Geoid Studies in Two Test Areas in Greece Using Different Geopotential Models towards the Estimation of a Reference Geopotential Value

Abstract

In the present study, we first examine the contribution of different, satellite-only or combined, global geopotential models in geoid computation employing the remove–compute–restore approach. For this reason, two test areas of about 100 km² each, one in northern and one in central Greece, were selected, and gravity measurements were conducted. These new gravity measurements were used along with the selected geopotential models to produce the reduced gravity field for the study areas. The classical and spectral residual terrain modeling effect is also removed to derive the residual gravity field. The latter is used for geoid computation using the 1D fast Fourier transform. The validation of the geoid models is carried out with gravity/GNSS/leveling measurements, which were conducted in two traverses located in the study areas. Special attention is given to the tidal approach, the geoid separation term as well as the coordinate reference system. Next, the northern study area is extended by incorporating gravity measurements obtained during the last five decades, and geoid models are recomputed. Lastly, using the geoid models computed, reference geopotential values are computed for both areas. From the results achieved for both study areas, the combined model XGM2019 provides the best overall statistical results with differences of 0.065 m and 0.036 m in terms of root mean square error. The incorporation of not recent data into the solutions leads to a degradation in accuracy by about 1.2 cm in terms of standard deviation. Lastly, the computed reference geopotential values present discrepancies between the two study areas, revealing network inconsistencies as well as the dependency on the geopotential model used for the geoid computations.

1. Introduction

In the last decade, significant efforts have been made towards the realization of an International Height Reference System (IHRS) [1]. As each country has one or more local/national height reference systems, it is necessary to determine their difference with respect to the IHRS. This may be achieved by determining the local/national zero-height potential level (reference geopotential value) and comparing it with the IHRF conventional geopotential value (International Association of Geodesy Resolution No. 1/2015—[2]). Numerous studies have been made to determine the zero-height potential level, especially for Greece [3,4,5,6,7]. One of the most common methods used for obtaining the reference geopotential value requires the use of GNSS, leveling, and gravity measurements at stations or benchmarks in an area with a known geoid model or global geopotential model; therefore, the computation of a geoid model also has an important part in the whole procedure.

The use of global geopotential models (GGMs) in geoid modeling for computing gravity-related quantities (e.g., gravity anomalies, geoid heights/height anomalies, and deflections of the vertical) is a common practice in geodesy. GGMs are usually exploited in the frame of a remove–compute–restore (RCR) approach. In this approach, the aim is to provide computations a smoother and less variable gravity field with respect to the original one. This is achieved by first removing the contribution of a GGM (reduced gravity field) and then the contribution of the topography (residual gravity field). One of the most common reductions used for removing the effects of the topography is the residual terrain modeling (RTM) technique. In its original form (classical RTM – cRTM) [8], it is computed with the aid of two digital terrain models (DTMs), one serving as a reference surface and the other as a detailed model of the Earth’s topography. The reference DTM is computed from the detailed DTM, and its resolution depends on the maximum expansion degree of the GGM used in the gravity reductions. In another approach, the so-called spectral RTM (sRTM) effect [9] is computed directly from a spherical harmonic expansion without the need to derive a reference DTM, but by choosing a minimum degree of expansion equal to the maximum degree of the GGM. Having obtained the residual field, the next step is the computation of the quasi-geoid model. As this is based on a residual field, the final step is to restore the contribution of the GGM and the RTM effects that were initially subtracted, in order to obtain the final height anomalies.

During the last decades, starting from the early 1960s, numerous GGMs have been compiled using different data types. The data may comprise satellite measurements (obtained from satellite gravimetry, satellite altimetry, and satellite laser ranging), terrestrial gravity measurements (obtained on land, at sea, or airborne), topographic data (including digital elevation and bathymetry models), or even previously computed GGMs. Based on the data used, two categories may be formed, i.e., the satellite-only and the combined models. The most recent satellite-only models rely mostly on data from the satellite missions of GRACE (Gravity Recovery And Climate Experiment) [10,11]/GRACE Follow-On [12,13], GOCE (Gravity recovery and steady-state Ocean Circulation Explorer), and LAGEOS (LAser GEOdynamics Satellite) 1/2 [14,15]. On the other hand, the combined models may also include satellite altimetry and terrestrial gravity data, while most recently topography has been included for taking into account the high frequencies of the Earth’s gravity field spectrum. The GGMs are provided as a set of spherical harmonic coefficients and seldom as ellipsoidal harmonic coefficients. Different models have different maximum degrees and orders (d/o) of expansion depending on their source data and the approach followed by the researchers that developed them.

Several studies have been conducted for the validation of the available GGMs and the assessment of their accuracy, spectral content, and contribution to modeling the Earth’s gravity field. Usually, a global study is carried out by the developers themselves, but independent research groups provide their own results at the national or local level. These studies contribute to the further improvement of the models. Regarding the validation of GGMs in geoid modeling, it is mentioned, e.g., [16] that studied over Greece the contribution of satellite-only models (GOCE- and GRACE-based models–4th release) after being spectrally enhanced by a combined model (EGM2008 [17]) for the medium and high frequencies of the gravity field spectrum using gravity, GNSS/leveling, and deflections of the vertical data. The authors of [18] performed an evaluation in the region of Fagnano Lake in the southern Andes using GPS buoy and GPS/leveling data for combined GGMs (EGM2008), especially those based on the fifth release of GOCE data (EIGEN6C4 [19], GECO [20], GGM05C [21], and GOCO05C [22]). The authors of [23] used gravity data based on absolute gravity measurements and GNSS/leveling data over Poland to validate 10 satellite-only (GOCE releases 4 and 5, GRACE- and GOCE–GRACE-based models) and 3 combined models. In the South China Sea, [24] used airborne gravity and marine gravity data for evaluating eight combined GGMs with high degrees of expansion. The authors of [25] studied more than 30 GGMs, combined and satellite-only, including the most recent ones for the area of Kenya, using GNSS/leveling and gravity data but with accuracies of the order of several decimeters in terms of standard deviation. In Turkey, the authors of [26] evaluated GOCE-based GGMs (all releases) spectrally enhanced by a combined model (EGM2008) using gravity and GNSS/leveling data. From the above, it may be seen that in many recent studies, numerous models are being compared, and this is due to the fact that even more research groups are involved in the development of GGMs.

As it was previously mentioned, some studies evaluating GGMs contain the computation of a geoid model using the RCR approach, which is compared to GNSS/leveling-derived geoid heights; however, this approach is affected by the topographic reductions used since the latter are mandatory in geoid modeling. In this study, we use and evaluate the latest GGMs in gravity and geoid modeling following the RCR approach in two test areas, one in northern and one in central Greece (Figure 1). In the two test areas, reliable and accurate data were recently obtained, including gravity values as well as GNSS/leveling-derived geoid heights. In order to achieve more accurate evaluation results, both the cRTM and sRTM are used for producing the residual gravity field, while the 1D spherical FFT [27] is employed for computing height anomalies. Finally, the geoid separation term is also considered for computing geoid heights and thus allowing their validation. The aforementioned steps will lead to the assessment of the contribution of the GGMs in high-accuracy local geoid modeling, i.e., whether it is better to use satellite-only or combined GGMs, and whether geoid heights derived from high-resolution GGMs may substitute local geoid models.

Additionally, as the two study areas are limited, in contrast to other geoidal studies in larger areas (see, e.g., [28]), additional data from a previously compiled gravity database are incorporated in the geoid solution of the northern area, thus extending the borders of the study area. In this way, the contribution of the additional data is investigated. Lastly, reference geopotential values are obtained for the two study areas to examine possible inconsistencies in the vertical datum as well as to compare with previously computed values and the IHRF conventional value.

2. Materials and Methods

2.1. Available Data

2.1.1. Geopotential Models

The examined GGMs in this study are listed in

Table 1. Global geopotential models examined in the present study.

Model	Max Degree	Source Data	Reference
EGM2008	2190	A, G, S (GRACE)	[17]
EIGEN-6C4	2190	A, G, S (GOCE, GRACE, Lageos)	[19]
GOCO06S	300	S (GOCE, GRACE, etc.)	[29,30]
GO CONS GCF 2 DIR R6	300	S (GOCE, GRACE, Lageos)	[31]
GO CONS GCF 2 TIM R6	300	S (GOCE)	[32]
GO CONS GCF 2 TIM R6e	300	G (Polar regions), S (GOCE)	[33]
IGGT R1C	240	G, S (GOCE, GRACE)	[34]
ITSG_GRACE2018S	200	S (GRACE)	[35,36]
SGG-UGM-2	2190	A, EGM2008, S (GOCE, GRACE)	[37]
Tongji-GMMG2021S	300	S (GOCE, GRACE)	[38]
TongjiGRACE02K	180	S (GRACE)	[39]
XGM2019	760	A, G, S (GOCO06S)	[40,41]
XGM2019e	5540	A, G, S (GOCO06S), Τ
XGM2019e_2159	2190	A, G, S (GOCO06S), Τ

A: satellite altimetry, G: terrestrial gravity, S: satellite data, Τ: topography.

. These include both satellite-only and combined ones, recent and published about a decade ago, such as the EGM2008 and EIGEN-6C4, which are still widely used, as well as the XGM2019e model that has the highest d/o of expansion (5540). Another remark that should be made is the use of previous models for deriving a new one. For example, in the XGM2019 group of models their lower degree part derives from the satellite-only model GOCO06S. On the other hand, models of higher degree and order of expansion, such as SGG-UGM-2 and EIGEN-6C4, rely for land areas on gravity values obtained from EGM2008, while XGM2019 models on a 15 arcmin resolution global gravity grid. Finally, among the examined models, XGM2019e and XGM2019e_2159 also contain gravity values derived from topography to model the high-frequency part of the gravity field spectrum.

2.1.2. Digital Elevation and Bathymetry Model

The digital elevation and bathymetry model (DEBM) used in this study (Figure 1) was derived by combining Copernicus digital elevation model [42] and the Greek Seas digital terrain model [43]. The choice was made following the validation results for different DEBM in the study areas, as they are presented in [44]. The two models have different spatial analyses (1 arcsec and 15 arcsec, respectively). Therefore, we first upsampled the DTM of the Greek Seas to 1 arcsec resolution using a bilinear interpolation technique. Then, we combined the two models by keeping land values (heights) from the Copernicus DEM and marine values (depths) from the Greek Seas DTM. All voids were filled using an inverse distance to power of 2 interpolation scheme. This may have led to a linear smoothing of the model in coastal areas, but this was not necessarily a disadvantage as it is described below.

The reference system used for the horizontal position of the aforementioned models is WGS84. Regarding heights from the Copernicus DEM, these refer to the EGM2008 geoid, while depths from Greek Seas DTM are considered to refer to the mean sea level, although for the latter there is no documentation available. The lack of documentation or in general the uncertainty in defining the reference system of depth data is unfortunately common among digital depth models, especially those that are derived from echo sounding measurements. Therefore, the use of an interpolation scheme for filling voids, especially in coastal areas, may have also smoothed any differences between the two models.

2.1.3. Gravity, GNSS, and Leveling Data

Gravity measurements were carried out in the two pilot areas covering an area of about 10 × 11 km in northern (Figure 2) and 10 × 10 km in central Greece (Figure 3). Significant effort was made to cover each area in a consistent way to have the same spatial density of values. This was not fully achieved as some parts of the areas were not accessible but still the distribution of the values is satisfactory. A total of 141 gravity values were measured in the northern and 92 in the central area. All values were measured with a Scintrex CG5 relative gravimeter and both campaigns have as reference a point located at the Laboratory of Gravity Field Research and Applications (GravLab) at the Aristotle University of Thessaloniki, in northern Greece. The reference point gravity value was obtained from measurements with a Microg-Lacoste A10-#027 absolute gravimeter. The position of the gravity values was determined with GNSS receivers. Therefore, only geometric heights are available for the point values, while their orthometric heights were derived through interpolation from the DEBM (described in the previous section). The processing of the gravity measurements was carried out as in [45]. Hence, the resulting gravity data follow the zero-tide concept [46], referring, regarding their position, to the International Terrestrial Reference Frame 2014 (ITRF2014) and have an accuracy of about 0.03 mGal.

Moreover, in both test areas, GNSS and spirit leveling measurements were carried out along two traverses (Figure 2 and Figure 3), one in each area. For the northern area, the loop closure error for the spirit leveling measurements was 14 mm/$\sqrt{\mathrm{k}\mathrm{m}}$ [44] and 175 geoid height values were obtained from the GNSS/leveling combination. For the central area the loop closure error was 13 mm/$\sqrt{\mathrm{k}\mathrm{m}}$ [44] and 138 geoid height values were computed. For both areas the accuracy of the measured geometric heights is estimated to be between 2 and 4 cm [44]. The GNSS/leveling-derived geoid heights are used in the present study for the validation of the computed geoid models. Therefore, out of these values, 161 and 104 were used for the northern and central areas, respectively, to avoid edge effects, when comparing with the geoid models computed, and to remove points with more than 3 cm accuracy in geometric height. The estimated accuracy of the GNSS/leveling-derived geoid heights is about 1–3 cm.

In the northern area and at a buffer zone of 0.25° from the area covered by the new measurements, 1470 additional gravity values (Figure 4) were selected to be used. The data was taken from the gravity database prepared by [47], originating mostly from measurements that were carried out at least 20 years ago and lack proper documentation. As the data refer to the national gravity network, a mixed linear model with polynomial of second degree determined by [45] was applied to make them compatible with the recent gravity measurements, tied to the absolute gravity station, and remove a trend present in the existing network (for a detailed explanation the interested reader may consult [45]). The values used follow the zero-tide concept.

2.2. Methodology

Two kinds of validation are used in the present study. The first examines how well the GGMs reduce the available gravity data by taking into account the residual terrain modeling effect. More specifically, we first compute surface gravity anomalies (Equations (8)–(9) in [48]), derived from the measured gravity values $g$ and the corresponding horizontal coordinates of the measured points (geodetic latitude $\phi$ and geodetic longitude $\lambda$):

$$\mathsf{\Delta }{g}_s=g-\gamma \left[1-2\left(1+f+m-2f\mathrm{s}\mathrm{i}{\mathrm{n}}^2\phi \right)\frac{H^{*}}{\alpha }+3{\left(\frac{H^{*}}{\alpha }\right)}^2\right]$$

(1)

where $\gamma$ is the normal gravity on the reference ellipsoid (GRS80) obtained from Somigliana’s formula (Equations (2)–(78) in [48]), $f$ is the geometrical flattening of the ellipsoid, $m$ is the ratio of centrifugal and gravity acceleration at its equator and α is the radius at the equator. The normal height $H^{*}$ was computed from the following equation:

$$H^{*}=H+N$$

(2)

where H is the orthometric height of the measured point and Ν-ζ is the geoid separation term. This term was computed from Equation (24) in [49] as follows:

$${\rm N}-\zeta =\frac{\mathsf{\Delta }{g}_{B}H+\left(V_{Po}-V_P\right)}{\stackrel{-}{\gamma }}$$

(3)

where $P$ is the computation point on the Earth’s surface and $Po$ its projection point on the geoid, $\mathsf{\Delta }{g}_B$ is the complete Bouguer anomaly, $H$ is the orthometric height, $\stackrel{-}{\gamma }$ is the mean normal gravity and $V_{Po}$ and $V_P$ are the gravitational potential of the topographic masses on the Earth’s surface and on the geoid, respectively.

Using these surface gravity anomalies, we subtract the contribution of the GGMs (see Equation (17) in [50]) from the surface anomalies and then, depending on the models’ maximum degree of expansion, we remove the sRTM effect or the cRTM effect. From the above computations three gravity fields are computed, one reduced and two residual fields depending on the RTM effect removed. The cRTM is computed following [8], while the sRTM is obtained from the precomputed model Earth2014 [51] for d/o up to 2160 and from the ERTM2160 gravity model [52] from d/o 2160 up to 96,000. As mentioned previously, for the cRTM, two DEBM are required, a detailed and a reference one. The reference field was obtained in two ways, i.e., (a) directly from the detailed without any filtering technique and (b) after smoothing using a moving average filter. The resolution $r$ of the reference models was computed from the approximate formula [53], which may be expressed in km as follows:

$$r=\frac{\pi R\mathrm{c}\mathrm{o}\mathrm{s}\left({\phi }_m\right)}{n}$$

(4)

where $n$ is the d/o of the GGM expansion, $R$ is the mean Earth’s radius and ${\phi }_m$ is the mean geocentric latitude of the study area. It should be noted that for the RTM effects the density of the land masses was assumed to be equal to 2.67 g·cm⁻³ and for marine areas equal to 1.03 g·cm⁻³. The assessment of the obtained residual gravity fields is carried out by examining the statistical results of each field, i.e., their mean, range or minimum and maximum value and standard deviation. The closer the values of the statistics to zero are, the better are the reductions. Of course, all statistical results are affected also by possible errors in the data, where a large mean value might imply that there is a systematic bias in the data and a large standard deviation might show the presence of large random errors. The validation described before provides a first indication for each model’s performance.

The second validation procedure involves geoid modeling. The residual fields obtained are used for the computation of residual quasi-geoid models. The residual height anomalies are obtained by using the 1D spherical FFT with 100% zero padding towards all directions. To obtain height anomalies we restored the effect of the GGMs (Equations (3) and (11) in [50]) and RTM effect. Then, the geoid heights were derived by adding the geoid separation term.

Before proceeding to the evaluation of the results using the GNSS/leveling-derived geoid heights, it is mandatory to have all values follow the same tidal concept. As the orthometric heights follow the mean-tide concept, all other values, geoid heights and geometric heights, need to be converted. As mentioned previously, the geoid models computed adopt the zero-tide concept due to the gravity data used. The conversion of the geoid heights from the zero-tide ($N_z$) to the mean-tide ($N_M$) concept was carried out using Equation (15) in [54] as follows:

$${{\rm N}}_M-N_Z=\left(9.9-29.6{\sin }^2{\phi }^{\prime }\right)\frac{1}{100}\left[m\right]$$

(5)

where ${\phi }^{\prime }$ is the geocentric latitude. The GNSS derived geometric heights follow the tide-free concept. To convert them to a mean-tide concept we added the following term ($\Delta h$) computed by Equation (23) from [46] as follows:

$$\mathsf{\Delta }h=(60.34-179.01{\sin }^2\phi -1.82{\sin }^4\phi )\frac{1}{1000}[m]$$

(6)

After having all values follow the same tidal concept, the statistical results of the differences between the computed geoid heights and the GNSS/leveling-derived geoid heights are obtained. Similar to the examination of the residual field, the closer the statistical results to zero the better the model.

The determination of the zero-height geopotential value $W_o$ for the two study areas was carried out using only points where GNSS, leveling and gravity measurements were conducted. As all computations for the local geoid models were made with the GRS80 ellipsoid as reference, the zero-degree term of the geoid heights ($N_o$) may be used for determining $W_o$. More specifically, by assuming that the reference ellipsoid has the same mass as the Earth, Equation (2.182) in [48] may be rewritten as follows:

$$W_P={U_Q-N}_o\stackrel{-}{\gamma }$$

(7)

where $W_P$ is the unknown potential on the geoid at a point $P$, $U_Q$ is the normal potential on the surface of the reference ellipsoid at the corresponding point $Q$, $\stackrel{-}{\gamma }$ is the mean gravity over Earth and $N_o$ is computed from the following relation:

$$N_o=N^{GNSSlev}-N^G$$

(8)

where $N^{GNSSlev}$ is the geoid height obtained from GNSS/leveling measurements and $N^G$ is the geoid height from the computed gravimetric geoid. By computing $W_P$ from the above equations for all points, $W_o$ may be determined by averaging the latter values.

3. Results

3.1. Gravity Field Modeling

Table 2. Reduced gravity field statistical results for different GGMs. [mGal].

		Northern Area							Central Area
	d/o	180	200	240	280	300	760	2190	180	200	240	280	300	760	2190
EGM2008	mean	46.44	43.44	39.11	44.39	52.22	11.12	−4.09	7.93	17.49	18.31	16.96	8.59	−7.72	−4.01
	stdev	7.76	7.63	8.38	8.05	8.51	8.06	7.16	6.04	5.78	5.49	5.98	5.99	5.58	4.38
	range	38.05	37.09	38.97	37.74	37.70	36.82	32.38	30.24	29.08	27.20	28.42	28.72	28.17	24.39
EIGEN-6C4	mean	47.41	44.17	39.79	44.96	52.97	11.85	−3.42	9.40	18.82	19.98	18.44	10.02	−6.17	−2.84
	stdev	7.74	7.60	8.34	8.01	8.47	8.05	7.21	6.04	5.79	5.51	6.00	6.02	5.64	4.35
	range	37.81	36.85	38.75	37.56	37.53	36.68	32.66	30.18	29.13	27.26	28.47	28.81	28.18	24.29
SGG-UGM-2	mean	47.85	43.87	39.44	45.20	53.18	11.55	−3.43	7.93	18.72	19.47	17.32	8.71	−8.18	−6.35
	stdev	7.74	7.61	8.36	8.05	8.51	8.08	7.28	6.04	5.80	5.48	5.97	5.96	5.64	4.28
	range	37.85	36.93	38.81	37.51	37.44	36.55	32.97	30.31	29.32	27.32	28.55	28.45	28.66	23.30
XGM2019e 2159	mean	47.61	43.86	38.52	44.54	52.15	12.03	−3.37	9.00	18.32	18.53	17.46	9.26	−5.50	−2.66
	stdev	7.75	7.61	8.41	8.07	8.54	8.09	8.32	6.01	5.79	5.51	6.06	6.01	5.29	4.99
	range	37.87	36.95	38.99	37.68	37.54	36.54	38.89	30.15	29.18	27.32	28.68	28.80	27.10	28.16
XGM2019e	mean	47.61	43.86	38.52	44.54	52.15	12.03	−3.33	9.00	18.32	18.53	17.46	9.26	−5.50	−3.44
	stdev	7.75	7.61	8.41	8.07	8.54	8.09	8.55	6.01	5.79	5.51	6.06	6.01	5.29	4.58
	range	37.87	36.95	38.99	37.68	37.54	36.54	38.40	30.15	29.18	27.32	28.68	28.80	27.10	25.67
XGM2019	mean	47.61	43.86	38.52	44.54	52.15	20.57		9.00	18.32	18.53	17.46	9.26	−0.35
	stdev	7.75	7.61	8.41	8.07	8.54	8.50		6.01	5.79	5.51	6.06	6.01	5.36
	range	37.87	36.95	38.99	37.68	37.54	39.06		30.15	29.18	27.32	28.68	28.80	27.04
GOCO06S	mean	47.55	43.43	36.29	31.27	28.28			9.41	19.12	18.95	14.62	18.12
	stdev	7.75	7.61	8.26	7.87	7.62			6.03	5.80	5.46	5.66	5.68
	range	37.86	37.04	39.64	38.28	38.43			30.19	29.26	27.03	28.59	28.69
GO CONS GCF 2 DIR R6	mean	47.70	43.27	36.14	31.17	28.25			9.43	19.01	19.12	14.19	17.51
	stdev	7.74	7.60	8.23	7.93	7.76			6.03	5.80	5.47	5.65	5.66
	range	37.84	37.03	39.42	38.49	38.70			30.18	29.25	27.10	28.56	28.54
GO CONS GCF 2 TIM R6	mean	47.66	43.47	36.39	31.58	28.71			9.51	19.26	19.10	15.04	18.37
	stdev	7.74	7.61	8.25	7.88	7.65			6.03	5.80	5.45	5.64	5.66
	range	37.86	37.03	39.61	38.33	38.46			30.18	29.25	27.01	28.53	28.61
GO CONS GCF 2 TIM R6e	mean	47.66	43.52	36.40	31.54	29.24			9.52	19.23	19.14	15.10	17.92
	stdev	7.75	7.60	8.25	7.87	7.65			6.02	5.80	5.45	5.64	5.65
	range	37.86	37.03	39.61	38.33	38.46			30.17	29.24	27.01	28.53	28.61
Tongji-GMMG2021S	mean	47.49	43.40	36.25	31.78	27.85			9.71	19.40	18.74	12.78	14.72
	stdev	7.74	7.60	8.21	8.06	7.90			6.02	5.79	5.41	5.50	5.40
	range	37.84	36.99	39.23	38.36	38.33			30.17	29.26	27.17	27.80	27.46
IGGT R1C	mean	49.07	42.02	33.68					6.70	19.65	10.20
	stdev	7.73	7.74	8.02					6.02	5.62	5.50
	range	37.47	37.10	37.99					30.19	28.80	28.70
ITSG GRACE2018S	mean	47.67	46.47						7.52	17.20
	stdev	7.75	7.57						6.07	5.73
	range	37.85	37.01						30.43	29.38
Tongji GRACE02K	mean	48.91							9.16
	stdev	7.75							6.11
	range	38.03							30.49

and

Table 3. Residual gravity field statistical results for different GGMs with the sRTM approach. [mGal].

		Northern Area							Central Area
	d/o	180	200	240	280	300	760	2190	180	200	240	280	300	760	2190
EGM2008	mean	15.17	13.88	8.35	−0.47	−1.44	−1.33	−1.95	−14.44	−7.65	5.71	11.95	10.29	0.39	0.60
	stdev	6.27	6.58	6.17	5.53	6.40	6.85	1.77	1.78	1.75	1.47	1.55	1.65	1.81	1.42
	range	28.58	29.89	28.15	25.41	29.10	30.87	8.84	7.00	7.23	6.10	6.07	6.74	7.10	7.90
EIGEN-6C4	mean	16.15	14.61	9.03	0.11	−0.70	−0.59	−1.27	−12.97	−6.32	7.38	13.42	11.72	1.93	1.78
	stdev	6.18	6.46	6.07	5.43	6.30	6.77	1.81	1.80	1.73	1.46	1.57	1.66	2.00	1.47
	range	28.16	29.38	27.74	24.96	28.67	30.51	9.06	7.08	7.15	6.07	6.12	6.78	7.90	8.12
SGG-UGM-2	mean	16.58	14.31	8.68	0.34	−0.49	−0.89	−1.28	−14.44	−6.43	6.87	12.30	10.41	−0.07	−1.73
	stdev	6.19	6.50	6.11	5.46	6.32	6.75	1.83	1.75	1.67	1.41	1.48	1.56	1.61	2.76
	range	28.21	29.55	27.90	25.11	28.75	30.42	9.05	6.92	6.95	5.84	5.72	6.34	6.60	11.39
XGM2019e 2159	mean	16.35	14.30	7.76	−0.32	−1.52	−0.42	−1.22	−13.36	−6.82	5.92	12.45	10.96	2.61	1.95
	stdev	6.21	6.50	6.19	5.54	6.38	7.08	7.17	1.80	1.71	1.44	1.57	1.66	1.63	1.57
	range	28.31	29.58	28.24	25.44	28.98	31.81	32.33	7.09	7.10	5.99	6.01	6.81	6.87	7.09
XGM2019e	mean	16.35	14.30	7.76	−0.32	−1.52	−0.42	−1.18	−13.36	−6.82	5.92	12.45	10.96	2.61	1.17
	stdev	6.21	6.50	6.19	5.54	6.38	7.08	7.59	1.80	1.71	1.44	1.57	1.66	1.63	1.50
	range	28.31	29.58	28.24	25.44	28.98	31.81	32.57	7.09	7.10	5.99	6.01	6.81	6.87	7.36
XGM2019	mean	16.35	14.30	7.76	−0.32	−1.52	8.12		−13.21	−13.36	−6.82	5.92	12.45	1.34
	stdev	6.21	6.50	6.19	5.54	6.38	8.86		1.71	1.80	1.71	1.44	1.57	1.61
	range	28.31	29.58	28.24	25.44	28.98	38.69		6.78	7.09	7.10	5.99	6.01	6.50
GOCO06S	mean	16.28	13.87	5.53	−13.59	−25.39			−12.95	−6.03	6.34	9.60	19.83
	stdev	6.20	6.52	6.13	5.41	5.43			1.79	1.69	1.52	2.07	2.15
	range	28.26	29.64	27.92	24.71	24.60			7.06	7.01	6.28	8.14	8.17
GO CONS GCF 2 DIR R6	mean	16.43	13.72	5.38	−13.69	−25.42			−12.94	−6.13	6.52	9.18	19.21
	stdev	6.19	6.51	6.06	5.54	5.66			1.80	1.69	1.50	2.07	2.11
	range	28.22	29.61	27.64	25.24	25.63			7.07	7.02	6.23	8.10	8.07
GO CONS GCF 2 TIM R6	mean	16.39	13.91	5.63	−13.28	−24.95			−12.86	−5.89	6.49	10.03	20.07
	stdev	6.20	6.51	6.11	5.43	5.48			1.80	1.69	1.52	2.06	2.15
	range	28.24	29.62	27.86	24.80	24.85			7.07	7.02	6.29	8.08	8.18
GO CONS GCF 2 TIM R6e	mean	16.39	13.96	5.64	−13.32	−24.43			−12.85	−5.92	6.54	10.09	19.62
	stdev	6.20	6.51	6.11	5.43	5.48			1.80	1.70	1.53	2.06	2.17
	range	28.26	29.62	27.84	24.77	24.85			7.08	7.04	6.29	8.08	8.24
Tongji-GMMG2021S	mean	16.23	13.84	5.49	−13.07	−25.82			−12.66	−5.74	6.13	7.76	16.43
	stdev	6.19	6.49	6.01	5.66	5.73			1.80	1.69	1.43	1.99	2.26
	range	28.21	29.53	27.43	25.85	26.15			7.07	7.02	5.94	7.89	8.85
IGGT R1C	mean	17.80	12.46	2.92					−15.67	−5.49	−2.40
	stdev	6.09	6.66	5.50					1.80	1.88	1.33
	range	27.76	30.26	25.29					7.08	7.72	6.12
ITSG GRACE2018S	mean	16.40	16.92						−14.85	−7.95
	stdev	6.20	6.47						1.71	1.70
	range	28.27	29.42						6.80	7.06
Tongji GRACE02K	mean	17.64							−13.21
	stdev	6.25							1.71
	range	28.47							6.78

present the statistical results of the reduced fields per model and per d/o for both test areas. Where applicable, computations were made for all maximum d/o of the models to make the results consistent. For example, XGM2019 was computed for d/o 180, 200, 240, 280, 300, and 719, where the latter is its maximum d/o.

From the results of Table 2, we first examine the standard deviation (stdev) of the results. All models provide similar results (the results differ to about 0.1 mGal) when we do not compare the maximum d/o of each model. To make this clearer, we take, as an example, GOCO06S. We observe that for the lower d/o, the results are similar to the rest of the models. When we examine the degrees closer to the maximum d/o of the model, we notice larger differences, especially with models that have a higher maximum d/o of expansion. A similar behavior is also observed for the range, although it varies more (about up to 1 mGal). On the other hand, the mean value is the one that varies independently of the d/o. When examining only the models for d/o 2190, we observe that EIGEN-6C4 presents the best combination in terms of the mean value, standard deviation, and range. A last remark refers to the reduced fields in the northern area, which is a semi-mountainous area, where a higher standard deviation is found than that of the central area at a level of about 2 cm; thus, we next examine the residual fields after removing the RTM effect.

Table 3 shows the statistical results of the residual fields after removing the sRTM effect from the reduced fields. Again, the results are provided for different d/o. Similar remarks may be made regarding the differences among the models with respect to the reduced field. There are, however, some significant differences that need to be noticed. For a d/o equal to 2190, we observe that XGM2019e and XGM2019e2159 have a significantly larger standard deviation for the northern area at the level of about 5 mGal; moreover, the range of the differences is also larger by about 20 mGal, this does not occur in the central area. Another remark that can be made is that the reduced fields based on the satellite models, apart from the one with IGGT R1C, present high mean values. This may be attributed to the lack of medium frequencies of the gravity field spectrum and, therefore, are not adequately modeled by the removed topography. For all models, the inadequate reductions or large mean values may be due to possible inconsistencies with the sRTM effects. The latter is further investigated when examining the results with the classical RTM approach. For the sRTM, EGM2008 and EIGEN-6C4 provide the smoothest fields with the lowest mean values for both areas.

For the computation of the residual fields following the cRTM approach, different reference DEBM models were used with different resolutions and with and without filtering. The residual fields without filtering led to worse results in geoid computations, so we chose to provide only the results for the reference models after filtering. These are presented in

Table 4. Residual gravity field statistical results for different GGMs with the cRTM approach. [mGal].

		Northern Area							Central Area
	d/o	180	200	240	280	300	760	2190	180	200	240	280	300	760	2190
EGM2008	mean	19.28	12.75	7.12	11.79	22.70	1.90	−5.34	−3.04	6.36	9.39	8.76	0.97	−7.06	−0.57
	stdev	6.20	5.69	7.07	5.47	6.22	5.01	4.03	1.42	1.66	1.69	2.19	2.33	1.44	2.77
	range	27.22	24.75	31.03	23.85	27.48	22.18	14.75	5.59	6.40	6.55	8.40	8.86	7.00	10.73
EIGEN-6C4	mean	20.25	13.48	7.80	12.36	23.45	2.64	−4.66	−1.57	7.69	11.06	10.23	2.41	−5.51	0.60
	stdev	6.09	5.55	6.97	5.36	6.12	4.92	4.08	1.44	1.66	1.71	2.23	2.37	1.47	2.75
	range	26.79	24.23	30.62	23.40	27.05	21.84	14.81	5.66	6.33	6.60	8.53	8.99	7.16	10.68
SGG-UGM-2	mean	20.69	13.17	7.45	12.59	23.66	2.34	−4.68	−3.05	7.59	10.55	9.11	1.10	−7.52	−2.91
	stdev	6.10	5.60	7.01	5.37	6.14	4.88	4.12	1.39	1.57	1.59	2.13	2.24	1.72	3.13
	range	26.84	24.40	30.79	23.55	27.12	21.70	15.35	5.49	6.11	6.28	8.15	8.51	8.19	14.17
XGM2019e 2159	mean	20.45	13.17	6.53	11.94	22.63	2.81	−4.62	−1.97	7.19	9.61	9.26	1.64	−4.83	0.78
	stdev	6.12	5.61	7.09	5.46	6.20	5.21	6.32	1.42	1.63	1.68	2.27	2.36	1.20	3.66
	range	26.95	24.44	31.13	23.88	27.35	22.92	26.10	5.56	6.27	6.54	8.62	8.96	5.52	14.64
XGM2019e	mean	20.45	13.17	6.53	11.94	22.63	2.81	−4.57	−1.97	7.19	9.61	9.26	1.64	−4.83	0.00
	stdev	6.12	5.61	7.09	5.46	6.20	5.21	6.56	1.42	1.63	1.68	2.27	2.36	1.20	2.97
	range	26.95	24.44	31.13	23.88	27.35	22.92	26.34	5.56	6.27	6.54	8.62	8.96	5.52	11.10
XGM2019	mean	20.45	13.17	6.53	11.94	22.63	11.35		−1.97	7.19	9.61	9.26	1.64	0.31
	stdev	6.12	5.61	7.09	5.46	6.20	7.13		1.42	1.63	1.68	2.27	2.36	1.72
	range	26.95	24.44	31.13	23.88	27.35	29.79		5.56	6.27	6.54	8.62	8.96	6.71
GOCO06S	mean	20.39	12.74	4.29	−1.34	−1.24			−1.56	7.99	10.03	6.41	10.51
	stdev	6.11	5.63	7.08	5.43	5.28			1.42	1.60	1.72	2.42	2.41
	range	26.89	24.50	30.81	23.15	22.98			5.59	6.17	6.63	9.39	9.58
GO CONS GCF 2 DIR R6	mean	20.54	12.58	4.15	−1.43	−1.27			−1.54	7.88	10.20	5.99	9.90
	stdev	6.10	5.62	7.00	5.56	5.51			1.43	1.61	1.72	2.41	2.34
	range	26.85	24.47	30.53	23.68	24.01			5.60	6.19	6.64	9.35	9.30
GO CONS GCF 2 TIM R6	mean	20.50	12.77	4.39	−1.03	−0.81			−1.46	8.13	10.18	6.84	10.75
	stdev	6.11	5.63	7.06	5.46	5.33			1.43	1.60	1.72	2.40	2.39
	range	26.87	24.48	30.74	23.24	23.23			5.59	6.19	6.62	9.33	9.50
GO CONS GCF 2 TIM R6e	mean	20.50	12.82	4.41	−1.06	−0.28			−1.45	8.10	10.22	6.90	10.30
	stdev	6.11	5.63	7.06	5.45	5.33			1.43	1.61	1.72	2.40	2.40
	range	26.89	24.47	30.73	23.21	23.23			5.59	6.20	6.62	9.32	9.53
Tongji-GMMG2021S	mean	20.33	12.70	4.26	−0.82	−1.67			−1.26	8.27	9.82	4.57	7.11
	stdev	6.10	5.60	6.95	5.63	5.57			1.42	1.59	1.52	2.19	2.13
	range	26.84	24.39	30.31	24.29	24.52			5.57	6.18	5.94	8.41	8.33
IGGT R1C	mean	21.91	11.33	1.69					−4.27	8.52	1.28
	stdev	5.97	5.73	6.39					1.42	1.64	1.06
	range	26.39	25.11	28.17					5.55	6.71	4.57
ITSG GRACE2018S	mean	20.51	15.78						−3.45	6.07
	stdev	6.11	5.59						1.38	1.48
	range	26.90	24.27						5.53	6.20
Tongji GRACE02K	mean	21.75							−1.81
	stdev	6.17							1.42
	range	27.11							5.66

. Up to d/o 300, there is no significant difference in the standard deviation of the results when compared to those in Table 3 (residual field with the sRTM approach). On the other hand, a significant improvement is noticed in the mean value of the satellite-only based models, apart from ITSG_GRACE2018S and TongjiGRACE02K, compared to the results of Table 3. For d/o 760 and 2190, we observe that the reductions are better for d/o 760 compared to those of d/o 2190 in terms of standard deviation, while when compared to Table 3, there is no clear pattern depicting which is better than the other. Consequently, the next step of the validation, i.e., geoid computation and assessment of the results, is decisive in the evaluation.

3.2. Geoid Modeling

From the previously presented residual gravity fields, we computed geoid solutions for both test areas. Each geoid solution was compared to the geoid heights derived from the available GNSS/leveling data. Before presenting the results of the comparison, we first compare the geoid heights obtained from the GGMs that have maximum d/o 2190 and 5140. The results are given in

Table 5. Comparison of GNSS/leveling-derived geoid heights with those obtained from the GGMs. [m].

Northern	EGM2008	XGM2019e_2159	XGM2019e	EIGEN-6C4	SSG-UGM-2
mean	0.000	0.023	0.030	0.065	0.063
stdev	0.074	0.051	0.055	0.077	0.077
rmse	0.074	0.056	0.062	0.101	0.099
range	0.325	0.285	0.287	0.337	0.335
Central	EGM2008	XGM2019e_2159	XGM2019e	EIGEN-6C4	SSG-UGM-2
mean	0.157	0.189	0.190	0.206	0.145
stdev	0.037	0.036	0.037	0.036	0.043
rmse	0.161	0.192	0.194	0.209	0.151
range	0.220	0.217	0.216	0.214	0.239

. It is observed that for the northern area, the best results are obtained from XGM2019e_2159, while for the central area, both EGM2008 and SSG-UGM-2 could be chosen as the best solutions when considering the root mean square error (RMSE) of the comparisons. It is noted that the lower standard deviation of the differences is 5.1 cm for the northern area and 3.6 cm for the central. The mean value for the central area is significantly high and no less than 14.5 cm.

Table 6. Geoid height differences between geoid models with the sRTM approach and GNSS/leveling-derived geoid heights [m].

		Northern Area							Central Area
	d/o	180	200	240	280	300	760	2190	180	200	240	280	300	760	2190
EGM2008	mean	0.380	0.338	0.193	0.035	0.002	−0.001	0.024	−0.380	−0.193	0.152	0.295	0.262	0.174	0.168
	stdev	0.049	0.045	0.048	0.055	0.049	0.047	0.062	0.033	0.035	0.037	0.035	0.033	0.034	0.034
	range	0.265	0.244	0.269	0.301	0.276	0.263	0.324	0.156	0.166	0.169	0.160	0.154	0.157	0.159
EIGEN-6C4	mean	0.450	0.400	0.255	0.094	0.064	0.061	0.086	−0.341	−0.158	0.196	0.334	0.301	0.217	0.211
	stdev	0.051	0.046	0.050	0.058	0.050	0.047	0.065	0.032	0.035	0.037	0.035	0.033	0.033	0.033
	range	0.274	0.254	0.276	0.309	0.285	0.270	0.331	0.151	0.164	0.166	0.157	0.153	0.153	0.155
SGG-UGM-2	mean	0.461	0.390	0.243	0.093	0.063	0.058	0.084	−0.390	−0.166	0.176	0.303	0.265	0.178	0.176
	stdev	0.051	0.045	0.049	0.057	0.050	0.047	0.065	0.034	0.037	0.039	0.037	0.035	0.038	0.038
	range	0.272	0.249	0.275	0.307	0.284	0.269	0.330	0.158	0.172	0.175	0.166	0.158	0.171	0.172
XGM2019e 2159	mean	0.456	0.391	0.223	0.080	0.043	0.048	0.045	−0.354	−0.173	0.158	0.309	0.278	0.195	0.192
	stdev	0.050	0.045	0.049	0.056	0.050	0.047	0.047	0.033	0.036	0.038	0.035	0.033	0.034	0.034
	range	0.271	0.249	0.271	0.303	0.281	0.262	0.263	0.154	0.168	0.170	0.160	0.153	0.155	0.157
XGM2019e	mean	0.456	0.391	0.223	0.080	0.043	0.048	0.044	−0.354	−0.173	0.158	0.309	0.278	0.195	0.194
	stdev	0.050	0.045	0.049	0.056	0.050	0.047	0.047	0.033	0.036	0.038	0.035	0.033	0.034	0.034
	range	0.271	0.249	0.271	0.303	0.281	0.262	0.262	0.154	0.168	0.170	0.160	0.153	0.155	0.156
XGM2019	mean	0.456	0.391	0.223	0.080	0.043	0.073		−0.354	−0.173	0.158	0.309	0.278	0.216
	stdev	0.050	0.045	0.049	0.056	0.050	0.046		0.033	0.036	0.038	0.035	0.033	0.031
	range	0.271	0.249	0.271	0.303	0.281	0.236		0.154	0.168	0.170	0.160	0.153	0.147
GOCO06S	mean	0.454	0.378	0.171	−0.191	−0.396			−0.338	−0.145	0.176	0.254	0.429
	stdev	0.050	0.045	0.048	0.056	0.057			0.033	0.036	0.036	0.030	0.030
	range	0.271	0.248	0.271	0.316	0.327			0.153	0.169	0.166	0.144	0.140
GO CONS GCF 2 DIR R6	mean	0.460	0.376	0.171	−0.192	−0.399			−0.337	−0.148	0.180	0.246	0.417
	stdev	0.051	0.045	0.049	0.055	0.055			0.033	0.036	0.036	0.030	0.030
	range	0.273	0.250	0.275	0.312	0.318			0.152	0.168	0.166	0.144	0.142
GO CONS GCF 2 TIM R6	mean	0.460	0.382	0.177	−0.181	−0.385			−0.334	−0.140	0.180	0.264	0.435
	stdev	0.051	0.045	0.048	0.056	0.056			0.033	0.036	0.036	0.030	0.030
	range	0.272	0.249	0.271	0.316	0.325			0.153	0.168	0.166	0.144	0.141
GO CONS GCF 2 TIM R6e	mean	0.459	0.384	0.177	−0.182	−0.377			−0.334	−0.141	0.181	0.265	0.427
	stdev	0.051	0.045	0.048	0.056	0.056			0.033	0.036	0.036	0.030	0.030
	range	0.272	0.249	0.271	0.316	0.324			0.153	0.168	0.166	0.144	0.141
Tongji-GMMG2021S	mean	0.454	0.379	0.174	−0.185	−0.410			−0.328	−0.136	0.169	0.209	0.352
	stdev	0.051	0.045	0.050	0.054	0.055			0.033	0.036	0.040	0.032	0.031
	range	0.272	0.250	0.278	0.309	0.319			0.153	0.169	0.177	0.151	0.147
IGGT R1C	mean	0.511	0.350	0.124					−0.436	−0.159	−0.069
	stdev	0.054	0.045	0.060					0.033	0.037	0.056
	range	0.282	0.245	0.312					0.156	0.170	0.222
ITSG GRACE2018S	mean	0.456	0.467						−0.403	−0.217
	stdev	0.050	0.045						0.034	0.040
	range	0.272	0.248						−0.473	−0.301
Tongji GRACE02K	mean	0.511							−0.352
	stdev	0.049							0.033
	range	0.266							0.157

depicts the statistical results of the differences between the geoid heights obtained from the sRTM approach fields and the GNSS/leveling-derived geoid heights, while

Table 7. Geoid height differences between geoid models with the cRTM approach (with filtering) and GNSS/leveling-derived geoid heights. [m].

		Northern Area							Central Area
	d/o	180	200	240	280	300	760	2190	180	200	240	280	300	760	2190
EGM2008	mean	0.389	0.270	0.071	0.157	0.338	−0.012	0.018	−0.692	−0.345	−0.173	−0.104	−0.234	−0.055	0.116
	stdev	0.056	0.065	0.044	0.050	0.046	0.048	0.063	0.062	0.051	0.040	0.038	0.036	0.038	0.035
	range	0.282	0.316	0.241	0.272	0.246	0.278	0.325	0.238	0.207	0.175	0.169	0.166	0.175	0.163
EIGEN-6C4	mean	0.459	0.332	0.132	0.217	0.401	0.050	0.080	−0.653	−0.310	−0.129	−0.065	−0.195	−0.012	0.158
	stdev	0.059	0.070	0.045	0.052	0.047	0.050	0.065	0.060	0.051	0.039	0.037	0.036	0.037	0.034
	range	0.291	0.329	0.249	0.280	0.253	0.285	0.332	0.233	0.205	0.173	0.165	0.164	0.169	0.158
SGG-UGM-2	mean	0.471	0.322	0.121	0.216	0.399	0.046	0.078	−0.702	−0.318	−0.149	−0.096	−0.230	−0.051	0.124
	stdev	0.059	0.068	0.044	0.051	0.047	0.050	0.065	0.063	0.054	0.042	0.039	0.038	0.043	0.039
	range	0.288	0.325	0.246	0.277	0.252	0.284	0.330	0.241	0.214	0.182	0.174	0.173	0.190	0.176
XGM2019e 2159	mean	0.465	0.323	0.100	0.202	0.379	0.036	0.039	−0.666	−0.325	−0.167	−0.090	−0.217	−0.034	0.140
	stdev	0.058	0.068	0.044	0.050	0.047	0.048	0.046	0.061	0.052	0.040	0.038	0.036	0.038	0.035
	range	0.287	0.324	0.243	0.274	0.251	0.278	0.263	0.234	0.208	0.176	0.168	0.165	0.173	0.160
XGM2019e	mean	0.465	0.323	0.100	0.202	0.379	0.036	0.038	−0.666	−0.325	−0.167	−0.090	−0.217	−0.034	0.141
	stdev	0.058	0.068	0.044	0.050	0.047	0.048	0.047	0.061	0.052	0.040	0.038	0.036	0.038	0.035
	range	0.287	0.324	0.243	0.274	0.251	0.278	0.263	0.234	0.208	0.176	0.168	0.165	0.173	0.161
XGM2019	mean	0.465	0.323	0.100	0.202	0.379	0.061		−0.666	−0.325	−0.167	−0.090	−0.217	−0.013
	stdev	0.058	0.068	0.044	0.050	0.047	0.045		0.061	0.052	0.040	0.038	0.036	0.034
	range	0.287	0.324	0.243	0.274	0.251	0.251		0.234	0.208	0.176	0.168	0.165	0.157
GOCO06S	mean	0.463	0.311	0.049	−0.068	−0.060			−0.650	−0.298	−0.149	−0.145	−0.067
	stdev	0.059	0.067	0.044	0.051	0.053			0.061	0.053	0.039	0.031	0.031
	range	0.288	0.323	0.242	0.288	0.296			0.236	0.210	0.173	0.150	0.148
GO CONS GCF 2 DIR R6	mean	0.469	0.308	0.048	−0.070	−0.063			−0.649	−0.301	−0.145	−0.153	−0.079
	stdev	0.059	0.068	0.044	0.050	0.051			0.061	0.052	0.039	0.032	0.031
	range	0.288	0.323	0.247	0.283	0.288			0.234	0.209	0.172	0.150	0.149
GO CONS GCF 2 TIM R6	mean	0.469	0.315	0.055	−0.059	−0.049			−0.646	−0.293	−0.144	−0.135	−0.060
	stdev	0.059	0.067	0.044	0.051	0.052			0.061	0.052	0.039	0.031	0.031
	range	0.288	0.323	0.244	0.286	0.294			0.234	0.209	0.173	0.150	0.149
GO CONS GCF 2 TIM R6e	mean	0.469	0.316	0.055	−0.059	−0.040			−0.646	−0.293	−0.144	−0.134	−0.068
	stdev	0.059	0.068	0.044	0.051	0.052			0.061	0.052	0.039	0.031	0.031
	range	0.287	0.323	0.244	0.287	0.292			0.234	0.209	0.172	0.150	0.148
Tongji-GMMG2021S	mean	0.463	0.311	0.052	−0.063	−0.073			−0.641	−0.289	−0.156	−0.190	−0.143
	stdev	0.059	0.068	0.045	0.049	0.051			0.061	0.053	0.043	0.034	0.033
	range	0.288	0.325	0.251	0.280	0.289			0.234	0.210	0.184	0.157	0.156
IGGT R1C	mean	0.520	0.282	0.002					−0.748	−0.311	−0.393
	stdev	0.063	0.065	0.051					0.061	0.053	0.060
	range	0.302	0.316	0.284					0.236	0.211	0.231
ITSG GRACE2018S	mean	0.465	0.399						−0.715	−0.370
	stdev	0.059	0.069						0.064	0.058
	range	0.288	0.325						0.244	0.228
Tongji GRACE02K	mean	0.521							−0.664
	stdev	0.057							0.062
	range	0.282							0.240

shows the corresponding differences with the cRTM approach fields (with filtering). Statistical results were also computed for the differences with the cRTM approach without filtering. These results led to higher mean values of differences for the lower degrees of expansion and, therefore, are not further examined. From the statistical results of Table 6, we may notice that most solutions have a large mean value. When compared to the results of Table 7, the standard deviation is either the same or better with the sRTM approach. For a better overview of the results, the standard deviation of the geoid height differences given in Table 7 is also depicted in Figure 5. Figure 6 shows the geoid solutions based on XGM2019 (d/o 760) with cRTM.

Due to the limited size of the test areas, an additional experiment was carried out in the northern area. By extending the area by 0.25° towards all directions (see Figure 4), additional gravity values were incorporated into a new geoid solution using XGM2019 up to d/o 760 along with the cRTM. As described in Section 2.1, these values were obtained many decades ago while every effort was made to make them compatible with the recently acquired ones; moreover, the distribution of the values used in the extended area is not as even as in the initial study area.

Table 8. Statistics of the residual and reduced gravity field for the extended northern study area using XGM2019 (d/o 760) and cRTM [mGal] and statistics of the comparison with the corresponding geoid heights with the ones from GNSS/leveling [m].

Gravity Field	Mean	Stdev	Range
Original	26.20	36.68	147.32
Reduced	–16.33	30.40	132.84
Residual	–7.46	23.45	94.13
Geoid
Comparison	0.064	0.057	0.311

provides the reduced and residual gravity fields as well as the results of the comparison of the new geoid model with the GNSS/leveling-derived geoid heights. The residual field has a significantly higher standard deviation than that of the original study area (7.13 mGal). Furthermore, the standard deviation of the compared geoid heights is worse than the one computed for the original area (0.045 m), while the corresponding range of the differences increased by 6 cm.

3.3. Reference Geopotential Value

As far as the computation of a reference geopotential value is concerned, it has been recommended by [2] to perform all computations in the zero-tide concept. Although the reference value should theoretically be independent of the tidal concept, it has been demonstrated that both the geopotential models used in the computations as well as the adopted tidal concept affect the final result [2]. In the present study, as all the computations were made following the zero-tide concept, $W_o$ was estimated for the northern and central area according to Equations (7) and (8), where the gravimetric geoid heights were obtained from the solution using XGM2019 up to d/o 760 and cRTM, whose results were presented in the previous section.

Table 9. Reference geopotential values for northern and central area, the IHRF conventional value and estimates from previous studies and their comparison [m²·s^–2].

Study	${\mathit{W}}_{\mathit{o}}$	Difference	Value
Northern area	62636860.25	Northern-Central	−0.73
Central area	62636860.98	Northern-Conventional	6.85
IHRF Conventional [2]	62636853.40	Central-Conventional	7.58
Mainland Greece [3]	62636859.66/62636859.81/ 62636859.86	Northern-average of Mainland values [3]	0.47
Mainland Greece [4]	62636860.27	Central-average of Mainland values [3]	1.20
		Northern-Mainland [4]	−0.02
		Central-Mainland [4]	0.71

presents the $W_o$ estimates of the present study, the adopted by the International Association of Geodesy’s conventional value for the IHRF, three estimates for the Greek mainland by [3], which were averaged to perform comparisons, and one estimate by [4], also for the Greek mainland. Additionally, Table 9 lists the differences between the conventional value and previous estimates with the values of the present study. It is easier to assess the presented differences if we consider that a difference of 1 m²·s^–2 corresponds to approximately 10 cm in height level difference. From the results, it is seen that northern and central areas have a height level difference of about −7 cm. With the IHRF conventional value, the differences rise up to 70 cm, while with the previous study [3] the differences are between 5 and 12 cm. It is to be noticed that the value of [4] is consistent with the northern area, although it was computed for all the mainland and not for a specific area, as in this study.

4. Discussion and Conclusions

Certain remarks may be made regarding the results presented in the previous section. First, we notice that the mean value of the differences has an opposite sign in the lower degrees for the northern area versus the central one. As we examine higher degrees, this difference diminishes, while for degree 2190 and the cRTM case, we observe that this increases again. As the central area presents larger mean values than the northern area, we further investigated the cause of these discrepancies. Apart from the large differences, the smaller ones (less than 10 cm) may be attributed to errors in the reference orthometric heights. While the height differences from spirit leveling in the present study were obtained with high accuracy, it was found that the leveling that connects the benchmarks of the central area to the national reference point had a loop closure error of about 6 cm [55]. Regarding the large differences, these may be attributed to the correspondence of the d/o of the sRTM models used for the reductions with the d/o of the GGMs.

The GOCO06S, GO CONS GCF 2 DIR R6, GO CONS GCF 2 TIM R6/R6e GGMs present a very similar behavior when examining the standard deviation of the differences (between 3 and 5 cm). The variation in the differences between these models is more or less the same (up to 2 mm), while for the central area its values are the lowest compared to the other solutions (3.1 to 3.3 cm). Regarding their mean value, the cRTM with the smoothed reference DEBM provides better results in terms of standard deviation, especially for d/o 300 (between 4 and 8 cm).

For models with d/o 760 and 2190 and sRTM, we notice that, although the XGM2019 group of models has a lower standard deviation in the differences in most cases, their mean values are larger, especially for the central area. This does not apply when considering cRTM, where for the central area the improvement in the mean value may reach 15 cm.

Out of all the solutions, we consider XGM2019 (d/o 760) with cRTM to provide the best statistical results for both test areas concurrently. The standard deviation of the differences is 4.5 and 3.4 cm for the northern and central areas, respectively. Compared to the results of Table 5, where the geoid heights from the GGMs with d/o 2190 and 5140 are given, we notice that for the northern area, there is an improvement from 0.6 up to 3.2 cm. For the central area, the improvement is much less, varying from 0.2 to 0.9 mm in terms of standard deviation. There are many possible reasons for these differences, especially for the highest d/o models that would be expected to provide the best results. The first possible cause is that the gravity data used by the higher d/o models have errors that affect the higher frequencies of the gravity field spectrum. This is highly possible for Greece, as the most commonly used gravity data were measured at least 25 years ago. Therefore, although XGM2019 uses these gravity data, it avoids higher frequencies, as it is expanded, up to d/o 760. On the other hand, this could be attributed to the RTM effects and the way the reference model resolution is selected. More specifically, it could be possible that the higher frequencies are affected much more by inconsistencies in the selection of the reference DEBM used for the computation of the RTM effects.

Another interesting remark refers to the comparison of older models (EGM2008 and EIGEN6C4) to the newer ones, based on the reprocessed GOCE and GRACE data, like the XGM2019 group of models and SGG-UGM-2. The improvement of the satellite data used for the lower frequencies of the gravity field spectrum in the two test areas does not have a significant impact in the geoid height differences examined, especially in the higher degrees. Moreover, no improvement in the results could be detected in the present study for SGG-UGM-2, whose results are comparable, but not significantly better, to those of the other models.

Regarding the combination of newer gravity data with data obtained more than two decades ago, the results from geoid modeling showed that this resulted in a degradation of accuracy. In our case, in the northern area, the results were worse by 1.2 cm in terms of standard deviation. We believe that the gravity values measured decades ago are not the ones responsible for this but rather the position and height that accompany these values. If we consider that the position was mainly estimated from maps using different projections/reference systems, as well as that the height was derived from maps or barometric altimeters, these may lead to erroneous computations of corrections and reductions.

Lastly, the computation of a reference geopotential value for the two study areas revealed a discrepancy of about 7 cm in height level between the two study areas. This difference depicts that there may be inherent inconsistencies in the national leveling network. But, as it has been shown in many studies (see, e.g., [2,3]), the computation of $W_o$ depends on the geopotential model used for or in the computations for the geoid heights of the study area; therefore, if in our study, we used a different model out of the many presented, the results would be different. The same conclusions are drawn from the comparison of the results of the present study with previous ones (see Table 9). Consequently, at present, it is up to the geoid modeler to select the proper geopotential model for the computations by taking into account appropriate comparisons and assessments, as was demonstrated in the present study. Another remark goes to the differences with the IHRF conventional value. The differences correspond to about 70 cm in height level. As these differences may be considered significant, they should be further investigated, not only for Greece but globally in the context of what is the physical meaning of the adoption of a conventional value in practical issues, like, for example, when defining the land to sea boundary in terms of height.

In conclusion, from our extended numerical investigation and coming back to the questions raised at the end of the Introduction, for local geoid modeling, the use of a combined model leads to better results in terms of mean value. The highest d/o models present larger values in both standard deviation and mean values. The local geoid models provide better statistical results when compared to GNSS/leveling-derived geoid heights than geoid heights obtained from the GGMs in terms of mean value, while the improvement in the standard deviation is of the order of some mm. The combination of XGM2019 up to d/o 760 with cRTM, along with a smoothed reference model, led to the best solution for both test areas of the present study. This result could be generalized to areas with similar topography and geographic features. On the other hand, the combination of recent data with data acquired in the past decades may degrade the results obtained. A challenging future issue can be a gravimetric geoid solution for the entire Hellenic area, taking advantage of the conclusions of the present research. Finally, the computation of a reference geopotential value depends on the geopotential model used in geoid modeling, and it significantly affects the results. Further studies should be carried out to minimizes this dependence.