The Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity satellite mission was a part of ESA’s (European Space Agency) Living Planet program [1] launched in the period between March 2009 and October 2013. Since begin of 2010, consecutive highly accurate GOCE-based (GOCE-only and combined) static Global Geopotential Models (GGMs) were developed using GOCE observations. The latest releases of GOCE-based GGMs have been made available to the user community in the mid of 2014, which incorporate the complete mission data set exhibiting a significant higher spatial resolution [2] . Up to now, GOCE-based GGMs represent the best estimate of the gravity field anomalies and mean quasi geoid heights observed from the space with a precision of about 1 mGal and 1 - 2 cm, respectively.

Three types of GOCE-based GGMs are developed through the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF) [3] [4] , processed mainly by three approaches: direct (DIR) [5] , time-wise (TIM) [6] and space-wise (SPW) [7] . For more description, the readers are referred to [4] . The maximum achi- eved spatial resolution of GOCE-based GGMs of types SPW, TIM and DIR ranges from 83.5 km, 71.5 km and 66.8 km, respectively based on the formula:

where is the maximum spherical harmonics (SH) degree; is the mean radius of the Earth at the reference ellipsoid (≈6378.137 km); D is the spatial resolution (i.e. half of wavelength) in km.

Moreover, other solutions of GOCE-based GGMs have been developed by different research groups such as the Gravity Observation Combination (GOCO) models [8] , the European Improved Gravity models of the Earth by New techniques (EIGEN) model series [9] [10] [11] , the Delft Gravity Model (DGM) solution [12] and the Bonn (Institute of Geodesy and Geoinformation, University of Bonn) solution (ITG-GOCE02) [13] .

To ensure the best accuracy of GOCE-based GGMs, especially for the local/regional geoid modeling issue, evaluation with the use of ground-based gravity data is quite important. Thus, our goal here is to examine which of GOCE-based GGMs approximate closely the gravity field over Egypt and in which spectral bands they do deliver improved information. We have to mention here that several studies concerning the validation as well as the accuracy assessment of GOCE-based GGMs have been performed in different regions of the world by different research teams, e.g. [14] [15] [16] world- wide; [17] [18] in Germany; [19] in Central Europe: [20] in Poland; [21] in Norway; [22] in Brazil; [23] [24] in Sudan and [25] in Saudi Arabia.

In the following, the gravity field functionals in terms of quasi geoid heights and free-air anomalies as determined from different of GOCE-based GGMs compared with the corresponding ones from the Earth Global Model 2008 (EGM2008, [26] ) at both global and regional scales over Egypt are examined. In addition, a comparison between the ground-based (terrestrial) free-air anomalies and the corresponding one obtained from GOCE-based anomalies is performed. For consistent validation in the later comparison, maximum d/o of GOCE-based GGMs will be completed using EGM2008 up to d/o 2190. In addition, the gravity signals induced beyond 2190 are compensated from local topography information which is considered as a remaining omission error.

The datasets used in this study consist of: 1) GOCE-based GGMs and EGM2008; 2) terrestrial free-air gravity anomalies and GPS/levelling data collected over the Egyptian region; and 3) High-resolution topographic data from the SRTM30_PLUS (Shuttle Radar Topography Mission) digital terrain model.

Quasi geoid heights computed from GOCE-based GGMs and the corresponding ones obtained from EGM2008 on both global and regional scales are now compared. On the global scale, degree variances are often used to quantify the power of signal and error in the gravity field estimates at various spectral wavelengths to quantify the powers of signal and error in the gravity field estimates at various spatial wavelengths as ( [35] , pp. 98)

with and are the differences between SH coefficients of GOCE-based

GGMs and the corresponding ones from the EGM2008 and are degree and order of spherical harmonics, respectively. They read as follows

Accordingly, the difference degree variances of geoid heights as shown in Figure 3 were computed here as:

With is the mean radius of the Earth at the reference ellipsoid (≈6378.137 km), the fully normalized spherical harmonics, and, are a preferred tool for many theoretical and practical applications in geodesy, especially for the representation of the Earth’s gravitational potential V [36]

where G is the geocentric gravitational constant; M is the mass of the Earth; are the spherical coordinates: geocentric radius, co-latitude and longitude of the computation point, respectively; is the fully normalized associated Legendre function; is the normal gravity at point; is the maximum SH degree of geopotential model applied. Therefore, most of the existing gravity field models

are formulated in terms of SH coefficients. In the geodesy representation, the related quantities such as gravity anomalies and quasi geoid heights computed with the following formulae ( [37] , p. 272):

As seen from Figure 3, some points can be addressed here: 1) the GOCE-based direct solutions (DIR) outperform the time-wise (TIM) and space-wise (SPW) ones, particularly at the long spectrum of the gravity wavelength, where the observations of the Gravity Recovery And Climate Experiment (GRACE) mission contribute the DIR solutions. 2) Higher errors of the TIM models are provided at lower harmonics. The reason may be back to that no a-priori gravity information, neither as reference model, were considered within the processing strategies of the TIM releases. However, the TIM releases provide outperformed accuracy than the DIR solutions in such cases (cf. releases 4 and 5 of TIM and DIR) at the short wavelength of the gravity spectrum due to the contribution of the satellite gravity gradiometer (SGG) only measurements. 3) The space-wise solution, SPW_R4 model, provides high variances at lower harmonics because EGM2008 was used to model the degree variances and to calibrate the error of the estimated gravitational potential along track. 4) The Eigen-6C4 solution has the least variances especially at higher spherical harmonic degrees because of the contribution of EGM2008 altimetry/terrestrial data into Eigen-6C series (Eigen-6C3stat and -6C4).

On a regional scale, the spectral consistency between GOCE-based GGMs and EGM- 2008 should be taken in to consideration toward a reliable comparison. This is due to the fact that GOCE-based GGMs differ in resolution, besides their processing strategies with respect to EGM2008. Accordingly, the solution of each model has been restricted from SH d/o 2 up to (10 d/o step) on grids from 1.8˚ × 1.8˚, 1.64˚ × 1.64˚・・・ 0.66˚ × 0.66˚, 0.64˚ × 0.64˚, respectively. We have focused in our study only on SH degrees from 100 to 280 for two main reasons; first to examine which of GOCE-based GGMs approximate closely the gravity field over Egypt especially at the medium to short wavelength of the gravity spectral bands, where the contribution of GOCE is significant. Second, to investigate at which spectral bands they deliver improved information.

The GOCE-based GGMs specified in section 3 have been chosen as a subject for the evaluation over the study area of Egypt using EGM2008. Since the investigated GGMs differ in their data processing methodology and spatial resolution as well (see section 1), the spectral consistency between them should be taken into account. In other words, the GOCE-based GGMs and EGM2008 are examined at the same SH spectral bands. Thus, each model has been restricted from SH d/o 2 up to (10 d/o step), on grids from 1.8˚ × 1.8˚, 1.64˚ × 1.64˚・・・ 0.62˚ × 0.62˚, 0.6˚ × 0.6˚, respectively.

Figure 4 and Figure 5 show the standard deviations (std.) of differences between the free-air gravity anomalies δ∆g (Equation 9) and quasi geoid heights (Equation 10) obtained from GOCE-based GGMs and the corresponding ones from EGM2008 at the same maximum d/o of their spherical harmonics:

The results, represented in Figure 4 and Figure 5, indicate that beside GOCE data the contribution of terrestrial/altimetry data to the Eigen-6C3 and Eigen-6C4 are clearly visible from d/o 170 onward. This is because the reason mentioned in section 3 that Eigen-6C series have been developed with the use of altimetry/terrestrial data obtained from EGM2008 and DUT12 GGM. From d/o 200 onward, the std. have increased rapidly for all the GOCE-based GGMs solutions, since the coefficients beyond d/o 180 - 200 were estimated with the use of Kaula’s rule [4] [5] [38] and because above d/o 200 noise starts to dominate signals [39] .

Except that, the contribution of GOCE at higher harmonics is clearly visible. For instance, comparing the DIR_R4 with DIR_R5 of Figure 4 at the same spectral domain (e.g. SH d/o 210), we find that the former model estimates free-air gravity anomalies as

well as quasi geoid heights better than the latter one. From d/o 200 onward, the std. for each of the relatively recent GOCE-based GGM releases outperform the former release of the same type. This can be obviously seen from Table 2, which shows the statistics of

the differences in terms of standard deviations between free-air gravity anomalies δ∆g and quasi geoid heights. The std. differences of quasi geoid heights and free-air gravity anomalies between GOCE-based GGMs and EGM2008 are at the level of about 29 - 32 cm and 6 - 7 mGal for d/o 200, respectively, of about 28 - 37 cm and 6 - 9 mGal for d/o 240 and of about 38 cm and 10.7 mGal for d/o 280.

To provide a reliable external validation and accuracy assessment of the investigated GOCE-based GGMs, the use of ground-truth data (terrestrial gravity and/or GPS/le- velling) is quite important. In the following, the validation strategies of both free-air gravity anomalies and quasi geoid heights obtained from GOCE-based GGMs using the corresponding ones obtained from terrestrial gravity and GPS/levelling data over Egypt region are discussed.

The results indicated in section 5.2 are represented in terms of std. of the differences between GPS/levelling data and GOCE-based GGMs. However, it should be noted that the mean values of these differences are at the level of 0.30 m. This can be interpreted as a bias which may be attributed to the offset between two equipotential surfaces formed by the gravimetric geoid model derived by GOCE and Egypt vertical datum (NED-95, see section 2.2.2).

In order to minimize this bias and to determine a better fit of geoid models determined from GOCE-based GGMs and the GPS/levelling data over Egypt, we have applied the parameter transformation models of corrector surface similar to [24] and [25] . Based on this, three types of parameter transformation models have been investigated; the four-, five- and seven-parameter transformation models. The four-parameter transformation model reads [36] :

While the five-parameter transformation model reads [42] :

And the seven-parameter transformation model reads e.g. [43] :

where are the transformation parameters between two datums; (φ_i, λ_i) are the latitude and longitude of the point; v_i is the residual of random noise term and e is the ellipsoid eccentricity.

Table 6 gives the statistics of the differences between quasi geoid heights [m] as obtained from GPS/levelling data and the corresponding of GOCE-based GGMs of types DIR_ R5, TIM_R5 and SPW_R4 before and after removing the biases effect applying Equations 17-19. The results presented in Table 6 indicate that applying the 4-, 5-, and 7-parameter transformation models removes the systematic error (mean values in Table 6). The assessed accuracy of quasi geoid heights differences in terms of std. after fitting gravimetric geoid models obtained from the GGMs investigated to the corresponding ones from GPS/levelling data is at the level of about 32 cm. Table 6 indicates that the difference among the three applied transformation models in terms of their resulting statistics provide consistent results. This may indicate that practically all transformation models implemented in this study are suitable for fitting the gravimetric geoid model to the geometrical one obtained from GPS/levelling data.

This paper investigated the gravity field solutions in terms of free-air anomalies and

quasi geoid heights as derived from the ground-based data and EGM2008 over the area of Egypt as a validation strategy of the GOCE-based GGMs. To validate the GOCE- based GGMs with ground-based data, two strategies have been applied: the spectral enhancement method and low-pass filter approach.

First, gravity functionals in term of free-air gravity anomalies and quasi geoid heights derived from GOCE-based GGMs were compared with the corresponding ones calculated from EGM2008 within the spectral range of the gravitational wavelength from d/o 100 to 300. Our findings indicate some deviations of the quasi geoid heights and the free-air gravity anomalies obtained from GOCE-based GGMs and the corresponding ones from EGM2008 within the level of 29 - 32 cm and 6 - 7 mGal, respectively, at SH d/o 200, at the level of 28 - 37 cm and 6 - 9 mGal at d/o 240 and at the level of 38 cm and 10.7 mGal at d/o 280.

Second, free-air gravity anomalies obtained from GOCE-based GGMs were compared with the corresponding ones from terrestrial gravity data using the spectral enhancement method and low pass filter approach. In the former method, the free-air gravity anomalies obtained from GOCE-based GGMs agree with the corresponding terrestrial ones within the range of 14.1 mGal at SH d/o 200 providing slight improvement than EGM2008. This may back to poor coverage of gravity data in the area of Egypt and neighboring countries when developing EGM2008. In case of quasi geoid heights, a considerable reduction of about 29% in terms of standard deviations of quasi geoid heights differences is observed after implementing the spectral enhancement method. This means that the SEM strategy helps much in improving the quality of GOCE-based GGMs.

Regarding the low pass filter strategy, GOCE-based GGMs exhibit improvement with respect to EGM08 by about 0.69 mGal in terms of the std. of their differences. The use of LPF approach results in about 14% reduction in std. of the differences for the investigated GOCE-based GGMs and the terrestrial gravity data.

Finally, we have applied three types of transformation models, namely four-, five- and seven-parameters transformation model, in order to deal with the problem of datum inconsistencies between different datums of both datasets (i.e. geoid models obtained from GOCE-based GGMs and from GPS/levelling data). Our findings showed that the tree applied models exhibit all most consistent results. They suggest that the standard deviations of vertical datum over the Egyptian territory are at level of about 32 cm.

To sum up, the GOCE-based SPW_R4 (up to SH d/o 200) model is suggested to be used as a reference model when modeling the local quasi-geoid heights from the terrestrial gravity data, since it approximates the gravity field well over Egypt. This is a subject of our future consideration.

The spatial representations of the results have been plotted using the GMT5 (Generic Mapping Tools) software [44] . The computations of the topography potential effect in the geoid were carried out using the TC software and RTM procedure [40] .

El-Ashquer, M., Elsaka, B. and El-Fiky, G. (2016) On the Accuracy Assessment of the Latest Releases of GOCE Satellite-Based Geopotential Models with EGM2008 and Terrestrial GPS/Levelling and Gravity Data over Egypt. International Journal of Geosciences, 7, 1323-1344. http://dx.doi.org/10.4236/ijg.2016.711097