ABSTRACT IZVLEČEK 
KLJUČNE BESEDE KEY WORDS
Geoid, quasigeoid, geopotential, vertical datum, disturbing 
potential, height anomaly
geoid, kvazigeoid, težnostni potencial, višinski datum, moteči 
potencial, anomalija višine
Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
ESTIMATION OF VERTICAL DATUM OFFSET FOR THE SOUTH AFRICAN VERTICAL DATUM, IN RELATION TO THE INTERNATIONAL HEIGHT REFERENCE SYSTEM | 282-297 |
UDK:  528.242
 Klasifikacija prispevka po COBISS.SI: 1.01
Prispelo: 24. 2. 2021
Sprejeto: 28. 4. 2021
DOI: 10.15292/geodetski-vestnik.2021.02.282-297
SCIENTIFIC ARTICLE
Received: 24. 2. 2021
Accepted: 28. 4. 2021
Matthews Siphiwe Mphuthi, Patroba Achola Odera
OCENA RAZLIK MED 
VIŠINSKIM DATUMOM 
JUŽNE AFRIKE IN DATUMOM 
MEDNARODNEGA 
VIŠINSKEGA REFERENČNEGA 
SISTEMA
ESTIMATION OF VERTICAL 
DATUM OFFSET FOR THE 
SOUTH AFRICAN VERTICAL 
DATUM, IN RELATION TO THE 
INTERNATIONAL HEIGHT 
REFERENCE SYSTEM
The vertical offset and the geopotential value over South 
Africa is estimated on the four fundamental benchmarks in 
relation to the international height reference system (IHRS). 
It is estimated to obtain discrepancies between the South 
African local vertical datum (W_P) and the global vertical 
datum (W_0). A single-point-based geodetic boundary 
value problem (GBVP) approach was used following 
Molodensky theory for estimating the height anomalies from 
the disturbing potential (T_P) using Bruns’s formula. The 
gravity potential at each tide gauge benchmark (TGBM) 
in South Africa deviates from the potential of the global 
reference surface by0.589,-1.993,-2.593 and 2.154 m
2
s
-2 
for Cape T own, Port Elizabeth, East London and Durban, 
respectively. The corresponding vertical datum offsets 
between the international height reference system and 
the four fundamental benchmarks over South Africa are 
6.013, -20.347, -26.478, and 21.996 cm for Cape T own, 
Port Elizabeth, East London and Durban, respectively. 
These offsets can be used for the unification of the South 
African vertical datum at the four tide gauge benchmarks 
in a manner that is consistent to the international height 
reference system.
Razlika in vrednost težnostnega potenciala v Južni Afriki 
se ocenjujeta na štirih temeljnih referenčnih točkah, in 
sicer s primerjavo z mednarodnim višinskim referenčnim 
sistemom IHRS. Predvideva se, da obstajajo razlike med 
višinskim datumom Južne Afrike (W
P
) in globalnim 
višinskim datumom (W
0
). V raziskavi je bil uporabljen 
pristop rešitve problema geodetskega robnega pogoja za eno 
točko (angl. geodetic boundary value problem – GBVP ), 
kjer smo z Brunsovo enačbo ocenili vrednost anomalije 
višine po teoriji Molodenskega iz tako imenovanega 
motečega potenciala (T
P
). Na obravnavanih mareografih 
v Južni Afriki je težnostni potencial odstopal od globalnih 
referenčnih vrednostih za naslednje vrednosti: Cape Town  
0,589 m
2
s
-2
, Port Elizabeth-1,993 m
2
s
-2
, East London 
-2,593 m
2
s
-2
, Durban2,154 m
2
s
-2
. Odmik višinskega 
datuma na obravnavanih točkah glede na mednarodni 
višinski referenčni sistem je tako 6,013 cenitmetra v Cape 
Townu, -20,347 cenitmetra v Port Elisabethu, -26,478 
cenitmetra v East Londonu in 21,996 cenitmetra v 
Durbanu. Ugotovljene razlike se lahko uporabljajo za 
uskladitev višinskega datuma Južne Afrike z mednarodnim 
višinskim referenčnim sistemom.
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V
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Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
ESTIMATION OF VERTICAL DATUM OFFSET FOR THE SOUTH AFRICAN VERTICAL DATUM, IN RELATION TO THE INTERNATIONAL HEIGHT REFERENCE SYSTEM | 282-297 |
1 INTRODUCTION
The resolution for the development of an international height reference system (IHRS) was released by 
the International Association of Geodesy (IAG) in July 2015 (IAG, 2015). The IHRS was developed to 
provide a global vertical reference system of high precision. This will provide support in monitoring global 
changes, geohazards, and prediction of several Earth’s science phenomena. The IHRS is defined by an 
equipotential surface of the Earth’s gravity field realised by a conventional value, W
0
  62,636,853.4 m
2
s
-2
 
(Burša et al., 2001; 2004; Sánchez et al., 2016; Sánchez and Sideris, 2017)Australian Height Datum 
1971 (AHD 71. However, a number of recent researches have shown that this value may have increased 
by 1  2 m
2
s
-2
 (Rülke et al., 2013; Albarici et al., 2019). The value of  in practice depends on the realisa -
tion of the vertical datum (Amjadiparvar et al., 2013)we compute the offsets of three height datums in 
North America (NAVD88, CGVD28 and Nov07.
The South African land levelling datum (LLD) has been providing the reference frame for a variety 
of practical applications, such as the construction of roads, the development of infrastructures and a 
variety of developmental activities in the country. The South African LLD was realized over a century 
ago, based on mean sea level (MSL) observations from four tide gauge stations (situated in Cape T own, 
Port Elizabeth, East London, Durban). It was connected to the national benchmark network by primary 
levelling networks, which were adjusted independently. In addition, heights measured above LLD are 
classified as spheroidal orthometric height system. This height system provides a poor approximation 
of the true orthometric height system. However, it is estimated to be more closer to the normal height 
system (Merry, 1985).
In this height system, the spheroidal orthometric correction was applied to all the height differences 
from the primary levelling networks, computed from normal gravity. However, the orthometric cor -
rection was computed for only four levelling loops around Cape Town, meaning that the actual gravity 
measurements were taken for only those loops (Merry, 1977, 1985; Wonnacott & Merry, 2011). The 
spheropotential number is used in this height system instead of the geopotential number which is derived 
from the normal gravity (Odumosu et al., 2015).
The South African vertical datum suffer from a number of problems such as; numerous errors from the 
levelling networks and tide gauge sea level measurements, instability due to high MSL variability, and 
it was established from inconsistent levelling networks, just to mention a few. In addition, it has been 
established by Merry (2003) that the South African LLD is 15–20 cm below the mean sea level. Therefore, 
in order for South Africa to meet the standards of the global vertical datum, the South African vertical 
datum must be unified and also be defined by a gravity potential value. This will provide South Africa 
with a modernised vertical datum. 
To achieve this, the South African vertical datum should be defined by means of a geoid model; this 
approach will solve some of the problems associated with the LLD. The main focus of this study is to 
estimate the vertical datum offset for the South African vertical datum, at the four fundamental bench -
marks, in relation to the IHRS. The national primary levelling networks can be adjusted using geopotential 
difference instead of height differences. This is conducted by studying the relationship between gravity 
potential and height in the vertical datum definition and realisation. The growing need for a global refer -
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ence surface requires a unification of all existing vertical datums around the world, which is a scientific 
problem of high practical significant (Sánchez et al., 2018).
Unification of height systems requires the determination of the transformation parameters or datum 
offsets between existing vertical datums, each of which is defined with a fundamental surface of zero 
elevation. V ertical datum offset is an existing discrepancy between datums; it can be estimated from GPS/
levelling data of benchmarks on land, GPS/levelling data of tide gauge stations, Global Geopotential 
Models (GGM) and a precise geoid model (Singh, 2018). Presently, it is common practice for a vertical 
reference surface to be defined by a gravimetric geoid model. 
Traditionally, national and regional height datums were defined with respect to a selected network of 
tide gauge stations; and height networks were established by terrestrial techniques such as spirit levelling. 
Height differences (dH) measured during levelling are scaled by gravity (g) to determine the difference 
in gravity potential (dW, also known as a change in gravity potential), this relationship can be expressed 
as follows (Heiskanen & Moritz, 1967):
 dW  g  dH (1)
The difference in gravity potential is known as geopotential number ( C
P
), in this study, it is defined as 
the difference between the constant gravity potential at the global geoid ( W
0
) and the gravity potential 
at the point P on the local geoid ( W
P
)  it can be expressed as follows (Heiskanen & Moritz, 1967):
 
P
0
00
,
P
PP
C W W dW gdH =−= −= −
∫∫
 (2)
If W
P
 and W
0
 could be measured and defined respectively, an ideal height system could be determined. 
The negative sign in the equation above indicates that an increase in height invokes a decrease in grav -
ity potential. It should also be noted that over a short or in regions of low gravitational variation, the 
geopotential number will be insignificant (Heiskanen and Moritz, 1967).
2 VERTICAL DATUM
The geoid is commonly known as the surface of equal geopotential; the numerical value of the geopotential 
of the global geoid has been determined from analysis of satellite tracking data, GPS/levelling data and 
satellite altimetry measurements. A vertical reference frame is a reference network consisting of a set of 
physical reference points, whose vertical coordinates refer to the reference system measured within that 
frame. Meanwhile, the vertical datum is defined as the zero-level surface (Sánchez and Sideris, 2017; 
Zhang et al., 2020). A local vertical datum is usually defined by a fundamental benchmark/s or point/s 
of origin, related to the mean sea level at tide gauge station(s). 
Over the years, many different types of vertical datums have been used. To name a few examples of da -
tums and their related height system, heights derived from GPS observations are referred to as ellipsoidal 
heights (h) have as a datum the ellipsoidal surface, the orthometric height ( H
0
) derived from traditional 
spirit levelling measured above a geoidal surface, the normal height ( H
N
) measured above a quasigeoid 
surface, and the spheroidal orthometric height ( H
LLD
) measured above the land levelling datum (LLD), 
as depicted in Figure 1. 
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Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
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Figure 1: Relationship between common height systems.
Ellipsoidal heights are geometric quantities with no physical meaning; it is only practical if the informa -
tion of the geoid undulation ( N) is available and also it is measured solely from space-based instruments. 
However, orthometric and normal heights are the most commonly used height systems, in which height 
differences can be represented in potential differences, as defined in section 1. These height systems can 
be expressed as follows (Heiskanen & Moritz, 1967):
 
00
; 
ON PP PP
WW WW CC
HH
gg γγ
−−
= = = =
 (3)
where g  and γ  are mean actual and normal gravity along actual and normal plumb-lines through 
point P (on the Earth surface), respectively. In general, national vertical datums are defined by selecting 
fundamental Benchmark/s at coastal tide gauge stations and setting N  0, W
0
  W
P
, and then they are 
connected to the national levelling network. 
In this study, vertical datum offsets are estimated using a single-point-based Geodetic Boundary Value 
Problem (GBVP) approach following Molodensky’s theory for estimating the height anomalies from the 
disturbing potential using Bruns’s formula. The vertical datum offset is only estimated at the four fun -
damental benchmarks to be able to unify the South African vertical datum to the global vertical datum.
3 THEORETICAL BACKGROUND
The relationship between the gravity potential (W) and the corresponding normal potential  of the 
reference ellipsoid can be determined from estimating the disturbing potential (, this can be expressed 
as follows (Heiskanen and Moritz, 1967):
 W
P
  U
P
  T
P 
(4)
The normal potential at a point  on the Earth surface is determined as follows (Heiskanen & Moritz, 1967): 
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0
0
,
PP
U
UU h
h
∂
= +
∂
 (5)
Where h
p
 represents an ellipsoidal height at the point P, U
0
 is the normal gravity potential obtained 
directly from the World Geodetic System 1984 (WGS84) reference ellipsoid and 
0
U
h
∂
∂
 is the gradient of 
normal gravity potential. In this study, a single-point-based GBVP approach is employed to determine 
the vertical datum offset for height system unification. This is done by following Molodensky theory for 
estimating the height anomalies from the disturbing potential using Bruns’s formula. 
The disturbing potential at the point P is computed from the spherical harmonic coefficients of the latest 
Gravity field and steady-state Ocean Circulation Explorer (GOCE) based GGM (to degree 300) TIM6 
(Zingerle et al., 2019)”abstract”:”TIM_R6e is an extended version of the satellity-only global gravity 
field model TIM_R6 (Brockmann et al., 2019. According to Odera (2019), the GOCE-based GGM, 
especially the timewise solution (TIM) has the best agreement with the latest gravimetric quasigeoid 
model over South Africa. It was integrated with the residual gravity anomalies ( ∆g  ∆g
GGM
) using Stokes’s 
integral while residual terrain model (RTM) was used to cater for the contribution of short wavelength 
component. This was done by evaluating Stokes’s integral of the gravity anomalies combined with the 
Molodensky G
1
 term. The solution to the GBVP at point P in terms of the disturbing potential is expressed 
as follows (Torge & Muller, 2012), this is usually referred to as the remove-compute-restore procedure,
 
( )
1
  ()
4
P GGM GGM RTM
R
TT gg G SdT
σ
ψσ
π
= + ∆ −∆ + × +
∫∫
 (6)
Where, R  a mean radius of the Earth, ∆g  free-air gravity anomalies,  ∆g
GGM
  gravity anomalies gener-
ated by the GGM, ψ  geocentric angle/ spherical distance, dσ  an infinitesimal surface element of the 
unit sphere σ (corresponding to ellipsoidal coordinates), S( ψ) Stokes’s function, T
GGM
  long-wavelength 
component of the disturbing potential. The Stokes’s Kernel function can be computed as expressed by 
equation (7) (Heiskanen & Moritz, 1967), the Stokes’ integral in equation (6) was evaluated using the 
technique described in detail by Yun (1999) (see section 3, eq. 6), a brief elaboration of the technique 
for computer programming was given by Bracewell (1978).
 
( )
2
1
 6 sin 1 5 cos 3 cos . ln sin sin .
2 22
sin( )
2
S
ψ ψψ
ψ ψψ
ψ

= − +− − +


 (7)
The contribution of the RTM to the disturbing potential ( T
RTM
) was evaluated as follows (Forsberg, 1985): 
 
( )
33
2
2
3
  
6
P
RTM P
HH
GR
T GH d
σ
ρ
πρ σ
−
= −−
∫∫

 (8)
Where H and H
P
 are the heights of roving point and computation point, respectively, G  is the Newtons 
gravitational constant, ρ  is the topographic mass density distribution, ρ  2670 kg.m
-3
, and l  is the 
planar distance between the computational point and the roving points. The residual gravity anomalies 
are in principle converted into residual disturbing potential, using 2D Fourier transform with a spheri -
cal approximation of the RTM terrain correction integration (Yun, 1999). Moreover, a digital elevation 
model (DEM) from the Shuttle Radar T opography Mission (SRTM) at 90 m spacing was used to evaluate 
the Molodensky G
1
 term (see equation (6)), this can be expressed as follow (McCubbine et al., 2018):
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Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
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( )
1P 33
11
  .H 
2
G Hg g
ϕλ
π
∆∆ 
= ∆ × − ∆×


 

 (9)
Where ∆ ϕ and ∆ λ are the differences in latitude and longitude, respectively. The G
1
 term contribution 
was only computed for the central 1 0  10 grid of the 40  40 gravity data grid in order to handle/reduce 
any edge effect. It is used as a terrain correction on the computed height anomaly. Furthermore, it is 
more significant in mountainous regions and relies heavily upon a detailed, accurate DEM (McCub -
bine et al., 2018). A computer program designed from python was used for this computation. Therefore 
substituting equations (5) and (6) into equation (4) to determine the gravity potential at a point P on 
the local vertical datum yields,
 
( )
0
01
 ()
4
P P GGM GGM RTM
U R
WU hT g g GS d T
h
σ
ψσ
π
∂
= + + + ∆ −∆ + × +
∂
∫∫
 (10)
Hence the gravity potential difference between global and local vertical reference (LLD) surfaces at a 
point P can be expressed as,
 
( )
0
0 00 1
[ ( ) ],
4
P P P GGM GGM RTM
U R
WW WWU hT gg GS dT
h
σ
δ ψσ
π
∂
= − = − + + + ∆ −∆ + × +
∂
∫∫
 (11)
The h
P
, in this case, refers to the ellipsoidal height at the tide gauge benchmark (TGBM). The height 
anomaly at the TGBM is estimated from Bruns’s formula, and the gradient of the normal potential gives 
an approximation of the normal gravity value 
0
  
U
h
γ
∂

≈

∂

. Therefore, equation (11) can be expressed as: 
 
0 00
( ) ( ),
LLD
P P P P GGM res RTM
W WW WU hH δ γ ζζζ =−=−+ − −−−
 (12)
where ζ
GGM
 gives the contribution of the GGM, expressed as, 
 ( ) ( )
0 , ,,
20
 cos  sin sin
max
n
n n
gg
G G M nm nm nm
nm
GM a
C m S mP
rr
ζζ λ λ ϕ
γ
= =

= + ∆ +∆ ×


∑∑
 (13)
where GM
g
  is the gravity mass constant of the geopotential model in m
3
s
-2
 defined from the geodetic 
model, γ  is normal gravity in ms
-2
, r  is radial distance to the computational point in m, a
g
  is the 
semi-major axis of the geopotential model, ∆C 
n,m
  is the difference between the fully normalised har-
monic coefficient C 
n,m
 and harmonic coefficient generated by the normal gravity field C*
n,m
, ∆S 
n,m
  is 
the difference between the fully normalised spherical harmonic coefficient S 
n,m
 and the harmonic coef -
ficient generated by the normal gravity field S*
n,m
,  n and m are the degree and order for a geopotential 
model, P 
n,m
  is the fully normalised harmonics Legendre function, ϕ  is geocentric latitude of the 
computation point, λ  is the geodetic longitude of the computation point. 
The ζ
0
 represents a zero-degree harmonic term to the GGM geoid undulations with respect to a specific 
reference ellipsoid, 
0
0
0
 
g
O
GM GM
WU
R
ζ
γγ
−
−
= − (Heiskanen & Moritz, 1967). The contribution of re -
sidual gravity anomalies ( ∆g) with the effect of the GGM and the terrain removed ( ζ
res
) is expressed as,
 
( )
1
  ()
4
res GGM
R
gg GS d
σ
ζ ψσ
πγ
= ∆ −∆ + ×
∫∫
 (14)
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Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
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The contribution of the indirect effect on the height anomaly at the point P( ζ
RTM
) is given by Amos 
(2007) as,
 
( )
33
2 2
3
   
6
P
P
RTM
HH
GH GR
d
σ
πρ ρ
ζσ
γγ
−
−
= −
∫∫

 (15)
After estimation of the local gravity potential value W
P
, using equation (10), the vertical datum offsets 
on the South African vertical datum in relation to the IHRS was computed as,
 
 .
P
P
P
W δ
δζ
γ
=
 (16)
This offset will provide an adjustment factor for the South African vertical datum to the IHRS. A unified 
vertical datum will provide a reference surface for engineering projects across countries, flooding control 
initiatives, plate tectonic movements determination and analysis, coastal hazard studies, unification of 
national gravity anomaly database, and improvement of the continental geoid, amongst other applications. 
4 DATA AND METHODS 
Several different data set were made available for the purpose of this study. The land and marine gravity 
data over South Africa were provided by the South African Council for Geoscience (SACGS) and Bureau 
Gravimetrique Internationale (BGI). However, the marine gravity data was coarse; it was supplemented 
with a global marine gravity model from CryoSat-2 and Jason-1 (Sandwell et al., 2014). Moreover, both 
the horizontal and vertical coordinates associated with the gravity data from SACGS and BGI are of 
low accuracy, as they have been interpolated from a 1:50000 map; this will introduce distortions on the 
resulting gravity anomalies. 
The gravity data was screened for duplications using Golden Surfer software; the free-air gravity anomalies 
on the land gravity data range between 101.3 and 129.3 mGal with a mean and standard deviation of 
16.3 and 31.1 mGal, respectively. The free-air gravity anomalies from the land gravity data were com -
pared to the set of free-air gravity anomalies generated using the GOCE-based GGM (TIM6) harmonic 
coefficients (up to degree and order 300). Thereafter, a mean difference of 2.1 mGal with a standard 
deviation of 10.7 mGal was obtained.
The free-air gravity anomalies on the marine gravity data range between 97.5 and 115.7 mGal with a 
mean and standard deviation of 2.4 mGal and 30.4 mGal, respectively. The free-air gravity anomalies 
from the marine gravity data were compared to the set of free-air gravity anomalies generated using the 
GOCE-based GGM (TIM6) harmonic coefficients (up to degree and order 300). Thereafter, a mean 
difference of 11.8 mGal with a standard deviation of 17.5 mGal was obtained. The gravity data was 
limited to a 40  40 grid around each TGBM to reduce computation time, as depicted in Figure 2. 
The first-order gravity data have a maximum uncertainty of 1 mGal while the accuracy of the first-
order levelling network in South Africa is estimated at 1.9 √Lmm , L being the distance of a levelling line 
in km. The GPS measurements of the TGBMs were collected by the Nation Geo-Spatial Information 
(NGI), South African government agency. The heights were determined using differential carrier-phase 
GPS measurements linked to the national network of permanent GPS stations, TrigNet. The coordi -
nates are in the ITRF2008(20016.2) reference frame and refer to the WGS84 ellipsoid.  The internal 
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Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
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accuracy of GPS coordinates is approximately 1 and 2 cm on the horizontal and vertical position, 
respectively (Odera, 2019). The differences between ellipsoidal and spheroidal heights are considered as 
height anomalies, as the South African LLD provides heights that are closer to the normal height system 
(Merry, 1985; Odera, 2019). 
The SRTM data at 3 arc-second (90 m resolution) DEM was used for computation of the terrain effect 
(G
1
 term). The DEM is uniform on the specified grid (40  40) around each TGBM, as depicted in 
Figure 3 – Figure 6. The remove-restore method is used to compute the height anomalies of the TG -
BMs. The long-wavelength component of the disturbing potential was determined from the spherical 
harmonic coefficients of the latest GOCE-based GGM (TIM6 up to 300 degrees and order), and the 
medium wavelength component was determined from the gravity data residuals, using Stokes’s integral 
as described in the previous subsection. The residual terrain model (RTM) was used to cater for the con -
tribution of the short-wavelength component. A computer program designed from python was used for 
this computation. The four fundamental tide gauge benchmarks located in Cape T own (TGBM_CPT), 
Port Elizabeth (TGBM_PEL), East London (TGBM_ELN), and Durban (TGBM_DBN) over South 
Africa are shown in Figure 2.  
Figure 2: Distribution of the fundamental benchmarks over South Africa.
The elevation map round each TGBM was generated using DEM from SRTM90 to provide a terrain 
visualisation, as depicted in Figure 3 - Figure 6. A kriging interpolation method was used to generate 
contour maps because it is statistically more sophisticated and it allows identifying distortions in the 
data. Moreover, it was used to evaluate the contribution of the indirect effect on the height anomaly. 
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Figure 3: Elevation around Cape Town TGBM.
Figure 4: Elevation around Port Elizabeth TGBM.
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Figure 5: Elevation around East London TGBM.
Figure 6: Elevation around Durban TGBM.
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5 RESULTS AND DISCUSSION
The derived physical constant of the normal gravity potential for the WGS84 reference ellipsoid is U
0
  
62636851.7146 m
2
s
-2
 as given by the International Earth Rotation and Reference Systems Service (IERS). 
The gravity residuals used on Stokes integral, as expressed in equation (6), was determined from the 
gravity anomalies computed from the observed gravity data ( ∆g), the gravity anomalies generated by the 
coefficients of the spherical harmonics ( ∆g
GGM
), from the GOCE based GGM and the Molodensky G
1
 
term determined from a convolution of heights with gravity anomalies. The residual gravity anomalies 
around each TGBM are as depicted in Figure 7 – Figure 11. A kriging gridding method on the Golden 
Sufer software was used to generate the contour maps, it produces a more accurate grid file, and it is a 
very flexible gridding method. 
Figure 7: Residual gravity anomalies around Cape Town TGBM (units are in mGal).
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Figure 8: Residual gravity anomalies around Port Elizabeth TGBM (units are in mGal).
Figure 9: Residual gravity anomalies around East London TGBM (units are in mGal).
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Figure 10: Residual gravity anomalies around Durban TGBM (units are in mGal).
As discussed in the previous section that the South African vertical datum is constrained to four TG -
BMs, the vertical datum offset is determined on the four fundamental benchmarks in relation to the 
IHRS, the estimated potential discrepancies are as depicted in Figure 11. The vertical datum offset at 
each TGBM was evaluated using equation (16), and the potential difference between the local and the 
global reference surface was evaluated using equation (12). The components involved in the computa -
tion of the vertical datum offset at each TGBM, are as illustrated in Table 1. Results of estimated offsets 
are also included in Table 1.
Table 1:  Vertical datum offset parameters and estimated offset at each TGBM.
TGBM h
P
(m) H
P
LLD
(m) ζ
GGM 
(m) ζ
res
(m) ζ
RTM
(m) W
P
(m
2
s
-2
) δW
P
(m
2
s
-2
)
CPT 34.423 3.6281 31.996 0.085 -1.519 62636852.811 0.589
PEL 31.487 4.2233 29.276 0.016 -1.997 62636855.393 -1.993
ELN 33.823 4.4153 30.642 0.018 -1.160 62636855.993 -2.593
DBN 32.678 4.3076 28.465 -0.010 -0.477 62636851.246 2.154
The gravity potential at each TGBM in South Africa deviates from the potential of the global reference 
surface by 0.589, 1.993, 2.593 and 2.154 m
2
s
-2
 for Cape Town, Port Elizabeth, East London and 
Durban, respectively. These deviations are as depicted in Figure 11.
The corresponding vertical datum offset between the international height reference system and the four 
fundamental benchmarks over South Africa are 6.013, 20.347, 26.478, and 21.996 cm for Cape 
T own, Port Elizabeth, East London and Durban, respectively. These offsets can be used for the unification 
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of the South African datum at the four TGBMs in a manner that is consistent with the international 
height reference system. The estimated gravity potential on the four fundamental benchmarks are as 
illustrated in Table 1.
Figure 11: Vertical datum offset on the four TGBM in relation to the global vertical datum.
This forms part of the datum parameters, and it should be as reliable as possible. The quality of the 
fundamental benchmarks can be improved by being connected to the gravity data networks. The desired 
physical heights system can be deduced from geopotential values using equation (3). The advantages of 
using geopotential value for height determination is that there is no need to compute orthometric or 
normal corrections to the measured height differences, thus avoiding any approximations in the cor -
rections and it is very easy to convert between height systems, as one does not have to compute a new 
set of corrections.
6 CONCLUSION
The vertical datum offset on the South African vertical datum in relation to the IHRS, has been estimated 
using the single-point-based GBVP solution at four TGBMs. The gravity data on a 4 0  40 grid around 
each fundamental benchmark was selected for the purpose of estimating their disturbing potential; this 
was performed in combination with the spherical harmonics coefficients from the GOCE based GGM, 
TIM-R6 (complete to 300 degrees and order). 
The gravity potential at each TGBM in South Africa deviates from the potential of the global reference 
surface by 0.589, 1.993, 2.593 and 2.154 m
2
s
-2
 for Cape Town, Port Elizabeth, East London and 
Durban, respectively. The corresponding vertical datum offset between the international height refer -
ence system and the four fundamental benchmarks over South Africa are 6.013, 20.347, 26.478, and 
21.996 cm for Cape T own, Port Elizabeth, East London and Durban, respectively. This evaluation pro -
vides South Africa with a direct link to the IHRS and a positive step towards the South African vertical 
datum realisation and unification.
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Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
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Acknowledgements
We are grateful to the Department of Rural Development and Land Reform (DRDLR) for the finan -
cial support granted through a postgraduate scholarship. The following organisations are appreciated 
for directly providing or making relevant data freely available on their websites; National Geo-Spatial 
Information (GPS/levelling and SAGEOID data), Bureau Gravimétrique International (land and marine 
gravity data), South African Council for Geoscience (terrestrial gravity data), United States Geological 
Survey (SRTM data). We are also grateful to the reviewers, for their comments and suggestions that have 
been used to improve the quality of the paper.
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Corresponding author: Matthews Siphiwe Mphuth
University of Cape Town, School of Architecture, Planning and 
Geomatics, Division of Geomatics
Private Bag X3, Rondebosch 7701, South Africa
e-mail: mphmat021@myuct.ac.za
Patroba Achola Odera
University of Cape Town, School of Architecture, Planning and 
Geomatics, Division of Geomatics
Private Bag X3, Rondebosch 7701, South Africa
Siphiwe Mphuthi M., Achola Odera P . (2021). Estimation of vertical datum offset for the South African vertical datum, in relation to the international height 
reference system, 65 (2), 282-297. 
DOI: https://doi.org/10.15292/geodetski-vestnik.2021.02.282-297
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Matthews Siphiwe Mphuthi, Patroba Achola Odera | OCENA RAZLIK MED VIŠINSKIM DATUMOM JUŽNE AFRIKE IN DATUMOM MEDNARODNEGA VIŠINSKEGA REFERENČNEGA SISTEMA | 
ESTIMATION OF VERTICAL DATUM OFFSET FOR THE SOUTH AFRICAN VERTICAL DATUM, IN RELATION TO THE INTERNATIONAL HEIGHT REFERENCE SYSTEM | 282-297 |