08 (Earth Gravitational Model 2008), znaša približno 7–10 cm.

In this paper we present the concepts and realization of the shallow-layer Shen geoid determination method, which is quite different from the classical geoid modeling methods (Stokes' method, Molodenskii's method, etc.), for determining the global or regional geoids. This method takes full advantage of the precise Earth gravity field model EGM2008, digital topographic model DTM2006.0 and global crust model CRUST2.0 of a shallow layer, a layer from the Earth's surface to a depth. As a case study this method is applied to the determination of a 5´ × 5´ geoid over the Xinjiang and Tibetan regions, which ranges from latitude 25 to 50°N and longitude 70 to 100°E. The modeled 5´ × 5´ regional geoid is compared with the EGM2008 geoid model in the same study area and validated by 21 GPS/leveling benchmarks (GPSBMs) distributed sparsely in the Xinjiang area. The results show that the regional geoid reaches an accuracy of ~18 cm and agrees with the GPSBMs better than the EGM2008 geoid in the Xinjiang region.

GPS technology has penetrated into all fields of surveying and mapping disciplines,and has been widely used in leveling measurement .By studying the feasibility of the refining of region quasi-geoid based on the existing quasi-geoid,this paper shows a new method which is a combination of the Earth's gravity field model and the GPS leveling fitting method to determine the region quasi-geoid and provide the specific ideas and calculation steps and do analysis and discussion about the feasibility and superiority of this method using actual data.This new method makes full use of the advantages of the high resolution of the gravity geoid and the high-precision of the GPS geoid to realize the complementary strengths.