Accuracy Evaluation of Geoid Heights in the National Control Points of South Korea Using High-Degree Geopotential Model

applied sciences

Article Accuracy Evaluation of Geoid Heights in the National Control Points of South Korea Using High-Degree Geopotential Model

Kwang Bae Kim 1,* , Hong Sik Yun 1 and Ha Jung Choi 2

1 School of Civil, Architectural, and Environmental System Engineering, Sungkyunkwan University, Suwon 16419, Korea; [email protected] 2 Interdisciplinary Program for Crisis, Disaster, and Risk Management, Sungkyunkwan University, Suwon 16419, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-31-299-4251

Received: 9 December 2019; Accepted: 18 February 2020; Published: 21 February 2020

Abstract: Precise geoid heights are not as important for understanding Earth’s gravity field, but they are important to geodesy itself, since the vertical datum is defined as geoid in a cm-level accuracy. Several high-degree geopotential models have been derived lately by using satellite tracking data such as those from Gravity Recovery and Climate Experiment (GRACE) and Gravity Field and Steady-State Ocean Circulation Explorer (GOCE), satellite altimeter data, and terrestrial and airborne gravity data. The Korean national geoid (KNGeoid) models of the National Geographic Information Institute (NGII) were developed using the latest global geopotential models (GGMs), which are combinations of gravity data from satellites and land gravity data. In this study, geoid heights calculated from the latest high-degree GGMs were used to evaluate the accuracy of the three GGMs (European Improved Gravity model of Earth by New techniques (EIGEN)-6C4, Earth Gravitational Model 2008 (EGM2008), and GOCE-EGM2008 combined model (GECO)) by comparing them with the geoid heights derived from the Global Navigation Satellite System (GNSS)/leveling of the 1182 unified control points (UCPs) that have been installed by NGII in South Korea since 2008. In addition, the geoid heights derived from the KNGeoid models were compared with the geoid heights derived from the GNSS/leveling of the 1182 UCPs to assess the accuracy of the KNGeoid models in terms of relative geoid heights for further gravimetric geoid determination studies in South Korea. As a result, the EGM2008 model could be selected as the suitable GGM from among the three GGMs for determining a gravimetric geoid model for South Korea.

Keywords: Korean national geoid; GNSS/leveling-derived geoid height; uniﬁed control points; global geopotential model

1. Introduction Geoid, which approximates the mean sea level (MSL), is a reference surface for determining topographic heights and ocean depths. It is necessary to establish a geoid model based on continuous gravity observations, because geoid represents the Earth’s shape resulting from the mass distributions above and below the Earth’s surface. The global geopotential models (GGMs) derived from the data obtained by Gravity Recovery and Climate Experiment (GRACE) launched in 2002 have been released. Earth Gravitational Model 96 (EGM96) [1], European Improved Gravity model of Earth by New techniques (EIGEN)-CG01C [2], EIGEN-CG03C [3], and Earth Gravitational Model 2008 (EGM2008) [4] were used for geoid model establishment in South Korea, and the accuracy of the EGM2008 model was evaluated using the Global Navigation Satellite System (GNSS)/leveling data [5–9]. Assessment of

Appl. Sci. 2020, 10, 1466; doi:10.3390/app10041466 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 1466 2 of 11 geoid height accuracy is necessary for determining the high-degree GGMs that have the best fit to GNSS/leveling data in national control points over South Korea. The recently released GGMs were based on gravity data obtained from the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) gravitational exploration satellite launched in 2009. Several GGMs have been calculated using satellite tracking data from the GRACE and GOCE gravity missions [10,11]. High-degree geopotential models of spherical harmonic coefficients are used for modeling the Earth’s exterior gravity field. Such coefficients are derived from satellite tracking data, altimeter data, and terrestrial and airborne gravity data. Hundreds of thousands of coefficients and standard deviation values for these coefficients are estimated through millions of measurements. The number and distribution of measurements, as well as the type of measurement, affect the accuracy of a geopotential model. Since the 1960s, satellite gravity field missions have provided accurate data for forming geopotential models and these models have been evaluated based on the data provided by the satellite gravity field missions. The National Geographic Information Institute (NGII) has developed three Korean national geoid (KNGeoid) models—KNGeoid13, KNGeoid14, and KNGeoid18—using the Earth’s gravity field model, satellite-altimetry-derived gravity data, land and ocean gravity data, airborne gravity data, and digital elevation model (DEM) data. The KNGeoid models are hybrid geoid models developed by NGII to improve the height measurement accuracy based on GNSS surveying. First, the KNGeoid13 model, developed by NGII for the land part, was based on gravity data obtained from several national control points (unified control point (UCP), benchmark (BM), triangulation point), airborne gravity data obtained since 2008, DTU10 satellite altimeter data [12], EGM2008 data, and 5 m gridded topographic data. The standard deviation of residuals of the KNGeoid13 model are 3.41 cm [13]. Second, the KNGeoid14 model was developed using the same method utilized for developing the KNGeoid13 model, by adding the gravity data obtained in 2014 and shipborne gravity data from the Korea Hydrographic and Oceanographic Agency (KHOA, http://www.khoa.go.kr) to the gravity data constructed on KNGeoid13, and its standard deviation of residuals was evaluated as approximately 3.3 cm [14]. Finally, the KNGeoid18 model was developed using Experimental Gravity Field Model 2016 (XGM2016) based on the GOCE gravity data [15] generated as the initial version of Earth Gravitation Model 2020 (EGM2020), and its standard deviation of residuals is 2.33 cm [16]. The development of geoid model based on the longer wavelength part of the geoid derived from the EGM2008 has been performed. Several studies have implemented accuracy evaluation of geoid heights derived from GGMs by comparing them with the GNSS/leveling geoid height data obtained in the national control points (BM and UCP) of South Korea [5–9]. The EGM2008 is a spherical harmonic model of the Earth’s gravitational potential developed through a least-squares combination of the ITG-GRACE03S gravitational model and its associated error covariance matrix, with the gravitational information obtained from a global set of area-mean free-air gravity anomalies defined on a 5’ equiangular grid [4]. The EGM2008 is complete to degree and order 2159 and contains additional coefficients up to degree 2190 and order 2159. The EIGEN-6C4 [17], the newest ultra-high-degree global gravity field model, is the latest release in the EIGEN-6C series, containing the complete satellite gravity gradiometry (SGG) data of the GOCE mission. The GOCE-EGM2008 combined model (GECO) [18] is a global gravity model which is computed by incorporating the GOCE-only TIM-R5 solution, which is the fifth release (R5) of the time-wise (TIM) model, into the EGM2008. The EGM2008 geoid undulations are computed on a global spherical grid with a 0.5◦ resolution by synthesizing the EGM2008 coefficients up to degree 359. The GOCE geoid on the same grid is computed by synthesizing the TIM-R5 coefficients, up to degree 250. Finally, the GECO spherical harmonic coefficients are computed by analyzing the combined global geoid grid up to degree 359 (consistently with the 0.5◦ resolution). From degrees 360 to 2190, the GECO coefficients are the same as those of EGM2008. The GECO coefficient errors are computed as the weighted average of the coefficient errors of EGM2008 and the TIM-R5 solution [18]. Appl. Sci. 2020, 10, 1466 3 of 11

Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 11 Once the latest high-degree global gravitational ﬁeld model has been developed and published, precisionOnce analysis the latest is performed high-degree to determineglobal gravitational its utility. field The Internationalmodel has been Center developed for Global and Earthpublished, Models (ICGEM,precision http: analysis//icgem.gfz-potsdam.de is performed to determine/home), operated its utility. by GFZThe International in Germany, receivesCenter for the Global GNSS /Earthleveling dataModels from Australia,(ICGEM, http://icgem.gfz Brazil, Canada,-potsdam.de/home), Europe, Japan, and operated USA, and by performs GFZ in Germany,a local precision receives analysis. the TableGNS1 S/lerepresentsveling data the precisionfrom Australia, levels Brazil, of the Canada, three GGMs Europe, that Japan, were usedand USA, in this and study. performs The precisiona local levelsprecision are root-mean-square-error analysis. Table 1 represents of the the mean precision diﬀerences levels betweenof the three the GGMs GNSS /thatlevelling-derived were used in this geoid heightsstudy. and The the precision model-computed levels are geoid root- heights.mean-square It is- openerror to of the the public. mean differences between the GNSS/levelling-derived geoid heights and the model-computed geoid heights. It is open to the public. Table 1. Precision levels of the GGMs used in this study [19]. Table 1. Precision levels of the GGMs used in this study [19]. Global Geopotential Models (Nmax) Country/Region GNSS/Leveling Data Points Global Geopotential Models (Nmax) Country/region GNSS/Leveling Data Points EGM2008 (2190) EIGEN-6C4 (2190) GECO (2190) EGM2008 (2190) EIGEN-6C4 (2190) GECO (2190) Australia 201 21.7 cm 21.2 cm 21.6 cm Australia 201 21.7 cm 21.2 cm 21.6 cm Brazil 1112 46.0 cm 44.6 cm 45.1 cm CanadaBrazil 1112 2691 46.0 12.8 cm cm 44.6 12.6 cm 45.1 13.1 cm cm CanadaEurope 2691 1047 12.8 12.5 cm cm 12.6 12.1 cm 13.1 12.3 cm cm EuropeJapan 1047 816 12.5 8.3 cm cm 12.1 7.9 cmcm 12.3 8.0 cm cm JapanUSA 816 6169 8.3 24.8 cm cm 24.77.9 cm cm 8.0 24.6 cm cm USAAll 6169 12,036 24.8 23.97 cm cm 23.6124.7 cm cm 24.6 23.71 cm cm All 12,036 23.97 cm 23.61 cm 23.71 cm The aim of this study was to evaluate the accuracy of the geoid heights derived from the latest high-degreeThe aim GGMs of this developed study was based to evaluate on satellite the accuracy gravity data of the such geoid as thoseheights from derived GRACE from and the GOCE latest as high-degree GGMs developed based on satellite gravity data such as those from GRACE and GOCE compared with the GNSS/leveling-derived geoid heights of the 1182 UCPs that have been installed by as compared with the GNSS/leveling-derived geoid heights of the 1182 UCPs that have been installed NGII in South Korea since 2008, as shown in Figure1. In addition, the geoid heights derived from the by NGII in South Korea since 2008, as shown in Figure 1. In addition, the geoid heights derived from KNGeoid models were compared with the GNSS/leveling-derived geoid heights of the 1182 UCPs to the KNGeoid models were compared with the GNSS/leveling-derived geoid heights of the 1182 UCPs assess the accuracy of the gravimetric geoid models in terms of relative geoid heights to GNSS/leveling to assess the accuracy of the gravimetric geoid models in terms of relative geoid heights to dataGNSS/leveling for further gravimetric data for further geoid gravimetric determination geoid studiesdetermination in South studies Korea. in South Korea.

FigureFigure 1. 1.Location Location map map of of thethe 11821182 GNSSGNSS/leveling/leveling UCPs UCPs in in South South Korea Korea that that were were used used in inthis this study study forfor the the evaluation evaluation of of the the geoid geoid heights heights inin thethe studystudy area area..

2. Materials and Methods Appl. Sci. 2020, 10, 1466 4 of 11

2. Materials and Methods

2.1. Study Area In the ﬁrst UCP construction project of NGII between 2008 and 2010, 1196 points were installed at 10 km intervals across the country. GNSS surveying, precise leveling, and gravity surveying were performed to provide the necessary three-dimensional survey results. A UCP is a new national control point built to maximize the surveying eﬃciency, such as the convenience, by integrating the national control point functions that were installed and managed individually (triangulation point, BM, gravity point, etc.). In this study, the geoid heights derived from the high-degree GGMs and KNGeoid models were evaluated by utilizing the geometric geoid heights obtained from the GNSS/leveling on 1182 of the 1196 UCPs installed by NGII between 2008 and 2010, in South Korea. Figure1 shows the location map of the 1182 GNSS/leveling UCPs that were used for the evaluation of the geoid heights in the study area (125◦ to 131◦ E, 33◦ to 39◦ N), which includes the inland and islands of South Korea.

2.2. Geoid Height Computation from GGMs For global gravity ﬁeld analysis, a spherical harmonic series can represent the geoid heights (N). The longer wavelength geoid heights (N) contributed by the geopotential models were computed at position (ϕP, λP) using Equation (1) [20]:

XL Xl h i N(ϕP, λP) = R ClmcosmλP + SlmsinmλP Plm(sinϕP) (1) l=2 m=0 where R is the Earth’s mean radius; Clm and Slm are the fully normalized spherical harmonic coefficients; Plm(sin ϕP) is the fully normalized associated Legendre function; and l and m are the degree and order, respectively. The finite series is usually truncated at the maximum degree of expansion l = L. The series coefficients allow the determination of geoid heights using Equation (1). Table2 summarizes the main characteristics of the three high-degree GGMs that were utilized in this study. In Table2, S stands for satellite (e.g., GRACE, GOCE, and LAGEOS), A for altimetry, and G for ground data (e.g., terrestrial, shipborne, and airborne measurements).

Table 2. Characteristics of the GGMs used in this study.

Geopotential Model Year Max. Degree Data Type GECO 2016 2190 S (GOCE), EGM2008 EIGEN-6C4 2014 2190 S (GOCE, GRACE, LAGEOS), A, G EGM2008 2008 2190 S (GRACE), A, G

The GGMs that were used in this study, as shown in Table2, were provided by ICGEM in the zero-tide system with respect to a geometrically ﬁxed GRS80 reference ellipsoid, and were calculated from GRACE and GOCE gravitational satellite observations, satellite-altimetry-derived gravity data, land gravity data, etc. For the geoid height accuracy evaluation in this study, high-degree GGMs developed from the combined contributions of various types of satellite tracking data, gravity data, and satellite altimetry data were considered, as shown in Table2. The geoid heights (NGGM) obtained through spherical harmonic synthesis from high-degree GGMs were interpolated into the 1182 GNSS/leveling UCPs to assess the accuracy of geoid heights derived from the three diﬀerent GGMs with maximum degree 2190. The GNSS/leveling geoid heights NGNSS/leveling at the 1182 UCPs were computed by subtracting the known orthometric height (Hleveling) calculated from the precise leveling between the BM and the Appl. Sci. 2020, 10, 1466 5 of 11

UCP from the known ellipsoidal height (hGNSS) obtained through GNSS observations, as given in EquationAppl. Sci. 20 (2).20, 10, x FOR PEER REVIEW 5 of 11 NGNSS/leveling = hGNSS Hleveling. (2) The GNSS-derived geoid heights are invariably different− from the values interpolated from a gravimetricallyThe GNSS-derived derived geoidgeoid heightsmodel, and are invariablyare influenced diff erentby the from datum the inconsistencies, values interpolated biases, from and a gravimetricallyerrors associated derived with the geoid independently model, and are derived influenced ellipsoidal by the and datum orthometric inconsistencies, height biases,data [21 and–24 errors]. An associatedexternal evaluation with the independently of the quality derived of a gravimetric ellipsoidal and geoid orthometric model or height GGM data can [ 21 be– 24performed]. An external by evaluation of the quality of a gravimetric푀표푑푒푙 geoid model or GGM can be performed by comparing its comparing its interpolated values (푁푖 ) at the 1182 UCPs with the corresponding GNSS/leveling- interpolated values NModel퐺푁푆푆at/푙푒푣푒푙푖푛푔 the 1182 UCPs with the corresponding GNSS/leveling-derived geoid derived geoid heightsi (푁푖 ). Such a comparison is conventionally based on the following GNSS/leveling heightsmodel, asNi given in Equation. Such (3) a comparison [22,23]: is conventionally based on the following model, as given in Equation (3) [22,23]: ∆푁 = 푁퐺푁푆푆/푙푒푣푒푙푖푛푔 − 푁푀표푑푒푙 = ℎ − 퐻 − 푁 푖 푖 GNSS/leveling 푖Model 푖 푖 푖, (3) ∆Ni = N N = hi Hi Ni, (3) i − i − − 퐺푁푆푆/푙푒푣푒푙푖푛푔 where ∆푁푖 stands for the relative geoid heights at the i-th GNSS/leveling points; 푁 for the 푖 GNSS/leveling where ∆N stands for the relative geoid heights at푀표푑푒푙 the i-th GNSS/leveling points; N for GNSS/levelingi geoid heights at the i-th points; 푁푖 for the geoid heights interpolatedi into the Model theGNSS/leveling GNSS/leveling points geoid from heights the gravimetric at the i-th points;geoid modelsNi andfor theGGMs; geoid ℎ푖 heights, 퐻푖, and 푁 interpolated푖 for the ellipsoidal into the GNSSheight,/leveling orthometric points height, from the and gravimetric geoid height, geoid respectively models and, at GGMs;the i-th hGNSS/levelingi, Hi, and Ni for points the. ellipsoidal height, orthometric height, and geoid height, respectively, at the i-th GNSS/leveling points. 3. Results and Discussions 3. Results and Discussions In this study, the geoid heights in South Korea were first calculated from the three KNGeoid modelsIn this and study, three the GGMs geoid to heights evaluat ine South the accuracy Korea were of the first geoid calculated models from on the the three 1182 KNGeoid GNSS/leveling models andUCPs three installed GGMs toby evaluate NGII, as the shown accuracy in Figure of the geoid1. Figure modelss 2 and on the3 show 1182 GNSSthe 1′ /griddedleveling UCPsgeoid installedheights byderived NGII, asfrom shown the inGGMs Figure (max1. Figuresimum degree2 and3 show2190) theand 1 from0 gridded the KNGeoid geoid heights models derived developed from theby NGII, GGMs (maximumrespectively. degree The 2190)1’ gridded and from geoid the height KNGeoid maps models were computed developed using by NGII, the “surface” respectively. routine The, 1’ which gridded is geoida continuous height maps curvature were computed surface using gridding the “surface” algorithm, routine, of which Generic is a continuous Mapping curvature Tools (GMT, surface griddinghtpp://gmt.soest.hawaii.edu) algorithm, of Generic [25 Mapping]. The geoid Tools heights (GMT, derived htpp:// fromgmt.soest.hawaii.edu the three GGMs ranged)[25]. Thefrom geoid 18.0 heightsto 33.4 m, derived as shown from in the Figure three 2. GGMsIn Figure ranged 3, the fromgeoid18.0 heights to 33.4 derived m, as from shown the inthree Figure KNGeoid2. In Figure models3, theare geoid distributed heights between derived 17.8 from and the 33.3 three m. KNGeoid models are distributed between 17.8 and 33.3 m.

FigureFigure 2. 2. TheThe 110′ gridded geoid geoid height height (푁(N퐺퐺푀GGM) derived) derived from from the the three three GGMs GGMs (max. (max. degree degree 2190) 2190).. (a) (aGECO) GECO;; (b ()b EIGEN) EIGEN-6C4;-6C4; and and (c ()c EGM2008.) EGM2008. The The attributes attributes listed listed for for this this and and subsequent subsequent maps maps include include thethe amplitude amplitude range range (AR (AR= = minimumminimum andand maximummaximum values),values), amplitude mean mean (AM), (AM), and and amplitude amplitude standardstandard deviation deviation (ASD). (ASD). The Thered reddots dots representrepresent thethe locationslocations ofof thethe 11821182 GNSSGNSS/leveling/leveling UCPs.

The three GGMs were interpolated into the GNSS/leveling data locations of the 1182 UCPs, which are shown as red dots in Figure1, to evaluate the accuracy of the geoid heights derived from the three GGMs. Figure4 shows the 1 0 gridded residual surface maps between the geoid heights derived from Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 11

Appl. Sci. 2020, 10, 1466 6 of 11 the GNSS/leveling on the 1182 UCPs and those derived from the three GGMs (GECO, EIGEN-6C4, and EGM2008). The statistics of the residual surface maps between the geoid heights NGNSS/leveling derived from the GNSS/leveling on the 1182 UCPs and those (NGGM) derived from the three GGMs, shown in Figure4a–c, respectively, are summarized in Table3. According to the summarized statistics in Table3, the geoid heights derived from the three GGMs as compared with those derived from the GNSS/leveling on the 1182 UCPs installed along the western coastal areas of South Korea, shown in Figure4a–c, respectively, had a maximum di ﬀerence of 1.837 m (126.425 E, 37.391 N), in GECO. − ◦ ◦ The RMSE of the residual surface maps between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from EGM2008, shown in Table3, is smaller than that of the residual surface maps between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from GECO, and between the geoid heights derived from the GNSS/leveling on the 1182 UCPs andAppl. those Sci. 20 derived20, 10, x FOR from PEER EIGEN-6C4. REVIEW 6 of 11

Figure 3. The 1′ gridded geoid heights (푁퐾푁퐺푒표푖푑) derived from the three KNGeoid models developed by NGII. (a) KNGeoid13; (b) KNGeoid14; and (c) KNGeoid18. The red dots represent the locations of the 1182 GNSS/leveling UCPs.

The three GGMs were interpolated into the GNSS/leveling data locations of the 1182 UCPs, which are shown as red dots in Figure 1, to evaluate the accuracy of the geoid heights derived from the three GGMs. Figure 4 shows the 1′ gridded residual surface maps between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from the three GGMs (GECO, EIGEN-6C4, and EGM2008). The statistics of the residual surface maps between the geoid heights

(푁퐺푁푆푆/푙푒푣푒푙푖푛푔) derived from the GNSS/leveling on the 1182 UCPs and those (푁퐺퐺푀) derived from the three GGMs, shown in Figure 4a–c, respectively, are summarized in Table 3. According to the summarized statistics in Table 3, the geoid heights derived from the three GGMs as compared with those derived from the GNSS/leveling on the 1182 UCPs installed along the western coastal areas of South Korea, shown in Figure 4a–c, respectively, had a maximum difference of −1.837 m (126.425° E, 37.391° N), in GECO. The RMSE of the residual surface maps between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from EGM2008, shown in Table 3, is smaller than that of the residual surface maps between the geoid heights derived from the GNSS/leveling on FigureFigure 3. 3.The The 1 01′gridded gridded geoidgeoid heights ((푁N퐾푁퐺푒표푖푑KNGeoid)) derivedderived from from the the three three KNGeoid KNGeoid models models developed developed theby 1182by NGII. NGII. UCPs (a ()a KNGeoid13;) KNGeoid13and those ;( b derived(b)) KNGeoid14; KNGeoid14 from ; GECO and ( c)), KNGeoid18 KNGeoid18. and between. The The the red red geoiddots dots represent represent heights the derived the locations locations from of of the GNSS/levelingthethe 1182 1182 GNSS GNSS/leveling on/leveling the 1182 UCPs. UCPs. UCPs and those derived from EIGEN-6C4.

The three GGMs were interpolated into the GNSS/leveling data locations of the 1182 UCPs, which are shown as red dots in Figure 1, to evaluate the accuracy of the geoid heights derived from the three GGMs. Figure 4 shows the 1′ gridded residual surface maps between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from the three GGMs (GECO, EIGEN-6C4, and EGM2008). The statistics of the residual surface maps between the geoid heights

(푁퐺푁푆푆/푙푒푣푒푙푖푛푔) derived from the GNSS/leveling on the 1182 UCPs and those (푁퐺퐺푀) derived from the three GGMs, shown in Figure 4a–c, respectively, are summarized in Table 3. According to the summarized statistics in Table 3, the geoid heights derived from the three GGMs as compared with those derived from the GNSS/leveling on the 1182 UCPs installed along the western coastal areas of South Korea, shown in Figure 4a–c, respectively, had a maximum difference of −1.837 m (126.425° E, 37.391° N), in GECO. The RMSE of the residual surface maps between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from EGM2008, shown in Table 3, is smaller than that of the residual surface maps between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from GECO, and between the geoid heights derived from the GNSS/leveling on the 1182 UCPs and those derived from EIGEN-6C4. Figure 4. Residual surface maps between the GNSS/leveling-derived geoid heights NGNSS/leveling on the 1182 UCPs and those (NGGM) derived from the three GGMs. (a) GECO; (b) EIGEN-6C4; and (c) EGM2008. The red dots represent the locations of the 1182 GNSS/leveling UCPs.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 11

Appl. Sci.Figure2020, 4.10 Residual, 1466 surface maps between the GNSS/leveling-derived geoid heights (푁퐺푁푆푆/푙푒푣푒푙푖푛푔) on 7 of 11 the 1182 UCPs and those (푁퐺퐺푀) derived from the three GGMs. (a) GECO; (b) EIGEN-6C4; and (c) EGM2008. The red dots represent the locations of the 1182 GNSS/leveling UCPs. Table 3. Statistics of the diﬀerences between the geoid heights NGNSS/leveling derived from the GNSSTable/ leveling 3. Statistics on the of 1182 the differences UCPs and those between(NGGM the) geoidderived heights from the(푁퐺푁푆푆 three/푙푒푣푒푙푖푛푔 GGMs) derived (unit, m). from the

GNSS/leveling on the 1182 UCPs and those (푁퐺퐺푀) derived from the three GGMs (unit, m). Min. Max. Mean Std. Dev. RMSE Min. Max. Mean Std. Dev. RMSE NGNSS/leveling NGECO 1.837 1.025 0.199 0.126 0.236 푁 − −푁 − − N퐺푁푆푆GNSS/푙푒푣푒푙푖푛푔/leveling NEIGEN퐺퐸퐶푂 6C4 −1.8371.817 1.0251.010 −0.1990.191 0.1120.126 0.2210.236 − − − − 푁퐺푁푆푆N/푙푒푣푒푙푖푛푔 − 푁퐸퐼퐺퐸푁NEGM−6퐶4 −1.8171.801 1.1.012010 −0.1910.184 0.1130.112 0.2160.221 GNSS/leveling − 2008 − − 푁퐺푁푆푆/푙푒푣푒푙푖푛푔 − 푁퐸퐺푀2008 −1.801 1.012 −0.184 0.113 0.216 From the histograms of the differences between the GNSS/leveling-derived geoid heights on From the histograms of the differences between the GNSS/leveling-derived geoid heights on the the 1182 UCPs and the three GGMs-derived geoid heights presented in Figure5a–c, the percentage 1182 UCPs and the three GGMs-derived geoid heights presented in Figure 5a–c, the percentage of of the absolute values of differences between the GNSS/leveling geoid heights on the 1182 UCPs the absolute values of differences between the GNSS/leveling geoid heights on the 1182 UCPs and and the geoid heights derived from the three GGMs was calculated, as shown in Table4. As can the geoid heights derived from the three GGMs was calculated, as shown in Table 4. As can be seen be seen in Table4, the agreement between the geoid heights derived from the GNSS /leveling on the in Table 4, the agreement between the geoid heights derived from the GNSS/leveling on the 1182 1182 UCPs and the EGM2008-derived geoid heights was 62.0% when the absolute values of differences UCPs and the EGM2008-derived geoid heights was 62.0% when the absolute values of differences were 20 cm, whereas for the other GGMs, the consistency level was 49.7% and 56.9% for GECO were 20 cm, whereas for the other GGMs, the consistency level was 49.7% and 56.9% for GECO and and EIGEN-6C4, respectively. Moreover, more than 90% of the 1182 UCPs showed an agreement EIGEN-6C4, respectively. Moreover, more than 90% of the 1182 UCPs showed an agreement between between the EIGEN-6C4- and GNSS/leveling-derived geoid heights and between the EGM2008- and the EIGEN-6C4- and GNSS/leveling-derived geoid heights and between the EGM2008- and GNSSGNSS/leveling/leveling-derived-derived geoidgeoid heightsheights thatthat is better than 30 30 cm cm as as compared compared with with 84.9% 84.9% for for the the GECO GECO GGM.GGM. AsAs aa result,result, wewe concludedconcluded that the EGM EGM20082008 model model is is the the more more suitable suitable GGM GGM from from among among the the threethree GGMsGGMs forfor determiningdetermining aa suitablesuitable gravimetric geoid model model for for South South Korea. Korea.

Figure 5. Histograms of the differences between the geoid heights (푁 ) derived from Figure 5. Histograms of the diﬀerences between the geoid heights N퐺푁푆푆GNSS/푙푒푣푒푙푖푛푔/leveling derived from (푁 ) GNSSGNSS//levelingleveling on on the the 11821182 UCPsUCPs and and the the geoid geoid heights heights (N퐺퐺푀GGM) derivedderived from from the the three three KNGeoid KNGeoid models.models. ((aa)) GECO;GECO; ((b)) EIGEN-6C4;EIGEN-6C4; and ( c)) EGM2008 EGM2008..

TableTable 4. 4.Percentage Percentage ofof thethe 1182 UCPs whose absolute values values of of differences diﬀerences between between the the geoid geoid height heightss derivedderived fromfrom thethe GNSSGNSS/leveling/leveling andand thosethose derivedderived from the three GGM GGMs.s.

Model 0.05 m 0.10 m 0.20 m 0.30 m 0.40 m 0.50 m Model ≤≤0.05 m ≤0.10≤ m ≤0.20≤ m ≤0.30≤ m ≤0.40≤ m ≤0.50≤ m GECO 71 (6.0%) 158 (13.4%) 587 (49.7%) 1003 (84.9%) 1142 (96.6%) 1173 (99.2%) EIGEN-6C4GECO 71 25 (6.0%) (2.1%) 158 86(13.4%) (7.3%) 587 673(49.7%) (56.9%) 1003 1108 (84.9%) (93.7%) 1142 1150 (96.6%) (97.3%) 1173 1168 (99.2%) (98.8%) EGM2008 40 (3.4%) 122 (10.3%) 733 (62.0%) 1086 (91.9%) 1152 (97.5%) 1172 (99.1%) EIGEN-6C4 25 (2.1%) 86 (7.3%) 673 (56.9%) 1108 (93.7%) 1150 (97.3%) 1168 (98.8%) In addition, for the assessment of the geoid heights of the KNGeoid models, the three KNGeoid EGM2008 40 (3.4%) 122 (10.3%) 733 (62.0%) 1086 (91.9%) 1152 (97.5%) 1172 (99.1%) models were interpolated into the GNSS/leveling data locations of the 1182 UCPs, which are shown as red dots in Figure1. Figure6 shows the 1 gridded residual surface maps between the GNSS/leveling 0 geoid heights NGNSS/leveling of the 1182 UCPs and the geoid heights (NKNGeoid) derived from the three KNGeoid models (KNGeoid13, KNGeoid14, and KNGeoid18). Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 11

In addition, for the assessment of the geoid heights of the KNGeoid models, the three KNGeoid models were interpolated into the GNSS/leveling data locations of the 1182 UCPs, which are shown as red dots in Figure 1. Figure 6 shows the 1′ gridded residual surface maps between the Appl.GNSS/ Sci. 2020leveling, 10, 1466 geoid heights (푁퐺푁푆푆/푙푒푣푒푙푖푛푔) of the 1182 UCPs and the geoid heights (푁퐾푁퐺푒표푖푑8 of) 11 derived from the three KNGeoid models (KNGeoid13, KNGeoid14, and KNGeoid18).

Figure 6. Residual surface maps between the GNSS/leveling-derived geoid heights (푁 ) on Figure 6. Residual surface maps between the GNSS/leveling-derived geoid heights퐺푁푆푆(N/푙푒푣푒푙푖푛푔GNSS/leveling) the 1182 UCPs and those (푁 ) derived from the three KNGeoid models. (a) KNGeoid13; (b) on the 1182 UCPs and those (N퐾푁퐺푒표푖푑KNGeoid) derived from the three KNGeoid models. (a) KNGeoid13; (b) KNGeoid14;KNGeoid14 and; and (c )(c KNGeoid18.) KNGeoid18 The. The red red dots dots representrepresent the locations of of the the 1182 1182 GNSS/leveling GNSS/leveling UCPs. UCPs.

TheThe statistics statistics of the of ditheﬀerences differences between between the geometric the geometric geoid heights geoid derived heights from derived GNSS /leveling from onGNSS/leveling the 1182 UCPs on and the the 1182 geoid UCPs heights and the derived geoid heights from the derived three from KNGeoid the three models KNGeoid are represented models are as shownrepresented in Table as5. shown The di ﬀinerences Table 5. between The differences the GNSS between/leveling the geoid GNSS/leveling heights and geoid the KNGeoid18heights and geoidthe heightsKNGeoid18 represent geoid good heights agreement represent with good the agreement smallest RMSE with the (10.3 smallest cm) in RMSE Table 5(10.3. cm) in Table 5.

Table 5. Statistics of the differences between the geoid heights (푁퐺푁푆푆/푙푒푣푒푙푖푛푔) derived from the Table 5. Statistics of the diﬀerences between the geoid heights (NGNSS/leveling) derived from the GNSS/leveling on the 1182 UCPs and those (푁퐾푁퐺푒표푖푑) derived from the three KNGeoid models (unit: GNSS/leveling on the 1182 UCPs and those (NKNGeoid) derived from the three KNGeoid models m). (unit: m). Min. Max. Mean Std. dev. RMSE Min. Max. Mean Std. dev. RMSE 푁퐺푁푆푆/푙푒푣푒푙푖푛푔 − 푁퐾푁퐺푒표푖푑13 −1.749 1.061 −0.033 0.107 0.112 N N 1.749 1.061 0.033 0.107 0.112 GNSS/leveling − KNGeoid13 − − 푁N퐺푁푆푆/푙푒푣푒푙푖푛푔 − N푁퐾푁퐺푒표푖푑14 −1.7071.707 1.0971.097 −0.0010.001 0.1060.106 0.106 0.106 GNSS/leveling − KNGeoid14 − − 푁NGNSS/leveling − N푁KNGeoid18 1.770 1.066 0.015 0.102 0.103 퐺푁푆푆/푙푒푣푒푙푖푛푔 − 퐾푁퐺푒표푖푑18 −−1.770 1.066 −0.015 0.102 0.103

TableTable6 presents 6 presents the the percentage percentage of of the the absolute absolute valuesvalues ofof didifferencesfferences between between the the GNSS/leveling GNSS/leveling geoidgeoid heights heights on on the the 1182 1182 UCPs UCPs and and the the geoidgeoid heightsheights derived from from the the three three KNGeoid KNGeoid models. models. From From thethe histograms histograms of of the the di differencesfferences between between thethe GNSSGNSS/leveling/leveling-derived-derived geoid geoid heights heights on on the the 1182 1182 UCPs UCPs andand the the three three KNGeoid KNGeoid model-derived model-derived geoid geoid heightsheights presented in in Figure Figure 77aa–c,–c, it it is is k knownnown that that 88.6% 88.6% of theof the absolute absolute values values of of di differencesfferences between between thethe GNSSGNSS/leveling/leveling-derived-derived geoid geoid heights heights on on the the 1182 1182 UCPs and the KNGeoid13-derived geoid heights were below 10.0 cm, whereas 91.3% and 93.7% of UCPs and the KNGeoid13-derived geoid heights were below 10.0 cm, whereas 91.3% and 93.7% of the the differences between the GNSS/leveling-derived geoid heights on the 1182 UCPs and the differences between the GNSS/leveling-derived geoid heights on the 1182 UCPs and the KNGeoid14- KNGeoid14- and KNGeoid18-derived geoid heights, respectively, were less than 10.0 cm. On the and KNGeoid18-derived geoid heights, respectively, were less than 10.0 cm. On the basis of the small basis of the small differences between the GNSS/leveling-derived geoid heights on the 1182 UCPs differences between the GNSS/leveling-derived geoid heights on the 1182 UCPs and the KNGeoid18 and the KNGeoid18 model-derived geoid heights, it was concluded that the most recently developed model-derivedKNGeoid18 model geoid heights, is better it than was the concluded KNGeoid13 that the and most KNGeoid14 recently models developed as a KNGeoid18 gravimetric model geoid is bettermodel than for the South KNGeoid13 Korea. and KNGeoid14 models as a gravimetric geoid model for South Korea.

Table 6. Percentage of the 1182 UCPs whose absolute values of diﬀerences between the geoid heights derived from the GNSS/leveling and those derived from the three KNGeoid models.

Model 0.02 m 0.05 m 0.10 m 0.15 m 0.20 m 0.50 m ≤ ≤ ≤ ≤ ≤ ≤ KNGeoid13 352 (29.8%) 764 (64.6%) 1047 (88.6%) 1107 (93.7%) 1128 (95.4%) 1174 (99.3%) KNGeoid14 471 (39.8%) 899 (76.1%) 1079 (91.3%) 1119 (94.7%) 1133 (95.9%) 1173 (99.2%) KNGeoid18 549 (46.4%) 950 (80.4%) 1108 (93.7%) 1131 (95.7%) 1141 (96.5%) 1173 (99.2%) Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 11

Table 6. Percentage of the 1182 UCPs whose absolute values of differences between the geoid heights derived from the GNSS/leveling and those derived from the three KNGeoid models.

Model ≤0.02 m ≤0.05 m ≤0.10 m ≤0.15 m ≤0.20 m ≤0.50 m

KNGeoid13 352 (29.8%) 764 (64.6%) 1047 (88.6%) 1107 (93.7%) 1128 (95.4%) 1174 (99.3%)

KNGeoid14 471 (39.8%) 899 (76.1%) 1079 (91.3%) 1119 (94.7%) 1133 (95.9%) 1173 (99.2%)

Appl. Sci. 2020, 10, 1466 9 of 11 KNGeoid18 549 (46.4%) 950 (80.4%) 1108 (93.7%) 1131 (95.7%) 1141 (96.5%) 1173 (99.2%)

Figure 7. Histograms of the differences between the geoid heights (푁 ) derived from Figure 7. Histograms of the diﬀerences between the geoid heights (N퐺푁푆푆GNSS/푙푒푣푒푙푖푛푔/leveling) derived from GNSS/leveling on the 1182 UCPs and the geoid heights (푁 ) derived from the three KNGeoid GNSS/leveling on the 1182 UCPs and the geoid heights (N퐾푁퐺푒표푖푑KNGeoid) derived from the three KNGeoid models.models. (a)) KNGeoid13;KNGeoid13; ( b) KNGeoid14 KNGeoid14;; and and ( (cc)) KNGeoid18 KNGeoid18..

4.4. ConclusionsConclusions InIn thisthis study,study, thethe accuracyaccuracy of the geoid heights heights derived derived from from the the recently recently released released Earth Earth gravity gravity modelmodel basedbased on gravity data calculated calculated from from the the Gravity Gravity Recovery Recovery and and Climate Climate Experiment Experiment (GRACE) (GRACE) andand thethe GravityGravity FieldField and and Steady-State Steady-State Ocean Ocean Circulation Circulation Explorer Explorer (GOCE) (GOCE) satellites satellites was was evaluated evaluated by comparingby comparing it with it the with geoid the heights geoid derived heights from derived the Global from Navigation the Global Satellite Navigation System Satellite (GNSS) System/leveling on(GNSS)/leveling the 1182 unified on control the 1182 points unified (UCPs) control installed points by (UCPs) the National installed Geographic by the National Information Geographic Institute (NGII)Information all over Institute South Korea.(NGII) Theall over following South conclusionsKorea. The following were obtained conclusions in this study:were obtained in this studyFirst: The geoid heights derived from the three high-degree global geopotential models (GGMs) and theFirst three The Korean geoid heights National derived Geoid from (KNGeoid) the three models high- presenteddegree global similar geopotential distributions models ranging (GGMs) from 17and to the33m three around Korean South National Korea. Geoid (KNGeoid) models presented similar distributions ranging fromSecond 17 to 33The m around EGM2008 South model Korea. showed a rather stable result for the root-mean-square-error (RMSE) Second The EGM2008GNSS/leveling modelModel showed a rather stable result for the root-mean-square-error of the residuals Ni Ni that were considered in this study in terms of relative geoid (RMSE) of the residuals (푁퐺푁푆푆− /푙푒푣푒푙푖푛푔 − 푁푀표푑푒푙) that were considered in this study in terms of heights (∆Ni) accuracy. Thus,푖 the EGM2008푖 model is a more suitable model than the GECO and EIGEN-6C4relative geoid models heights as ( compared∆푁푖) accuracy. with Thus, the GNSS the EGM2008/leveling model geoid heightsis a more all suitable over South model Korea. than Asthe a result,GECO theand EGM2008 EIGEN-6C4 model models could as becompared selected with as the the suitable GNSS/leveling GGM from geoid among heights the threeall over GGMs South for determiningKorea. As a result, a gravimetric the EGM2008 geoid model could for South be selected Korea. as the suitable GGM from among the three GGMsThird for determiningAmong the three a gravimetric KNGeoid geoid models, model the mostfor South recently Korea developed. KNGeoid18 model showed betterThird results A asmong a gravimetric the three geoid KNGeoid model models, all over the South most Korea recently than the developed KNGeoid13 KNGeoid18 and KNGeoid14 model modelsshowed as better compared results with as athe gravimetric GNSS/leveling-derived geoid model all geoid over heights.South Korea than the KNGeoid13 and KNGeoid14For further models study as compared of the new with gravimetric the GNSS/leveling geoid model,-derived it is geoid necessary heights to. perform an accuracy assessmentFor further of the study geoid of model the new all gr overavimetric South Koreageoid model, by adding it is gravitynecessary data to andperform by collocating an accuracy the GNSSassessment/leveling-derived of the geoid geoid model heights. all over South Korea by adding gravity data and by collocating the GNSS/leveling-derived geoid heights. Author Contributions: Original idea and methodology, writing, and data analysis, K.B.K.; project administration and supervision, H.S.Y.; review and editing, H.J.C. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by a National Research Foundation of Korea (NRF) grant funded by the South Korean government (MSIT) (NRF-2019R1C1C1005590). Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2020, 10, 1466 10 of 11

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