AST OGEODETIC NETWQ ____ ""'-
OF SLOVENIA D GEOID 
Asst.Prof Dr. Bojan Stopar, Dr. Miran Kuhar 
Faculty of Civil Engineering and Geodesy-Department of 
Geodesy, Ljubljana 
Receivedfor publication: 24April 1997 
Prepared for publication: 7 July 1997 
Abstract 
The astrogeodetic geoid was computed for the area covered 
by the astrogeodetic nelwork of Slovenia. With dejleclions of 
the vertical and with geoida! heights, the observed directions 
and distances can be correctly reduced onto the reference 
ellipsoid. The goal of this paper was to establish how rcduced 
obscrvations change the positions of points in thc network 
and how they improvc network accuracy. 
Keywords: astrogeodetic network, geoid, overall accuracy of 
geodetic network 
1 SHAPE AND SIZE OF THE ASTROGEODETIC NE'IWORK OF SLOVENIA 
The astrogeodetic network of Slovenia covers the territory of the Republic of Slovenia. As regards its shape, it is a standard trigonometric network (Figure 1 ). 
Due to the requirements of standard geodesy, trigonometric points are located on 
hills and stabilised by short or long concrete beams. When discussing the 
astrogeodetic network of Slovenia, the first series of trigonometric points located in 
the territory of the Republic of Croatia is often also taken into account. This network 
covers an area of approximately 260 km x 180 km. Together with points in the 
territory of the Republic of Croatia, the astrogeodetic nctwork of Slovenia comprises 
46 points which make 66 triangles. Due to the fact that at present the territory of the 
network also comprises the state boundary between the Republics of Slovenia and 
Croatia, only the points located in the territory of Slovenia were kcpt in one version 
of the network. The network in the territory of Slovenia comprises 34 first order 
trigonometric points. For the needs of this paper, it was assumed that the 375 
Gorjanci trigonometric point is also part of the national geodetic network of 
Slovenia. For this reason, the astrogeodetic network of Slovenia is discussed in this 
paper as a network of 35 points which make 46 triangles. The network covers and 
arca of 230 km x 140 km. 
2 RENOVATION OF THE ASTROGEODETIC NE'IWORK OF SLOVENIA 
Por historical reasons, the position of the astrogeodetic network on the reference ellipsoid is incorrect; the network has large scale deformations and its accuracy is 
not homogeneous (Jenko, 1986). Work for the rcnovation of the Slovenian part of 
the astrogeodetic network of the formcr Yugoslavia began after 1974. Renovation 
was performed on the astrogeodetic network of Slovenia, with the first series of 
Geodetski vestnik 41 (1997) 2 
points in Croatia. The greatest emphasis was put on the ineasurement of lengths in 
the network and determination of the scale of the official national geodetic network. 
In addition to length rncasurements, the heights above sea level were determined 
anew for many points, such that they have been determined for all points (Jenko, 
1986). 
enovation resulted in the fina! tria! adjustment of the national geodetic network 
in the local coordinate system on thc Gauss-Krucger projection plane without 
the introduction of any conditions or li.nks into adjustment This adjustment included 
<lata from observations of direction from the period 1963 - 1966 and newly measured 
lengths. In order to determine the positions of 46 points, 222 directions and 49 
lengths were used. The values of reference standard deviations determined a priori, 
i.e. CTOs = 0,45" for directions and CTOd = 0,038 m for lengths, were used in the 
adjustment The value of the reference standard deviation dctermined a posteriori is 
A • 
c,0 = 1,0106 (Jenko, 1986). On the basis of this adjustment, thc scalc of the official 
network was determined by comparing the official point coordinates with point 
coordinates estimated in the tria! adjustment The result of these comparisons 
confirmed the bypothesis that the network, in addition to being displaced and 
rotated, also exhibits strong scalc dcforrnations. Lincar scale deformations range 
within the interval of -43,7 mm/km to 11,0 mm/km, which mcans that the official 
nctwork is quitc nonhomogeneous with respect to scale (Jenko, 1986). 
372 
21' 
202 
OBSERVE0 D!RECT!ON 
OBSERVED OIRECTION ANO LENCTH 
·~------ OGSERVED LENCTH 
182 
183 193 
Figure 1 
To complete the network renovation, a trial astronomic-geodetic oricntation of the 
national geodetic network was also performed. Four celestial longitudes, six celestial 
Geodetski vestnik 41 ( 1997) 2 
Jatitudes, six astronomical azimuths measured at six Laplace points, and twelve geoid 
points with given astronomical coordinates were used as orientation. The gcodetic 
coordinates of the network points were the result of orientation of the network. The 
values of displacements of individual points in the official national geodetic network 
were determined from the comparison of geodetic coordinates obtained after 
orientation of the geodetic network and officially valid coordinates. These ranged 
from -335,3 to -341,9 min the direction of the y axis and from -84,8 m to -91,7 min 
the direction of the x axis (Jenko, 1986). Due to a lack of knowledge about the geoid 
during network renovation, the renovatcd nctwork obtained after tria] adjustment 
suffered from two shortcomings: 
o in the observed directions, the influence of deflections of the vertical was not 
corrected, 
o lengths were reduced onto the reference ellipsoid on the basis of the heights 
above sea leve! and not heights of points in the ellipsoid. 
3 DETERMINATION OF THE GEOID PLANE IN THE TERRITORY COVERED BY 
THE ASTROGEODETIC NETWORK OF SLOVENIA 
A fter the completion of !atest research projects involving the national geodetic 
J-\..network, the relative geoid was determined for the tcrritory of Slovenia and a 
part of Croatia (Čolic et al., 1992). The relative astrogeodetic geoid available at that 
tirne was determined on the basis of astronomic coordinates of 42 network points 
and geodetic coordinates which werc obtained as results of trial astronomic-geodetic 
orientation of the network. The calculation of the geoid yieldcd relative undulations 
of the geoid which represent the relative shape of the geoid plane (Figures 2 and 3). 
In addition to the relative shape of the geoid plane, the authors also wished to 
achieve an absolute orientation of the geoid. This was done using the OSU91A 
geopotential model (Ohio State University 91A). Absolute undulations of the geoid 
ranged within the interval from 44,3 to 48,5 m. Accuracy in determining geoid 
undulations was 1 dm (Čolic et al., 1992). 
On the basis of conclusions from the renovation of the astrogeodetic network of Slovenia, it is assumed that the officially valid astrogeodetic network of Slovenia 
has been discussed in detail and it will not be further discussed in this paper. W ~ will 
only be concerned with the fina! tria! adjustment of the astrogeodetic network which 
was performed within the framework of the renovation programme for the 
astrogeodetic network of Slovenia. 
4 CORRECTIONS OF TERRESTRIAL OBSERVATIONS 
In reducing observations onto thc surface of the reference ellipsoid, two groups of corrections need to be considercd. The first one consists of corrections which take 
into account the influence of the Earth's gravitational field on the observations. The 
second one includes geometrical corrections which arisc from the geometry of the 
rotational e!lipsoid. Let us assumc that the geomctrical corrections of observations 
were performed correctly. Since the values of corrections of observations from both 
groups of corrections are added, the corrections of observations which have not yet 
been calculated can be calculated frorn the known parameters of thc geoid. The 
relative geoid heights and relative deflections of thc vertical are currently available 
Geodetski vestnik 41 ( 1997) 2 
and they can be used to reduce observations onto the surface of the reference 
ellipsoid. 
5150000.0 
5400000.00 5450000.00 5500000.00 5600000.00 
Figure 2 
Figure 3 
4.1 Reduction of observations in the astrogeodetic netwmrk of Slovenia 
The observed direction Sij between points Pi and Pj must be reduced from the vertical onto the normal in point Pi by the influence of components of the 
deflection of the vertical in the station point ~i and T]i, or corrected by the value of 
C2;i (Sideris, 1990): 
Geodetski vestnik 41 (1997) 2 
C2;i = -(~i sinaij - 'fli cosaij) cotZij (4.1-1) 
where OCij is the geodetic azimuth and Zij is the zenith distance between points Pi and 
Pj. If the zenith distance is not observed, it can be determined from the following 
expression: 
cotZij = 
- hi 
E 
D .. 
IJ 
E D„ 
_I_J 
2Rm' 
where hi and hj are ellipsoid heights of points Pi in Pj, D~ is the length of the 
IJ 
(4.1-2) 
geodetic line between the points, and Rm is the mean radius of curvature of the 
ellipsoid in azimuth OCij between points Pi and Pj. The influence of the deflection of 
the vertical on the value of zenith distance can be neglected. It can be established on 
the basis of expressions for geometrical reduction of observations that the lack of 
knowledge about geoid undulations affects the valuc of azimuthal reduction. Since 
the difference of geoid undulations between neighbouring points in the network is 
always smaller than 2 m (i.e. lower than 0,0001"), this correction can be neglected. 
As can be seen from the above equations, the corrections of observed directions depend on the magnitude of deflection of the vertical at the station point, the 
azimuth of the observed direction and the zenith distance from the observed point. 
In absolute terms, the greatest correction of observed direction in the astrogeodetic 
network of Slovenia is in point 518 Korada towards point 202 Kanin and it amounts 
to C2 = 0,5231". However, data on the deflection of the vertical in points 179 
Mangart, 515 Košuta, 202 Kanin and 194 Privis are not available. With regard to the 
fact that points 179, 515 and 202 are located at a large height above sea level, and 
that the observed directions towards the neighbouring points which also lie at a large 
height are located in these points, it can bc expected that in absolute terms there are 
no greater corrections of observed directions in the entire network. 
he components of deflection of the vertical in points Pi and Pj do not affect the 
value of reduced length, but they do affect the value of geoid undulations Ni and 
Nj. Since heights above sea leve! were used for length reduction onto the surface of 
the reference ellipsoid instead of ellipsoidal heights, thc reduced values should be 
reduced (algebraically) by the following value (Sideris, 1990): 
E N-Nj E 
Lillij = 2Rm Dij ' ( 4.1-3) 
where Rm is the mean radius of curvature of thc ellipsoid in azimuth Olij between 
points Pi and Pj. In this manner the actual lengths of geodetic lines on the reference 
ellipsoid are obtained. The absolutely greatest correction of observed length is the 
length correction between points 170 Rodica and 202 Kanin and it amounts to 
E ,0,.D.. = 0,0102 m. It is of interest that the longest measured length in the network, 
11 
Geodetski vestnik 41 (1997) 2 
i.e. the length betwecn points 373 Mrzlica and 214 Donačka Gora requires thc 
smallest correction, LiD~ = -0,0001 m 
IJ 
ecause the reductions of observations wcre made correctly, in addition to the values 
of observed quantitics, the positions of points in the network also change. Reducecl 
obscrvations were therefore readjusted in thc free network. The new positions of points 
differ from the old ones by up to 4 cm. Point 519 Kamenek changes its position most: by 
Ll.y= +0,0295 m and l'.lx= +0,0275 m, which means a difference in position of Ll.P = 
0,0403 m. Figure 4 presents changes in the positions of points in the network as a result 
of reduction of obscrvations onto the reference ellipsoid. 
~ 
~ \ 
~ / /- I 
/ 
r 
\ r 
o o 
? 
°"' / i 'o ,,fi _,,0 
l 
DISPLACEMENT SCALE 
L____J 
o 2cm 
---8 
Figure 4 
5 THE GEOID'S INFLUENCE ON THE ACCURACY OF THE ASTROGEODETIC 
NETWORK OF SLOVENIA 
In addition to changes in the position of points in the network, it was also desired to establish whether the accuracy of the astrogeodetic network of Slovenia changes due 
to correctly reduced observations onto the reference computation plane. The following 
overnll accuracy criteria were applied to establish changcs in network accuracy: 
✓ sled ( L;: 
o mcan standard deviation s; = 
u - d 
Geodetski vestnik 41 (1997) 2 
u-d -~---
• generalised standard deviation s Q = ✓ det ( :E~) 
• maximum own value Amax of the covariance matrix L( and 
• homogeneity of the astrogeodetic network, i.e. the value of Amin . 
Amax 
The position of the network in thc coordinate system is determined by the position of 
point 173 Kucelj. The network's scalc and orientation are determined in the same 
manner as for the free network. The covariance matrix of the network is singular in 
ali cases, with a defect of d = 1, or with a rank equal to (~) = u - d = u - 1 , 
where u is the number of coordinate unknowns in thc network. The determinant and 
the trace of the covariance matrix refcr only to the coordinate part of thc covariance 
matrix LQ . All criteria for overall accuracy collected in Table l are therefore directly 
comparable. Since the covariance matrix LQ is singular, the value of the determinant 
of the covariance matrix was taken to be the product of multiplication of all own 
values of the covariance matrix I:~ which differed from zero. The minimum own 
value given in Table 1 is the smallest own value of the covariance matrix I:~ which 
differs from zero. The determinant of the covariance matrix I:~ is proportional to thc 
volume of a hyperellipsoid which is represented by the quadratic equation 
(x - ;? I:~ (x - ;) - x
1 
2 
(u) . In the same manner as thc hyperellipsoid 
- O( 
equation depends on all these parts of the covariant matrix, its determinant is a 
scalar which depends on all pai:_!§ of the covariance matrix. For this reason, 
generalised standard deviation s; was tak.en as the most important criterion of 
network accuracy for the comparison of individual versions of adjustment. 
As was mentioned above, the entire astrogeodetic network includcd in trial adjustment during network renovation (Jenko, 1986) is discussed separately, and 
the network located in the territory of the Rcpublic of Slovenia separately. In the 
entire network, 49 lengths and 222 directions are observed on 46 points, while on 9 
points of the network no lengths are observed. In the network which covers only the 
territory of the Republic of Slovenia, 40 lengths and 160 directions are observed. No 
lengths are observed on 5 points of the network. The ratio of the number of observed 
directions to lengths is more favourable for the network covering the territory of 
Slovenia; i.e. 4 : 1, than that of the entire network, which amounts to 4,5 : l. The 
ratio of the number of points which are not connected with lengths to the number of 
all points of the network is also more favourable for the network in the territory of 
Slovenia. The ratio of the number of observations to the number of unknowns in the 
network is equal and sufficiently favourable for both networks. 
Trial adjustment refcrs to: 
1) adjustment of nonreduced observations in the entire network with the values of 
reference standard deviations of lengths of CTOd = 0,038 m and for directions 
aos = 0,45", and this represents adjustment with equal weights of observations as in 
the tria! fina! adjustment during network renovation; 
Geodetski vestnik 41 ( 1997) 2 
2) adjustment of observations reduced onto the reference ellipsoid in the entire 
network; 
3) adjustment of nonreduced observations in the network in the tcrritory of Slovenia; 
4) adjustment of observations reduced onto the reference ellipsoid in the network in 
the territory of Slovenia. 
As regards the value of the generalised reference variance s; , the network in the territory of Slovenia which was adjustcd on the basis of nonreduced 
observations ( case 3) is the one with the highest accuracy. In both cases and taking 
into account all accuracy criteria, the network adjusted on the basis of nonredueed 
observations is better than that in the case of adjustment of reduced observations. 
The requirements for homogeneous accuracy are not fulfilled to any greater extent 
neither by the entire network nor by the network in the territory of Slovenia. 
- = 
Amin Amax 
Amin A2 
Adjustment s; s~ -- Go 
Amax 
1) 0,0511 0,0282 0,000031 0,058199 0,000532 1,00523 
2) 0,0530 0,0292 0,000033 0,062545 0,000532 1,04209 
3) 0,0533 0,0272 0,000033 0,076472 0,000436 1,04755 
4) 0,0552 0,0282 0,000036 0,082120 0,000436 1,08555 
Table 1 
If the results of the first and second adjustments are analysed together, and those of 
the third and fourth adjustments togethcr, it can be established that the differences 
in network accuracy for networks adjusted on the basis of reduced and nonreduced 
observations are caused by different values of the reference variance ~~ estimated a 
posteriori, since the reference variance ~3 is used for the calculation of the 
covariance matrix 1:; = ~~ (A TPAt of estimated coordinate points of the network. 
The covariance matrix depends on the matrix of coefficients in correction equations, 
A, which is practically equal for both cases, from the matrix of observation weights P 
and the value of the reference variance ~~ estimated a posteriori. H the value of the 
reference variance ;~ is taken to bc equal for ali cases, the overall and local accuracy 
criteria are obtained which are completely equal in the first and second adjustments 
and the third and fourth adjustments. This means that the differences in 
astrogeodetic network accuracy which would result from the disregard of reductions 
of observations onto the reference ellipsoid cannot be estimated. In our opinion it is 
impossible to state that network accuracy changes due to the reduction of 
observations onto the reference ellipsoid, or that the reduction of observations 
means an improvement in geodetic network accuracy. 
If the influence of the reference variance I; is removed from the value of the generalised standard deviation s ( and mean standard deviation d~, and the results 
of adjustment of the en tire network are compared with the results of adjustment of 
Geodetski vestnik 41 ( 1997) 2 
the part of the network in the territory of Slovenia, it can be seen that the network 
in the territory of Slovenia is slightly more_ accurate than the entire network. 
Differences in the accuracy of individual adjustments therefore occur due to differcnt 
values of the reference variance a~. The values of the reference variance a~ exceed 1 
in ali adjustments. This means that at least for one of the values of the reference 
standard deviations G0, and Go<1, the taken value is too small, which means that thc 
weights of at least one type of observations are excessive. 
The fact that in the first adjustment a~ ; 1 follows from the ratio of the reference 
standard deviations of directions and lengths which is determined on the basis of 
extensive analyses for the geoid's influence in nonreduced observations. In. these 
analyses, only the accuracy of observed directions in individual triangles is analyscd, 
and the total accuracy of the entire network, and the accuracy of observed lengths is 
observed separately (Jenko, 1986). On the basis of these analyses, the ratio of the 
reference standard deviations of direction and length cr0, and CToc1 is determined. 
If one wished to establish the ratio direction and length accuracy after reduction for the values of deflections of the vertical and the values of geoid heights, a detailed 
analysis such as was performed in (Jenko, 1986) would need to be performed again. 
However, it is clear that the estimated length accuracy would not change at all - only 
the accuracy of observed directions could change. The original angular observations 
were not available during the preparation of this paper, and the analysis of accuracy 
of observed directions was therefore not performed. The values determined by 
(Jenko, 1986) were taken for the calculation of the ratio of the reference standard 
deviations of directions and lengths G0, and God· 
In order to obtain an objective estimate of the ratio of weights of observed directions and lengths, a posteriori estimate of weights of observed quantities 
could be made. Reference variances of reasonably composed groups of observations 
are the starting point for the procedure of a posteriori estimation of weights. In our 
case such groups are the groups of observed directions and lengths. For a correct a 
posteriori estimate of weights of observations, a s_ufficient number of correctly 
distributed observations are required, in order to be ab!e to estimate the coordinates 
of network points only on the basis of one type of observations. This means that the 
number of each type of observations must be higher and that the observations of 
each type must be uniformly distributed over the entire network (Kogoj, 1992). In 
the astrogeodctic network of Slovenia, these requirements are not fulfilled. The 
number of observed lcngths is too small to obtain a correct estimate of weights for 
groups of observations. 
By comparing the results of adjustment of the entire network and the results of adjustment of the network in the territory of Slovenia, and disregarding the 
reference variance a~ in the calculation of the covariance matrix of adjusted network 
point coordinates, it can be seen that the nctwork in the territory of Slovenia is 
slightly more accurate than the entire network. But sincc the value of the reference 
matrix for the network in the territory of Slovenia is ;~ > 1, it can be concluded that 
Geodetski vestnik 41 ( 1997) 2 
too high values were taken for weights of observations. It is impossible to predict 
now what changed weights of observations would mean for network accuracy. It can 
only be claimed that by reducing the its overall accuracy is not reduced. The 
adjustments of observations were pcrformed using the GEM 3 computer program 
(Ambrožič, 1988) which was slightly modificd for the needs of preparing this paper. 
6 CONCLUSIONS 
he adjustment of reduced observations in the national geodetic network was 
performed with thc intention of comparing this network with the renovated 
astrogeodetic network treated in the trial as part of the renovation of the 
astrogeodetic network of Slovenia (Jenko, 1986). The objective of the comparison of 
the results of adjustments of the entire network with the results of adjustment of its 
part in the territory of the Republic of Slovenia was to establish whether network 
reduction affects network accuracy. On the basis of the above results it can be 
claimcd that network accuracy is not improved with a correctly performed reduction 
of observations for values of deflections of the vcrtical and geoid heights, at least not 
to a detectable extent, and that the reduction of the network only to the territory of 
Slovenia does not reduce the overall accuracy of the geodetic network. 
The authors gratefully acknowledge the assistance of the Surveying and Mapping 
Authority of the Republic of Slovenia in providing the required data. 
Literntllllre: 
Ambrožič, T., Izdelava programa za izravnavo ravninske mreže za ATARJ in IBM. Diplomska 
naloga. FAGG OGG, Ljubljana, 1988 
Čolič, K et al., New geoid solution for Slovenia and a pari of Croatia. Proceeding JAG First 
Continental Workshop on the Geoid in Europe. Praga, 1992, 158-165 
Jenko, M., Dela na astronomsko-geodetski mreži v letih 1975-1982. Ljubljana, Inštitut GZ SRS, 
1986 
Kogoj, D., Izbira najprimemejše metode a posteriori ocene uteži merjenih količin geodetskih mrež. 
Doktorska disertacija. Ljubljana, FAGG OGG, 1992 
Kuhar, M., Raziskave ploskve geoida v Sloveniji. Doktorska disertacija. Ljubljana, FGG-Oddelek 
za geodezijo, 1996 
Niemeier, W, Netzqualitaet und Optimierung, Geodaetische Netze in Landes und 
Ingenicun,ermcssung II. Konrad Wittwer, Stuttgart, 1985 
Sideris, M., The role of the geoid in one-, two- and three- dimenensional network adjustments. 
Canadian Surveyor, 1990, No. l,p. 9-18 
Stopar, B., Sanacija astrogeodetske mreže Slovenije z GPS opazovanji. Doktorska disertacija. 
Ljubljana, FGG-Oddelek za geodezijo, 1995 
Review: Ma1jan Jenko 
Dušan Miškovič 
Geodetski vestnik 41 (1997) 2