Estimation of Vertical Datum Parameters Using the GBVP Approach Based on the Combined Global Geopotential Models

remote sensing

Article Estimation of Vertical Datum Parameters Using the GBVP Approach Based on the Combined Global Geopotential Models

Panpan Zhang 1,2, Lifeng Bao 1,2,* , Dongmei Guo 1,2, Lin Wu 1,2 , Qianqian Li 1,2, Hui Liu 1,2, Zhixin Xue 1,2 and Zhicai Li 3

1 State Key Laboratory of Geodesy and Earth’s Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China; [email protected] (P.Z.); [email protected] (D.G.); [email protected] (L.W.); [email protected] (Q.L.); [email protected] (H.L.); [email protected] (Z.X.) 2 University of the Chinese Academy of Sciences, Beijing 100049, China 3 National Geomatics Center of China, Beijing 100830, China; [email protected] * Correspondence: [email protected]

Received: 11 November 2020; Accepted: 13 December 2020; Published: 17 December 2020

Abstract: Unification of the global vertical datum has been a key problem to be solved for geodesy over a long period, and the main challenge for a unified vertical datum system is to determine the vertical offset between the local vertical datum and the global vertical datum. For this purpose, the geodetic boundary value problem (GBVP) approach based on the remove-compute-restore (RCR) technique is used to determine the vertical datum parameters in this paper. In the RCR technique, a global geopotential model (GGM) is required to remove and restore the long wavelengths of the gravity field. The satellite missions of the GRACE (Gravity Recovery and Climate Experiment) and GOCE (Gravity field and steady-state Ocean Circulation Exploration) offer high accuracy medium–long gravity filed information, but GRACE/GOCE-based GGMs are restricted to medium–long wavelengths because the maximum degree of their spherical harmonic representation is limited, which is known as an omission error. To compensate for the omission error of GRACE/GOCE-based GGM, a weighting method is used to determine the combined GGM by combining the high-resolution EGM2008 model (Earth Gravitational Model 2008) and GRACE/GOCE-based GGM to effectively bridge the spectral gap between satellite and terrestrial data. An additional consideration for the high-frequency gravity signals is induced by the topography, and the residual terrain model (RTM) is used to recover the omission errors effect of the combined GGM. In addition, to facilitate practical implementation of the GBVP approach, the effects of the indirect bias term, the spectral accuracy of the GGM, and the systematic levelling errors and distortions in estimations of the vertical datum parameters are investigated in this study. Finally, as a result of the GBVP solution based on the combined DIR_R6/EGM2008 model, RTM, and residual gravity, the geopotential values of the North American Vertical Datum of 1988 (NAVD88), the Australian Height Datum (AHD), and the Hong Kong Principal Datum (HKPD) are estimated to be equal to 62636861.31 0.96, 62653852.60 0.95 and 62636860.55 0.29 m2s 2, ± ± ± − respectively. The vertical offsets of NAVD88, AHD, and HKPD with respect to the global geoid are estimated as 0.809 0.090, 0.082 0.093, and 0.731 0.030 m, respectively. − ± ± − ± Keywords: geodetic boundary value problem; weighting method; the combined GGM; residual terrain model; geopotential value; vertical offset

Remote Sens. 2020, 12, 4137; doi:10.3390/rs12244137 www.mdpi.com/journal/remotesensing Remote Sens. 2020, 12, 4137 2 of 23

1. Introduction The vertical datum system is a reference used for determining the physical heights of a country or region. A local vertical datum (LVD) is usually determined by the local mean sea level (MSL) observed by single or multiple tide gauge stations. However, the local mean sea level does not exactly coincide with the global geoid, which makes the vertical datums of many countries and regions contain offsets [1,2]. It is essential to realize a unified vertical datum, and there are many applications that require a unified vertical datum system, such as engineering construction, environmental monitoring and scientific research. Unification of the existing vertical datum systems around the world is one of the major tasks of the Global Geodetic Observing System (GGOS) of the International Association of Geodesy (IAG). The IAG released a resolution for the definition and realization of an International Height Reference System (IHRS) during the 2015 International Union of Geodesy and Geophysics (IUGG) General Assembly, where the global equipotential reference surface was fixed by the geopotential 2 2 value of W0 = 6,2636,853.4 m s− [3,4]. According to this resolution, the existing local vertical datum systems can be integrated into the IHRS, which will ensure the consistency of the global vertical datum systems. At present, three strategies have been extensively discussed for vertical datum unification: the geodetic levelling approach, the oceanographic approach, and the gravity field approach. The geodetic levelling approach involves directly measuring the potential differences between two or more LVDs via spirit levelling in combination with gravity measurements. This approach provides sub-millimeter relative accuracy over short distances, but the absolute accuracy (relative to the global vertical datum) is about 2 m [5]. If the two LVDs cannot be connected by spirit levelling, ± the approach is restricted. The oceanographic approach is used to determine the datum offsets between different LVDs through the mean dynamic topography (MDT), whose accuracy depends on the quality and spatial resolution of the ocean model [6,7]. However, the ocean model has low quality around tide gauge stations. The gravity field approach estimates the vertical datum offset by comparing the discrepancies between the biased geometric geoid height determined by GNSS/levelling (Global Navigation Satellite System/levelling) and the unbiased gravimetric geoid height determined by the gravity field data. This approach mainly includes two strategies: (1) GNSS/levelling combined with the global geopotential model (GGM) method [8–10] and (2) the geodetic boundary value problem (GBVP) method [11–15]. With the development of satellite-based GGMs, GNSS/levelling combined with the GGM method provides absolute accuracy from centimeters to decimeters [5,16], but the GGMs do not completely represent the signals of the Earth’s gravity field because their accuracy and resolution are limited. The GBVP approach is a rigorous method for LVD unification, which uses the gravity anomaly data of the local vertical datum to determine the gravimetric geoid by solving the geodetic boundary value problem. Compared with other approaches, the GBVP approach has obvious advantages. The key to determine the vertical datum offsets using the GBVP approach is to determine the gravimetric geoid based on the remove-compute-restore (RCR) technique [17,18]. In this technique, the GGM and digital terrain model (DTM) provide the long and short wavelength parts of the gravity field spectrum, and the residual gravity field spectrum is provided by the residual gravity. Although the RCR technique has many advantages, its results depend largely on the accuracy of the GGM. With the implementation of gravity satellite missions like GRACE (Gravity Recovery and Climate Experiment) and GOCE (Gravity field and steady-state Ocean Circulation Exploration), these satellite missions offer unprecedented high-accuracy information for the medium-frequency of the gravity field spectrum; moreover, the accuracy of the medium–long wavelength in the static gravity field has been improved by two to three orders of magnitude [19–21]. However, the maximum expansion degree of GRACE/GOCE-based GGMs is limited, and these models have certain omission errors due to the influence of residual gravity field signals. Sánchez [22] demonstrated that the global mean omission error is about 45 cm for GRACE/GOCE-based GGMs with a degree of 200. In addition ± to the omission errors of the GGMs, the factors that affect the estimation of vertical datum offsets Remote Sens. 2020, 12, 4137 3 of 23 also include the spectral accuracy of the GGM, the indirect bias term, and the systematic errors and distortions in the vertical control networks. Therefore, to determine high-accuracy vertical datum offsets, the aforementioned factors must be considered. The objective of this study is to determine the vertical datum parameters of the USA, Australia, and Hong Kong based on the GBVP approach. For practical implementation of the GBVP approach to unify the vertical datum, the omission errors of the GOCE/GRACE-based GGMs, the spectral accuracy of the GGM, the indirect bias term, and the systematic errors and distortions in the vertical networks are investigated in this paper. To reduce the geoid omission error effect of the GOCE/GRACE-based GGMs, we combine GOCE/GRACE-based GGMs and the EGM2008 model to determine the combined GGMs via the weighting method. The combined GGMs include the high-accuracy medium and long wavelength information of the GRACE/GOCE-based GGMs and retain the short wavelength Remote Sens. 2020, 12, x 5 of 23 information of EGM2008. However, the maximum degree and order of the combined GGMs is only 2190, which is equivalent to a spatial resolution of 5 5 . The combined GGMs, however, fail to theoretically, be a fixed constant. However, vertical0 ×offsets0 have certain differences due to the representinfluence a high-frequency of systematic errors geoid such signal as systematic with a spatial errors resolution and distortions greater in thanthe leveling50 50. network Considering and the × influencerandom of theerrors high-frequency in computation gravity of the ellipsoidal, field signal as abovewell as ageoidal degree heights. 2190, the To residualreduce the terrain random model techniqueand systematic [23–27] is errors, used to the recover vertical the datum high-frequency offset will be gravity estimated field by signal. applying Finally, a parametric the GBVP model approach based[32,33] on the. For combined each GNSS/leveling GGM, residual point terrainP, the observation model (RTM), equation and can residual be formulated gravity as: isused to estimate the vertical datumlhHNN parameters=− − of − thePP North − N America − N P=cosδ Vertical HaBBaLL +()() Datum − + of 1988 − (NAVD88), B the Australia P P P01020GGM RTM res P P P (10) Height Datum (AHD), and the Hong Kong Principal Datum (HKPD). δ () () where H is mean vertical offset, BLPP, is the geodetic coordinates, BL00, is the mean value 2. Methodsof the geodetic coordinates of all GNSS/levelling benchmarks in the local vertical datum zone, and a and a are the north–south tilt and east–west tilt, respectively. 2.1. Determination1 2 of the Vertical Datum Parameters Based on the Geodetic Boundary Value Problem (GBVP) Approach The vertical datum parameters can be expressed in the form of the vertical offset δ H , the δ LVD LVD Accordingpotential difference to the generalizedW0 , and geopotential Burns’ formula W0 [28of ],the for local a point verticalP, datum the geoid (see Figure height 1).N PThecan be expressedδ H can as: be estimated by means of least square adjustment (LSA) of the system in Equation (10). LVD According to the relationship between theT vertical∆W offsetδW and potential difference, the potential = P 0 + 0 δ LVD NP − (1) difference W0 can be expressed as: γ γ where ∆W0 is the difference between the geopotentialδδWHLVD = W0⋅γof the global geoid and the normal geopotential(11) LVD 0 U0 of the reference ellipsoid, δW0 is the potential difference between the geopotential W0 of the LVD globalwhere geoid γ and is the geopotentialmean value ofW 0the ofnormal thelocal gravity vertical of all datum the GNSS/levelling (see Figure1), benchmarks.γ is the normal gravity on the reference ellipsoid, and T is the disturbing potential.LVD In this study, the geopotential Consequently, we can further determineP the geopotential value W of the local vertical datum as: 2 2 0 W0 = 62636853.4 m s− of the global geoid released by the IHRS is used. The ellipsoid parameters LVD=−δ LVD used in this paper are based on the GRS80WWW ellipsoid000 [29]. (12)

Figure 1. Vertical datum parameters (left of red dotted line) and relations between the ellipsoidal Figure 1. Vertical datum parameters (left of red dotted line) and relations between the ellipsoidal heights, levelling height and geometric geoid height at benchmark P (right of the red dotted line). heights, levelling height and geometric geoid height at benchmark P (right of the red dotted line). NoteNote that thatP and P andP’ are P’ are the the same. same.

2.2. Determination of the Combined Global Geopotential Models (GGMs) by the Weighting Method

Remote Sens. 2020, 12, 4137 4 of 23

The uniﬁcation of the vertical datum requires the gravity anomalies ∆g based on the global datum, but we can only obtain the local vertical datum gravity anomalies ∆gl in the practice case. The relationship between the ∆g and the ∆gl can be expressed as [30]:

2 ∆g = ∆g + δWLVD (2) l R 0 where R is the mean Earth radius. The solution of the GBVP in terms of its disturbing potential is [14,31]

δGM R 2 T = + S(ψ) ∆g + δWLVD dσ (3) P R 4πx l R 0 Ω where δGM is the diﬀerence between the gravitational constant of the Earth and the gravitational constant of the reference ellipsoid, and S(ψ) is the stokes kernel function. By applying Equation (3) to (1) = ( ) = ( ) ( ) = LVD and using N0 δGM/Rγ ∆W0/γ , NStokes R/4πγ S ψ ∆gldσ, and δH δW0 /γ, − sΩ Equation (1) can be written as follows:

P 1 NP = N + δH + N + δH S(ψ)dσ (4) 0 Stokes 2πx · Ω where N0 is the zero-degree term of the geoid height, and δH is the vertical oﬀset between the LVD and the global vertical datum. P Using the RCR technique, NStokes in Equation (4) can be computed from the GGM and stokes integration with the residual gravity anomalies. To obtain high-frequency geoid information, the RTM is used to recover the short-wavelength geoid in this paper. Then, Equation (4) can be written as follows:

N = N + δH + NP + NP + R Sres(ψ)∆gresdσ+ P 0 GGM RTM 4πγ s Ωl 1 res (5) 2π δH S (ψ)dσ sΩ ·

P P where NGGM and NRTM are the GGM geoid height and the RTM geoid height, respectively, res ∆g = ∆gl ∆gGGM ∆gRTM is the residual gravity anomalies, and the fifth term on the right-hand − − P res side of Equation (5) is the residual geoid Nres. The modified Stokes kernel function S (ψ) can then be expressed as: XN res 2n + 1 S (ψ) = S(ψ) Pn(cos ψ) (6) − n 1 n=2 − where N is the maximum degree of the GGM, and Pn(cos ψ) is the Legendre polynomials. The sixth term in the right-hand side of Equation (5) is the indirect bias term. If there are k vertical datum zones, and the vertical offset δH is constant in each datum zone, the indirect bias term Nind can be expressed as:

ψ k k Z 2 1 1 X X N = δH Sres(ψ)dσ = δH Sres(ψ)dσ = δH Sres(ψ) sin(ψ)dψ (7) ind 2πx · 2π ix i Ω i=1 Ωi i=1 ψ1 where ψ1 is the spherical distance between the computation point and the vertical datum zone Ωi, ψ2 = ψ1 + ∆ψi, ∆ψi is the width of the datum zone, and δHi represent the average of all datum offsets integrated over azimuth for vertical datum zone i. The integration element in Equation (7) is in terms of polar coordinates and carried out the integration over azimuth. Then, they are left with integration over spherical distance. Gerlach and Rummel [30] and Amjadiparvar et al. [14] showed that the indirect bias term can be neglected when a GGM with a higher degree and order is used. The indirect bias term Remote Sens. 2020, 12, 4137 5 of 23 is also evaluated in Section 4.3 in this paper. The result shows that the effect of the indirect bias term is equal to 1 mm when higher degree GGMs (300 degree and order) are used, as the maximum degree of the GGM increases, the indirect bias effect becomes smaller. Therefore, the effect of the indirect bias term can be ignored. The geoid height NP is also computed from GNSS/levelling as:

NP = hP HP (8) − where hP is the ellipsoidal height, and HP is the levelling height (see Figure1). By combining Equations (5) and (8), the vertical oﬀset δHP for point P can be expressed as:

P P P δHP = hP HP N N N N (9) − − 0 − GGM − RTM − res The datum surface corresponding to the global or local vertical datum is a gravitational equipotential surface. Therefore, the vertical datum offset δHP determined by Equation (9) should, theoretically, be a fixed constant. However, vertical offsets have certain differences due to the influence of systematic errors such as systematic errors and distortions in the leveling network and random errors in computation of the ellipsoidal, as well as geoidal heights. To reduce the random and systematic errors, the vertical datum offset will be estimated by applying a parametric model [32,33]. For each GNSS/leveling point P, the observation equation can be formulated as:

P P P lP = hP HP N N N N = δH + a (BP B ) + a (LP L ) cos BP (10) − − 0 − GGM − RTM − res 1 − 0 2 − 0 where δH is mean vertical offset, (BP, LP) is the geodetic coordinates, (B0, L0) is the mean value of the geodetic coordinates of all GNSS/levelling benchmarks in the local vertical datum zone, and a1 and a2 are the north–south tilt and east–west tilt, respectively. The vertical datum parameters can be expressed in the form of the vertical offset δH, the potential LVD LVD difference δW0 , and geopotential W0 of the local vertical datum (see Figure1). The δH can be estimated by means of least square adjustment (LSA) of the system in Equation (10). According to the LVD relationship between the vertical offset and potential difference, the potential difference δW0 can be expressed as: δWLVD = δH γ (11) 0 · where γ is the mean value of the normal gravity of all the GNSS/levelling benchmarks. Consequently, LVD we can further determine the geopotential value W0 of the local vertical datum as:

WLVD = W δWLVD (12) 0 0 − 0 2.2. Determination of the Combined Global Geopotential Models (GGMs) by the Weighting Method

The NGGM in Equation (10) can be determined by the GRACE/GOCE-based GGMs. However, the omission errors of the GRACE/GOCE-based GGMs must be considered. EGM2008 makes full use of terrestrial, airborne and satellite altimetry gravity data [34], so this model can effectively represent the short wavelength gravity field signal. Consequently, the EGM2008 model can be used to extend GOCE/GRACE-based GGMs to compensate for omission errors. We combine GOCE/GRACE-based GGMs and the EGM2008 model to determine the combined GGMs via the weighting method [35–37]. The degree 0–N of the combined GGM (N is the maximum degree of the GOCE/GRACE-based GGMs) is determined by spectrum combination, and the degree from N+1 to 2190 is supplemented by the corresponding degree and order of the EGM2008 model. The combined GGMs can have the high-accuracy medium and long wavelength information of the GRACE/GOCE GGMs and retain the Remote Sens. 2020, 12, 4137 6 of 23 short wavelength information of EGM2008. The spherical harmonic coefficients combined using by the weighting method can be expressed as:   Combine C EGM08 C GOCE/GRACE  C = p C + 1 p C  nm nm nm − nm nm (13)  Combine EGM08 GOCE/GRACE  S = pS S + 1 pS S nm nm nm − nm nm

EGM08 EGM08 where Cnm and Snm are the EGM2008 spherical harmonic coefficients of degree n and order m, GOCE/GRACE GOCE/GRACE Cnm and Snm are the GOCE/GRACE-based GGMs’ spherical harmonic coefficients of C S degree n and order m, and pnm and pnm represents the spectral weight of the EGM2008 model, and they can be calculated as:  2 σGOCE/GRACE  C ( C )  pnm = 2 2  EGM08 + GOCE/GRACE  (σC ) (σC )  GOCE/GRACE 2 (14)  (σ )  S S  pnm = 2 2  EGM08 + GOCE/GRACE (σS ) (σS ) EGM08 EGM08 where σC and σS are the coefficient standard deviations of the EGM2008 model, GOCE/GRACE GOCE/GRACE and σC and σS are the coefficient standard deviations of GOCE/GRACE-based GGMs. According to the error propagation law, the coefficient standard deviations of the spectrum combination can be calculated as:  q  Combine C 2 EGM082 C 2 GOCE/GRACE2  σ = pnm σ + 1 pnm σ  C q C − C (15)  2 2 2 2  σCombine = pS σEGM08 + 1 pS σGOCE/GRACE S nm S − nm S 3. Data Sets This section will briefly introduce the data sources used. The main data required are: (1) global geopotential models; (2) GNSS/levelling data; (3) gravity data; and (4) high resolution topographic data.

3.1. Global Geopotential Models (GGMs) There are seven GGMs used in this study: DIR_R6 [38], TIM_R6 [39], GOSG01S [40], IfE_GOCE05s [41], IGGT_R1 [42], SPW_5 [43], and EGM2008 [34], as shown in Table1. Except for TIM_R6, which is based on the zero-tide system, the other models are based on a tide-free system. For the spherical harmonic coefficients, only C20 is affected by the tide system. Hence, the C20 coefficient of TIM_R6 is transformed from zero-tide to tide-free system according to Sánchez et al. [44]:

TF ZF 8 0.3 C20 = C20 + 3.1108 10− (16) · · √5

TF ZF where C20 is the spherical harmonic coeﬃcient under the tide-free system, and C20 is spherical harmonic coeﬃcient under the zero-tide system. Remote Sens. 2020, 12, x 7 of 23

Table 1. Global geopotential models (GGMs) used in this investigation.

Models D/O Data Tide System Released Date EGM2008 2190 S(Grace), G, A Tide-free 2008 DIR_R6 300 S(Goce, Grace, Lageos) Tide-free 2019 TIM_R6 300 S(Goce) Zero-tide 2019 GOSG01S 220 S(Goce) Tide-free 2018 IfE_GOCE05s 250 S(Goce) Tide-free 2017 Remote Sens. 2020, 12, 4137 7 of 23 IGGT_R1 240 S(Goce) Tide-free 2017 SPW_5 330 S(Goce) Tide-free 2017 S Table= satellite 1. Global data, G geopotential = ground data, models A = (GGMs)altimetry used data, in D/O this = investigation. degree/order.

3.2. GNSS/LevellingModels Data D/O Data Tide System Released Date EGM2008 2190 S(Grace), G, A Tide-free 2008 For mainlandDIR_R6 America, 300 a GNSS/levelling S(Goce, Grace, data Lageos) set consisting Tide-free of 23961 benchmarks 2019 was made available by TIM_R6the National Geodetic 300 Survey S(Goce) (NGS). These data Zero-tide include the ellipsoidal 2019 heights in NAD83 and GOSG01Slevelling heights 220 with respect to S(Goce) the North American Tide-free Vertical Datum 2018of 1988 (NAVD88). IfE_GOCE05s 250 S(Goce) Tide-free 2017 The AustralianIGGT_R1 State and Territory 240 geodetic S(Goce) agencies provide Tide-freea set of 7545 GNSS/levelling 2017 points, consisting of SPW_5GNSS-based ellipsoidal 330 heights S(Goce) in ITRF14 and levelling Tide-free heights with 2017 respect to the Australian Height DatumS = satellite (AHD). data, G We= ground used data, 122 A GN= altimetrySS/levelling data, D /benchmarksO = degree/order. evenly distributed in Hong Kong, the ellipsoidal heights are given in ITRF96, and the levelling heights determined by precise3.2. GNSS geometric/Levelling leveling Data or trigonometric leveling are given with respect to the HKPD. The spatial distributionFor mainland of the America,GNSS/levelling a GNSS benchmarks/levelling data for setMainland consisting America, of 23961 Mainland benchmarks Australia, was made and Hongavailable Kong by are the shown National in Figure Geodetic 2. Survey (NGS). These data include the ellipsoidal heights in NAD83To andensure levelling consistent heights calculations, with respect the to geometric the North coordinates American Vertical in mainland Datum America of 1988 (NAVD88).and Hong TheKong Australian are transformed State and into Territory ITRF2014. geodetic In this study, agencies the providetide-free asystem set of 7545will be GNSS consistently/levelling used points, for allconsisting quantities. of GNSS-basedThe levelling ellipsoidalheights are heightstransformed in ITRF14 from mean and levelling tide to the heights tide-free with system respect based to theon Australianthe formula Height below [45] Datum: (AHD). We used 122 GNSS/levelling benchmarks evenly distributed in Hong Kong, the ellipsoidal heightsHHTF=− are MF given+0.68 in() 0.099 ITRF96, 0.296 and sin the2 ϕ levelling heights determined(17) by precise geometric leveling or trigonometric leveling are given with respect to the HKPD. The spatial wheredistribution HTF is ofthe the tide-free GNSS /levellinglevelling height, benchmarks HMF is the for Mainlandmean tide America, levelling Mainlandheight, and Australia, ϕ is the and latitude Hong ofKong the areGNSS/levelling shown in Figure benchmarks.2.

Figure 2. Spatial distribution of GNSS/Levelling benchmarks in (a) mainland America, (b) mainland Australia and (c) Hong Kong.

To ensure consistent calculations, the geometric coordinates in mainland America and Hong Kong are transformed into ITRF2014. In this study, the tide-free system will be consistently used for all quantities. The levelling heights are transformed from mean tide to the tide-free system based on the formula below [45]: HTF = HMF + 0.68 0.099 0.296 sin2 ϕ (17) − Remote Sens. 2020, 12, x 8 of 23

Figure 2. Spatial distribution of GNSS/Levelling benchmarks in (a) mainland America, (b) mainland Australia and (c) Hong Kong.

3.3. Gravity Data In total, 1,633,499 gravity data in the USA were made available by the National Oceanic and Atmospheric Administration (NOAA). These data mainly include land gravity observations in Mainland America, Alaska, and Hawaii along with airborne gravity data near the coastline, as well as a small number of gravity observation data of Canada and Mexico. After removing gravity observation data from Alaska, Hawaii, and airborne gravity data, we selected 822,301 gravity observation data distributed across the land. For Australia, we used 1,835,358 gravity observation dataRemote from Sens. 2020the ,Geoscience12, 4137 Australia’s national gravity database; we also used 610 gravity points8 of in 23 Hong Kong. The spatial distribution of the gravity station points for the USA, Australia, and Hong Kong are shown in Figure 3a,c,e. where HTF is the tide-free levelling height, HMF is the mean tide levelling height, and ϕ is the latitude We calculated normal gravity via the closed form of the 1980 international gravity formula [29] of the GNSS/levelling benchmarks. and applied second-order free air and atmospheric corrections [28] to the point gravity values to obtain3.3. Gravity free-air Data gravity anomalies. The bicubic interpolation method was employed to interpolate the point gravity anomalies data into 11′′× gravity anomaly grid for the USA and Australia and 33′′× ′′ gravityIn total,anomaly 1,633,499 grid for gravity Hong dataKong. in Because the USA the were point made gravity available anomalies by the are National mainly Oceanic distributed and acrossAtmospheric the Administrationland, for (NOAA).offshore Thesegravity data mainlydata, includewe landchose gravity the observationslatest grav.img.28.1 in Mainland (ftp://topex.ucsd.edu/pub/global_grav_1min/)America, Alaska, and Hawaii along with airborne model gravity derived data from near multi-mission the coastline, satellite as well altimetry. as a small Fornumber USA and of gravity Australia, observation the land gravity data of grid Canada data we andreMexico. combined After with removingoffshore gravity gravity data observation to obtain unifieddata from resolution Alaska, (1 Hawaii, arc-minute) and airborne gravity gravityanomaly data, grid we data selected for land 822,301 and sea. gravity For observationHong Kong, data the bicubicdistributed interpolation across the method land. Forwas Australia,employed weto interpolate used 1,835,358 the gravity11′′× offshore observation gravity data data from into the a Geoscience33′′× ′′ grid, Australia’s then, the land national gravity gravity grid data database; were combined we also used with 610 the gravity33′′× ′′ points offshore in gravity Hong Kong. data toThe obtain spatial unified distribution resolution of the (3 gravityarc-second) station gravity points anomalies for the USA, grid Australia, data on land and Hongand sea. Kong Figure are shown3b,d,f showsin Figure the3 a,c,e.unified gravity anomalies grid in the USA, Australia, and Hong Kong, respectively.

Remote Sens. 2020, 12, x 9 of 23

Figure 3. Spatial distribution of land gravity observation stations in (a) the USA, (b) Australia, and (Figurec) Hong 3. Spatial Kong and distribution reconstructed of land free-air gravity gravity observation anomalies stations for in (d ()a) the the USA, USA, ((eb)) Australia,Australia, andand ( cf) Hong Kong.Kong and reconstructed free-air gravity anomalies for (d) the USA, (e) Australia, and (f) Hong Kong.

3.4. Topographic Data The topographic data were used to calculate the RTM effect. SRTM (Shuttle Radar Topographic Mission) data [46] with a spatial resolution of 7.5′′× 7.5 ′′ were used for USA and Australia over land, and SRTM with a spatial resolution of 33′′× ′′ was used for Hong Kong over land. In the sea area, the latest SRTM15_PLUS V2 topographic data [47] were used, with a spatial resolution of 15″×15″. RET2012 was used as a topographic reference model. The RET2012 reference terrain elevations are synthesized using [48]:

Nn ref =+()λλθ() HHCmHSmP nmcos nm sin nm cos (18) nm==00

where N = 2160 is the maximum degree, λ and θ are the longitude and the geocentric co-latitude ()θ of the computation points, respectively, and Pnm cos is the associated fully-normalized spherical Legendre functions of degree n and order m. In coastal areas, the influence of water depth on RTM cannot be ignored. Because the mass- density of sea water is inconsistent with the standard topographic mass density, to avoid the need to distinguish density changes in the calculation process, the rock-equivalent topography (RET) method was used to compress the water depth into the equivalent rock height [49,50], then, the SRTM and SRTM15_PLUS data were merged to determine the unified land and sea topographic data. This merging was accomplished in a two-step procedure. Firstly, the RET method was employed to process the sea water depths, and the bicubic interpolation method was employed to interpolate the data into a spatial resolution the same as that of the land topographic data; then, the land data were combined with the water depth data to obtain unified resolution topographic data on land and sea. Figure 4a,c,e shows the topography merged for the USA, Australia, and Hong Kong, respectively. The RTM is the difference between the topography merged and reference terrain. Figure 4(b, d, f) shows the RTM elevation for the USA, Australia, and Hong Kong, respectively.

Remote Sens. 2020, 12, 4137 9 of 23

We calculated normal gravity via the closed form of the 1980 international gravity formula [29] and applied second-order free air and atmospheric corrections [28] to the point gravity values to obtain free-air gravity anomalies. The bicubic interpolation method was employed to interpolate the point gravity anomalies data into 1 1 gravity anomaly grid for the USA and Australia and 300 300 gravity 0 × 0 × anomaly grid for Hong Kong. Because the point gravity anomalies are mainly distributed across the land, for offshore gravity data, we chose the latest grav.img.28.1(ftp://topex.ucsd.edu/pub/global_grav_1min/) model derived from multi-mission satellite altimetry. For USA and Australia, the land gravity grid data were combined with offshore gravity data to obtain unified resolution (1 arc-minute) gravity anomaly grid data for land and sea. For Hong Kong, the bicubic interpolation method was employed to interpolate the 1 1 offshore gravity data into a 300 300 grid, then, the land gravity grid data 0 × 0 × were combined with the 300 300 offshore gravity data to obtain unified resolution (3 arc-second) × gravity anomalies grid data on land and sea. Figure3b,d,f shows the unified gravity anomalies grid in the USA, Australia, and Hong Kong, respectively.

3.4. Topographic Data The topographic data were used to calculate the RTM eﬀect. SRTM (Shuttle Radar Topographic

Mission) data [46] with a spatial resolution of 7.500 7.500 were used for USA and Australia over land, × and SRTM with a spatial resolution of 300 300 was used for Hong Kong over land. In the sea area, × the latest SRTM15_PLUS V2 topographic data [47] were used, with a spatial resolution of 15” 15”. × RET2012 was used as a topographic reference model. The RET2012 reference terrain elevations are synthesized using [48]:

XN Xn re f H = (HCnm cos mλ + HSnm sin mλ)Pnm(cos θ) (18) n=0 m=0 where N = 2160 is the maximum degree, λ and θ are the longitude and the geocentric co-latitude of the computation points, respectively, and Pnm(cos θ) is the associated fully-normalized spherical Legendre functions of degree n and order m. In coastal areas, the influence of water depth on RTM cannot be ignored. Because the mass-density of sea water is inconsistent with the standard topographic mass density, to avoid the need to distinguish density changes in the calculation process, the rock-equivalent topography (RET) method was used to compress the water depth into the equivalent rock height [49,50], then, the SRTM and SRTM15_PLUS data were merged to determine the unified land and sea topographic data. This merging was accomplished in a two-step procedure. Firstly, the RET method was employed to process the sea water depths, and the bicubic interpolation method was employed to interpolate the data into a spatial resolution the same as that of the land topographic data; then, the land data were combined with the water depth data to obtain unified resolution topographic data on land and sea. Figure4a,c,e shows the topography merged for the USA, Australia, and Hong Kong, respectively. The RTM is the difference between the topography merged and reference terrain. Figure4b,d,f shows the RTM elevation for the USA, Australia, and Hong Kong, respectively. Remote Sens. 2020, 12, 4137 10 of 23 Remote Sens. 2020, 12, x 10 of 23

Figure 4. TheThe topography based based on on merged merged SRTM SRTM in in ( aa)) the the USA, ( c) Australia, and and ( e)) Hong Kong and residual terrain mode modell (RTM) elevations for (b) the USA, ((d)) Australia,Australia, andand ((ff)) HongHong Kong.Kong.

4. Results 4. Results 4.1. Spectral Accuracy Evaluation of the GGMs 4.1. Spectral Accuracy Evaluation of the GGMs In this study, the spectral accuracy of each GGM is evaluated according to its degree error and In this study, the spectral accuracy of each GGM is evaluated according to its degree error and signal-to-noise ratio (SNR). The degree error is computed as the square root of the error degree signal-to-noise ratio (SNR). The degree error is computed as the square root of the error degree variances, and the SNR represents the ratio of the geoid signal to the degree error. The calculation variances, and the SNR represents the ratio of the geoid signal to the degree error. The calculation formulas for degree error and SNR can be found in Ustun et al. [51]. formulas for degree error and SNR can be found in Ustun et al. [51]. Figure5a shows the degree errors of the applied GGMs in terms of geoid height. It can be seen Figure 5a shows the degree errors of the applied GGMs in terms of geoid height. It can be seen from the ﬁgure that the degree error of the EGM2008 model gradually becomes lower than that of from the figure that the degree error of the EGM2008 model gradually becomes lower than that of the GRACE/GOCE-based GGMs after about degree 220, indicating that the spectral accuracy of the the GRACE/GOCE-based GGMs after about degree 220, indicating that the spectral accuracy of the EGM2008 model is better than that of the GRACE/GOCE-based GGMs with a higher degree and order. EGM2008 model is better than that of the GRACE/GOCE-based GGMs with a higher degree and The degree errors of the DIR_R6, TIM_R6, GOSG01S, IfE_GOCE05s, and IGGT_R1 models before order. The degree errors of the DIR_R6, TIM_R6, GOSG01S, IfE_GOCE05s, and IGGT_R1 models about degrees 220, 203, 202, 209, 175, respectively, are lower than the degree errors of EGM2008 model. before about degrees 220, 203, 202, 209, 175, respectively, are lower than the degree errors of EGM2008 From about degrees 65 to 197, the degree error of the SPW_5 model is smaller than that of the EGM2008. model. From about degrees 65 to 197, the degree error of the SPW_5 model is smaller than that of the

Remote Sens. 2020, 12, x 11 of 23

EGM2008. The results indicate that the DIR_R6, TIM_R6, GOSG01S, IfE_GOCE05s, IGGT_R1, and SPW_5 models have better spectral accuracy in medium and long wavelengths. The geoid signal power of GGMs at different wavelengths could be expressed by SNR. Figure 5b shows the SNR of the GGMs. The SNRs of the DIR_R6, TIM_R6, GOSG01S, IfE_GOCE05s, and IGGT_R1 models are better than those of the EGM2008 model before about degrees 220, 204, 203, 209 and 175, respectively. The SNR of the SPW_5 model is better than that of the EGM2008 model from Remotedegrees Sens. 652020 to 198., 12, 4137The results show that the GRACE/GOCE-based GGMs have a stronger geoid 11signal of 23 in the medium and long wavelength, and the EGM2008 model has a stronger geoid signal in the short wavelength. From the perspective of the SNR and degree error, the spectral accuracy of the DIR_R6 The results indicate that the DIR_R6, TIM_R6, GOSG01S, IfE_GOCE05s, IGGT_R1, and SPW_5 models model is significantly better than that of the other GRACE/GOCE GGMs, because the DIR_R6 model have better spectral accuracy in medium and long wavelengths. takes advantage of GRACE gravity data and LAGEOS Satellite Laser Ranging (SLR) data.

Figure 5. (a) Geoid degree errors of the GGMs in terms of geoid height based on their formal errors and (b) the signal-to-noise ratios (SNRs)(SNRs) ofof thethe GGMs.GGMs.

4.2. OmissionThe geoid Errors signal of the power GGMs of GGMs at different wavelengths could be expressed by SNR. Figure5b shows the SNR of the GGMs. The SNRs of the DIR_R6, TIM_R6, GOSG01S, IfE_GOCE05s, and IGGT_R1The GRACE/GOCE-based models are better thanGGMs those have of high the EGM2008 medium modeland long before wavelength about degrees accuracy, 220, 204,but 203,the 209maximum and 175, degree respectively. of these The models SNR of is the limited, SPW_5 as model there is are better certain than omission that of the errors EGM2008 affected model by from the degreesroughness 65 toof 198.the Theremaining results show(residual) that thegravity GRACE fiel/dGOCE-based signals. To GGMsquantify have the a strongermagnitude geoid of signalthese inomission the medium errors, and we long assume wavelength, that degree and 300 the is EGM2008 the general model expansion has a stronger degree geoid of the signal GRACE/GOCE- in the short wavelength.based GGMs Fromand that the the perspective geoid represented of the SNR from and degree 301 error, to the2190 spectral in the EGM2008 accuracy ofmodel the DIR_R6 can be modelviewed is as significantly the omission better errors than of the that GRACE/GOCE-based of the other GRACE/ GOCEGGMs. GGMs, because the DIR_R6 model takesFigure advantage 6 shows of GRACE the spatial gravity distribution data and of LAGEOS the omission Satellite errors Laser for Ranging the USA, (SLR) Australia, data. and Hong Kong, the corresponding statistics are presented Table 2. It can be seen from the Table that the effect 4.2.of the Omission omission Errors errors of thereaches GGMs the dm level in the USA and Australia, while the impact of omission errors in Hong Kong is at the cm level. The omission errors of the largest amplitude with –159 cm The GRACE/GOCE-based GGMs have high medium and long wavelength accuracy, but the and 88 cm are reached in USA and Australia, respectively, and the mean omission errors in the USA maximum degree of these models is limited, as there are certain omission errors affected by the roughness and Australia reach –2.2 and –2.4 cm, respectively. In Hong Kong, the mean omission error is about of the remaining (residual) gravity field signals. To quantify the magnitude of these omission errors, 1 cm. In Figure 5, we can also see that the omission errors in the rugged regions (for example, the we assume that degree 300 is the general expansion degree of the GRACE/GOCE-based GGMs and that western region of the USA) are larger. Therefore, the omission errors of the GRACE/GOCE-based the geoid represented from degree 301 to 2190 in the EGM2008 model can be viewed as the omission GGMs must be considered for the purpose of the vertical datum offset estimation. errors of the GRACE/GOCE-based GGMs. Figure6 shows the spatial distribution of the omission errors for the USA, Australia, and Hong Kong, the corresponding statistics are presented Table2. It can be seen from the Table that the e ffect of the omission errors reaches the dm level in the USA and Australia, while the impact of omission errors in Hong Kong is at the cm level. The omission errors of the largest amplitude with 159 cm and − 88 cm are reached in USA and Australia, respectively, and the mean omission errors in the USA and Australia reach 2.2 and 2.4 cm, respectively. In Hong Kong, the mean omission error is about 1 cm. − − In Figure5, we can also see that the omission errors in the rugged regions (for example, the western region of the USA) are larger. Therefore, the omission errors of the GRACE/GOCE-based GGMs must be considered for the purpose of the vertical datum offset estimation.

Remote Sens. 2020, 12, 4137 12 of 23 Remote Sens. 2020, 12, x 12 of 23

Figure 6.6. OmissionOmission errors errors obtained obtained from from the geoidthe geoi representedd represented from degreefrom degree 301 to 2190 301 ofto the 2190 EGM2008 of the modelEGM2008 for (modela) the USA,for (a) ( bthe) Australia, USA, (b) andAustralia, (c) Hong and Kong. (c) Hong Kong. Table 2. Statistics of the omission errors obtained for the three study areas of the USA, Australia, Table 2. Statistics of the omission errors obtained for the three study areas of the USA, Australia, and and Hong Kong (Unit: m). Hong Kong (Unit: m). Study Areas Max Min Mean Std Study areas Max Min Mean Std USA 1.408 1.592 0.022 0.213 USA 1.408 −1.592− −0.022− 0.213 Australia 0.876 0.808 0.024 0.190 − − Hong KongAustralia 0.1280.876 −0.8080.152 −0.024 0.009 0.190 0.067 Hong Kong 0.128 −0.152− 0.009 0.067 4.3. Effect of the Indirect Bias Term 4.3. Effect of the Indirect Bias Term The effect of the indirect bias term is assessed by means of Equation (7). The indirect bias term The effect of the indirect bias term is assessed by means of Equation (7). The indirect bias term is not only closely related to the vertical offset δH and integral function, but it corresponds to δH is not only closely related to the vertical offset δ H and integral function, but it corresponds to δ H being integrated over the respective datum zone (weighted average using stokes function as distance being integrated over the respective datum zone (weighted average using stokes function as distance depending weights). The local mean sea level does not coincide with the global geoid, and the resulting depending weights). The local mean sea level does not coincide with the global geoid, and the discrepancies between the different local vertical datum zones cause global height datum offsets of resulting discrepancies between the different local vertical datum zones cause global height datum about 1~2 m. To evaluate the indirect bias term, we assume δH = 1 m. The datum zones are divided offsets of about 1~2 m. To evaluate the indirect bias term, we assume δH =1 m. The datum zones into near zone and far zone. The datum zone where the calculation point is located is called the near are divided into near zone and far zone. The datum zone where the calculation point is located is zone—that is, ψ1 = 0; the datum zone outside calculation point is called the far zone—that is, ψ1 , 0. called the near zone—that is, ψ =0 ; the datum zone outside calculation point is called the far zone— Then, the indirect bias term of the1 calculation point is divided into two aspects: one is the indirect bias ψ ≠ ethatffect is, of the1 near0 . Then, zone, the and indirect the other bias is theterm indirect of the calculation bias effect of point thefar is divided zone. into two aspects: one is theFigure indirect7a showsbias effect the indirectof the near bias zone, e ffect and as a th functione other is of the the indirect size ∆ψ ofbias the effect datum of zonethe far for zone. different maximumFigure degrees 7a showsN. Thisthe indirect figure reflects bias effect the indirect as a function bias term of etheffect size of the Δ nearψ of zone the to datum the calculation zone for point.different The maximum indirect biasdegrees effect N. reachesThis figure a maximum reflects the of indirect about 2 bias cm term for N effect= 50. of As the the near degree zone ofto the GGMcalculation increases, point. the The indirect indirect bias bias eff effectect gradually reaches a becomes maximum smaller, of about especially 2 cm for for N N= 50.= 300, As the where degree the maximumof the GGM indirect increases, bias the eff indirectect is only bias 0.1 effe cm.ct gradually becomes smaller, especially for N = 300, where the maximum indirect bias effect is only 0.1 cm. ψ Figure 7b shows the indirect bias effect as a function of the distance 1 between the far zone and the computation point for different maximum degrees N. This figure reflects the indirect bias term effect of the far zone to the calculation point. To simplify evaluation of the indirect bias effect,

Remote Sens. 2020, 12, x 13 of 23 we assume that the width of the near datum zone is fixed at Δψ =60 . The indirect bias effect reaches a maximum of about 2.5 cm for N = 50, and the maximum indirect bias effect is only 0.8 mm for N = 300. It can be seen from Figure 7b that the indirect bias term effect of the far zone to the calculation ψ ψ point gradually becomes smaller with an increase of 1 . Therefore, the corresponding 1 when the indirect bias term effect is the largest can be selected to evaluate the total indirect bias term effect of the far zone and the near zone to the calculation point. The total indirect bias term effect for different maximum degrees N is shown in Figure 7c, where the maximum total indirect bias effect is

Remoteonly 0.1 Sens. cm2020 for, 12 N, 4137 = 300. Therefore, the indirect bias term effect can be ignored by using GGMs13 of 23a high degree and order.

Figure 7. Indirect biasbias termterm eeffectsﬀects under under di diﬀfferenterent truncation truncation degrees degrees N N of of (a) ( thea) the near near zone, zone, (b) ( theb) the far zone,far zone, and and (c) a(c combination) a combination of the of the near near zone zone and and the the far zone.far zone.

Figure7b shows the indirect bias e ffect as a function of the distance between the far zone 4.4. Determination the Combined GGM ψ1 and the computation point for different maximum degrees N. This figure reflects the indirect bias termThe effect omission of the far errors zone of to the the GRACE/GOCE-based calculation point. To GGMs simplify must evaluation be considered of the indirectfor the purpose bias effect, of wevertical assume datum that theoffset width estimation. of the near The datum DIR_R6 zone ismodel fixed atremains∆ψ= 60 better◦. The spectral indirect biasaccuracy effect reachesthan the a maximumTIM_R6, GOSG01S, of about 2.5 IfE_GOCE05s, cm for N = 50, IGGT_R1, and the maximum and SPW_R5 indirect models bias e(seeffect Section is only 0.84.1). mm Therefore, for N = 300.we built Ita combined can be seen model from DIR_R6/EGM2008 Figure7b that the indirectby comb biasining term the DIR_R6 e ffect of model the far and zone the to EGM2008 the calculation model pointusing graduallythe weighting becomes method smaller (Section with 2.2). an increase of ψ1. Therefore, the corresponding ψ1 when the indirectFigure bias 8 term shows effect the is thegeoid largest height can signals be selected and togeoid evaluate errors the of total the indirect DIR_R6, bias EGM2008, term effect and of theDIR_R6/EGM2008 far zone and the models. near zone The to geoid the calculation height signal point. of The DIR_R6/EGM2008 total indirect bias is termso close effect to for that di ffoferent the maximumEGM2008 model degrees thatN is the shown two curves in Figure are7 c,identical where the after maximum degree 300. total The indirect geoid bias height e ffect signal is only provided 0.1 cm forby theN = DIR_R6300. Therefore, model shows the indirect no difference bias term with eff DIectR_R6/EGM2008 can be ignored before by using about GGMs degree of a220, high and degree after anddegree order. 220, it is lower than that of the DIR_R6/EGM2008 model. The geoid degree error of the

4.4. Determination the Combined GGM The omission errors of the GRACE/GOCE-based GGMs must be considered for the purpose of vertical datum offset estimation. The DIR_R6 model remains better spectral accuracy than the TIM_R6, GOSG01S, IfE_GOCE05s, IGGT_R1, and SPW_R5 models (see Section 4.1). Therefore, we built a combined model DIR_R6/EGM2008 by combining the DIR_R6 model and the EGM2008 model using the weighting method (Section 2.2). Figure8 shows the geoid height signals and geoid errors of the DIR_R6, EGM2008, and DIR_R6/EGM2008 models. The geoid height signal of DIR_R6/EGM2008 is so close to that Remote Sens. 2020, 12, x 14 of 23 Remote Sens. 2020, 12, 4137 14 of 23 DIR_R6/EGM2008 is approximately the same as that of the DIR_R6 model before degree 200, after which, the geoid degree error of DIR_R6/EGM2008 is significantly lower than that of the DIR_R6 of the EGM2008 model that the two curves are identical after degree 300. The geoid height signal model. The degree error of the DIR_R6/EGM2008 model is lower than that of the EGM2008 model provided by the DIR_R6 model shows no difference with DIR_R6/EGM2008 before about degree 220, before about degree 250. The cumulative degree error of the DIR_R6/EGM2008 model is consistent and after degree 220, it is lower than that of the DIR_R6/EGM2008 model. The geoid degree error with the DIR_R6 model before degree 150, and from degree 150 onwards, it becomes significantly of the DIR_R6/EGM2008 is approximately the same as that of the DIR_R6 model before degree 200, lower than the cumulative degree error of the DIR_R6 model. Compared with the EGM2008 model, after which, the geoid degree error of DIR_R6/EGM2008 is significantly lower than that of the DIR_R6 the cumulative degree error of DIR_R6/EGM2008 is significantly lower than that of the EGM2008 model. The degree error of the DIR_R6/EGM2008 model is lower than that of the EGM2008 model model over the entire spectral domain. before about degree 250. The cumulative degree error of the DIR_R6/EGM2008 model is consistent Therefore, the best combination of the EGM2008 model and DIR_R6 model can be achieved by with the DIR_R6 model before degree 150, and from degree 150 onwards, it becomes significantly using the weighting method. The DIR_R6/EGM2008 model offers stronger geoid signal and higher lower than the cumulative degree error of the DIR_R6 model. Compared with the EGM2008 model, accuracy compared to the EGM2008 and DIR_R6 models. The DIR_R6/EGM2008 model provides the the cumulative degree error of DIR_R6/EGM2008 is significantly lower than that of the EGM2008 high-accuracy medium and long wavelength information of the DIR_R6 model and retains the short model over the entire spectral domain. wavelength information of EGM2008.

Figure 8.8. GeoidGeoid height height signals signals and and the the geoid geoid errors errors of the of EGM2008, the EGM2008, DIR_R6, DIR_R6, and DIR_R6 and DIR_R6/EGM2008/EGM2008 models. models. Therefore, the best combination of the EGM2008 model and DIR_R6 model can be achieved by using4.5. Residual the weighting Gravity Anomalies method. The DIR_R6/EGM2008 model oﬀers stronger geoid signal and higher accuracy compared to the EGM2008 and DIR_R6 models. The DIR_R6/EGM2008 model provides the Δres Δ −Δ −Δ high-accuracyThe residual medium gravity and anomalies long wavelength ggg= informationl GGM of gthe RTM DIR_R6 are used model to calculate and retains the theresidual short wavelength information of EGM2008.Δ geoid Nres in Equation (9). The gGGM is determined by the DIR_R6/EGM2008 model. However, Δg 4.5.calculating Residual the Gravity high-resolution Anomalies GGM point by point will reduce computational efficiency. To

resolve the problem of numerical computationalres efficiency, Colombo et al. [52] proposed the lumped The residual gravity anomalies ∆g = ∆g ∆g ∆gRTM are used to calculate the residual coefficient approach (LCA), but this approach cannotl − GGM be applied− to irregular grid surfaces. The LCA geoid N in Equation (9). The ∆g is determined by the DIR_R6/EGM2008 model. However, approachres regards the calculated gridGGM surface as a regular surface (a sphere or ellipsoid) and does not calculating the high-resolution ∆g point by point will reduce computational efficiency. To resolve consider the influence of the actualGGM terrain surface. Hirt [53] presented a gradient approach. the problem of numerical computational efficiency, Colombo et al. [52] proposed the lumped coefficient According to this approach, the gravity field functions and its derivatives of chosen order can be approach (LCA), but this approach cannot be applied to irregular grid surfaces. The LCA approach computed on the sphere or the ellipsoid employing the LCA, and subsequently continued to an regards the calculated grid surface as a regular surface (a sphere or ellipsoid) and does not consider irregular surface. Therefore, we use the gradient approach to determine the Δg in this study. the influence of the actual terrain surface. Hirt [53] presented a gradient approach.GGM According to this approach,The RTM gravity the gravity anomalies field functions are estimated and its via derivatives prism integration of chosen order[54] based can be on computed planar approximation. on the sphere orBecause the ellipsoid the spatial employing resolution the of LCA, the andreference subsequently surface is continued equivalent to to an that irregular of the surface.DIR_R6/EGM2008 Therefore, model, the RTM gravity anomalies can represent the high-frequency gravity field signal with a spatial we use the gradient approach to determine the ∆gGGM in this study. The RTM gravity anomalies are estimatedresolution viaof less prism than integration 5 arc-minutes. [54] based on planar approximation. Because the spatial resolution of res the referenceFigure 9 surfaceshows the is equivalent residual gravity to that ofanomalies the DIR_R6 Δg/EGM2008 and the model,corresponding the RTM frequency gravity anomalies statistics canhistograms. represent The the resolution high-frequency of residual gravity gravity field signal anomalies with a in spatial the USA resolution and Australia of less than is 11 5′′× arc-minutes., and the res resolutionFigure of9 showsresidual the gravity residual anomalies gravity anomaliesin Hong Kong∆g is and33′′× the ′′ . correspondingIt can be seen from frequency the histograms statistics histograms.that the residual The resolution gravity anomalies of residual in gravity the USA, anomalies Australia, in the and USA Hong and AustraliaKong maintain is 10 1 0a, andnormal the ×

Remote Sens. 2020, 12, 4137 15 of 23

resolution of residual gravity anomalies in Hong Kong is 300 300 . It can be seen from the histograms that × theRemote residual Sens. 2020 gravity, 12, x anomalies in the USA, Australia, and Hong Kong maintain a normal distribution.15 of 23 The statistics of the original gravity anomalies data, each reduction term, and the final gravity anomalies distribution. The statistics of the original gravity anomalies data, each reduction term, and the final are summarized in Table3. In Table3, we can see that there are biases of 3.24 mGal, 0.27 mGal, gravity anomalies are summarized in Table 3. In Table 3, we can see that− there are biases− of –3.24 and 1.41 mGal after removing the reference field and RTM effect in the USA, Australia, and Hong mGal,− –0.27 mGal, and –1.41 mGal after removing the reference field and RTM effect in the USA, Kong, respectively. Australia, and Hong Kong, respectively.

Figure 9. Residual gravity anomalies grid for (a) the USA, (c) Australia, and (e) Hong Kong and their frequencyfrequency distributiondistribution for (b) thethe USA, (d)) Australia,Australia, andand ((ff)) HongHong Kong.Kong. Note that thethe limitslimits ofof thethe colorcolor barbar dodo notnot correspondcorrespond toto thethe overalloverall minimumminimum andand maximummaximum values.values.

Table 3. Statistics of the gravity anomalies and each reduction term as well as the final residuals for the three study areas of the USA, Australia, and Hong Kong. Unit: mGal (1mGal = 10−5 m/s2).

Region Reduction Max Min Mean Std Δ gOBS 259.17 −181.61 −9.38 27.99 Δ−Δ USA ggOBS GGM 305.58 −239.10 −4.28 17.86 Δ−Δ−Δ ggOBS GGM g RTM 297.96 −238.10 −3.24 16.56

Remote Sens. 2020, 12, 4137 16 of 23

Table 3. Statistics of the gravity anomalies and each reduction term as well as the ﬁnal residuals for the 5 2 three study areas of the USA, Australia, and Hong Kong. Unit: mGal (1 mGal = 10− m/s ).

Region Reduction Max Min Mean Std ∆g 259.17 181.61 9.38 27.99 OBS − − USA ∆g ∆g 305.58 239.10 4.28 17.86 OBS − GGM − − ∆g ∆g ∆g 297.96 238.10 3.24 16.56 OBS − GGM − RTM − − ∆g 317.37 200.95 3.56 29.80 OBS − − Australia ∆g ∆g 298.95 348.50 0.66 10.40 OBS − GGM − − ∆g ∆g ∆g 296.63 309.36 0.27 9.70 OBS − GGM − RTM − − ∆g 66.06 41.8 13.71 13.40 OBS − − Hong Kong ∆g ∆g 70.51 37.24 2.38 11.20 OBS − GGM − − ∆g ∆g ∆g 46.34 66.86 1.41 10.37 OBS − GGM − RTM − −

4.6. Estimation of Vertical Datum Parameters

2 2 The geopotential W0 = 62,636,853.4 m s− of the global vertical datum released by the IHRS is used here. Based on the GBVP approach, the vertical datum parameters of the USA, Australia, and Hong Kong are determined. The key to determine the vertical datum parameters via the GPVP approach is to use the RCR technique to determine the gravimetric geoid. The NGGM is determined by the DIR_R6/EGM2008 model, the RTM geoid height NRTM is estimated by prism integration [25], and the residual geoid height Nres is calculated by stokes integration using 1D-FFT (One Dimensional Fast Fourier Transform) approach. The stokes integration radius is as same as the computation area. The RTM integration radius depends on the gravity functional, as well as on the spectral energy of the RTM elevations, for RTM geoid height (the spatial scales shorter than 5 arc-min), an integration radius of ~200 km is suitable [50]. Figure 10a shows the geoid contribution of residual gravity and RTM in USA, with a maximum, minimum, mean, and standard deviation of 0.466 m, 0.430 m, 0.001 m, − and 0.02 m, respectively. Figure 10c shows the geoid contribution of residual gravity and RTM in Australia, with a maximum, minimum, mean, and standard deviation of 0.539 m, 0.372 m, 0.012 m, − and 0.030 m, respectively. Figure 10e shows the geoid contribution of residual gravity and RTM in Hong Kong, with a maximum, minimum, mean, and standard deviation of 0.073 m, 0.039 m, − 0.005 m, and 0.010 m, respectively. The geoid contribution of residual gravity and RTM contains most − its power in the higher frequencies of the gravity field, and larger geoid signals are strongly correlated with the topography. Figure 10b,d,f represents the difference between the geometric geoid heights obtained by GNSS/levelling and the gravimetric geoid determined by the GBVP approach based on DIR_R6/EGM2008, RTM, and residual gravity, thus providing the spatial distribution of the vertical datum offsets. Figure 10b,d,f shows that the vertical datum offsets have east–west or north–south discrepancies that are mainly caused by systematic tilt errors and distortions in the vertical networks. The influence of systematic tilt errors and distortions on the estimation of vertical datum offsets is evaluated according to Equation (10). The results with and without consideration of tilts and distortions are compared in the following. For readability, these two cases will be called “non-tilts” and “with-tilts” scenarios, respectively. For the three study areas, Table4 presents the numerical results of both scenarios under the different approaches. As shown in the Table4, considering the influence of the systematic tilt and distortions, the accuracy of the vertical offset for the USA determined by the GBVP approach was improved by 69.1% (from 29.1 cm to 9.0 cm) in terms of the standard deviation. Moreover, the accuracy of the vertical offset for Australia determined by the GBVP approach was improved by 44.6% (or from 16.8 cm to 9.3 cm) in terms of the standard deviation, and the accuracy of the vertical offset for Hong Kong determined by the GBVP approach was improved by 18.9% (from 3.7 cm to 3.0 cm) in terms of the standard deviation. Based on the above analysis, there are significant accuracy improvements on the estimation of vertical datum offsets when considering the systematic tilts and distortions. Remote Sens. 2020, 12, x 16 of 23

Δ gOBS 317.37 −200.95 −3.56 29.80 Δ−Δ Australia ggOBS GGM 298.95 −348.50 −0.66 10.40 Δ−Δ−Δ ggOBS GGM g RTM 296.63 −309.36 −0.27 9.70 Δ gOBS 66.06 −41.8 −13.71 13.40 Δ−Δ Hong Kong ggOBS GGM 70.51 −37.24 −2.38 11.20 Δ−Δ−Δ ggOBS GGM g RTM 46.34 −66.86 −1.41 10.37

4.6. Estimation of Vertical Datum Parameters

The geopotential W0 = 62,636,853.4 m2s−2 of the global vertical datum released by the IHRS is used here. Based on the GBVP approach, the vertical datum parameters of the USA, Australia, and Hong Kong are determined. The key to determine the vertical datum parameters via the GPVP approach is to use the RCR technique to determine the gravimetric geoid. The NGGM is determined by the

DIR_R6/EGM2008 model, the RTM geoid height NRTM is estimated by prism integration [25], and the residual geoid height Nres is calculated by stokes integration using 1D-FFT (One Dimensional Fast Fourier Transform) approach. The stokes integration radius is as same as the computation area. The RTM integration radius depends on the gravity functional, as well as on the spectral energy of the RTM elevations, for RTM geoid height (the spatial scales shorter than 5 arc-min), an integration radius of ~200 km is suitable [50]. Figure 10a shows the geoid contribution of residual gravity and RTM in USA, with a maximum, minimum, mean, and standard deviation of 0.466 m, –0.430 m, 0.001 m, and 0.02 m, respectively. Figure 10c shows the geoid contribution of residual gravity and RTM in Australia, with a maximum, minimum, mean, and standard deviation of 0.539 m, –0.372 m, 0.012 m, and 0.030 m, respectively. Figure 10e shows the geoid contribution of residual gravity and RTM in Hong Kong, with a maximum, minimum, mean, and standard deviation of 0.073 m, –0.039 m, –0.005 m, and 0.010 m, respectively. The geoid contribution of residual gravity and RTM contains most its power in the higher frequencies of the gravity field, and larger geoid signals are strongly correlated with the topography. Figure 10b,d,f represents the difference between the geometric geoid heights obtained by GNSS/levelling and the gravimetric geoid determined by the GBVP approach based on DIR_R6/EGM2008, RTM, and residual gravity, thus providing the spatial distribution of the vertical

Remotedatum Sens. offsets.2020, 12Figure, 4137 10b,d,f shows that the vertical datum offsets have east–west or north–south17 of 23 discrepancies that are mainly caused by systematic tilt errors and distortions in the vertical networks.

Remote Sens. 2020, 12, x 17 of 23

Figure 10. The local residual gravity anomalies and residualresidual terrain model (RTM) contributions to geoid height for (a) mainland America, ( c) mainland Australia, and ( e) Hong Kong as well as the didifferencesﬀerences betweenbetween the the geometric geometric geoid geoid height height and gravimetricand gravimetric geoid heightgeoid forheight (b) mainlandfor (b) mainland America, (America,d) mainland (d) mainland Australia andAustralia (f) Hong and Kong. (f) Hong Note Kong. that theNote limits that of the the limits color of bar the (a color,c) do bar not ( corresponda and c) do tonot the correspond overall minimum to the overall and maximum minimum values.and maximum values.

FromThe influence Table4, considering of systematic the tilt inﬂuence errors and of thedistor systematictions on tiltsthe estimation and distortions, of vertical the GBVP datum solution offsets featuresis evaluated an improvement according to of Equation 8.2% compared (10). The to the results DIR_R6 with/EGM2008 and without solution consideration in terms of theof standardtilts and deviationdistortions for are determining compared in the the vertical following. oﬀset For in the readability, USA. For Australia,these two thecases GBVP will solutionbe called provides “non-tilts” an 11.4%and “with-tilts” improvement scenarios, compared respectively. to the DIR_R6 For the/EGM2008 three study solution areas, in termsTable of4 presents the standard the numerical deviation. Inresults Hong of Kong, both thescenarios GBVP under solution the experiences different appr a 14.3%oaches. improvement As shown compared in the Table the 4, DIR_R6 considering/EGM2008 the solutioninfluence in of terms the systematic of the standard tilt and deviation. distortions, Therefore, the accuracy the accuracy of the of vertical the GBVP offset solution for the is betterUSA thandetermined that using by the only GBVP DIR_R6 approach/EGM2008 was solution.improved thisby 69.1% is because (from the 29.1 RTM cm to and 9.0 residual cm) in terms gravity of canthe compensatestandard deviation. for the omission Moreover, errors the ofaccuracy the DIR_R6 of the/EGM2008 vertical model.offset for Australia determined by the GBVP approach was improved by 44.6% (or from 16.8 cm to 9.3 cm) in terms of the standard deviation, and the accuracy of the vertical offset for Hong Kong determined by the GBVP approach was improved by 18.9% (from 3.7 cm to 3.0 cm) in terms of the standard deviation. Based on the above analysis, there are significant accuracy improvements on the estimation of vertical datum offsets when considering the systematic tilts and distortions. From Table 4, considering the influence of the systematic tilts and distortions, the GBVP solution features an improvement of 8.2% compared to the DIR_R6/EGM2008 solution in terms of the standard deviation for determining the vertical offset in the USA. For Australia, the GBVP solution provides an 11.4% improvement compared to the DIR_R6/EGM2008 solution in terms of the standard deviation. In Hong Kong, the GBVP solution experiences a 14.3% improvement compared the DIR_R6/EGM2008 solution in terms of the standard deviation. Therefore, the accuracy of the GBVP solution is better than that using only DIR_R6/EGM2008 solution. this is because the RTM and residual gravity can compensate for the omission errors of the DIR_R6/EGM2008 model.

Remote Sens. 2020, 12, 4137 18 of 23

Table 4. Descriptive statistics of the non-tilts and with-tilts scenarios in terms of the estimated height datum oﬀset δH based on diﬀerent solution. The results are shown for the three study areas of the USA, Australia, and Hong Kong, respectively.

Region Scenario Solution Max Min Mean STD DIR_R6/EGM2008 0.049 2.071 0.799 0.300 non-tilts − − − GBVP 0.048 2.013 0.804 0.291 USA − − − DIR_R6/EGM2008 0.375 1.855 0.801 0.098 with-tilts − − − GBVP 0.367 1.807 0.809 0.090 − − − DIR_R6/EGM2008 0.820 0.563 0.086 0.181 non-tilts − GBVP 0.843 0.529 0.087 0.168 Australia − DIR_R6/EGM2008 0.588 0.507 0.086 0.105 with-tilts − GBVP 0.605 0.537 0.082 0.093 − DIR_R6/EGM2008 0.625 0.847 0.737 0.042 non-tilts − − − GBVP 0.652 0.841 0.731 0.037 Hong Kong − − − DIR_R6/EGM2008 0.643 0.817 0.737 0.035 with-tilts − − − GBVP 0.659 0.811 0.731 0.030 − − −

Considering the influence of the systematic tilts and distortions, the vertical datum parameters are estimated according to Equations (10) to (12). Table5 shows the vertical datum parameters for the USA, Australia, and Hong Kong. The geopotential value of the NAVD88 is estimated as equal to 62636861.31 0.96m2s 2, and the vertical offset with respect to geoid is 0.809 0.090 m, ± − − ± which means that the NAVD88 is about 80.9 cm below the geoid. The geopotential value of the AHD is 62653852.60 0.95 m2s 2, and the vertical offset with respect to geoid is 0.082 0.093 m, ± − ± which means that the AHD is about 8.2 cm above the geoid. The geopotential value of the HKPD is 62636860.55 0.29 m2s 2, and the vertical offset with respect to geoid is 0.731 0.030 m, which means ± − − ± that the HKPD is about 73.1 cm below the geoid.

Table 5. Vertical datum parameters in the USA, Australia, and Hong Kong with respect to the global 2 2 reference level released with the conventional value of W0 = 62,636,853.4 m s− .

2 2 2 2 Vertical Datum Vertical Oﬀsets (m) Potential Diﬀerences (m s− ) Reference Potential (m s− ) NAVD88 –0.809 0.090 7.91 0.96 62,636,861.31 0.96 ± − ± ± AHD 0.082 0.093 0.80 0.95 62,653,852.60 0.95 ± ± ± HKPD –0.731 0.030 –7.15 0.29 62,636,860.55 0.29 ± ± ±

Prior to this work, Amjadiparvar et al. [55] found vertical offset of 47.5 cm for NAVD88 with 2 2 − respect to the W0 = 62,636,856.0 m s− , which means that the geopotential value of NAVD88 should 2 2 2 2 be around 62,636,860.6 m s− , which fits very well to our value of 62636861.3 m s− considering the 2 2 standard deviation of our value being around 1 m s− . Burša et al. [56] reported the geopotential values of 62,636,861.51 and 62636851.83 for NAVD88 and AHD respectively, which fits well to our results. Few the geopotential value of the Hong Kong were presented in the literature. Zhang et al. [13] used the linear fixed GBVP approach to calculate the potential difference between the 1985 vertical datum of China and the HKPD as 8.78 0.63 m2s 2. The geopotential value of the − ± − HKPD determined in present study is 62,636,860.55 0.29 m2s 2. Therefore, the geopotential value ± − of China’s 1985 vertical datum can be estimated as 62,636,851.77 0.69 m2s 2, and the vertical offset ± − between China’s 1985 vertical datum and the global vertical datum is 1.63 0.69 m2s 2. Sánchez et al.[4] ± − used the GBVP approach to determine the vertical datum parameters of South America. We chose Brazil and Argentina as representative in South America. The potential differences between the Brazil and Argentina vertical datum and global geoid are 3.79 0.18 m2s 2 and 6.51 0.49 m2s2, respectively. ± − ± North America can use the same vertical datum of NAVD88. Figure 11 shows the potential differences between the North America, Australia, Brazil, Argentina, and Hong Kong vertical datums and the global geoid. By determining the potential differences between the global existing LVDs and the global Remote Sens. 2020, 12, 4137 19 of 23

Remote Sens. 2020, 12, x 19 of 23 geoid (as shown in Figure 11), the LVDs can be incorporated into the IHRS to ensure the global vertical existingdatum system’s LVDs and consistency. the global geoid (as shown in Figure 11), the LVDs can be incorporated into the IHRS to ensure the global vertical datum system’s consistency.

Figure 11.11. VisualizationVisualization of of the the vertical vertical datum datum potential potential diff erencesdifferences in North in North America, America, Australia, Australia, Brazil, Argentina,Brazil, Argentina, and Hong and Kong Hong (sub-graph Kong (sub-graph of southwest of corner)southwest with corner) respect with to the respect global referenceto the global level released with the conventional value of W = 62,636,853.4 m2s 2. reference level released with the conventional0 value of W0 = 62,636,853.4− m2s−2. 5. Conclusions 5. Conclusions The unification of the global vertical datum is a key problem to be solved for geodesy for a long The unification of the global vertical datum is a key problem to be solved for geodesy for a long time, the main goal of which is to determine the vertical datum parameters for local vertical datum. time, the main goal of which is to determine the vertical datum parameters for local vertical datum. In this study, we evaluate the vertical datum parameters of the USA, Australia, and Hong Kong using In this study, we evaluate the vertical datum parameters of the USA, Australia, and Hong Kong using the GBVP approach. Meanwhile, the geoid omission errors of the GGM, the spectral accuracy of the the GBVP approach. Meanwhile, the geoid omission errors of the GGM, the spectral accuracy of the GGM, the indirect bias term, and the systematic errors and distortions on the estimation of the vertical GGM, the indirect bias term, and the systematic errors and distortions on the estimation of the vertical datum parameters are analyzed. These factors should be considered for the implementation of the datum parameters are analyzed. These factors should be considered for the implementation of the GBVP approach to unify the vertical datum. GBVP approach to unify the vertical datum. The effect of the indirect bias term is equal to 1 mm when higher degree GGMs (300 degree The effect of the indirect bias term is equal to 1 mm when higher degree GGMs (300 degree and and order) are used. Therefore, this effect can be neglected safely when implementing the GBVP order) are used. Therefore, this effect can be neglected safely when implementing the GBVP approach. To analyze the influence of the systematic tilts and distortions on the estimation of the approach. To analyze the influence of the systematic tilts and distortions on the estimation of the vertical datum parameters, the parametric model in Equation (10) is used to absorb the existing vertical datum parameters, the parametric model in Equation (10) is used to absorb the existing systematic errors and distortions. The results show that there are significant accuracy improvements systematic errors and distortions. The results show that there are significant accuracy improvements on the estimation of the vertical datum offsets when considering the systematic tilts and distortions in on the estimation of the vertical datum offsets when considering the systematic tilts and distortions vertical networks. The GRACE/GOCE-based GGMs have lower degree errors and stronger geoid signal in vertical networks. The GRACE/GOCE-based GGMs have lower degree errors and stronger geoid in the medium–long wavelength, but the maximum expansion degree of the GRACE/GOCE-based signal in the medium–long wavelength, but the maximum expansion degree of the GRACE/GOCE- GGMs is limited. These models have certain omission errors. The effect of omission errors reaches the based GGMs is limited. These models have certain omission errors. The effect of omission errors decimeter level in the USA and Australia, while the impact of the omission error is at the centimeter reaches the decimeter level in the USA and Australia, while the impact of the omission error is at the level in Hong Kong. centimeter level in Hong Kong. The omission errors of GOCE/GRACE-based GGMs cannot be ignored when unifying the vertical The omission errors of GOCE/GRACE-based GGMs cannot be ignored when unifying the datum. To compensate for the omission errors of GOCE/GRACE-based GGMs, the EGM2008 model vertical datum. To compensate for the omission errors of GOCE/GRACE-based GGMs, the EGM2008 is used to extend the GOCE/GRACE-based GGMs to determine the combined GGMs. Because the model is used to extend the GOCE/GRACE-based GGMs to determine the combined GGMs. Because spectral accuracy of the DIR_R6 model is significantly better than other GRACE/GOCE-based GGMs, the spectral accuracy of the DIR_R6 model is significantly better than other GRACE/GOCE-based we built a combined model, DIR_R6/EGM2008, by combining the DIR_R6 model and the EGM2008 GGMs, we built a combined model, DIR_R6/EGM2008, by combining the DIR_R6 model and the model using the weighting method. The combined DIR_R6/EGM2008 model provides a stronger EGM2008 model using the weighting method. The combined DIR_R6/EGM2008 model provides a geoid signal and a lower geoid degree errors than the EGM2008 and DIR_R6 models. To obtain stronger geoid signal and a lower geoid degree errors than the EGM2008 and DIR_R6 models. To high-frequency geoid information caused by the Earth’s topographic masses, the RTM is used to recover obtain high-frequency geoid information caused by the Earth’s topographic masses, the RTM is used the short-wavelength geoid in this paper. The reference terrain model with a resolution of 5 5 and a to recover the short-wavelength geoid in this paper. The reference terrain model with a resolution0 × 0 of 55′′× and a modified stokes kernel function are used to allow the contributions of the RTM and

Remote Sens. 2020, 12, 4137 20 of 23 modified stokes kernel function are used to allow the contributions of the RTM and residual gravity to the geoid further compensate for the omission errors of the DIR_R6/EGM2008 model. Finally, the vertical datum parameters of the USA, Australia and Hong Kong are determined using the GBVP approach based on DIR_R6/EGM2008, RTM and residual gravity. The geopotential value of the NAVD88 is estimated as equal to 62,636,861.31 0.96 m2s 2, and the vertical offset with ± − respect to the global geoid is 0.809 0.090 m, which means that the NAVD88 is about 80.9 cm below − ± the geoid. The geopotential value of the AHD is 62,653,852.60 0.95 m2s 2, and the vertical offset with ± − respect to the global geoid is 0.082 0.093m, which means that the AHD is about 8.2 cm above the ± geoid. The geopotential value of the HKPD is 62,636,860.55 0.29 m2s 2, and the vertical offset with ± − respect to the global geoid is 0.731 0.030 m, which means that the HKPD is about 73.1 cm below − ± the geoid. With the vertical datum parameters, we can thus connect the local height system with the global height system to ensure the global vertical datum system’s consistency.

Author Contributions: Conceptualization, All; methodology, P.Z.and L.B.; software, P.Z.and D.G.; validation, P.Z.; formal analysis, P.Z.and L.B.; investigation, All; resources, D.G.; data curation, P.Z.and L.B.; writing—original draft preparation, P.Z.; writing—review and editing, L.B., D.G. and L.W.; visualization, P.Z. and Q.L.; supervision, L.W., Z.X. and H.L.; project administration, L.B.; funding acquisition, L.B., Z.L. and D.G. All authors have read and agreed to the published version of the manuscript. Funding: This work is funded by the National Natural Science Foundation of China (No. 41774022) and the Basic Frontier Science Research Program of Chinese Academy of Sciences (No. ZDBS-LY-DQC028). Acknowledgments: We are grateful for the platform provided by the University of Chinese Academy of Sciences. We are also grateful the National Geodetic Survey (NGS) and The Australian State and Territory geodetic agencies for providing us with GNSS/levelling data. We thank the Geoscience Australia’s national gravity database and National Oceanic and Atmospheric Administration (NOAA) for providing us with gravity data. We thank International Centre for Global Earth Models (ICGEMs) and NASA Shuttle Radar Topographic Mission (SRTM) for providing us with the Global Geopotential Models and the digital terrain model data, respectively. We thank Christian Gerlach for providing codes about stokes integration. All the authors appreciate the excellent review work of the anonymous reviewers. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Filmer, M.S.; Hughes, C.W.; Woodworth, P.L.; Featherstone, W.E.; Bingham, R.J. Comparison between geodetic and oceanographic approaches to estimate mean dynamic topography for vertical datum unification: Evaluation at Australian tide gauges. J. Geod. 2018, 92, 1–25. [CrossRef] 2. Vu, D.T.; Bruinsma, S.; Bonvalot, S.; Remy, D.; Vergos, G.S. A Quasigeoid-Derived Transformation Model Accounting for Land Subsidence in the Mekong Delta towards Height System Unification in Vietnam. Remote Sens. 2020, 12, 817. [CrossRef] 3. Drewes, H.; Kuglitsch, F.; Adám, J.; Rózsa, S. The Geodesist’s Handbook 2016. J. Geod. 2016, 90, 907–1205. [CrossRef] 4. Ihde, J.; Sánchez, L.; Barzaghi, R.; Drewes, H.; Foerste, C.; Gruber, T.; Liebsch, G.; Marti, U.; Pail, R.; Sideris, M. Definition and proposed realization of the international height reference system (IHRS). Surv. Geophys. 2017, 38, 549–570. [CrossRef] 5. Sánchez, L.; Sideris, M.G. Vertical datum unification for the International Height Reference System (IHRS). Geophys. J. Int. 2017, 209, 570–586. [CrossRef] 6. Thompson, K.R.; Huang, J.; Véronneau, M.; Wright, D.G.; Lu, Y. Mean surface topography of the northwest Atlantic: Comparison of estimates based on satellite, terrestrial gravity, and oceanographic observations. J. Geophys. Res. 2009, 114, C07015. [CrossRef] 7. Woodworth, P.L.; Hughes, C.W.; Bingham, R.J.; Gruber, T. Towards worldwide height system unification using ocean information. J. Geod. Sci. 2012, 2, 302–318. [CrossRef] 8. Gomez, M.E.; Pereira, R.A.D.; Ferreira, V.G.; Del Cogliano, D.; Luz, R.T.; De Freitas, S.R.C.; Farias, C.; Perdomo, R.; Tocho, C.; Lauria, E.; et al. Analysis of the Discrepancies Between the Vertical Reference Frames of Argentina and Brazil. In IAG 150 Years; Rizos, C., Willis, P., Eds.; Springer International Publishing: Cham, Switzerland, 2015; Volume 143, pp. 289–295. Remote Sens. 2020, 12, 4137 21 of 23

9. Tocho, C.; Vergos, G.S. Estimation of the Geopotential Value W0 for the Local Vertical Datum of Argentina Using EGM2008 and GPS/Levelling Data. In IAG 150 Years; Rizos, C., Willis, P., Eds.; Springer International Publishing: Cham, Switzerland, 2015; Volume 143, pp. 271–279. 10. He, L.; Chu, Y.H.; Xu, X.Y.; TengXu, Z. Evaluation of the GRACE/GOCE Global Geopotential Model on estimation of the geopotential value for the China vertical datum of 1985. Chin. J. Geophys. 2019, 62, 2016–2026. 11. Colombo, O. A World Vertical Network. OSU Report No. 296; The Ohio State University: Columbus, OH, USA, 1980. 12. Rummel, R.; Teunissen, P. Height datum definition, height datum connection and the role of the geodetic boundary value problem. Bull. Geod. 1988, 62, 477–498. [CrossRef] 13. Zhang, L.; Li, F.; Chen, W.; Zhang, C. Height datum unification between Shenzhen and Hong Kong using the solution of the linearized fixed-gravimetric boundary value problem. J. Geod. 2009, 83, 411–417. [CrossRef] 14. Amjadiparvar, B.; Rangelova, E.; Sideris, M.G. The GBVP approach for vertical datum unification: Recent results in North America. J. Geod. 2016, 90, 45–63. [CrossRef] 15. Ebadi, A.; Ardalan, A.; Karimi, R. The Iranian height datum offset from the GBVP solution and spirit-leveling/gravimetry data. J. Geod. 2019, 93, 1207–1225. [CrossRef] 16. Rülke, A.; Liebsch, G.; Sacher, M.; Schäfer, U.; Schirmer, U.; Ihde, J. Unification of European height system realizations. J. Geod. Sci. 2012, 2, 343–354. [CrossRef] 17. Sansò, F.; Sideris, M.G. Geoid Determination; Springer: Berlin/Heidelberg, Germany, 2013. 18. Barzaghi, B. The Remove-Restore Method. In Encyclopedia of Geodesy; Springer International Publishing: Cham, Switzerland, 2015; pp. 1–4. 19. Gruber, T.; Visser, P.; Ackermann, C.; Hosse, M. Validation of GOCE gravity field models by means of orbit residuals and geoid comparisons. J. Geod. 2011, 85, 845–860. [CrossRef] 20. Rummel, R. Height unification using GOCE. J. Geod. Sci. 2012, 2, 355–362. [CrossRef] 21. Tziavos, I.N.; Vergos, G.S.; Grigoriadis, V.N.; Tzanou, E.A.; Natsiopoulos, D.A. Validation of GOCE/GRACE Satellite Only and Combined Global Geopotential Models Over Greece in the Frame of the GOCESeaComb Project. In IAG 150 Years; Rizos, C., Willis, P., Eds.; Springer International Publishing: Cham, Switzerland, 2015; Volume 143, pp. 297–304. 22. Sánchez, L.; Sideris, M.; Ihde, J. Activities and Plans of the GGOS Focus Area Unified Height System; IUGG General Assembly: Montreal, QC, Canada, 2019. 23. Forsberg, R.; Tscherning, C. The use of height data in gravity field approximation by collocation. J. Geophys. Res. 1981, 86, 7843–7854. [CrossRef] 24. Forsberg, R. A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling; Technical report; The Ohio State University: Colombus, OH, USA, 1984. 25. Hirt, C.; Featherstone, W.E.; Marti, U. Combining EGM2008 and SRTM/DTM2006.0 Residual Terrain Model Data to improve Quasigeoid Computations in Mountainous Areas Devoid of Gravity Data. J. Geod. 2010, 84, 557–567. [CrossRef] 26. Rexer, M.; Hirt, C.; Bucha, B.; Holmes, S. Solution to the spectral filter problem of residual terrain modelling (RTM). J. Geod. 2017, 92, 675–690. [CrossRef] 27. Hirt, C.; Bucha, M.; Yang, M.; Kuhn, M. A numerical study of residual terrain modelling (RTM) techniques and the harmonic correction using ultra-high degree spectral gravity modelling. J. Geod. 2019, 93, 1469–1486. [CrossRef] 28. Heiskanen, W.A.; Moritz, H. Physical Geodesy; W.H. Freeman and Company: San Francisco, CA, USA, 1967. 29. Moritz, H. Geodetic Reference System 1980. J. Geod. 2000, 74, 128–133. [CrossRef] 30. Gerlach, C.; Rummel, R. Global height system unification with GOCE: A simulation study on the indirect bias term in the GBVP approach. J. Geod. 2013, 87, 57–67. [CrossRef] 31. Grombein, T.; Seitz, K.; Heck, B. Height system unification based on the fixed GBVP approach. In IAG 150 Years; Rizos, C., Willis, P., Eds.; Springer International Publishing: Cham, Switzerland, 2015; Volume 143, pp. 305–311. 32. Hayden, T.; Amjadiparvar, B.; Rangelova, E.; Sideris, M.G. Estimating Canadian vertical datum offsets using GNSS/levelling benchmark information and GOCE global geopotential models. J. Geod. Sci. 2012, 2, 257–269. [CrossRef] Remote Sens. 2020, 12, 4137 22 of 23

33. Grombein, T.; Seitz, K.; Heck, B. On High-Frequency Topography-Implied Gravity Signals For a Height System Unification Using GOCE-based Global Geopotential Models. Surv. Geophys. 2016, 38, 1–35. [CrossRef] 34. Pavlis, N.K.; Holmes, S.A.; Kenyon, S.C.; Factor, J.K. The development and evaluation of the earth gravitational model 2008 (EGM2008). J. Geophys. Res. 2012, 117, B04406. [CrossRef] 35. Wenzel, H.G. Geoid computation by least squares spectral combination using integral formulas. In Proceedings of the IAG General Meeting, Tokyo, Japan, 7–15 May 1982; pp. 438–453. 36. Gilardoni, R.; Reguzzoni, M.; Sampietro, D. GECO: A global gravity model by locally combining GOCE data and EGM2008. Stud. Geophys Geod. 2016, 60, 228–247. [CrossRef] 37. Gerlach, C.; Ophaug, V. Accuracy of Regional Geoid Modelling with GOCE. In International Association of Geodesy Symposia; Springer: Cham, Switzerland, 2017; Volume 148, pp. 17–23. 38. Förste, C.; Abrykosov, O.; Bruinsma, S.; Dahle, C.; König, R.; Lemoine, J.M. ESA’s Release 6 GOCE Gravity Field Model by Means of the Direct Approach Based on Improved Filtering of the Reprocessed Gradients of the Entire Mission (GO_CONS_GCF_2_DIR_R6). Available online: https://doi.org/10.5880/ICGEM.2019.004 (accessed on 14 May 2020). 39. Brockmann, J.M.; Schubert, T.; Torsten, M.G.; Schuh, W.D. The Earth’s Gravity Field as Seen by the GOCE Satellite-An Improved Sixth Release Derived with the Time-Wise Approach (GO_CONS_GCF_2_TIM_R6). Available online: http://doi.org/10.5880/ICGEM.2019.003 (accessed on 14 May 2020). 40. Xu, X.; Zhao, Y.; Reubelt, T.; Robert, T. A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models. Geod. Geodyn. 2018, 8, 260–272. [CrossRef] 41. Wu, H.; Müller, J.; Brieden, P. The IfE global gravity field model from GOCE-only observations. In Proceedings of the International Symposium on Gravity, Geoid and Height Systems, Thessaloníki, Greece, 19–23 September 2016. 42. Lu, B.; Luo, Z.; Zhong, B.; Zhou, H.; Flechtner, F.; Förste, C.; Barthelmes, F.; Zhou, R. The gravity field model IGGT_R1 based on the second invariant of the GOCE gravitational gradient tensor. J. Geod. 2017, 92, 561–572. [CrossRef] 43. Gatti, A.; Reguzzoni, M.; Migliaccio, F.; Sansò, F. Computation and assessment of the fifth release of the GOCE-only space-wise solution. In Proceedings of the 1st Joint Commission 2 and IGFS Meeting, Thessaloníki, Greece, 19–23 September 2016. 44. Sánchez, L.; Ågren, J.; Huang, J.; Wang, Y.M.; Forsberg, R. Basic Agreements for the Computation of Station Potential Values as IHRS Coordinates, Geoid Undulations and Height Anomalies within the Colorado 1 cm Geoid Experiment. In Proceedings of the International Symposium on Gravity, Geoid and Height Systems 2018 (GGHS2018), Copenhagen, Denmark, 17–21 September 2018. 45. Ekman, M. Impacts of geodynamic phenomena on systems for height and gravity. Bull. Geod. 1989, 63, 281–296. [CrossRef] 46. Jarvis, A.; Reuter, H.I.; Nelson, A.; Guevara, E. Hole-Filled SRTM for the Globe Version 4, Available from the CGIAR-SXI SRTM 90m Database. 2008. Available online: http://srtm.csi.cgiar.org (accessed on 1 June 2020). 47. Tozer, B.; Sandwell, D.T.; Smith, W.H.F.; Olson, C.; Beale, J.R.; Wessel, P. Global Bathymetry and Topography at 15 Arc Sec: SRTM15+. Earth Space Sci. 2019, 6, 1847–1864. [CrossRef] 48. Hirt, C.; Kuhn, M.; Claessens, S.; Pail, R.; Seitz, K.; Gruber, T. Study of the Earth’s short-scale gravity field using the ERTM2160 gravity model. Comput. Geosci. 2014, 73, 71–80. [CrossRef] 49. Rummel, R.; Rapp, R.H.; Sünkel, H.; Tscherning, C.C. Comparisons of Global Topographic/Isostatic Models to the Earth’s Observed Gravity Field; Report No 388; Ohio State University: Columbus, OH, USA, 1988. 50. Hirt, C. RTM gravity forward-modeling using topography/bathymetry data to improve high-degree global geopotential models in the coastal zone. Mar. Geod. 2013, 36, 1–20. [CrossRef] 51. Ustun, A.; Abbak, R. On global and regional spectral evaluation of global geopotential models. J. Geophys. Eng. 2010, 7, 369. [CrossRef] 52. Colombo, O. Numerical Methods for Harmonic Analysis on the Sphere; Report No. 310; The Ohio State University: Columbus, OH, USA, 1981. 53. Hirt, C. Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach. J. Geod. 2012, 86, 729–744. [CrossRef] 54. Forsberg, R. Terrain Effects in Geoid Computations. In Lecture Notes, International School for the Determination and Use of the Geoid; International Geoid Service: Milan, Italy, 2008. Remote Sens. 2020, 12, 4137 23 of 23

55. Amjadiparvar, B.; Rangelova, E.V.; Sideris, M.G.; Véronneau, M. North American height datums and their offsets: The effect of GOCE omission errors and systematic levelling effects. J. Appl. Geodesy. 2013, 7, 39–50. [CrossRef] 56. Burša, M.; Kouba, J.; Müller, A.; Radˇej,K.; True, S.A.; Vatrt, V.; Vojtíšková, M. Determination of geopotential differences between local vertical datums and realization of a world height system. Studia Geophys. Geod. 2001, 45, 127–132. [CrossRef]

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