GPS MEASUREMENTS OF THE GEOTECTONIC RECENT
MOVEMENTS IN EAST SLOVAKIA
Prof. Vladimir SEDLAK, Slovakia
Key words: GPS, 3D transformations, topocentric
coordinate system, free adjustment, adjustment with constraints,
deformation teststatistics.
Abstract
The paper deals with transformation procedures
observed GPS (Global Positioning System) data from the
World Geodetic System WGS-84 into the national geodetic grid
datum S-UTCN (System of United Trigonometric Cadaster
Network) and Baa.(The Baltic Sea after adjustment). GPS
measurements are situated into the geodetic network in the Košice-Valley
for a purpose of deformation surveying geotectonic recent movements in
the East-Slovak regions. Adjustment with constraints and free
adjustment are applied at determining coordinates of the geodetic
network points.
Transformation from WGS-84 into S-UTCN is
the most frequently by means of using the 7-element Helmert
transformation with using three identical points. Geodetic network was
adjusted by two ways. In a case when datum parameters are absolutely
accuracy then an adjustment with constraints is considered; in a case
when datum parameters are determined with a concrete accuracy, what
has also an influence on an accuracy of adjusted parameters except on
measured accuracy, then a free adjustment is considered.
The GPS measurements are realised on points
of the geodetic network (GN) localised in the Kosice-Valley
(Slovakia). The aim of these measurements is determining recent
geotectonic movements in the urban agglomeration of Kosice-city. 3D
coordinates of the network points determined from satellite navigation
present a realisation result of the solved scientific project at the
Department of Geodesy and Geophysics of the Technical University of
Kosice since 1997.
GPS measurements are periodically realised
twice a year (spring and autumn). Altogether, 17 points of GN
are measured by means of using the GPS static method. A
priority of the chosen static method for our measurements is above all
a high accuracy in determining point position, which is conditioned by
longer period of measurement on a determined point (cca 45 minutes).
The determined GN points are solved by double GPS vector
technology always regarding two reference points, i.e. three GPS receivers
are used for measurements. These points are placed so that the
territories in which some geotectonic movements are presupposed
according to geologists. The main tectonic fault in the Kosice-Valley,
according to which two expressive geological faults of the Earth
ground blocks should move, is assumed in the north-south direction
along the river Hornad. The secondary tectonic faults of smaller
extent are in the direction perpendicular to the Hornad fault, i.e. in
the east-west direction. These secondary tectonic faults are mutually
parallel.
Three double-frequency GPS receivers Sokkia
GRS 2100 were used to measurement. Adjustment of observed data was
realised by the firm software Prism ver 2.1 Sokkia. Coordinates of all
points in GN were transformed from WGS-84 into a plane
coordinate system S-UTCN, which is obligatory coordinate system
for realisation of geodetic works in Slovakia. The non-linear rotary
matrix method was applied to the adjustment. After transformation, the
coordinates were consecutively adjusted by an adjustment with
constraints.
For a purpose of deformation consideration in the
monitoring network the coordinate differences are subjected to the
teststatistic hypotheses. A size of deformations is presented by the
deformation vectors on the individual network points. Modelling
deformations in the Kosice-Valley is based on GIS data by means of
using the MicroSoft and Kokes software.
Introduction
The GPS measurements are realised on points
of the geodetic network (GN) localised in the Kosice-Valley
(Slovakia) (Fig. 1). The aim of these measurements is determining
recent geotectonic movements in the urban agglomeration of Kosice-city
(Sedlak et al., 1998). 3D coordinates of the network points determined
from satellite navigation present a realisation result of the solved
scientific project at the Department of Geodesy and Geophysics of the
Technical University of Kosice since 1997.
GPS
measurements in the KoSice-Valley
GPS measurements are periodically realised
twice a year (spring and autumn). Altogether, 17 points of GN
are measured by means of using the GPS kinematic method.
Priority of the chosen kinematic method for our measurements is above
all a high accuracy in determining point positions which is
conditioned by the period of measurement on a determined point (cca 5
minutes). The accuracy of the kinematic methods is same like at the
static method (Hofmann-Wellenholf et al., 1993; Seeber, 1993; Leick
1995). The determined GN points are solved by double GPS vector
technology always regarding two reference points, i.e. three GPS receivers
are used for measurements. These points are placed so that the
territories in which some geotectonic movements are presupposed
according to geologists. The main tectonic fault in the Kosice-Valley,
according to which two expressive geological faults of the Earth
ground blocks should move, is assumed in the north-south direction
along the river Hornad. The secondary tectonic faults of smaller
extent are in the direction perpendicular to the Hornad fault, i.e. in
the east-west direction. These secondary tectonic faults are mutually
parallel (Jacko, 1997).
Figure1. Positioning the Kosice-Valley
GN (1:100 000).
Three double-frequency GPS receivers Sokkia
GRS 2100 were used to measurement. Adjustment of observed data was
realised by the firm software Prism ver 2.1 Sokkia. Coordinates of all
points in GN were transformed from WGS-84 into a plane
coordinate system S-UTCN, which is obligatory coordinate system
for realisation of geodetic works in Slovakia. The non-linear rotary
matrix method was applied to the adjustment (Melicher and Flassik,
1998). After transformation, the coordinates were consecutively
adjusted by an adjustment with constraints.
COORDINATE TRANSFORMATION OF THE GN POINTS FROM
WGS-84 INTO S-UTCN
System NAVSTAR GPS uses WGS-84 with
the purpose of expressing the position anywhere in the earth and
space. The reality that the GPS system determines a position in
global dimensions is its priority. However, the disadvantage for
surveying is its limitation in a plane rectangular system that is our
national geodetic grid S-UTCN (system S).
Transformation of coordinates from WGS-84 into a
topocentric horizontal system
The coordinate axes (X, Y, Z)WGS-84
with an origin in the centre of ellipsoid create the system SWGS-84
(Fig. 2). The coordinate axes X´´, Y´´, Z´´
create the topocentric horizontal coordinate system S´´ (Melicher
and Flasik, 1998; Hurcikova, 1998). Its origin lies in the point D.
Point D is one point belonging to points of a local network.
This point is situated approximately in its centre. To assume that the
geodetic horizon in D is a parallel plane to S-UTCN is
only possible in a case of local GN with a small dimension (if
distances between network points are not longer than some units of
kilometres). Table 1 presents coordinates of the point D. Axes X´´
and Y´´, which lie in a geodetic horizon of the point D,
while the axis X´´ is oriented into the south branch of a
meridian. The axis +Z´´ lies in a normal line and is directed
into the geodetic zenith and the axis +Y´´ creates with the
mentioned axes a left-hand system.
Figure 2. SWGS-84
and S´´.
Table 1. Coordinates of the point D
in WGS-84 and S-UTCN.
Transformation of coordinates from the system SWGS-84
into the system S´´ is a possible according to the following
equation
(1)
where index D means that the tangent point
is considered, D,(D)
is ellipsoidal geocentric latitude, (longitude) of point D,
RY (90° - D ),
RZ(D ) are non-linear
rotation matrices according to the following equations (Melicher et
al., 1993)
(2)
(3)
The coordinates of the GN points in WGS-84
are obtained by a convenient adjustment of measurements, which
were realised by the system NAVSTAR GPS. The right-hand system
is changed into left-hand which is preferred in geodesy. This change
can be reached by multiplying a diagonal matrix with the diagonal (1,
-1, 1). The point D has the coordinates (X, Y, Z)T=(0,
0, 0)T in the system S´´.
The system S´´ is also possible to obtain
by using the following equation
(4)
where D X is
distance of the normal from the centre of the ellipsoid, D
Z is a displacement of the plane XY into the point D in
the normal direction.
In the sense of Figure 3, the quantities D
X and D Z are derived
according the equations
 (5)
where A(D) is the
transverse radius of curvature in the point D, e is the
numerical eccentricity, HD is the ellipsoid height
of the point D.
A transition between the local and the commonly
used national system should be the simplest in their contact point for
a purpose of using the state network. It means that the coordinates of
the local network should not much differ from S-UTCN. It can be
reached by turning the system S´´ in the point D about
the meridian convergence C (Fig. 2) of S-UTCN and by
displacement of the origin of the system S´´ round the
rectangular coordinates XD, YD in S-UTCN.
The mentioned transformation is expressed by the following equation
(6)
where RZ(-C) is the
rotation matrix determined by the equation
(7)
and hD is the over-sea level
height of the point D in the Baltic sea elevation system after
adjustment (Baa.).
In this way we obtained the topocentric horizontal
coordinate system whose the coordinate axes X´, Y´ lie in the
geocentric horizon of the point of normal intersection of the point D
with the geoid. Because the point D has the coordinates (0,0,0)T
in the system S´´, then this point will obtain identical
coordinates with the coordinates in S-UTCN by adding the vector
(xD,yD,hD)TS-UTCN
.
Figure 3. Transformation from the
geocentric into the topocentric system.
Transformation of coordinates into the local
coordinate system
It is not possible to calculate directly the values
of elements which would harmonise with the values measured in a
terrain using the coordinates of points in the system S´.
These coordinates are influenced by the Earth curvature and also by a
relative difference in elevation of point over the horizon plane.
Regarding to a network dimension, we can substitute the ellipsoid by
the reference sphere whose radius is equal to the mean radius of the
Earth curvature R in the point D according to the
equation
(8)
where M is the meridian and N the
transversal radius of curvature of the ellipsoid, which are described
by the following equations (Mervart and Cimbalnik, 1997)
(9)
(10)
where a and e2 are the
constants of the used ellipsoid (the semimajor axis and square of the
1st numerical eccentricity) (the Bessel ellipsoid).
An influence of the difference of elevation is
possible to eliminate if the coordinates X´, Y´ are reduced
into the intersection of the normal with the tangent plane, or with
the basic plane. The reduction from a relative difference of elevation
is possible to influence significantly by moving the geodetic horizon
into the basic plane of GN in the height z0.
This height equals to approximate mean elevation value in which
geodetic measurements are realised. The reduction of the coordinates
X´, Y´ of the GN points in the system S´ into the
intersection of the normal with the basic plane equals to a gnomic
projection which regarding to the network dimension is considered a
conform projection.
The presented method has several priorities. Above
all, it is the fact that a high relative accuracy in determining point
positions by means of using NAVSTAR GPS technology is not lost.
Similarly a measurement on one identical point is only enough instead
of three identical points, by that a transmission of some errors at
the transformation can be reduced. Reductions from elevation and
cartographic distortion are needed in S-UTCN where only
reduction from a relative height or elevation is considered in some
local system. This reduction is also minimised by a convenient choice
of transformation parameters (Melicher and Flassik, 1998).
Adjustment with constraints of 2D geodetic network
Geodetic networks can be adjusted by two ways. If
we consider datum parameters as absolutely accurate and we do not
include them into an adjustment process, the adjustment with
constraints is considered in this case. In fact that datum parameters
are also determined with a concrete accuracy that has an influence on
an accuracy of adjustment parameters except for measurement accuracy.
In this case a network can be adjusted by a free adjustment with
consideration of datum parameters (Mervart, 1994). Regarding the
applied confinement adjustment in the Kosice-Valley GN a theoretic
procedure of this adjustment is presented, which is the most
convenient for our national geodetic grid S-UTCN.
The least mean square method (LMSM) is
chosen as an estimate principle, and the inverse solution is chosen as
a mathematical principle, which is a standard procedure in an
adjustment of GN. After adjustment the position and form
of GN are changed but the datum point positions are not changed
(datum points are considered as absolutely accurate). This fact is
presented so that the configuration matrix A and also
the matrix N of GN will be regular; the rank of
matrices h(A)=k, h(N)=k, where k is
a number of determined parameters. For the adjustment the following
four vectors and matrices are necessary to be:
is the
vector of approximate coordinates of the network points which are
calculated from measured quantities and approximate coordinates of the
reference points, instead of the coordinates obtained after
transformation;
 is the vector of
approximate values of measured observations which are calculated from
the approximate coordinates of the GN points on a base of the
model equations (i.e. common mathematical equations for calculation of
geodetic elements, for example lengths, angles, etc.);
 is the
configuration matrix of GN. Terms of this matrix are determined
by partial derivatives of the model equations L
according to the studied parameters. For a check it is possible to
spread this matrix for the datum (object) points too, by this way we
can get a global configuration matrix. A sum of the terms in a row of
the constructed matrix must be equalled to zero. However, we only
consider a submatrix containing the determined points at calculations,
where n is a number of measurements and k is a
number of the determined parameters;
 is the cofactor
matrix of the measured quantities.
It is the matrix in which cofactors of the measured
quantities are occurred. These cofactors can be calculated according
to the equation
(11)
where  is the
standard deviation of measurement, while the variance factor 
(a priori variance factor) is determined by the equation
(12)
Solving equations of the estimate statistic model
by means of using MSM we will get the following the linear
equation
AT. .A. -AT. .dl=0
(13)
where dl=l-lo is
the vector of reduced observations, while l is the
vector of the observed quantities and lo
is the vector of the approximate values of the measured quantities.
If we indicate N=AT..A
and n = AT..dl,
we will get the following equation for the vector of the adjusted
coordinate complements
=N-1.n
= (AT.
.A)-1 .AT.
.dl .
(14)
After adding to
the vector of the approximate coordinates of points we will obtain the
adjusted coordinates  of points
(Tab. 2) according to the equation
=Co+.
(15)
The quality of the adjusted network is universally
characterised by two matrices:
- the cofactor matrix of the estimates 
of coordinates
=(AT.
 .A)-1=N-1
, (16)
- the covariance matrix of the estimates 
of coordinates
 , (17)
(18)
where  is empirical
variance factor determined by the equation
(19)
in which a numerator expresses the quadratic form
of corrections W and a denominator
expresses the number of superfluous measurements (redundantion of a
network). A presumption of better quality of GN will increase
together with increasing the difference n-k. The vector of
corrections v is determined by the equation
v=A.  -dl
(20)
Table 2. The adjusted coordinates
of the GN points in S-UTCN.
The covariance estimates of the coordinates are
situated on a diagonal of the covariance matrix in a direction of
individual axes. The adjusted values of the measured terms  =l+v
are also determined in a frame of an adjustment.
Deformation vector d was estimated by
a simple way, i.e. algebraic calculations in rectangular triangles in
the plane of S-UTCN. Position deformation vector presents
deformations in a plane of X,Y axes and height deformation
vector presents deformations (subsidences) in the Baltic sea level
system as a difference between the heights on the GN points.
Table 3 presents the deformation vectors with standard deviations of
the GN points. Because all deformation vectors are in limits of
the error circles, we did not presuppose any recent geotectonic
movements in the Kosice-Valley.
Table 3. The deformation vector d
(in position and height) and standard deviations of the GN
points.
CONCLUSIONS
The results of measurements by GPS technology
confirm a typical event of using this satellite measurement in GN with
a spread application in geodesy. The applied kinematic method of GPS
measurements shows on a high accuracy of satellite measurements
which is also acceptable for some other geodetic measurements, for
example: a deformation surveying the earth surface and engineering
structures. The reached results of the presented transformation
procedures refer to the adaptability of transformations from WGS-84
into the national geodetic grid S-UTCN and Baa. The
chosen confinement adjustment by means of using the Gauss-Markov model
is demonstrated as the most suitable mathematical model in an
adjustment of GN in the Kosice-Valley locality (Hurcikova,
1998). The presupposed possible recent geotectnonic movements in the
direction of north-south along the Hornad river are not confirmed.
The paper followed out from the research Project
No. 1/7335/20: Deformation modelling geotectonic recent movements in
Kosice Valley.
References
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Havlice,K., Sadera,M., Gasinec,J. & Jacko, S. (1998) Deformacne
modelovanie geotektonickych recentnych pohybov vo vychodoslovenskom
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is associate professor on
geodesy and engineering surveying at the Technical University of
Kosice, Department of Geodesy and Geophysics, Slovakia. He graduated
on mine surveying and geodesy at TU Kosice in 1980. In 1987 he
obtained PhD. on mine surveying. He is member of FIG (Commission 4),
Presidium ISM and General Assembly CLGE.
CONTACT
Assoc. Prof. Vladimir Sedlak, MSc., PhD.
Technical University
Department of Geodesy and Geophysics
Park Komenskeho 19
SK-043 84 Kosice
SLOVAKIA
Tel. + 421 95 6022 916
Fax + 421 95 6336 618
E-mail: sedlak@tuke.sk
14 April 2001
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