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G 201
7
GEODETSKI VESTNIK | letn. / Vol. 61 | št. / No. 4 |
IZVLEČEK 
KLJUČNE BESEDE KEY WORDS
The paper presents the new joint adjustment of the Croatian 
First Order Gravity Network, for the first time adjusted 
as a whole. The adjustment involves absolute and relative 
gravity measurements, latter performed in the course of 
four survey stages. Firstly, the measurements are concisely 
described. Revision of the absolute and pre-processing of the 
relative measurements are briefly presented. The applied 
adjustment model is described. Accordingly, the gravity 
values of all stations (absolute and relative), corrections of 
linear calibration coefficient and linear drift coefficients are 
included in the functional model as unknown parameters. 
The absolute measurements are included in the adjustment 
as observations. The new adjustment resulted in significantly 
different gravity values as compared to previous adjustments 
(of individual stages of the network). The differences in 
gravity values are an order of magnitude greater than the 
expected accuracy. It is shown that the differences are mainly 
due to the errors in the gravimeters’ calibration constants, 
which were neglected in the previous adjustments. Because 
of the significant differences, the new linear transformation 
function from the Potsdam to the Croatian Gravity System 
is determined.
V prispevku je predstavljen postopek izravnave hrvaške 
gravimetrične mreže 1. reda kot celote. V izravnavo so 
vključene absolutne in relativne gravimetrične meritve, ki 
so bile izvedene v štirih fazah. Podan je podroben opis vseh 
meritev, pa tudi ponovna obdelava absolutnih meritev in 
predhodna obdelava relativnih gravimetričnih meritev. 
Opisan je funkcionalni model izravnave, kjer kot neznanke 
nastopajo absolutne in relativne vrednosti težnega pospeška, 
popravki linearne kalibracijske konstante ter popravki 
linearne funkcije hoda gravimetra. V modelu nastopajo 
absolutne meritve težnega  pospeška kot opazovanja. 
Rezultat izravnave so vrednosti težnega pospeška, ki precej 
odstopajo od rezultatov predhodnih izravnav, opravljenih 
v štirih fazah. Razlike presegajo velikost pričakovane 
natančnosti, kar je posledica pogreškov kalibracijskih 
konstant gravimetra, ki prej niso bili dovolj obravnavani. 
Zaradi tega smo določili novo linearno transformacijsko 
funkcijo prehoda iz Potsdamskega sistema težnosti v hrvaški 
gravimetrični referenčni sistem.
DOI: 10.15292/geodetski-vestnik.2017.04.630-648
SCIENTIFIC ARTICLE
Received: 20. 4. 2017
Accepted: 20. 10. 2017
SI 
 |  
EN
gravity network adjustment, fundamental gravity network, 
Croatia, calibration constants
izravnava meritev v gravimetrični mreži, osnovna gravimetrična 
mreža, Hrvaška, kalibracijska konstanta
NEW ADJUSTMENT Of THE 
CROATIAN fIRST ORDER 
GRAVITY NETWORK 
ABSTRACT 
UDK: 528.56(497.5) 
Klasifikacija prispevka po COBISS.SI: 1.01
Prispelo: 20. 4. 2017
Sprejeto: 20. 10. 2017
Marija Repanić
NOVA IZRAVNAVA HRVAŠKE 
GRAVIMETRČNE MREŽE
1. REDA
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1 inTroduCTion
In the last few decades, the availability of portable absolute gravimeters and their high accuracy made 
possible the establishment of national fundamental gravity networks based on absolute gravity mea-
surements, which serve as a reference system (Wilmes, Richter, and Falk, 2003; Vitushkin, 2007). Mo-
reover, the IAG adopted the Resolution no 2 (2015) for the establishment of the new Global Absolute 
Gravity Reference System based on the International Comparisons of Absolute Gravimeters (ICAG), 
which realisation also utilises measurements of superconducting gravimeters, in order to enable the 
interpolation between different epochs of measurements (Wilmes et al., 2016). Cyclic re-observations 
of national zero order absolute stations with properly maintained and regularly compared absolute gra-
vimeters are usually carried out in order to provide up to date gravity values and to facilitate analysis of 
different geodynamic processes. Often, all available relative gravity measurements are then revised and 
re-adjusted but constrained with recent absolute measurements. For example, the new cycles of absolute 
measurements are carried out in Czech Republic, Slovakia, Hungary (Pálinkáš et al., 2013) and Slovenia 
(Medved et al., 2015). Re-adjustments of some fundamental networks followed (Csapó and Koppán, 
2013; Lederer and Nesvadba, 2015). On the other hand, some European countries just established 
contemporary national gravity networks: e.g. Serbia (Odalović et al., 2012), Bosnia and Herzegovina 
(Abaza, 2014), or established zero-order stations which shall serve as a basis for fundamental networks: 
Montenegro, Kosovo and Albania (Mitterschiffthaler et al., 2016). Recently, field absolute gravimeters 
are also used for densification of zero order networks instead of, or as a complement to, traditionally 
used relative gravimeters. For example, national gravity networks of Poland and Finland are completely 
modernised and now comprise a smaller number of in-door stations, measured by a high precision FG5 
absolute gravimeter, and a larger number of field stations measured by a field absolute gravimeter A10 
(Makinen, Sękowski and Kryński, 2010; Bosy and Krynski, 2015). Field absolute gravimeters are also 
utilised in other European countries: Germany (Wziontek, Falk and Wilmes, 2015), France (Duquenne, 
Duquenne and Gattacceca, 2005), Sweden, Norway, Denmark (Krynski, 2015; Kempe et al., 2017), 
etc. In addition, there is a regional and global trend of integration of gravity networks with positioning 
and vertical networks, which is also followed on national scales.
In Croatia, a new cycle of absolute measurements is planned. However, before re-observation and 
re-adjustment of the Croatian Fundamental Gravity Network, significant improvement in accuracy of 
gravity values is possible based on existing data, which can also serve as a good basis for analysis of the 
two cycles of measurements and give an insight into the drawbacks of the existing measurements, which 
should be overcome in the future.
The Croatian Fundamental Gravity Network comprises the zero, the first and the second order networks 
(Pravilnik o načinu izvođenja osnovnih geodetskih radova, 2009). The Zero Order Gravity Network 
(ZOGN) initially comprised six stations determined by absolute gravimetry (Figure 1). The stations of 
the ZOGN were established in the course of two international projects in the period from 1996 to 2000. 
In place of the devastated station in Makarska (AGT05), its eccentric station has been included in the 
ZOGN (Bašić, Markovinović and Rezo, 2006).
The First Order Gravity Network (FOGN) initially covered the land part of the country and comprised 
36 stations. The original network was established by the Faculty of Geodesy, University of Zagreb (FGUZ) 
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in 2003 in the course of a project contracted with the Croatian State Geodetic Administration (CSGA) 
(Bašić, Markovinović and Rezo, 2006). The network was extended in three stages (in 2007, 2008 and 
2009) by the Croatian Geodetic Institute (CGI) and now also covers all major Croatian islands and 
comprises 59 stations (Figure 1, Repanić, Grgić and Bašić, 2014).
Figure 1: Stations of the Croatian Fundamental Gravity Network. Absolute gravity stations of the ZOGN are designated with 
AGT, while stations of the FOGN are designated with GT.
In 2008 the CGI started the establishment of the Second Order Gravity Network (SOGN). Due to the 
affiliation of the CGI to the CSGA in 2010, the latter continued the activities on the establishment of 
the SOGN. The measurements were completed in 2015 by the final, 11th stage. The network comprises 
193 second order gravity stations. The processing and adjustment of the SOGN are in progress.
Up to now, the adjustment of the FOGN has been carried out separately for each of the four stages. The 
adjustment models applied for different stages are presented in Bašić, Markovinović and Rezo (2004; 
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2006); Markovinović (2009) and Repanić et al. (2010). Since the gravity values at the FOGN stations 
have been obtained from adjustments of the four individual parts of the network (corresponding to 
the four stages of the FOGN establishment) and because of different adjustment models applied, the 
need for a uniform joint adjustment of the whole network emerged. Thereby applied adjustment model 
should account for already perceived instrumental error influences. This paper presents the method 
and results of the new joint adjustment of the FOGN and comparison of its results with results of the 
previous adjustments.
2 MeaSureMenTS
2.1 absolute gravity measurements in the Zero order Gravity network
The stations of the ZOGN were established in the course of two projects: the Connection of the Republic 
of Croatia to International Absolute Gravity Basestation Network in 1996 and the Unification of Grav-
ity Systems in Central Europe (UNIGRACE). The projects in Croatia were coordinated by the FGUZ. 
Absolute gravity measurements, determination of vertical gravity gradients and relative connections to 
eccentric stations were carried out by the former German Institut for Applied Geodesy (IfAG), later 
German Federal Agency for Cartography and Geodesy (BKG) and French School and Observatory of 
Earth Sciences (EOST) (Table 1). The measurements have been documented in a number of technical 
reports, publications and other materials, which have been reviewed and summarised in the Report on 
data of the Croatian Absolute Gravity Network (compiled in the CGI by Ž. Hećimović). The vertical 
gravity gradients on stations AGT02 and AGT03 were also determined by the CSGA in 2013 and 2014 
(Repanić, Kuhar and Malović, 2015).
Table 1: Overview of absolute gravity measurements at the ZOGN stations. For AGT01, the second measurement and AGT06, 
both measurements the data for actual height of measurement is not available.
Station Measured by
Date of 
measurement
Instrument
g
[nms-2]
σg
[nms-2]
Reference height 
[m]
AGT01 BKG 7.-8.8.2000. FG5-101 9806582866 27 1.3038
AGT01 EOST 27.-29.11.2000. FG5-206 9806583739 13 1.00
AGT02 IfAG 4.-5.6.1996. FG5-101 9806618330 37 1.3148
AGT03 IfAG 6.-7.6.1996. FG5-101 9805099032 10 1.3148
AGT04 IfAG 9.-10.6.1996. FG5-101 9806070091 20 1.3107
AGT05 IfAG 11.-13.6.1996. FG5-101 9804069258 27 1.3190
AGT06 EOST 26.-29.8.1999. FG5-206 9803693491 24 1.00
AGT06 BKG 6.-8.4.2000. FG5-101 9803692759 75 1.25
2.2 relative gravity measurements in the First order Gravity network
The FOGN comprises 59 stations established in four stages (Figure 1). Stations GT101 to GT136 were 
established in stage 1, which covers the land part of the country and stations GT137 to GT145, GT146 
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to GT150 and GT151 to GT159 in stages 2, 3 and 4, which cover the north, the central and the south 
Adriatic islands, respectively. More detail information on the selection of locations, network design and 
measurement practice applied in the course of the FOGN survey is available in Bašić, Markovinović 
and Rezo (2004; 2006); Markovinović (2009) and Repanić et al. (2010). Below, only the data relevant 
for the network adjustment is given.
Measurements in all four stages were carried out using the same three Scintrex gravimeters (Table 2). 
Every station was connected to adjacent points by at least two connections, each measured twice. Howe-
ver, all three stages of the extension to the Adriatic islands significantly differ from the initial land part 
of the network as regards distances between stations and means of transportation. Specifically, besides 
car transportation, the extensions involved transportation by ferry boat and, in case of stage 2, even fast 
boat (catamaran) and speedboat. Furthermore, although stages 1 and 2 have been designed as a network 
of triangles, due to specific terrain configuration, available ferry lines and financial resources, network 
configurations of stages 3 and 4 have been much weaker. All three gravimeters were transported together 
and, if the station monumentation allowed, the readings of all three gravimeters were taken simulta-
neously. Stage 1 of the survey comprised 37 days of measurements, each involving from three to nine 
station occupations. Thereby, from one to five station occupations have been redundant to provide for 
at least linear daily drift determination. Stages 2, 3 and 4 involved 14, 5 and 9 days of measurements, 
respectively. In each day there have been two or three redundant station occupations.
Table 2:  Relative gravimeters used for the survey of the FOGN.
Owner Manufacturer Model Serial number
CSGA Scintrex CG-3M 4373
CSGA Scintrex CG-3M 4372
FGUZ Scintrex CG-5 10012
The calibration of the relative gravimeters was usually carried out before and after each stage of the survey 
on the auxiliary vertical calibration line (comprising only two stations AGT02 and AGT03) with the 
gravity range of approximately 1500 µms-2 and the height difference of almost 850 m. An exception is 
the calibration of gravimeter CG-5 in stage 1, which was calibrated on the calibration line in Orangeville, 
Canada (Markovinović, 2009). Thus, for each gravimeter, a different calibration constant was used for 
every stage (Table 3). For the CG-5 gravimeter, the data on processing method and reached precision of 
calibration constants is unavailable. Figure 2 depicts all determinations of the calibration constants for 
the two CG-3M gravimeters on the auxiliary vertical calibration line. The measurement scheme, dura-
tion of the observation series, data processing and consequently reached accuracy significantly differs 
for different determinations. In general, the last six determinations can be considered the most accurate. 
Nevertheless, given a long time span, Figure 2 provides a good insight into the behaviour of the calibra-
tion constants for the two gravimeters. One can observe that the calibration constants were significantly 
changing during the first three years. Since about a year elapsed between the determination of the cali-
bration constants of gravimeters 4372 and 4373 and stage 1 of the FOGN survey, the measurements 
could be significantly affected by the change in the calibration constants. Estimated relative change of 
about 3.5·10-4 year-1 during the corresponding period could result in significant errors in the range of 
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1.52 µms-2 over the range of the FOGN of 4335 µms-2. In addition, there is a question of accuracy of the 
determined calibration constants. The standard deviations for determinations used during the FOGN 
survey amount up to 6 µms-2/CU (CU stands for counter units), or relatively 10-4. In addition, because 
of relatively small calibration range with respect to the range of the FOGN, even the environmental ef-
fects can significantly affect the calibration constants determined on the auxiliary vertical calibration line 
(Repanić and Kuhar, 2017). Therefore, the corrections of the calibration constants have been included 
in the adjustment of the FOGN as unknown parameters (see section 3).
Table 3:  Calibration constants used for specific stages of the FOGN survey in µms-2/CU.
Campaign Gravimeter
4372 4373 10012
GCAL1 σGCAL1 GCAL1 σGCAL1 GCAL1
Stage 1 (2003) 60553.42 0.41 61828.17 0.62 84272.88
Stage 2 (2007) 60623.34 1.32 61905.03 0.61 84255.93
Stage 3 (2008) 60620.94 6.35 61910.84 4.93 84285.76
Stage 4 (2009) 60627.12 5.63 61919.15 1.06 84280.17
Figure 2: Determined values of linear calibration constants for gravimeter 4372 (left) and 4373 (right).
Besides the uncertainty of the calibration constants, additional weaknesses of the relative gravity data are 
the significant hysteresis effects after transport in the upright position in the measurements of the two 
CG-3M gravimeters (Repanić and Kuhar, 2017), presence of the tilting effects caused by the transport in 
tilted position (Reudink et al., 2014) and insufficient drift control. The analysis of the long observations 
series used for the purposes of calibration of the two CG-3M gravimeters from 2012 revealed significant 
and not completely uniform hysteresis effects, but surprisingly uniform daily drift behaviour through 
all days (Figure 3). Because relative gravity measurements of the FOGN comprise only five 60-second 
readings for each occupation, which were taken after the instruments were stabilising for 10 minutes, 
it is not possible to model or eliminate hysteresis effects, which are still significant and not completely 
homogenous. Consequently, the hysteresis and even more inhomogeneous tilting effects are superimposed 
to nonlinear drift. Specifically, because of practical reasons, the two CG-3M gravimeters during stage 
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1 and gravimeter 10012 at least during stages 2, 3 and 4 were not transported in the upright position, 
but on the back seat of the car, which caused the tilting effects in readings. The analysis of the hysteresis 
effects for gravimeter 10012 is not available, but the data of the FOGN suggest that this instrument is 
not so sensitive to the hysteresis effect after transport in the upright position and that it often exhibits a 
linear drift. Still, there are indications of the tilting effects.
Figure 3: Typical hysteresis effects after transport in the upright position and daily drift approximated based on the mean value 
of observation series for gravimeters 4372 (left) and 4373 (right) for 30-minutes observation series.
3 neW JoinT adJuSTMenT oF THe FirST order GraviTY neTWorK
In order to carry out the joint adjustment of the FOGN, available data on absolute gravity measurements 
has been revised again. For certain stations, new gravity values at the ground level have been calculated. 
Furthermore, uniform pre-processing of all four stages of the relative gravity measurements have been 
carried out.
3.1 revision of the absolute gravity measurements
The revision of the absolute gravity measurements comprised all materials on basis of which the Report 
on data of the Croatian Absolute Gravity Network was compiled, as well as the report itself. Since the 
measurements at stations AGT01 and AGT06 for the purpose of vertical gravity gradient determination 
were carried out at three heights (0.3, 0.8 and 1.3 m), the vertical gravity gradient has been determined 
using the second order polynomial. Accordingly, new gravity values at the ground level have been cal-
culated for each absolute gravity measurement at the two gravity stations as well as the linear vertical 
gravity gradient between the ground level and the height of 0.25 m for the purpose of reduction of rela-
tive measurements (Table 4). The differences between the new mean gravity values at the ground level 
and the values given in the Report on data of the Croatian Absolute Gravity Network are significant 
and amount to -67 and 68 nms-2, for stations AGT01 and AGT06, respectively. Such significant differ-
ences are probably due to the fact that, in the report, the gravity values were reduced from the height 
of 1 m to the ground level using linear gravity gradients corresponding to the heights of 0.3 and 1.3 
m. Furthermore, new gravity values at the ground level have been calculated for stations AGT02 and 
AGT03 based on more precise vertical gravity gradients determined by Repanić, Kuhar and Malović 
(2015). However, the differences between these values and the previous values are not significant and 
amount to 11 and -5 nms-2.
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Table 4: Revised data on absolute gravity stations.
Station Instrument
g(h)
[nms-2]
h
[m]
σg(h)
[nms-2]
g(0)
[nms-2]
Wzz
[ns-2]
ug(0.25)
[nms-2]
AGT01 FG5-101 9806582866 1.3038 27 9806586587 -2967 58
FG5-206 9806583739 1.00 13 9806586622 -2967 53
mean - - - 9806586605 -2967 39
AGT02 FG5-101 9806618330 1.3148 37 9806622579 -3232 59
AGT03 FG5-101 9805099032 1.3148 10 9805104402 -4084 47
AGT04 FG5-101 9806070091 1.3107 20 9806073900 -2906 51
AGT05 FG5-101 9804069258 1.3190 27 9804072611 -2542 54
AGT06 FG5-206 9803693491 1.00 24 9803696363 -2786 56
FG5-101 9803692759 1.25 75 9803696379 -2786 91
mean - - - 9803696371 -2786 53
Besides the new gravity values at the ground level, the uncertainties of gravity values at the height of 
0.25 m have been determined for all absolute measurements (Table 4). Although all adjusted gravity 
values refer to the ground level, the height of 0.25 m, which is an average sensor height of Scintrex 
CG-3M and CG-5 gravimeters, represents the effective height for the network adjustment. Specifically, 
the relative measurements at the absolute stations have been reduced using the same vertical gradients. 
Thus, the errors caused by the height reduction between 0.25 m and the ground level have been can-
celled out. The uncertainties were determined analogously to Vitushkin et al. (2002) according to the 
following expression:
 
2 2 2 2
(0.25) ( ) ( )g g h ins dg hu u uσ= + + , (1)
where σg(h) represents the standard deviation of absolute gravity measurement (at the height of measure-
ment), uins the instrumental uncertainty of the absolute measurement due to systematic errors (value of 
40 nms-2 is taken as determined by Vitushkin et al. (2002)) and udg(h) is the uncertainty of the reduction 
to the height of 0.25 m.
In order to evaluate the instrumental uncertainties and check for existence of significant offsets in the 
absolute measurements, an analysis of the deviations of measurements of gravimeters FG5-101 and 
FG5-206 from the reference values of the ICAGs have been made based on the results of the ICAGs 
from 1994 to 2013 (Marson et al. 1995; Robertsson et al. 2001; Vitushkin et al., 2002; Jiang et al., 
2011; Jiang et al., 2012; Francis et al., 2015). The deviations presented in Figure 4, in most cases, are 
not significant for gravimeter FG5-101 with respect to their uncertainties, while for both gravimeters 
the deviations are not consistent in time. In addition, the deviations are in accordance with the applied 
value of uins (equation (1)). One can draw the same conclusion from available data on regional and 
bilateral comparisons close in time to the absolute measurements in Croatia (Wilmes, Richter and Falk, 
2003; Van Camp et al., 2003). Therefore, the corrections for the deviations have not been reduced 
from absolute gravity measurements. In addition, neither global nor local hydrological influences have 
been reduced.
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Figure 4:  Deviations from the reference value (DoE) of respective ICAG for gravimeters FG5-101 and FG5-206 with their un-
certainty (1 σ).
3.2 Pre-processing of the relative gravity measurements
Prior to the adjustment of the gravity network, the observation data files from all stages have been uni-
formly analysed and processed. Firstly, the errors in the data were corrected, such as incorrect calibrations 
constant applied during a day, stationary drift constant or time parameters. Thereby, if applied, the cor-
rections of gravity readings were made based on the formulas for signal processing given in CG-3/3M 
Autograv Automated Gravity Meter Operator Manual (1989) and CG-5 Autograv System Operational 
Manual (2006). Analysis of the simultaneous measurements of the three gravimeters revealed significant 
clock drift for gravimeter 10012 (CG-5). The clock drift amounted to about 14 s day–1 during 2003 
and September 2007, while during November 2007, 2008 and 2009 it amounted to about 4 s day–1. 
Hence, the start time of each reading of gravimeter 10012 was corrected for the clock drift to provide 
for the accurate calculation of the Earth tide reduction. For comparison, the clock drift of gravimeter 
4373 with respect to gravimeter 4372 amounted to 0 s day–1, during all four stages.
Since field records for gravimeter 10012 have not been available for 16 days of measurements from stage 1, it 
was necessary to approximate the instrument heights. Firstly, the average differences of instrument heights of 
each of the two CG-3M gravimeters and gravimeter 10012 were determined for the days with field records. 
Then, two sets of heights for gravimeter 10012 were calculated, each from the average height difference and 
the heights of corresponding CG-3M gravimeter. The final heights were determined as the mean values of 
the two sets, between which a good agreement has been reached (average difference amounts to 2 ± 3 mm).
The Scintrex gravimeters automatically apply certain instrument corrections (for a stationary drift, a tilt 
of the sensor and a change of the sensor temperature) as well as the Earth tide reductions according to 
Longman (1959) formula. After the corrections of the data, the Earth tide reductions were recalculated 
using the PREDICT software (Wenzel, 1996) based on synthetic tidal parameters interpolated from 
Timmen’s and Wenzel’s (1995) regular grid with utilization of Tamura’s (1987) tidal potential catalo-
gue. Also, the polar motion reductions were calculated using the PREDICT software. Next, the mean 
readings were calculated as the weighted mean of the five readings taken during each occupation, with 
weights inversely proportional to the readings’ variances. In addition, the reduction for the variation in 
atmospheric pressure and reduction to the ground level have been applied.
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Although the analysis of the daily drift has been performed in the phase of pre-processing, the drift is 
determined in the course of the adjustment. Because of a small number of redundant occupations du-
ring a day and due to the superimposed hysteresis and tilting effects (see section 2.2), which sometimes 
yield unrealistic quadratic drift coefficients, only the linear drift coefficients have been included in the 
adjustment model as parameters.
3.3 network adjustment
The absolute gravity measurements at the ZOGN stations have been introduced in the network adjustment 
as observations (or quasi-observations, since derived from original observations of time and distance) 
with associated uncertainties (Torge, 1989). There are a few examples in the literature of introduction 
of absolute measurements (or a priori given gravity values) as pseudo-observations in gravity network 
adjustment, e.g. Hwang, Wang and Lee (2002) and Medved et al. (2009). The model applied in this 
study is somewhat different and represents a generalisation of the adjustment with pseudo-observations, 
since more than one absolute measurement per station can be introduced.
3.3.1 adjustment model
The network adjustment comprises the relative gravity measurements from the four stages of the FOGN 
survey, as well as the absolute gravity measurements, which define the network datum. For relative me-
asurements, the following functional model has been applied:
 , , 4 ,0 1 1 0' 10 ( )
c T d gr gr k d gr
i i i iz v g N y z d t t
−+ = − − ⋅ + −
 , , 41 1' 10
gr k gr ky y= ⋅  (2)
where zi
c is ith relative gravity observation (the weighted mean, corrected and reduced as described in 
section 3.2), vi its residual and zi
 the raw reading (uncorrected and unreduced); gT, No
d,gr, yi
gr,k and di
d.gr  are 
the adjusted unknown parameters (the gravity value at a station T, instrument level, correction of linear 
calibration coefficient and linear drift coefficient, respectively, for day d, gravimeter gr and stage k); and 
finally ti
 and t0 are the time of ith reading and reference time, respectively. Substitution for the correction 
of calibration coefficient (y'i
gr,k) has been introduced to provide for computations’ numerical stability.
Absolute measurements have been represented by the equation (Torge, 1989):
 c Tj jL v g+ = , (3)
where Lj
c is corrected and reduced jth absolute gravity measurement.
Thus, the observation equations have been compiled from two sets: relative and absolute observations:
 R R R R
A A A A
       
+ =       
       
L v A P
x
L v A P

 + =L v A x P  (4)
where L and v are vectors of observations and their corrections, A the design matrix, x the vector of un-
knowns and P associated diagonal weight matrix. Weights for ith relative and jth absolute measurements 
have been defined according to (Feil, 1989):
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2
2
,
,
R
i
i
A
j
j
c
p
c
p
u
σ
=
=
 (5)
where cR and cA are appropriate constants, σi
2 is the variance of the relative and uj
2 squared uncertainty 
of the absolute gravity measurement.
Such observation model represents a combined adjustment by parameters of indirect and direct obser-
vations (Feil, 1989), but if for every station no more than one absolute measurement is introduced, it 
also corresponds to adjustment with pseudo-observations (e.g. Fan, 1997; Niemeier, 2002).
3.3.2 adjustment course and results
In order to estimate the accuracy of the relative measurements and to screen the measurements for 
outliers, firstly the adjustment of only relative measurements was carried out. Thereby, the weights 
were defined according to the equation (5), with σi
2 and cR equal the variance of the weighted mean 
(see section 3.2) and the mean value of all variances, respectively. The gravity values at two ZOGN 
stations: AGT01 and AGT06 were held fixed in order to define the network’s origin and scale. All 
other unknown parameters were free for determination. Outlier detection was carried out according 
to Pope’s Tau-test (Pope, 1976; Kavouras, 1982). Only two outliers were detected, both form stage 2, 
one with gravimeter 10012 and one with gravimeter 4373 from measurement days involving trans-
portation by fast boat (catamaran) and speed-boat, respectively. The a posteriori reference standard 
deviation of 0.17 µms-2 was obtained.
In addition to the described stochastic model, adjustment with equal weights (P = I) was carried 
out. However, such stochastic model resulted in more outliers (six) and larger standard deviations 
of the adjusted gravity values and corrections of calibration coefficients. The differences of the grav-
ity values between the results of the two applied stochastic models are in the range from -0.18 to 
0.24 µms-2. The reasoning for choosing a stochastic model with weights inversely proportional to 
observations’ variances, besides experimental results, was that the observations’ variances rather well 
reflect the influence of hysteresis effect after transport, which is considerable for the two CG-3M 
gravimeters, especially gravimeter 4372 (Figure 3). However, due to the same hysteresis effect, the 
observations’ variances do not provide reliable information on accuracy, but only on the relative 
ratios among them.
After the adjustment of the relative measurements and elimination of outliers, the combined 
adjustment of relative and absolute measurements was carried out. The absolute measurements 
comprised 7 measurements at 5 stations (without AGT05, devastated before relative campaigns). 
All unknown parameters were free for determination. Weights of the relative measurements were 
defined same as in the first adjustment. In order to balance the weights of absolute measurements 
with respect to the relative ones, the constant cA in the equation (5) was set to the a posteriori vari-
ance from the adjustment of only relative measurements. The adjustment resulted in the reference 
standard deviation of 0.17 µms-2 and the standard deviations of gravity values from 0.03 to 0.05 
µms-2 for the absolute stations, and from 0.05 to 0.18 µms-2 for the relative stations. However, 
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analysis of the results reviled poor accuracy of the adjusted corrections of calibration coefficients 
for stage 3 (due to the limited gravity range of involved stations, Table 5), but a rather good agree-
ment among the obtained calibration coefficients for the last three stages. In addition, frequent 
calibration of the two CG-3M gravimeters on the auxiliary calibration line (stations AGT02 and 
AGT03), indicated that apparent variations in the calibration constant from 2007, are mostly 
caused by instrumental error effects and external hydrological and barometric influences (Figure 
2). Therefore, the mean calibration constant was determined for each gravimeter for the period 
comprising the last three stages and the observation data files were re-calculated to correspond 
the common calibration constants.
Table 5:  Gravity range at stations involved in the specific campaign.
Campaign Stations with the extreme values
(min – max)
Gravity range
[µms-2]
Absolute measurements AGT06 – AGT02 2926
Stage 1 (2003) GT127 – GT112 4335
Stage 2 (2007) GT145 – GT122 2342
Stage 3 (2008) GT123 – GT129 719
Stage 4 (2009) GT131 – GT132 1715
Stages 2–4 together GT131 – GT122 3481
Finally, the adjustment of relative and combined adjustment of absolute and relative measurements 
were repeated. The final adjusted gravity values are presented in Table 6. The repeated adjustments 
resulted in no significant difference as regards the outliers and accuracy assessment of measurements. 
The reference standard deviation again amounts to 0.17 µms-2. However, moderate improvement 
has been obtained in the precision of the adjusted gravity values for the stations involved in stage 
4 and considerable improvement in the precision of adjusted corrections of calibration coefficients 
for the last three stages together. Specifically, the standard deviations of the adjusted gravity values 
are now up to 0.16 µms-2 for relative stations (Table 6). However, the standard deviations for the 
majority of stations are up to 0.10 µms-2. The higher values generally correspond to stations with 
extremely small gravity values (GT127 and GT131) or to the stations determined in stage 4 with 
sparse connections. The differences in the gravity values after the repeated adjustment are not sig-
nificant (up to 0.06 µms-2). The differences over 0.02 µms-2, in general, correspond to the stations 
determined in stage 4.
The values of adjusted corrections of calibration coefficients and corresponding calibration constants 
are given in Table 7. Although the gravity range of the ZOGN stations is almost double the range of 
the auxiliary calibration line, it is evident that the precision of calibration constants (and other adjusted 
parameters) is impaired by the weakness of the relative gravity measurements. In addition, since the 
range of the FOGN stations exceeds that of the ZOGN for 1400 µms-2, the uncertainty of the calibra-
tion constants, which in fact reflects the uncertainty of the scale of the network, is propagated over the 
whole network. Consequently, the gravity stations with extreme gravity values have the largest standard 
deviations.
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Table 6:  Final adjusted gravity values with their standard deviations.
Station g
[µms-2]
σg
[µms-2]
Station g
[µms-2]
σg
[µms-2]
AGT01 9806586.642 0.034 GT127 9802841.24 0.13
AGT02 9806622.513 0.048 GT128 9803822.45 0.09
AGT03 9805104.365 0.041 GT129 9805366.48 0.06
AGT04 9806073.898 0.045 GT130 9804952.13 0.07
AGT06 9803696.384 0.044 GT131 9802934.03 0.11
AGT05 9804067.33 0.10 GT132 9804649.09 0.07
AGT05E 9804058.35 0.09 GT133 9803430.43 0.09
GT101 9806427.05 0.08 GT134 9803978.73 0.08
GT102 9806433.43 0.06 GT135 9803201.30 0.09
GT103 9806145.15 0.08 GT136 9803557.84 0.09
GT104 9806809.04 0.07 GT137 9805797.10 0.07
GT105 9806072.71 0.06 GT138 9806003.29 0.07
GT106 9806512.82 0.06 GT139 9805794.43 0.07
GT107 9806261.67 0.05 GT140 9805686.77 0.08
GT108 9806577.79 0.06 GT141 9805626.92 0.07
GT109 9805927.54 0.06 GT142 9805582.74 0.07
GT110 9806819.81 0.07 GT143 9805706.99 0.07
GT111 9806572.41 0.06 GT144 9805472.07 0.07
GT112 9807176.44 0.08 GT145 9804073.03 0.11
GT113 9806893.01 0.06 GT146 9805315.03 0.10
GT114 9806826.59 0.07 GT147 9805241.22 0.09
GT115 9806421.74 0.05 GT148 9805290.73 0.09
GT116 9805621.29 0.06 GT149 9805160.37 0.09
GT117 9806584.17 0.06 GT150 9805023.61 0.09
GT118 9806386.89 0.06 GT151 9804399.71 0.11
GT119 9805157.44 0.06 GT152 9803733.51 0.12
GT120 9804874.32 0.08 GT153 9804299.24 0.14
GT121 9804932.83 0.06 GT154 9803933.28 0.13
GT122 9806415.18 0.06 GT155 9804604.69 0.13
GT123 9806085.23 0.06 GT156 9804500.39 0.14
GT124 9805632.58 0.06 GT157 9804274.26 0.10
GT125 9804638.32 0.07 GT158 9804608.89 0.16
GT126 9805078.36 0.07 GT159 9804247.68 0.12
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Table 7:  Adjusted corrections of linear calibration coefficients (y1), calibration constants applied on data before adjustment 
(GCAL1) and corrected calibration constants after adjustment (GCAL1) with corresponding standard deviations.
Campaign Gravimeter y1˙10
4 σy1˙10
4 GCAL1
[µms-2/CU]
GCAL1
[µms-2/CU]
GCAL1σ
[µms-2/CU]
1 (2003) 4372 4.260 0.448 60553.42 60579.22 2.71
4373 6.287 0.497 61828.17 61867.04 3.07
10012 4.251 0.410 84272.88 84308.70 3.45
2–4 4372 0.033 0.880 60626.601 60626.80 5.34
(2007– 2009) 4373 0.029 0.702 61908.981 61909.16 4.35
10012 0.098 0.632 84291.261 84292.08 5.33
1 Average calibration constants determined in the course of preliminary adjustment.
4 CoMPariSon WiTH PreviouS adJuSTMenTS
The final adjusted gravity values: A (Table 6) have been compared with the results from previous adjust-
ments of separate stages, which also comprise the measurements of all three gravimeters:
B: Bašić, Markovinović and Rezo (2006) for stage 1 and separate adjustments of stages 2, 3 and 4 
carried out in the CGI according to the model described in Repanić et al. (2010), which lean on 
the gravity values of the FOGN stations from Bašić, Markovinović and Rezo (2006);
C: Markovinović (2009), involving separate adjustments of stages 1 and 2.
Figure 5: Differences of the gravity values between the result of adjustment B and A (top) as well as C and A (bottom) against 
the gravity values. Differences for the ZOGN stations with gravity values form B or C fixed in the adjustment are 
designated as red disks, and for the absolute station with free gravity value as a red circle.
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Figure 6: Spatial distribution of differences of gravity values between B and A in µms-2. The extreme negative differences are 
in the Dinaric Alps region with higher altitudes, for which the closest fixed gravity values are close to the sea level, 
while the extreme positive differences are in the northern part, where the closest fixed gravity value is at AGT03 on 
the mountain Medvednica.
The differences between the results form B or C and the results A are substantial and amount to form 
-1.09 to 0.64 µms-2 or from -1.16 to 0.47 µms-2, respectively (Figure 5). The results from B and C are 
in considerably better agreement. The differences are in the range of ±0.29 µms-2, with exception of the 
gravity value at station AGT02 (0.61 µms-2), which was not held fixed in Bašić, Markovinović and Rezo 
(2006) adjustment. Given the declared accuracy of Scintrex gravimeters of 0.05 µms-2 and estimated 
standard deviations of gravity values from all three adjustments, the comparisons imply considerable 
systematic influences, obviously the errors in the calibration constants, which were neglected in Bašić, 
Markovinović and Rezo (2006) and Markovinović (2009). Indeed, there is a significant correlation 
between the differences and the gravity values itself (Figure 5). The correlation coefficient amounts to 
0.8 and the regression coefficient 2.104·10-4 for both comparisons. The received regression coefficients 
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are significantly smaller than the adjusted corrections of calibration coefficients, since, in the previous 
adjustments, the network scale was partially adjusted by fixing the gravity values at the absolute stations. 
However, if corrections of the calibration coefficients are not included in the functional model, such 
procedure can introduce distortions. One can notice that the extreme values of the differences are for 
the gravity values, which are out of the range of the fixed gravity values. Though, there are also gravity 
values within this range with considerable differences, what can be explained with considerable difference 
in the gravity values as compared to the closest stations with fixed gravity values (Figure 6).
5 TranSForMaTion FroM THe PoTSdaM SYSTeM
Because of the considerable differences of the new gravity values of the FOGN stations as compared 
to the previous adjustments, parameters of the linear transformation has been determined again. The 
parameters of the linear transformation function according to Torge (1989):
 gHRGS03 = gPotsdam + a + b (gPotsdam − g0) (6)
have been determined based on 25 identical stations with the gravity values in Potsdam and the 
Croatian Gravity System (HGRS03). In equation (6) gHRGS03 is a gravity value in HGRS03, gPotsdam in 
Potsdam system and g0 the average value in Potsdam system of 9805584.01, all in µms
-2. Determined 
parameters of the linear transformation with their accuracy estimates are a = -151.314±0.249 µms-2 and 
b = (-1.254±0.201).10-3 (Figure 7). The reference standard deviation is 1.27 µms-2. The values in the 
Potsdam system for the 25 stations of fundamental gravity network of former Yugoslavia, which are now 
included in the FOGN, are taken form Bašić, Markovinović and Rezo (2006) and Markovinović (2009).
Figure 7:  Differences between the gravity values in the Potsdam system and the new gravity values with the linear transfor-
mation function.
6 ConCluSion and ouTlooK
For the first time, the Croatian First Order Gravity Network has been adjusted as a whole, according to 
the adjustment model, which accounts for the corrections of linear calibration coefficients. Prior to the 
adjustment, revision of the absolute and uniform pre-processing of the relative measurements has been 
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carried out. The adjustment resulted in significantly different gravity values, as compared to the previous 
adjustments. Accordingly, the differences are an order of magnitude greater than the expected accuracy. 
It is shown that the differences in gravity values are mainly due to the errors in gravimeters’ calibration 
constants, which were neglected in the previous adjustments. Therefore, it is advisable to include the 
correction of the linear calibration coefficients in the functional model at least in the first iteration of 
the fundamental gravity network adjustment. In contrary, adjusted gravity values could be significantly 
biased and their estimated precision too favourable.
In addition, it is not advisable to change gravimeter’s calibration constant for every campaign, but rather 
only to check their stability on suitable calibration line. That is especially important if the calibration 
constant cannot be determined with sufficient accuracy and if the initial period of several years have been 
passed, during which calibration constant of a gravimeter significantly changes (CG-3/3M Autograv 
Automated Gravity Meter Operator Manual, 1998).
In the Croatian FOGN, there is an evident problem of the scale of the network. Neither the existing 
auxiliary calibration line nor the ZOGN sufficiently covers the range of the FOGN. To overcome this 
problem, several options can be employed. For example, the relative gravimeters can be calibrated on 
foreign calibration lines of sufficient range, e.g. the German or Swiss with gravity ranges of more than 
5000 and 6000 µms-2, respectively (Timmen et al., 2006; Marti et al., 2015). Alternatively, a calibration 
line of greater range than the present auxiliary line can be established in Croatia. In addition, at least 
several first order stations could be observed with a field absolute gravimeter, e.g. in Dinaric Alps on the 
south where the extremely small gravity values occur and where the establishment of in-door zero order 
stations is not possible. Also, connections to absolute stations of neighbour countries could be realized.
Furthermore, treatment of other significant instrumental error influences in Sintrex’ measurements, such 
as hysteresis effect and transportation drift, in pre-processing and adjustment model should be further 
analysed in order to minimise their effects on results of the adjustment.
Since the adjusted gravity values are significantly different from the previous, the author proposed to 
make the results of the new adjustment official.
acknowledgments
The author would like to thank two anonymous reviewers for their valuable comments which helped to 
significantly improve the manuscript. M. Kuhar and P. Pavlovčič Prešeren are gratefully acknowledged 
for help and advice concerning the manuscript revision. The author would also like to thank M. Kuhar 
for the help with the PREDICT software and a fruitful discussion on gravity network adjustment.
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Marija Repanić, M.Sc.
Croatian State Geodetic Administration
Gruška 20, HR-10000 Zagreb, Croatia
e-mail: marijarepanic@yahoo.com
Repanić M. (2017).  New Adjustment of the Croatian First Order Gravity Network. Geodetski vestnik, 61 (4), 630-648.
DOI: 10.15292/geodetski-vestnik.2017.04.630-648
Marija Repanić | NOVA IZRAVNAVA hRVAŠKE GRAVIMETRČNE MREžE 1. REDA | NEW ADJUSTMENT OF ThE CROATIAN FIRST ORDER GRAVITy NETWORK | 630-648 |