Abstract: This paper briefly discusses the data and methodology used in the construction of a Hong Kong local geoid of centimeter accuracy HKGEOID-2000. The results have been published in the website of Department of Land obtain the Surveying and Geo-Informatics, the Hong Kong Polytechnic University for public use. Users can directly geoidal height of a point by inputting its geodetic coordinates (in ITRF96 reference frame) or the HK1980 grid coordinates. Examples are given to illustrate the procedures

Simon Kwok

Yong-Qi Chen

Zhicai Luo

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Journal of Geospatial Engineering, Vol. 5,  No.2 (December 2003), pp.35-41. 
Copyright   The Hong Kong Institution of Engineering Surveyors 
 
35
Precise Hong Kong Geoid HKGEOID-2000 
Yong-Qi Chen 1,  Zhicai Luo 1 and Simon Kwok 2 
1.Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University 
2 Geodetic Survey Section, Lands Department, HKSAR 
Abstract:  
This paper briefly discusses the data and methodology used in the construction of a Hong Kong 
local geoid of centimeter accuracy HKGEOID-2000. The results have been published in the 
website of Department of Land obtain the Surveying and Geo-Informatics, the Hong Kong 
Polytechnic University for public use. Users can directly geoidal height of a point by inputting its 
geodetic coordinates (in ITRF96 reference frame) or the HK1980 grid coordinates. Examples are 
given to illustrate the procedures.  
1 Introduction 
The heights of points on land are traditionally determined by leveling technique. The leveling is a 
time-consuming and labor-intensive operation process, though it provides precise information on 
the heights of points above the mean sea level. With the development of GPS survey technique the 
height of points can be determined with much ease. However, GPS derived heights are referred to 
WGS-84 ellipsoid, while the leveled heights are referred to geoid. Their difference is called 
geoidal height or geoid undulation. In practices, most of survey projects require leveled heights, 
for they have physical meaning. Therefore precise geoid must be available to transform GPS 
derived heights to leveled heights. 
With the support of Research Grant Council (RGC) of Hong Kong Government the authors 
conducted a project “precise determination of Hong Kong geoid using heterogeneous data”. Its 
main objective is to construct Hong Kong local geoid with centimeter accuracy using the existing 
data and information. The project has completed with a product HKGEOID-2000. HKGEOID-
2000 has one-kilometer resolution and covers the whole territory of Hong Kong, from 800km to 
850km in northing and 800km to 870km in easting in the HK19980 grid coordinate system. The 
tests showed that it can provide geoidal height at any point within the territory with accuracy better 
than 1.5 cm. The product is now ready for local professional community to use. 
This paper firstly outlines all aspects involved in the practical computation and evaluation of the 
HKGEOID-2000. Examples are then given to demonstrate the way to use the product. 
2 Data and Information Used for the Construction of HKGOID-2000 
2.1 Digital terrain model:  
To compute terrain effects DTM of Hong Kong and its neighboring region must be available. 16 
sheets of 1:20 000 digital topographical maps covering the whole territory of Hong Kong were 
used to generate DTMs with resolution 100m using software 3D Analyst of ArcView GIS version 
3.1. 100m resolution DTM of Shenzhen was also obtained.  
2.2 GPS/leveling data: 
55 high quality GPS/leveling stations in Hong Kong are available. Their ellipsoidal heights (h) are 
referenced to WGS84 ellipsoid (ITRF 96 at 1998:121), and their leveled heights (H) above the 
Hong Kong Principal Datum were determined with precise geometric leveling or trigonometric 
leveling. The geoidal height of a point, Nobs (treated as “observation”) can be computed from: 
HhN Obs −=                                                                                   (1) 
 Chen, YQ,  Luo,  ZC and Kwok, S. 
 36 
The estimated accuracy of so derived geoidal height is at level of 1~2 cm for 17 stations, 3~5 cm 
for 26 stations, and no better than 10 cm for 12 stations. Figure 1 shows the distribution of these 
GPS/leveling stations and their accuracy.  
 
Figure 1 Distribution of GPS/leveling stations and their accuracy 
2.3 Gravity data:  
Hong Kong has 640 discrete gravity observations with station spacing 2 km on land and 2-4 km on 
sea. They were collected using Lacoste and Romberg model ‘G’ land gravity meter and model 
‘H/U’ seabed gravity meter by Electronic and Geophysical Services Ltd. (EGS, 1988; 1991). The 
local gravity base is connected to International Gravity Standardization Net 1971 (IGSN 71) with 
accuracy of 0.03 mGal (Evans, 1990). The gravity measurements in neighboring region Shenzhen 
are available: 3609 points on land and 1262 points on sea with 1 km resolution. They were 
measured with Lacoste & Romberg model ‘G’ and ‘D’ land gravimeter and model ‘S’ sea 
gravimeter in 2001 (Ning, et al., 2003). We selected 2158 gravity measurements near Hong Kong 
territory together with 640 gravity measurements in Hong Kong for the determination of 
HKGEOID-2000. Figure 2 shows the distribution of these gravity measurements.  
Precise Hong Kong Geoid HKGEOID-2000 
 37
 
Figure 2 Distribution of gravity observations over Hong Kong and its neighboring region  
2.4 Selection of geo-potential model  
There are several geo-potential models in the world. After a detail evaluation of these models 
using the data in the region, we select WDM94 for the project which is best fitted into the data. 
For the detail evaluation of different models the readers are referred to (Luo and Chen, 2002). 
3 Outline of the Computation Procedure for HKGEOID-2000 
A hybrid approach was used in this project. For the detail the readers are referred to (Chen and 
Yang, 2001; Luo and Chen, 2002). 
3.1 Determination of a gravimetric geoid 
The remove-restore technique was used to construct the Hong Kong gravimetric geoid (Yang and 
Chen 2001). With the technique the long-medium wavelength component GMg∆ derived from a 
geopotential model and the short wavelength component c due to the topographic effect are 
removed mathematically from the observed gravity anomalies ∆g,   
),(),(),(),( λϕλϕλϕλϕδ cggg GM +∆−∆=                                 (2) 
 Chen, YQ,  Luo,  ZC and Kwok, S. 
 38 
And then the Stokes′ integration of the remaining parts of the gravity anomalies (called residual 
gravity anomaly) provides the residual geoid height N(ϕp, λp). The final geoid heights of points are 
obtained by restoring the previous removed components, i.e., the gravimetric geoid at point 
),( PPP λϕ , denoted as ),( PPGN λϕ , and topographic indirect effect on the geoid, 
denoted ),( PP
TN λϕ , as 
),(),(),(),( PP
T
PP
GM
PPPP
G NNNN λϕλϕλϕλϕ ++=            (3) 
3.2 Model systematic biases 
The gravimetric geoid may contain systematic biases with respect to GPS/leveling derived geoid. 
The biases may come from the difference between the gravimetric geoid and the leveling datum, 
and the long-wavelength systematic errors. To remove these systematic biases a least squares 
fitting was carried out with the following transformation model: 
xyayaxaayxNyxN GObs 3210),(),( ++++=                         (4) 
where x and y are grid coordinates of a point. 
3.3 Refine the constructed gravimetric geoid 
To improve the accuracy of the geoid, the GPS/Leveling data should be fully utilized. Therefore 
local Shepard surface interpolation method is used to interpolate the residuals at one-kilometer 
grid, i.e., 




=
≠⋅
= ∑∑ ==
0  ,                                           ),(
0  ,     )()(),(
),( 11
iii
i
n
i
m
i
n
i
m
iii
PP
rN
rrKrKN
N
λϕδ
λϕδ
λϕδ       (5) 
where Nδ  is the difference between the GPS/leveling derived geoid height and the transformed 
gravimetric geoid height, ir  is the distance between interpolating point ),( PP λϕ  and known point 
),( ii λϕ , m  the fitting power should be integer ),2,1( L=m , and )( irK  the kernel function or 
so-called weighting function, i.e., 






>
≤<−
≤<
=
Dr ,                          0
3D ,        )1(
4
27
3Dr0 ,                       1
)( 2 Dr
D
r
D
r
rK                                     (6) 
with D interpolating or searching radius. The final geoid was obtained by adding the residuals at 
one-kilometer grid to the above transformed geoid. Such derived geoid is called hybrid or 
combined geoid, which named HKGOID-2000. Figure 3 shows the HKGEOID-2000. 
Precise Hong Kong Geoid HKGEOID-2000 
 39
 
Figure 3 Contour map of HKGEOID-2000 at 1km grid  
(HK1980 Grid Coordinates System, contour interval 0.2m) 
4 Evaluation of HKGEOID-2000 
23 precise GPS/leveling stations, which not used in the construction of HKGEOID-2000, were 
employed to assess the accuracy of HKGEOID-2000.  The estimated accuracy of the geoid heights 
derived from these GPS/leveling data is better than 2 cm. Figure 4 shows the distribution of these 
GPS/leveling stations. The accuracy of HKGEOID-2000 is evaluated by comparing the geoid 
heights at these stations derived from the model NModel with that derived from GPS/Leveling Nobs in 
absolute or relative sense, i.e. 
ModelObs NNN −=∆                                                                     (7) 
and 
ModelObs NNN ∆−∆=δ                                                                 (8) 
The comparison shows that the root mean square (RMS) of N∆  is 2.2 cm, and that of δN is better 
than 1 ppm for base line length longer than 25 km. Taking into consideration of the geoid heights 
derived from the GPS/leveling data, the practical accuracy of HKGEOID-2000 should be better 
than 1.6 cm (assuming that the errors of the data and geoid model have similar size), and relative 
accuracy better than 1 ppm in 20 km. 
 Chen, YQ,  Luo,  ZC and Kwok, S. 
 40 
 
Figure 4 Distribution of 23 GPS/leveling check stations  
5 Application 
The constructed Hong Kong geoid has been published in the following website for public use. 
Users input the position of a point in either geodetic coordinates or the HK1980 grid coordinates 
and obtain geoid height as output. 
 http:///www.lsgi.polyu.edu.hk/project_HKGEOID-2000/index.htm. 
Example 1. Given the geodetic coordinates (ITRF96 reference frame) of a point: Latitude = 22 31 
36.77259, Longitude = 114 12 27.86855, Ellipsoidal height h = 35.413, compute the leveled height. 
 Go to the website and click “get the geoid “, then input the latitude and longitude values. It 
gives you geoid height N = – 2.972m. The leveled height of the point H is computable from 
 H = h – N = 35.413 – (-2.972) = 38.385 m 
Example 2. Given the HK1980 grid coordinates of a point as E = 850000.000m, N=830000.000m . 
Get its geiod height.  
 The result N = -2.344m 
Acknowledgements 
This research was supported by Research Grant Council of Hong Kong SAR government 
(B.34.37.Q328). The GPS/Leveling data of Hong Kong was provided by Geodetic Survey Section, 
Survey and Mapping Office, Lands Department, Hong Kong. And the gravity data of Shenzhen 
were supplied by Land Planning Department of Shenzhen, China. 
Precise Hong Kong Geoid HKGEOID-2000 
 41
References 
ChenYQ and Yang ZJ (2001). A hybrid method to determine the Hong Kong geoid, Presented in 
FIG Working Week, May, 2001, South Korea. 
Electronic and Geophysical Services Ltd, 1988, High density gravity survey--Sheet 6 NW 8, Final 
report, Job Number HK 36088, Hong Kong. 
Electronic and Geophysical Services Ltd., 1991, Regional gravity survey of Hong Kong, Final 
report, Job Number HK 50190, Hong Kong. 
Evans, R.B., 1990, Hong Kong gravity observations in July 1990 with BGS Lacoste and Romberg 
meter No. 97 and international connections to IGSN 71, British and Geology Survey 
Technical report WK/90/24R. 
Luo ZC and Chen YQ (2002) Precise determination of Hong Kong geoid using heterogeneous data, 
Proceeding of FIG XXII International Congress, Washington, D.C. USA, April 19-26 2002 
Luo ZC and Chen YQ (2002). Evaluation of Geo-potential Models EGM96, WDM94 and 
GPM98CR in Hong Kong and Shenzhen. Journal of Geospatial Engineering, Vol.4, No. 1, 21-
30, June 2002.  
Ning JS., Luo ZC, Yang ZJ, Chen YQ, and Zhang TJ (2003). Determination of Shenzhen geoid 
with 1-km resolution and centimeter accuracy.  Acta  Geodaetica et Cartographica Sinica (in 
press).  
Yang ZJ and  Chen YQ (2001). Determination of the Hong Kong gravimetric geoid. Survey 
Review, Vol. 36, No 279, pp. 23-34. 
 


Precise Hong Kong Geoid HKGEOID-2000

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Precise Hong Kong Geoid HKGEOID-2000

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