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GEODETSKI VESTNIK | letn. / Vol. 66 | št. / No. 3 |
SI 
 |  
EN
ABSTRACT IZVLEČEK 
KLJUČNE BESEDE KEY WORDS
Global Geopotential Model, height anomalies, quasigeoid, 
gravity, GNSS/dh
globalni geopotencialni model, anomalije višin, 
kvazigeoid, gravitacija, GNSS/dh
UDK: 528.21(497.11) 
Klasifikacija prispevka po COBISS.SI: 1.04
Prispelo: 28. 2. 2022
Sprejeto: 20. 7. 2022
DOI: 10.15292/geodetski-vestnik.2022.03.432-448
PROFESSIONAL ARTICLE 
Received: 28. 2. 2022
Accepted: 20. 7. 2022
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković
PRIMERJAVA URADNEGA 
KVAZIGEOIDNEGA 
MODELA SRBIJE Z 
NEKATERIMI GLOBALNIMI 
GEOPOTENCIALNIMI MODELI
VALIDATION AND 
COMPARISON OF SEVERAL 
GLOBAL GEOPOTENTIAL 
MODELS WITH AN OFFICIAL 
QUASIGEOID SOLUTION OF 
SERBIA
This study aims to analyze the quality of several local reference 
quasigeoid surfaces obtained from several Global Geopotential 
Models (GGM)  relative to the official quasigeoid solution 
of Serbia (SQM2011) and GNSS/dh observations for the 
territory of Serbia. Therefore, validation and comparison of 
the derived surfaces from the three GGM’s were made based 
on comparisons of height anomaly derived from GGM’s, 
SQM2011, and the GNSS/dh observations at the points 
of the high-precision leveling network. The selected publicly 
available GGM’s in this study are GOCO05c, SGG-
UGM-2, and XGM2019e. Primarily, at 1001 points of 
the high-precision leveling network, the differences between 
GGM and GNSS/dh height anomaly were calculated. The 
final translation parameters were calculated in the iteration 
procedure, which was then used to calculate the final values 
of the estimated height anomaly for all GGM’s at 143207 
points of the regular grid of spatial resolution 0.5' × 0.5' 
on the entire territory of Serbia. From the estimated height 
anomaly, three GGM-derived surfaces were modeled relative 
to the SQM2011. According to the results of the calculations, 
SGG-UGM-2 provides the best approximation of the quasi-
geoid SQM2011, where the remaining differences have a 
mean value of 0.01 m, a standard deviation of 0.06 m, and 
a span of 0.67 m.
V raziskavi smo primerjali uradni kvazigeoidni model 
Srbije (SQM2011) s tremi lokalnimi kvazigeoidnimi 
ploskvami, ki smo jih izračunali na osnovi treh globalnih 
geopotencialnih modelov (GGM): GOCO05c, SGG-
UGM-2 in XGM2019e. Izračunane anomalije višin smo 
primerjali na 1001 točki mreže preciznega nivelmana, 
na katerih so nam znane nadmorske in elipsoidne višine. 
Na osnovi izračunanih razlik na teh točkah smo določili 
transformacijski model, ki je omogočal izračun končnih 
vrednosti anomalij višin na 143.207 točkah gridne 
mreže v ločljivosti 0,5' × 0,5' na območju celotnega 
ozemlja Srbije. Analiza je pokazala najboljše ujemanje za 
globalni geopotencialni model SGG-UGM-2, pri čemer so 
statistični kazalci preostalih razlik: srednja vrednost 0,01 
m, standardni odklon 0,06 m in razpon razlik 0,67 m.
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
PRIMERJAVA URADNEGA KVAZIGEOIDNEGA MODELA SRBIJE Z NEKATERIMI GLOBALNIMI GEOPOTENCIALNIMI MODELI | 432-448 |
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Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
PRIMERJAVA URADNEGA KVAZIGEOIDNEGA MODELA SRBIJE Z NEKATERIMI GLOBALNIMI GEOPOTENCIALNIMI MODELI | 432-448 |
INTRODUCTION
This research aims to validate and compare GGM-derived local reference surfaces from three selected 
global geopotential models or abbreviated GGM’s (GOCO05c, SGG-UGM-2, and XGM2019e) with 
the SQM2011 and available data from GNSS/dh (GNSS-levelling) observations. The aim is to determine 
the achieved real quality of these surfaces, which directly depends on the quality of the used data available 
for the territory of Serbia and used GGM’s. After the conclusions, it is necessary to give a proposal and 
define the direction of development of a local reference surface model for the part or the entire territory 
of Serbia for the needs of geodesy, geophysical research, engineering projects, topographic surveying, 
height determination with GNSS methods, vertical datum definition and unification. Over time, GGM’s 
have significantly advanced in terms of the degree and order, which is primarily the result of defining 
GGM’s based on combined data sets obtained from a variety of sources such as altimetric observations 
over the seas and oceans, satellite observations, and tracking of satellite trajectories (such as GRACE, 
GOCE, and LAGEOS), terrestrial observations by various methods on land, seas, oceans, and from the 
air, and topography data. Combining these data sets from diverse sources in defining a single GGM 
provides higher quality, better territorial coverage, more homogeneous GGM data accuracy for most of 
the global territory, and a more refined model surface due to the higher degree and order of the model.
In defining SQM2011 (Ågren, J. et al., 2012) used the satellite-only model GOCO02s, which is of 
a much lesser degree and order compared to today’s GGM’s, and since it is a satellite-only model, the 
long-wavelength characteristic is limited due to the satellite-only data sources in GGM. On the other 
hand, data obtained from the GNSS/dh observations at the points of high precision leveling network 
on the territory of Serbia are accompanied by many problems because the data were obtained using 
different methods and instruments in different periods. There is also the problem of old available data 
and stability of some points. All this means that these data are of inhomogeneous and questionable 
quality. These mentioned problems are why a certain number of points were rejected or not used in the 
direct determination of SQM2011. These facts are more than enough to raise whether it is necessary to 
define a new height reference surface based on more modern and advanced GGM’s and new terrestrial 
data, but that is a question for some future scientific endeavors. The validation of GGM-derived local 
reference surfaces was done using three modern GGM’s relative to SQM2011 and combined with GNSS/
dh observations at the high precision leveling network points for which there is some security. Therefore, 
it is necessary to have reliable information that the points are stable and that the observations were made 
simultaneously, with the same accuracy and method. Then it can be claimed with some certainty that 
the results are of the same accuracy. Future local reference surfaces must be defined in the most modern 
ways of modeling geoids and quasigeoids based on terrestrial gravity and topographic data in combina-
tion with the most modern combined GGM. Significant improvements in modeling local or national 
models of geoids and quasigeoids depend on improved terrestrial data sets and the GGM’s based on 
modern satellite missions (GOCE and GRACE). Based on results of GNSS/dh observations, there are 
estimates of the accuracy of modern European gravimetric quasigeoids on a national (local) basis from 
1 cm to 2 cm and on the continental level from 2 cm to 4 cm, where the available input data must be 
of high quality and resolution in the territory of interest.
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2 SERBIAN QUASIGEOID MODEL 2011 – SQM2011
Since the reference surfaces are determined for the territory of Serbia, it is necessary in this chapter to briefly 
describe the used data sources and calculation procedure of SQM2011. For more detailed information, 
the reader is referred to the report of Ågren, J. et al., 2012. The report is not publicly available, but it 
can be obtained on request from the Republic Geodetic Authority in Serbia. In quasigeoid computing, 
there is a wide variety of approaches. When quasigeoid is derived from gravity data, a digital terrain 
model, and a global geopotential model, the remove-compute-restore (RCR) methodology is utilized 
most frequently. The goal is, after determining the gravimetric model, it should be adapted to the local 
(national) reference system using GNSS/dh height anomaly, which is defined as the differences between 
GNSS-determined heights above the ellipsoid and leveled normal heights at the same points on the 
physical surface of the Earth. Statistics properties of SQM2011-derived height anomaly 2011SQMζ  at the 
points of the high-precision leveling network are given in Table 1, and surface data is shown in Figure 1.
Table 1:  Characteristic values of SQM2011-derived height anomaly at 1001 points (unit: meter).
ζ minζ maxζ averageζ rangeζ ( )STDEVζσ
2011SQMζ 42.31 46.38 44.62 4.07 0.85
SQM2011 is based on the following data sets: gravity data on the territory of Serbia, normal heights 
of the stations, gravity data in the neighboring countries, digital terrain models (DTM’s), GGM’s 
(EGM2008, at the time, the most suitable model and satellite-only model GOCO02s), GNSS/dh geoid 
heights and heights above the ellipsoid (ETRS 89) (Ågren, J. et al., 2012). The data used in calculating 
the gravimetric model are the existing gravimetric data on the territory of Serbia. The method used is 
Least Squares Modification of Stokes’ formula with Additive corrections (LSMSA), i.e. (KTH method) 
of Sjöberg, L. E. (1991, 2003a, 2003b) and Sjöberg, L. E. et al., 2000, together with the latest satelli-
te-only global geopotential model GOCO02s. In border areas, EGM2008-derived gravity anomalies 
on the area-means grid of spatial resolution 5' × 5' were used to obtain information on high-frequency 
gravity anomalies for the border areas approximately from 200 km to 300 km in the territory of those 
neighboring countries where no data of gravity observations were available.
After computing the gravimetric model, a comparison and fitting with GNSS/dh observations at FR 
– (points of national leveling network) and SREF – (Serbian passive GNSS network) points was made 
(Veljković, Z. et al., 2011 and Vušović, N. et al., 2013). Comparison and fitting were made by a four-
parameter transformation model based on the following equation,
 ( ) ( ) ( ) ( ) ( )0 1 2 3cos cos cos sin sin
o
i i i i i i i ih H N b b b bϕ λ ϕ λ ϕ− − = + + − , (1)
where oiH  is the benchmark orthometric height, hi is the ellipsoidal (geometric) height derived from 
GPS measurements, Ni is the gravimetric geoid height (undulation) predicted at GPS/benchmarks, b0, 
b1, b2, b3 are the translation parameters of the model, and ϕ1, λi are the benchmarks’ coordinates (lati-
tudes and longitudes).
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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After all, a smooth residual surface based on least square collocation was created to transform ellipsoidal 
heights to normal height (Ågren, J. et al., 2012).
Figure 1:  The contour plot (map) of SQM2011 with points of GNSS/dh high precision leveling network (n=1001).
3 GLOBAL GEOPOTENTIAL MODELS GOCO05C, SGG-UGM-2, AND XGM2019E – PUBLICLY 
AVAILABLE DATA
Over the last 50 years, many researchers and scientific organizations have created GGMs. Their work 
has been published online and is available for research and work to the scientific and technical public. 
Five internet sites publish finished models with data, explanations, and other GGM-related material. 
The International Association of Geodesy coordinates these services (IAG). International Centre for 
Global Earth Models (ICGEM); Bureau Gravimetrique International (BGI); International Service for 
the Geoid (ISG); EOST’s International Geodynamics and Earth Tide Service (IGETS); ESRI’s Interna-
tional Digital Elevation Model Service (IDEMS). When drafting this research, ICGEM had 177 global 
geopotential models.
3.1 Mathematical background and application of GGM
The mathematical function that approximates the Earth’s gravity field in three-dimensional space outsi-
de the body of the Earth is called the global geopotential model, the gravity field model, or the global 
model (Barthelmes, 2014). As a function of spherical harmonics, the gravity potential of the Earth’s 
gravity can be represented as:
 
( ) ( ) ( ) ( ) ( )
1
2 2 2
0 0
1, , cos sin sin
2
n n
nm nm nm
n m
GM aW r C m S m P t r
a r
θ λ λ λ ω θ
+∞
= =
   = + +    
∑ ∑ , (2) 
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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and normal potential:
 
( )
( )
( )
( ) ( )
18
2 2 2
0 2
1, , sin
2
n
U
nm nm
n
GM aU r C P t r
a r
θ λ ω θ
+∞
=
 = + 
 
∑ , (3) 
where are: nmC and nmS are unknown coefficients, fundamental physical constants (G, M, ω) that can 
be considered already determined for applying the expression (2), and coordinates of points (r, θ, λ) 
that are relatively easy to determine today (Odalović, O., 2010 and Odalović, O. et al., 2012, 2018a, 
2018b, 2020). To apply the equations (2) and (3), coefficients must be determined to a specific degree 
n and order m, using satellite-only data sets or a combination of satellite-only and terrestrial data sets 
for most of the world’s territory, and a more refined model surface due to the higher degree and order of 
the model. Only satellite trajectory observation data can be used to estimate coefficients for satellite data 
groups. The relationship between spherical harmonic expansion coefficients and changes in Kepler satellite 
trajectory parameters (Heiskanen & Moritz, 1967) is used to study the Earth’s gravity field. Gravity field 
satellite missions include CHAMP (Reigber et al., 2003), GRACE (Adam, 2002), and GOCE (Gravity 
Field and Steady-State Ocean Circulation Explorer) (Rebhan et al., 2000). From satellite-only data 
sets, low-grade solutions can be defined. Satellite solutions are combined with terrestrial measurements, 
gravity anomalies, discrete height anomalies, or geoid undulations to create global geopotential models 
(Pavlis et al., 2012). All the essential data from several worldwide geopotential models are online as text 
files, and their archives are classified into Static Models and Temporal Models. For the purposes of this 
research, 25 GGM’s were tested, among which three GGM’s were singled out, which were found to 
best approximate the discrete values of height anomalies for the territory of Serbia. This paper employed 
three global models: GOCO05c (Fecher et al., 2015, 2017, 2020), XGM2019e_2159 (Zingerle, P. et 
al., 2020), and SGG-UGM-2 (Liang, W. et al., 2020). These worldwide models are summarized below.
4 THE GNSS/DH OBSERVATIONS
On the points of the GNSS/dh high-precision leveling network on the territory of Serbia, GNSS/dh 
height anomalies were calculated as differences between the heights determined by GNSS observations 
relative to the GRS80 ellipsoid in ETRF 2000 and leveled normal heights in the leveling network of high 
precision (Nivelman visoke tačnosti 2– NVT2) at the same observation points. Many network points 
are located in valleys and lowlands, with the rest at higher altitudes and near mountains. To determine 
SQM2011, the model’s authors used 1130 points (Ågren, J. et al., 2012). The ellipsoidal heights were 
obtained using several GNSS approaches. The static GNSS approach was employed in the limited number 
of observations to calculate SREF stations and NVT2 fundamental benchmarks. However, a significant 
number of GNSS/dh observations were made via Network RTK. Disadvantages that need to be emp-
hasized are: the normal heights of the NVT2 network points are determined differently, which is the 
reason for their inhomogeneous quality, there is no small uncertainty in the stability of the benchmarks, 
and the standard errors of the benchmark should also reflect the possibility of the influence of systematic 
errors of the height system at longer distances, and due to the impact of unmodelled systematic errors 
and time correlations, the actual standard errors are not corresponding to the formal standard errors 
(Ågren, J. et al., 2012). In validating GGM-derived local reference surfaces in this research, 1001 points 
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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of the high-precision leveling network were included, on which GNSS/dh observations were performed. 
The reason for using 1001 points instead of 1130 is because only those points were available from the 
Republic Geodetic Authority, and all the data for available points we received show a high agreement 
with the SQM2011 solution and there are no outliers. The remaining 129 points were not available to 
us. Table 2 contains statistical properties from height anomaly from 1001 points.
Table 2:  Statistical properties of GNSS-derived height anomaly at 1001 points (unit: meter).
ζ minζ maxζ averageζ rangeζ ( )STDEVζσ
/GNSS dhζ 42.33 46.41 44.64 4.08 0.85
Figure 2 presents the validation and comparison of GGM-derived local reference surfaces with SQM2011 
and GNSS/dh observations as an algorithm for more profound insight into the calculations processes 
and understanding of the research’s aim. Calculations are presented as tables and graphical maps showing 
height anomalies before and after two iterations and estimated differences in 1001 points of the high-pre-
cision leveling grid. The results are shown in tables and on a contour plots (maps) of estimated height 
anomaly from three GGM’s and SQM2011 calculated from the gridded residuals on 1001 points at 
143207 points on a regular grid (contour map).
5 NUMERICAL RESEARCH
5.1 Calculation procedure on GNSS/dh high-precision leveling network
The original calculations refer to a high-precision leveling network of 1001 points (Pi, i ∈ (1, 1001)). 
Coordinates latitudes ϕi, longitudes λi, normal heights Hi, GNSS/dh-derived height anomaly 
/GNSS dh
iζ
, and SQM2011-derived height anomaly 2011SOMiζ are available for all network points. Also, GGM height 
anomalies are derived at the same network points (Table 3). The derivation procedure of GGM height 
anomaly is based on the three selected GGM’s.
Table 3:  Characteristic values of GGMζ  at 1001 points (unit: meter).
GGMζ minζ maxζ averageζ rangeζ ( )STDEVζσ
05GOCO cζ 42.29 46.16 44.59 3.87 0.88
2SGG UGMζ − − 42.28 46.25 44.52 3.97 0.83
2019XGM eζ 42.25 46.17 44.51 3.92 0.82
At the points of a high-precision leveling network (GNSS/dh grid), the differences 05GOCO c 2SGGi
UGMt − −
and 2019XGM eit  (where is i ∈ (1, 1001)) between the values of the height anomaly of the GGM’s and 
GNSS/dh were calculated (for each model individually):
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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Figure 2:  Structure of SQM2011 validation and comparison algorithm.
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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/ hGGM GGM G
i i
NSS d
it ζ ζ= − , (4) 
where GGMit are the differences between height anomaly from GOCO05c, SGG-UGM-2, and XGM2019e 
models and GNSS/dh (Figure 3). Height anomaly GGMiζ can represent elements of three data sets who-
se values are GGM-derived from GOCO05c, SGG-UGM-2, and XGM2019e. The obtained values were 
statistically processed, and for each was obtained: average value , GGMaveraget minimum value 
GGM
mint , maximum 
value GGMmaxt , range 
GGM
ranget  , and standard deviation GGMtσ (Table 4).
Table 4:  Statistics of GGMt (unit: meter).
GGMt mint maxt averaget ranget ( )t STDEVσ
05GOCO ct -0.33 0.21 -0.05 0.54 0.10
2SGG UGMt − − -0.49 0.09 -0.12 0.58 0.07
2019XGM et -0.42 0.09 -0.13 0.51 0.08
Figure 3:  Contour map (plot) of GGMt for a) GOCO05c, b) SGG-UGM-2 and c) XGM2019e (n=1001).
Starting from three-parameter transformation (equation (5)):
 
( ) ( ) ( ) ( ) ( )ˆ ˆ ˆˆ cos cos cos sin sini i i i i it X Y Zϕ λ ϕ λ ϕ= ∆ + ∆ + ∆ , (5) 
and based on the coefficients from the equation (5) and latitudes iϕ and longitudes iλ  of the points Pi (i ∈ 
(1, 1001)), a design matrix An×3 and weight matrix Pn×n were formed according to the following principle:
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( )
1 1 1 1 1
3
3
cos cos cos sin sin
1 0 0 0
0 1 0 0
cos cos cos sin sin and
0 0 1
cos cos cos sin sin
n i i i i i n n
n n
n n n n n n
ϕ λ ϕ λ ϕ
ϕ λ ϕ λ ϕ
ϕ λ ϕ λ ϕ
× ×
×
×
 
  
  
  = =
  
  
  
 
A P
  
   
  

  
(6)
  
The vector of free terms fn×1 is given by subtracting the difference t from the mean 
GGM
averaget :
 
1
1
1
GGM GGM
average
GGM GGM
n average i
GGM GGM
average n n
t t
t t
t t
×
×
 −
 
 
 = −
 
 
 − 
f


, 
(7)
 
where n=1001. Design matrix N3×3, cofactor matrix Qx3×3, and vector n3×1 were obtained as:
 
T T 1
3×3 3× 3 3× ×3 3×3 3×3 3×1 3× × 1 3 1, and
T T
n n×n n× n n n n n n× ×n n
−
×= = = = =xN A P A A A Q N n A P f A f . (8)
Finally, the vector fn×1of the estimated parameters 3 1ˆ ×x is calculated as:
 
3×1 3×1 3×3
ˆ
ˆˆ
ˆ
X
Y
Z
 ∆
 
= − = ∆ 
 
∆  
xx n Q .
 
(9)
 
In order to simplify the method as much as possible, we used a three-parameter transformation, that is, 
only translations along the X, Y, and Z axes, and we did not use the residual surface because it is a step 
that significantly complicates the calculations. The obtained values of X̂∆ , Ŷ∆  and Ẑ∆  represent the 
translation parameters on all of the three axes, respectively. Hence, calculations in a network of highly 
precise leveling in the iterative procedure based on the latitudes iϕ and longitudes iλ  of the points Pi, 
and the differences ( 05GOCO c
it ,
2SGG UGM
it
− − , 2019XGM e
it ) between the height anomaly (
GOCO05c
iζ ,  
2SGG UMG
iζ
− −  , 
XGM2019e
iζ ) and 
GNSS/dh
iζ at the same points, the translation parameters for all three are calculated, where 
j is the number of iterations. The values in the zero iteration (j = 0) are given in Table 5.
Table 5:  GGM translation parameters in j = 0 iteration (unit: meter, n=1001).
GGM 0X̂∆ 0̂Y∆ 0Ẑ∆
GOCO05c 1.30 1.54 -1.82
SGG-UGM-2 -1.64 1.04 1.20
XGM2019e -1.64 1.58 1.00
Based on these values, it is possible to estimate the vector of differences t̂  , and according to equation 
(5), it is possible to obtain vector elements 
ît  representing a three-parameter transformation. Estimated 
differences 
ît in all 1001 points in the zero iteration for all three GGM were statistically processed, and 
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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for each was obtained: average value ˆGGM
averaget , minimum value ˆ
GGM
mint , maximum value ˆ
GGM
maxt , range ˆ
GGM
ranget  , 
and standard deviation GGMiσ (Table 6).
Table 6:  Statistics of the estimated ˆGGMt  (iteration j = 0) (unit: meter, n=1001).
ˆGGMt mint̂ maxt̂ âveraget r̂anget ( )t̂ STDEVσ
05ˆGOCO ct -0.11 0.08 0.00 0.19 0.04
2ˆSGG UGMt − − -0.05 0.04 0.00 0.09 0.02
2019ˆXGM et -0.05 0.05 0.00 0.10 0.02
In the next step are obtained values of height anomalies after the transformation (Table 7):
 
ˆ ˆGGM GGM GGM
i i itζ ζ= − , (10) 
where ˆGGM
iζ are estimated GGM-derived height anomaly from the GOCO05c, SGG-UGM-2, and 
XGDM2019e models, GGM
iζ  GGM-derived height anomaly from the GOCO05c, SGG-UGM-2, and 
XGDM2019e models, and ˆ M
i
GGt  are values of estimated differences from equation (5) for all three GGM’s.
Table 7: Statistics of the estimated ˆGGMζ  (iteration j = 0) (unit: meter, n=1001).
ˆGGMζ minζ̂ maxζ̂ ˆaverageζ ˆrangeζ ( )ˆ STDEVζσ
05ˆGOCO cζ 42.29 46.15 44.59 3.86 0.86
2ˆ SGG UGMζ − − 42.23 46.28 44.52 4.05 0.85
2019ˆ XGM eζ 42.20 46.21 44.51 4.01 0.84
The values of GNSS/dh-derived height anomaly / hGNSS d
iζ  is subtracted from the estimated value 
ˆGGM
iζ  , 
which gives differences (Table 8):
 
/ˆGGM GGM dhN
i i
S
i
G SR ζ ζ= − . (11) 
Table 8:  Statistics of the diffrences GGMiR  (iteration j = 0) (unit: meter, n=1001).
GGMR minR maxR averageR rangeR ( )R STDEVσ
05GOCO cR -0.32 0.18 -0.05 0.51 0.09
2SGG UGMR − − -0.49 0.12 -0.12 0.61 0.07
2019XGM eR -0.43 0.11 -0.13 0.54 0.08
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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At the end of each iteration, the GGM-derived estimated height anomaly from all three GGM’s and 
the SQM2011-derived height anomaly are compared at 1001 points. In other words, we compared 
GGM values at interpolated height anomalies from SQM2011. The differences between these values 
are statistically processed, and the characteristic values are given in Table 9.
Table 9:  Statistics of the differences ζ̂∆  (unit: meter, n=1001).
2011ˆ ˆGGM SQMζ ζ ζ∆ = − minζ̂∆ maxζ̂∆ ˆaverage ˆrangeζ∆ ( )ˆ STDEVζσ∆
05 2011ˆ ˆGOCO c SQMζ ζζ = −∆ -0.29 0.16 -0.03 0.45 0.08
2 2011ˆ ˆ SGG UGM SQMζ ζζ − −= −∆ -0.45 0.05 -0.10 0.50 0.06
2019 2011ˆ ˆ XGM e SQMζ ζ ζ= −∆ -0.40 0.03 -0.11 0.43 0.07
The average value of the differences thus obtained should be equal to zero. However, this is not the case, 
which is why it is necessary to reduce the values of the vector R so that the average value weighs toward 
zero. Therefore, the values of the vectors of the estimated parameters 3 1ˆ ×x , vector of free terms fn×1, and 
vector n n×1 belong to the zero iteration. Each subsequent iteration involves a change of vector fn×1:
 
( )
( )
( )
( )
11,
×1 ,
, 1
ˆ
ˆ
ˆ
GGM GGM
j
GGM GGM GGM
n ij i j
GGM GGM
nn j n
t t
t t
t t
×
 −
 
 
 = − 
 
 
 − 
f


, (12)
 
where are: ( ),ˆ
GGM
i jt – estimated differences from (j) iteration, and 
GGM
it  is the differences derived from 
equations (4). In the next iteration (first, j =1), the recalculation of previous results is continued to achi-
eve the best possible adjustment of the surfaces obtained from the GGM’s and GNSS/dh observations 
with the surface of the SQM2011. Therefore, after the iterative change, the vectors fn×1, n n×1 and 3 1ˆ ×x  
are recalculated, where the values of the vector 
3 1ˆ ×x in the new iteration (j = 1) represent increments 
(Table 10) that are added to the estimated parameters from the previous iteration thus obtaining the 
final translation parameters (Table 10), i.e.:
 
( )
( )
( )
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
ˆ1 0 1ˆ 1
ˆ ˆ3 1 1 1 1 0 1
ˆ 1 ˆ1 0 1
ˆ ˆ
ˆ ˆˆ
ˆ ˆ
GGM GGM GGMGGM
j jXjX
GGM GGM GGM GGM GGM
j j j jY Y
GGM GGM GGM GGM
jZ j jZ
X X dd
d Y Y d
d Z Z d
= =∆=∆
× = = = =∆ ∆
=∆ = =∆
∆ = ∆ + 
   = → ∆ = ∆ +
  
  ∆ = ∆ +  
x . (13) 
The procedure is repeated in the same way as in the zero iteration, so there is no need to repeat the 
explanation. Only the numerical (Tables 11, 12, 13, and 14) and graphical (Figure 4) results of the first 
iteration are given below. The whole procedure is described additionally in the algorithm on Figure 2. 
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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Table 10:  Increments and translation parameters in j = 1 iteration (unit: meter, n=1001).
GGM
Increments Translation parameters
1X̂
d
∆ 1̂Y
d
∆ 1Ẑ
d
∆ 1X̂∆ 1̂Y∆ 1Ẑ∆
GOCO05c -0.03 -0.01 -0.03 1.27 1.53 -1.86
SGG-UGM-2 -0.08 -0.03 -0.08 -1.71 1.01 1.12
XGM2019e -0.09 -0.03 -0.09 -1.73 1.55 0.91
Table 11:  Statistics of estimated ˆGGMt  (unit: meter, n=1001).
ˆGGMt mint̂ maxt̂ âveraget r̂anget ( )t̂ STDEVσ
05ˆGOCO ct -0.15 0.03 -0.05 0.19 0.04
2ˆSGG UGMt − − -0.16 -0.07 -0.12 0.09 0.02
2019ˆXGM et -0.18 -0.08 -0.13 0.10 0.02
Figure 4:  Contour plots of residuals ˆGGMt for a) GOCO05c, b) SGG-UGM-2 and c) XGM2019e (iteration j=1, n=1001).
Table 12:  Statistics of ˆGGMζ  (unit: meter, n=1001).
ˆGGMζ minζ̂ maxζ̂ ˆaverageζ ˆrangeζ ( )ˆ STDEVζσ
05ˆGOCO cζ 42.34 46.20 44.64 3.86 0.86
2ˆ SGG UGMζ − − 42.35 46.40 44.64 4.05 0.85
2019ˆ XGM eζ 42.33 46.34 44.64 4.01 0.84
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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Table 13: Statistics of GGMR (unit: meter, n=1001).
GGMR minR maxR averageR rangeR ( )R STDEVσ
05GOCO cR -0.27 0.23 0.00 0.51 0.09
2SGG UGMR − − -0.37 0.24 0.00 0.61 0.07
2019XGM eR -0.30 0.24 0.00 0.54 0.08
Table 14:  Characteristic values of ζ̂∆ (unit: meter, n=1001).
2011ˆ ˆGGM SQMζ ζ ζ∆ = − minζ̂∆ maxζ̂∆ ˆaverageζ∆ ˆrangeζ∆ ( )ˆ STDEVζσ∆
05 2011ˆ ˆGOCO c SQMζ ζζ = −∆ -0.24 0.21 0.02 0.45 0.08
2 2011ˆ ˆ SGG UGM SQMζ ζζ − −= −∆ -0.33 0.17 0.02 0.50 0.06
2019 2011ˆ ˆ XGM e SQMζ ζ ζ= −∆ -0.27 0.16 0.02 0.43 0.07
The iterative technique stops in the first iteration since the translation parameter increments are zero 
in the second iteration. Thus, all selected GGMs’ final translation parameters were derived from the 
first iteration. All three GGM and GNSS/dh surfaces at 1001 high-precision leveling points matched 
SQM2011. We treated iterations defining translation parameters based on three selected GGMs and 
GNSS/dh measurements at high-precision leveling points as zero and first. The purpose of these iterations 
is to define the final values of translation parameters, which we used in the next part of the calculation 
to adjust the surface of GGM’s with the surface of SQGM2011 on a grid of spatial resolution 1 km × 
1 km throughout Serbia using three-parameter transformation.
5.2 Calculation procedure on the regular grid on the entire territory of the Republic of 
Serbia
After the procedure of defining translation parameters and obtaining their final values for all three chosen 
GGM’s, the regular grid of mentioned resolution with 143207 points with known height anomaly GGMζ
was defined for the entire territory of Serbia (Table 15) for which the estimated values of differences 
(Table 16) were calculated by three-parameter transformation based on the latitude and longitude of the 
grid points and the final translation parameters. The values of all three GGM’s height anomalies on the 
same regular grid were modeled and corrected by the estimated differences. The obtained values are the 
estimated anomalous heights (Table 17). In this way, the surface derived from GGM’s is adjusted based 
on SQM2011 for the entire territory of Serbia (Figure 5). Standard deviations, minimum, maximum, 
and mean values were calculated for the estimated GGM differences and corrected values of GGM height 
anomaly. Finally, corrected GGM height anomalies are compared to the SQM2011 height anomaly 
(Figure 6 and Table 18).
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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Table 15:  Characteristic values GGMζ on a regular grid of 143207 points (unit: meter).
GGMζ minζ maxζ averageζ rangeζ ( )STDEVζσ
05GOCO cζ 40.97 46.35 44.57 5.39 0.93
2SGG UGMζ − − 41.00 46.35 44.57 5.35 0.94
2019XGM eζ 40.97 46.39 44.56 5.42 0.94
Table 16:  Characteristic values of ˆGGMt (unit: meter, n=143207).
ˆGGMt mint̂ maxt̂ âveraget r̂anget ( )t̂ STDEVσ
05ˆGOCO ct -0.16 0.05 -0.05 0.21 0.05
2ˆSGG UGMt − − -0.19 -0.07 -0.12 0.12 0.03
2019ˆXGM et -0.19 -0.07 -0.13 0.12 0.03
Table 17:  Characteristic values of ˆGGMζ (unit: meter, n=143207).
ˆGGMζ minζ̂ maxζ̂ ˆaverageζ ˆrangeζ ( )ˆ STDEVζσ
05ˆGOCO cζ 41.01 46.38 44.62 5.37 0.89
2ˆ SGG UGMζ − − 41.07 46.51 44.69 5.44 0.96
2019ˆ XGM eζ 41.04 46.55 44.69 5.51 0.96
Table 18:  Characteristic values of ˆGGMζ∆ (unit: meter, n=143207).
2011ˆ ˆGGM GGM SQMζ ζ ζ∆ = − minζ̂∆ maxζ̂∆ ˆaverageζ∆ ˆrangeζ∆ ( )ˆ STDEVζσ∆
05 05 2011ˆ ˆGOCO c GOCO c SQMζ ζ ζ−∆ = -0.85 0.53 -0.05 1.38 0.14
2 2 2011ˆ ˆSGG UGM SGG UGM SQMζ ζ ζ− − − −∆ = − -0.40 0.27 0.01 0.67 0.06
2019 2019 2011ˆ ˆXGM e XGM e SQMζ ζ ζ−∆ = -0.34 0.37 0.01 0.72 0.08
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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Figure 5:  Мodeled surfaces of ˆGGMζ for a) GOCO05c, b) SGG-UGM-2 and c) XGM2019e (n=143207).
Figure 6:  Мodeled surfaces of ˆGGMζ∆  for a) GOCO05c, b) SGG-UGM-2 and c) XGM2019e (n=143207).
6 CONCLUSION
This paper presents a study to evaluate and determine the quality of three GGM-derived reference 
surfaces using SQM2011 and GNSS/dh observations at 1001 points of the high-precision leveling 
network. The evaluation was performed using GGM (GOCO05c, SGG-UGM-2, and XGM2019e), 
SQM2011, and GNSS/dh-derived height anomaly, and calculated translation parameters based on a 
three-parameter transformation. In the evaluation process of local reference surface models derived from 
Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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GOCO05c, SGG-UGM-2, and XGM2019e for the territory of Serbia, the estimated anomalous heights 
are obtained in the following ranges. For the GOCO05c model, results go from a minimum of -0.85 
m to the maximum value of 0.53 m, with an average of -0.05 m and a standard deviation of 0.14 m; 
for the SGG-UGM-2 model, the same statistical data are, -0.40 m, 0.27 m, 0.01 m and, 0.06 m; and 
for the XGM2019e model, -0.34 m, 0.37 m, 0.01 m and, 0.08 m. A detailed analysis found that the 
quality of these GGM-derived local reference surfaces is of sufficient accuracy for the needs of topograp-
hic survey, works in engineering projects, and GNSS height determination methods. Also, due to the 
constant improvement and development of global geopotential models, public availability of their data, 
increasing global territory coverage, and increasing degree and order, this method of defining local and 
regional reference surfaces can be strongly recommended. This recommendation is also supported by the 
fact that the necessary terrestrial data on local geodetic networks required for this method are frequently 
maintained with new terrestrial observations and geodetic surveys. This method is also supported by the 
latest research in Europe and the world, which evaluates the model of gravimetric quasigeoids using data 
obtained from GNSS/dh observations, and gives accuracy at the national level in the range of 1 cm to 
2 cm and at the continental level in the range of 2 cm to 4 cm.
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Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
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Marko Stanković, M.Sc. Geodesy
Department of Geodesy and Geoinformatics, 
Faculty of Civil Engineering, University of Belgrade
Bul. kralja Aleksandra 73, 11120, Belgrade, Serbia
e-mail: stankovic.d.marko@gmail.com
Assoc. Prof. Oleg Odalović, Ph. D. Sci. Geod. Eng.
Department of Geodesy and Geoinformatics,
Faculty of Civil Engineering, University of Belgrade
Bul. kralja Aleksandra 73, 11120, Belgrade, Serbia
e-mail: odalovic@grf.bg.ac.rs
Miloš Marković, M.Sc. Geodesy.
Department of Geodesy and Geoinformatics, 
Faculty of Civil Engineering, University of Belgrade
Bul. kralja Aleksandra 73, 11120, Belgrade, Serbia
e-mail: milmarkovic85@gmail.com
Stanković M.,  Odalović O., Marković  M. (2022). Validation and comparison of several Global Geopotential Models with an official quasigeoid 
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Marko D. Stanković, Oleg R. Odalović, Miloš D. Marković| VALIDATION AND COMPARISON OF SEVERAL GLOBAL GEOPOTENTIAL MODELS WITH AN OFFICIAL QUASIGEOID SOLUTION OF SERBIA | 
PRIMERJAVA URADNEGA KVAZIGEOIDNEGA MODELA SRBIJE Z NEKATERIMI GLOBALNIMI GEOPOTENCIALNIMI MODELI | 432-448 |