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Figure 8 (2013), the 24,531 pairs were grouped into 12 bins, contain- ing approximately 2,000 pairs each. Then, the RMS values of height anomaly differences were computed for the 12 bins. The results are illustrated in Fig. 8. The five models in the best agreement are highlighted. The RMS values range from 2 to 3 cm for these five models at baseline lengths from 2 to 100 km, and then, they disagree up to 6 cm at longer baseline lengths. The CGS model presents RMS values between 2 and 3 cm in every baseline length.

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derived height anomalies. However, the comparison results of this experiment indicate a 5 cm accuracy (relative). This data set was distributed to the contributing groups with the inten- tion that it was used as initial reference for the quality control in the (quasi)geoid computations. The original ellipsoidal height was in the NAD83 system, but it was converted to the IGSO8 frame for this experiment. Out of 509 benchmarks, 467 benchmarks were from the NGS Integrated Database and the rest from the NGS OPUS-Share Tool (https://www. ngs.noaa.gov/OPUS/). The spatial distribution of the bench- marks is sparse due to the mountainous terrain (see Fig. 2). tions are given at flight altitude (~ 6,000 m) with positions ir the IGSO8 reference frame (Rebischung et al. 2012).

Stokes GBVP (e.g., Vaniéek and Martinec 1994; Huang and Véronneau 2013; Abd-Elmotaal and Kiihtreiber 2003, 2021), then converted geoid undulations into height anomalies using the geoid-quasigeoid separation term. Most of the solutions applied the remove-compute-restore (RCR) technique using different parameters for the modification of the Stokes inte- gration kernel or the spectral combination of the different input data (GGM, airborne and terrestrial gravity data, topog- raphy gravity signals). Four solutions follow the least squares modification of Stokes’s formula with additive corrections (LSMSA) algorithm (Sjoberg 1980,1981,1984,2003). Three solutions performed the data processing using spherical radial basis functions (SRBF), least-squares collocation (LSC) and residual LSC. The variety of parameter choices for the data processing is so large that it is impractical to provide a detailed description of the different computation strategies in this article. We concentrate therefore on the key characteristics considering three main computation stages: handling of topographic effects on gravity (Table 2), prepa-

Table 2 Methods for estimating the topographic gravity effects within the contributing solutions to the Colorado experiment (FFT — fast Fourie transformation, RTM - residual terrain modeling, DEM — digital elevation model) 4 Model comparison and validation ration of surface (terrestrial + airborne) gravity data (Table 3), and solution of the GBVP (Table 4). Table 4 also provides the corresponding bibliographical references describing the detailed computation strategies. All the groups delivered a geoid model on a 1’ I’ grid and geoid heights on the 223 GSVS17 marks; they also provided the geopotential values on the marks and the comparison results can be found in Sanchez et al. (2021). All the groups (except Minia) also provided a quasigeoid model on a 1’x 1’ grid and height anomalies on the GSVS17 marks. To simplify the analysis and to focus on the main results, we present in this

Table 4 continued where 7 is the mean normal gravity (GRS80) along the nor- mal plumb line, H is the orthometric height, N and ¢ are the geoid undulation and the height anomaly, respectively. paper the comparison between the quasigeoid models only; the comparison between the geoid models produces very similar results and conclusions. As mentioned, 11 groups delivered quasigeoid models as their primar y solutions, while three groups computed the geoid models directly (cf. Table 4). These three geoid models were converted into quasigeoid models by applying exactly the same geoid-quasigeoid con- version term (see Sect. 4.1). In this way, we avoid possible sources of discrepancy caused by the diff ferent conversion terms used in the converted models (cf. Table 4) and expect that the discrepancies between the results are mainly caused by the different data processing and GBVP solution strate- gies.

4.2 Quasigeoid area comparisons (grids of 1’ x 1’) limited to a one-degree cap. The complete GQS was com- puted for the experimental geoid 2020 (Wang et al., 2021) and the Colorado area is a part of it. Figure 3a, b show the GQS term computed using Eq. (2) and the differences com- puted using Eqs. (1) and (2), respectively. The differences reach 27.7 cm in the extreme case, and 2.1 cm in terms of root mean square (RMS) values (see Fig. 3b). The models provided by the contributing groups are given on a 1’x 1’ grid over the study area between latitudes 36° N to 39° N and longitudes 109° W to 103° W. This study area has been reduced by a one-degree border along each side with respect to the area covered by the gravity data (Fig. 1) to reduce edge effects in the quasigeoid models. As the grids are submitted in different formats (point values and lines of scan) and in different cell registrations (center and corner), the first step was to transform the grids to a common format, and then to interpolate the quasigeoid models onto I’ x 1’ grids with a common registration (hereafter called standardized grids). To show the precision of the models, the 14 quasigeoid models were compared with the group mean. In the end, the height anomalies were interpolated on the GSVS17 marks from the The models provided by the contributing groups are given on The improvement offered by Eq. (2) is implied by the fact that the CGS model (under the Helmert-Stokes scheme) gets closer to NGS model (under the Molodensky theory) when the complete separation term is used. The STD of the differ- ences between these two models is reduced from 3.1 cm to 2.2 cm. For this reason, the complete separation is applied to the three geoid models (CGS, Minia and GSI) in the follow- ing analysis. ences between these two models is reduced from 3.1 cm to

Table 5 Statistics of the differences among the individual quasigeoid models and the mean of all quasigeoid models at the 1’ x 1’ grid; 85 cm was added to Minia model. Units in cm

Figure 7 shows that the mean model has similar smaller features as the CGS model before the mark 160, and the mean model is 1.4 cm higher than the CGS model. The offset is consistent with the 1.1 cm standard deviation of the mean residuals shown in Table 6, which characterize the uncer- tainty due to methodology. Both sets of height anomalies decrease after mark 160, and the CGS model decreases less: the differences of the mean model have a 10-cm slope rel- ative to the GSVS17 data from mark 160 to 193, whereas the slope of the CGS model is 5 cm. The smaller slope of the CGS model after the mark 160 is reflected in the smaller STD of the differences (1.7 cm) than the STD values of the mean model (2.6 cm). Another notable feature is that both models agree well at the first peak of height 3310 m (mark

“Mean value excludes the Minia model. Values of three converted mod- els (CGS, Minia and GSI) are in italics Table 6 Statistics of the differences between the height anomalies pro- vided by the individual solutions and the height anomalies inferred from the leveling-based geopotential numbers and GPS positioning along the GSVS17 profile (Gmodel — SGsvs17), units in cm 86), where the elevation raises from 2200 to 3300 m in about 60 km distance. 4.4 GSVS17 differential quasigeoid (slope) comparisons

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