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Ŕ periodica polytechnica
Civil Engineering
58/2 (2014) 131–136
doi: 10.3311/PPci.7489
http://periodicapolytechnica.org/ci
Creative Commons Attribution
RESEARCH ARTICLE
Variations of the gravity field due to
excavations of the Budapest Metro4
subway line
Csaba Éget˝
o/Nikolett Rehány /Lóránt Földváry
Received 2013-04-25, revised 2013-06-10, accepted 2013-07-11
Abstract
The effect of the construction of the 4th subway line of Bu-
dapest (Metro4) on the potential surfaces of the gravity field has
been investigated from the aspects of monitoring vertical defor-
mation. In the study mass loss due to the excavation of the two
tunnels and of the stations has been considered. Practically, the
effect of the mass loss on leveling measurements was determined
at a level 1m above the ground, roughly simulating common in-
strument heights. The indirect effect of the actual deformations
of the physical surface on the leveling was not considered, so in
the investigation a rigid Earth has been assumed. In the study,
different arrangements of the leveling lines and of the excava-
tions were examined, furthermore, the steepness of the leveling
line and the density of the leveling points were analyzed. Ac-
cording to the results, under certain arrangements of the level-
ing line, the effect can reach the 0.05mm order of magnitude,
which is equivalent to the accuracy of the precise leveling.
Keywords
temporal variations of gravity ·deformation of potential sur-
faces ·precise leveling ·virtual veritcal motion ·prism mod-
elling
Csaba Éget˝
o
Department of Geodesy and Surveying, Budapest University of Technology and
Economics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary
e-mail: ecsaba@agt.bme.hu
Nikolett Rehány
Faculty of Civil Engineering, Budapest University of Technology and
Economics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary
Lóránt Földváry
Department of Geodesy and Surveying, Budapest University of Technology and
Economics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary
1 Introduction
Temporal variations of gravity field are of central interest in
geodesy and geophysics, since it reflects the mass redistributions
on the surface and within the planet Earth, providing a unique
tool for investigating its interior [14], [8]. Nowadays there is
an urge of such analyses in global and regional scales due to
the obvious signs of a climate change [11]. The state-of-the-
art gravity satellites turn to be very efficient tools for that [9],
[16], [12], [10]. As the knowledge of the temporal gravity in
global and regional scales improves notably, there is a demand
for the local gravity variations as well, since satellites are in-
sensitive for the temporal varying gravity in local scales. Thus,
it can only be determined independently by terrestrial measure-
ment techniques e.g. [3] to provide the short frequency part of
the spatial wavelength spectrum [15].
In the present study temporal variation of a local gravity fea-
ture due to a technogenic mass redistribution effect is modeled
and analyzed. In case of a huge industrial project, notable redis-
tribution of soil occurs due to the earthworks. It does affect the
gravity field [1]. As so, it also has an effect on the local hori-
zontal and vertical directions. As surveying instruments are set
up with respect to the horizontal or vertical, it does affect the ac-
companying surveying measurements. In the present study cer-
tain geodetic aspects of an actual surveying engineering applica-
tion is considered. The excavation of various layers of soil dur-
ing the construction of the new subway line of Budapest, Metro4
is investigated from the aspect of its effect on the simultaneously
performed vertical deformation measurements.
2 Theoretical Background and Formulation
2.1 The classical formulae of modelling
In case of excavating tunnels, deformations on the surface are
expected to be accompanied. Furthermore, the removal of a no-
table amount of soil changes the structure of the gravity field
too. Therefore, observed height variation, dH consists of two
components [1], [13]:
dh =dH +dN (1)
Variations of the gravity field due to excavations of the Budapest Metro4 subway line 1312014 58 2
where dh refers to the actual surface deformation, and dN is
change of the geoid undulation due to the mass loss. (Generally,
in this study the differential symbol, drefers to change in time
and not in space.) The change of the geoid undulation does af-
fect the height measurements, but has no effect on the neighbor-
ing structures, buildings. Such a height difference involved to
the geometrical leveling is rather the consequence of theoretical
looseness than actual vertical displacement, thus the confusion
of the two should be avoided.
Change of the geoid undulation by time, dN in Eq. (1) is usu-
ally interpreted as (cf. [2] and [17]):
dN =R
4πgZ Z dg ·S(ψ)dσ+R
4πgZ Z 2g
Rdh ·S(ψ)dσ
(2)
The terms on the right hand side of Eq. (2) refers to two inde-
pendent sources of mass variations:
1 deformation of the equipotential surface due to the mass loss
corresponding to the tunnelling,
2 deformation of the equipotential surface due to the actual ver-
tical displacement of the physical surface.
For every infinitely small area, dσ=cosϕdϕdλthe spherical
distance from the point of interest, Pis computed as
cosψ=cosθcosθP+sinθsinθPcos(λ−λP) (3)
Eq. (2) describes the temporal variation of geoid undulation as
a function of that of the gravity anomaly, dg, weighted by the
Stokes function, S(ψ). Considering a mass anomaly model, dm
as known input data, Eq. (2) can be solved by using the Newto-
nian gravitational law,
dg =kZdm
r2(4)
A more direct approach to obtain temporal variations of geoid
undulation from a mass variation model is to derive geoid un-
dulation variations from Newtonian potential using the Bruns’
equation.
dN =1
˜γkZdm
r(5)
In Eq. (5) ˜γrefers to the mean normal gravity between the point
and the reference ellipsoid along the plumb line. Note that
Eq. (5) is analogue with the first term of the right hand side of
Eq. (2).
In the followings a single height difference between stations A
and Bwith repeated levelling measurements performed at time
tand t1is considered (c.f. Fig. 1). Temporal variation of geoid
undulation between the two stations can be defined as the differ-
ence of the temporal changes in each point
dNAB =dNB(t1−t0)−dNA(t1−t0) (6)
The measured vertical displacement is interpreted as temporal
change of the height difference
dHAB =HAB (t1)−HAB (t0)(7)
Fig. 1. Visualization of the apperent vertical displacements due to the tem-
poral variation of gravity field
Then the relationship between change of geoid undulation and
observed vertical displacement can be derived from equations
above as follows (c.f. equation 212.5 of [1], on page 31):
dHAB =−dNAB −
˜gB(t1)−˜gB(t0)
˜gB(t1)HAB (t0).(8)
In Eq. (8) ˜gBrefers to the mean gravity between levels of Aand
Balong the plumb line through point B. This is approximately
equal to the gravity at the half of the HAB height difference.
2.2 Elaborating the formulation
In the present case, height of the benchmarks on the test re-
gion is available after the tunneling, after the excavation has
been performed. However, according to [1], HAB (t0)in Eq. (8)
refers to the original height difference, before the deformation
takes place. As so, a modified version of Eq. (8) is derived:
dHAB =−dNAB −
˜gB(t1)−˜gB(t0)
˜gB(t1)(HAB (t1)+dNAB)(9)
For details of the derivation see [5].
In the present study effect of the excavation on the level sur-
faces was estimated based on equations Eq. (5)- Eq. (9)). The
integral of equation Eq. (5) was approximated by a summation
over mass elements. A preceding study has already been pre-
sented by [4], where the non-deformational effects due to the
tunneling were investigated in a regular grid, and not along lev-
eling lines as it is to be presented here. In this study the effect of
the actual surface deformation on the gravity field, i.e. second
term of the right hand side of Eq. (2) has been neglected. The
effect due to actual surface deformations has been analyzed and
discussed by [6].
3 Data and Calculations
The calculations have made use of the following available
data:
1 longitudinal sections and cross sections of the tunnels pro-
vided by the DBR Metro Project Directorate
2 soil density along the tunnels provided by the Geovil Ltd
3 gravity anomaly data in the vicinity of the tunnels provided
by the Hungarian Geological and Geophysical Institute
Per. Pol. Civil Eng.132 Csaba Éget˝o /Nikolett Rehány /Lóránt Földváry
4 horizontal trace of the leveling lines provided by the Consor-
tium of the Soldata Ltd and Hungeod Ltd
The top view of the tunnels and the location of the surrounding
gravimetric data are displayed on Fig. 2. The used coordinates
are the local stereographic coordinates (BÖV, equivalent to Bu-
dapest Municipal Projection system).
Fig. 2. The tunnel (solid black line interrupted at the stations) and the gravi-
metric points (grey stars) with main structures of Budapest in the background.
The axis are local stereographic coordinates in m.
First a 3D geometry model of the tunnels and of the stations
was determined based on the longitudinal sections. The corre-
sponding mass model of the excavated soil has been derived by
using the soil density data, and the effect of the excavation on
geoid undulation was determined by Eq. (5). The resolution of
the mass model was 30cm, i.e. the masses were concentrated
into cubes with 30 cm long edges. Then the adequate formulas
of rectangular prisms (c.f. [7]) were simplified to simple point
masses, and Eq. (5) was used. The accuracy loss due to this sim-
plification was verified by [4], and it was found to cause negli-
gible difference. In case of the stations, the exact rectangular
prism formula was used. The mass removals due to excavation
in the tunnels and in the stations were added, and the result is
considered to be an estimate of the effect of the excavation with-
out any displacements assumed to occur.
Subsequently, gravity value was interpolated from the neigh-
boring gravity data to each point of evaluation, and for every
single height differences of the leveling line the effect of the tun-
neling on the every single height measurements was determined
by Eq. (8).
The effect of the excavation was evaluated at the points of ac-
tual vertical control leveling lines on the south-east part of the
subway line, c.f. Fig. 3 (note that the interruptions of the tun-
nels refer to the presence of stations). To the known heights of
the benchmarks 1 m has been added to simulate a typical instru-
ment height. Altogether there were 40 leveling lines involved in
the analysis. Among them, only some typical features are to be
presented in the following section.
4 Results
Generally, the effect of the excavation on the measurements
along the leveling lines is varying between 0 and 0.01mm.
Though it does not seem to be much, it is important to note
that effect of this theoretical error is in the same magnitude than
that of the precision of the precise level instruments. As so, it is
detected in the measurements. Furthermore, it could be detected
that under certain conditions it accumulates along the leveling
line. The accumulation of the error depends on the orientation
of the line, the distance of the instrument and the staff, and the
height difference.
4.1 Dependence on the relative orientation
The orientation of the leveling line relatively to the orienta-
tion of the tunnels makes notable impact on the result. The two
extremes of the orientation is when the tunnel and the leveling
line are parallel to each other and when they are perpendicularly
cross each other. There was no good example for the perpen-
dicular crossing (c.f. Fig. 3), but as an example, a nice example
for that is adopted from [4] on a different part of the same sub-
way line. The perpendicular crossing with very short distances
between the instrument and the staffis displayed in Fig. 4. The
figure shows the longitudinal section of the leveling line. The
ordinate is the distance from the midpoint of the two tunnels,
and the abscissa is the estimated error in µm. The horizontal
location of the tunnels is displayed by the two circles close to
the zero value of the ordinate. From the figure at the crossing a
sudden change of sign can obviously be detected. Due to that,
when the line is processed, the different sign of the error neutral-
izes the effect. So practically the closure error is compensated.
On the other hand, in cases, when the leveling line runs par-
allel to the tunnels, the error effect becomes similar for every
single height differences. Thus the error is accumulating along
the line. For such an example in Fig. 5 and Fig. 6 the line no.
34 is presented. Fig. 5 shows the horizontal arrangement of the
line, which is parallel to the tunnels, passing through at the up-
per left corner. The abscissa and the ordinate refer to the BÖV
coordinates in meter. On Fig. 6 the error estimate is shown along
with the longitudinal section of the leveling line. The zero or-
dinate refers to the starting point. The error on the abscissa is
displayed in µm.
As it can be seen, the error has consequently the same sign,
affecting all the measurements in the same extent. On Fig. 6 also
the back and forth errors are displayed. It is obvious that this er-
ror cannot be eliminated by averaging back- and forth measure-
ments. As so, in such a case the error remains in the adjusted
heights even in case of a thorough processing.
4.2 Dependence on the height difference
In the cases of the tested leveling lines, with the exception of
line no.14, the height difference between the first and the last
point is only 1-2 m. The last section of the line no. 14 climbs up
to a hill, resulting in a 13 m height difference along the line, cf.
Fig. 7 and Fig. 8. On Fig. 7 and Fig. 8 the axes are the same as
in the case of Fig. 5 and Fig. 6, respectively.
According to Fig. 8, the dependence of the error on the height
Variations of the gravity field due to excavations of the Budapest Metro4 subway line 1332014 58 2
Fig. 3. The location of the vertical control lines (thin lines) with respect to the tunnels (thick parallel lines).
Fig. 4. An example on the effect of perpendicularly crossing the tunnels.
Per. Pol. Civil Eng.134 Csaba Éget˝o /Nikolett Rehány /Lóránt Földváry
Fig. 5. Top view of the levelling line no. 34. dashed line: leveling line.
symbol ’o’: place of staff, symbol ’x’: assumed place of the instrument. The
two tunnels are marked by thin lines on the upper left part of the figure.
difference is obvious. Where the steep part comes, the error
suddenly increases with an order of magnitude. The result is
not surprising at all, as in Eq. (9) height difference appears as a
multiplier.
Fig. 6. The estimated error for every single height differences along level-
ling line no. 34 in µm. The error along the back- and forth levelling is deter-
mined independently, then the mean error was derived. At the bottom staff- and
instrument places are shown in the same manner as in Fig. 5.
4.3 Dependence on instrument-staff distance
Oddly, the dependence of the error on the distance of the level
instrument and the leveling staffcannot be demonstrated from
real measured data, like in the present case. Although it is obvi-
ous: measuring with short distances means a finer resolution of
the level surfaces, than long ones. Thus with shorter distances
the error can be decreased.
The reason it could not be demonstrated is a very practical
one: in case of an actual work, short distances are used only in
case of large height differences. So it is used for steep fields,
such as line no 14 on Fig. 6. However, in this case the unlike
effect of the height difference suppresses the effect of the short
distances.
5 Conclusions
Basically, the effect of the excavation on the measurements
along the leveling lines was found to have theoretical impor-
tance in most cases. Still, in such cases, when the leveling line
Fig. 7. Top view of the levelling line no. 14. dashed line: leveling line.
symbol ’o’: place of staff, symbol ’x’: assumed place of the instrument. The
two tunnels are marked by thin lines on the bottom of the figure.
Fig. 8. The estimated error for every single height differences along levelling
line no. 34 demonstrating a very steep levelling line at its final section. The error
along the back- and forth levelling is determined independently, then the mean
error was derived. At the bottom staff- and instrument places are shown in the
same manner as in Fig. 7.
Variations of the gravity field due to excavations of the Budapest Metro4 subway line 1352014 58 2
was running parallel to the tunnels, the cumulative error reached
the precision of precise leveling. Even though precise leveling
measurements are subsequently processed, dropping the random
errors out, in case of parallel orientation the error remains in the
adjusted results and becomes systematic.
Further relevance of the study is that Metro4 is just a subway
line. There are several other constructions resulting in huge ex-
cavations, removal of notable masses. It is important to keep in
mind that redistribution of mass does affect the gravity field. In
case of a large construction there is always a demand for sur-
veying control measurements in order to detect actual deforma-
tions. Thus it is worth to conclude that surveying control mea-
surements are infected by the temporal variation of the gravity
field as well, and that the two cannot be separated from the mea-
surement, only by theoretical considerations. It remains the case
as long surveying methods make use of the gravity field for ad-
justing the direction of the horizontal and the vertical.
Finally, it is worth to note that this study was considering ap-
parent deformations only. In reality, there are always real de-
formations as well. The actual deformations occur much closer
to the instrument than the tunnels are, it is just 1-1.5 m below
the instrument. These deformations also have their impact on
the gravity field. It means that real deformations have two ef-
fects on the control measurements: a direct effect, as they are
the deformations to be detected, and an indirect effect, as they
alter the gravity field. Even though these close-to-instrument
deformations can be small in magnitude, as the gravity vanishes
with square distance, they can affect the measurements in larger
extent. As a complementary research to the present study, it is
essential to quantify the magnitude of the indirect effect of real
deformations.
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