G
V
GEODETSKI VESTNIK | letn, / Vol, 59 | št. / No. 3 |
8
l 59/3|
identifikacija premikov identification of z uporabo različnih movements using geodetskih metod different geodetic deformacijske analize methods of
deformation analysis
Zoran Sušic, Mehmed Batilovic, Toša Ninkov, Ivan Aleksič, Vladimir Bulatovic
UDK: 528,02
Klasifikacija prispevka po COBISS.SI: 1,02 Prispelo: 20.2.2015 Sprejeto: 17.8.2015
DOl: 10,15292/geodetski-vestnik,2015,03,537-553 REVIEW ARTICLE Received: 20.2.2015 Accepted: 17.8.2015
IZVLEČEK ABSTRACT
V prispevku je predstavljena primerjava metod deformacijske analize geodetskih mrež na podlagi simulacije dvodimenzionalnih komponent vektorjev GNSS v dveh terminskih izmerah za potrebe odkrivanja nedvoumnih premikov v horizontalni ravnini. V članku primerjamo naslednje modele deformacijske analize: Pelzerjevo metodo (hannovrski postopek), metodo Karlsruhe, modificirano metodo Karlsruhe in metodo, kije izvedena sprostodostopno programsko rešitvijo JAG3D.
This paper is based on comparative analysis of applied analysis methods on geodetic networks deformation, based on two-dimensional components of GNSS baseline vectors simulated by two epochs, for the purpose of identifying significant movements in horizontal plane. The following models of deformation analysis have been applied: Pelzer (Hanover procedure) method, Karlsruhe method, Modified Karlsruhe method, and the method implemented in the JAG3D open-source software.
KLJUČNE BESEDE KEY WORDS
deformacijska analiza, Pelzerjeva metoda, metoda Karlsruhe, modificirana metoda Karlsruhe, JAG3D
Deformation analysis, Pelzer method, Karlsruhe Method, modified Karlsruhe Method, JAG3D
1	INTRODUCTION
Each point on the Earth's surface is subject to constant changes under the influence of various factors, such as tectonic influences, ground waters, landslides earthquakes and other. Constructions developed on the ground are susceptible to subsidence and deformations, occurring as a consequence of internal and external forces, such as the influence of wind, changes in temperature and ground water levels, tectonic and seismic influences, dynamic and static building load, etc. If the deformations had not been accounted for during the design or if they were not discovered timely, they may lead to disturbances in normal utilization, or even to a collapse of the entire construction. In order to prevent negative consequences, it is necessary to observe the construction by applying a geodetic method. Interest in the geodetic methods application for monitoring the construction behavior has suddenly risen after the catastrophic collapse of Gleno Dam in Italy in the year 1923 and of St. Francis Dam in California of 1928 (USA).
The Geodetic Deformation Analysis Methods, referring to the relative construction points' movements, have being applied since 1970's. The most familiar method have been named after development centers or authors: Hanover (Pelzer, 1971), Delft (Heck et al., 1982), Karlsruhe (Heck, 1983), Fredericton (Chrzanowski, 1981) and Munich (Welsch, 1981). Modern application of the methods above can be found with the following authors: (Setan and Singh, 2001; Jäger et al., 2006; Marjetic et al., 2012).
Application of geodetic methods in order to determine land and construction deformation is based on examining of temporal evolution of geodetic networks, realized through physically stabilized points. The proper design of geodetic networks is an integral part of the geodesist's task. Movements of points are being established by comparison of geodetic measurements, being performed in different time epochs. The geodetic network being used to determine movements and deformations consists of the reference points outside the construction (basic network) and of control points on the construction, which are interlinked through geodetic measurements (Caspary, 1987). Basic geodetic network points need to be stabilized outside the expected deformations zone, while the points on the construction need to provide the best possible spatial interpretation of the construction.
2	PELZER METHOD
The method is based on testing congruence of points' coordinates, obtained after network adjustment in the two epochs. Each epoch of measured values is being adjusted independently, with the assumption that the measured values contain only random errors, with a normal distribution.
2.1 Homogenous Accuracy of Measurement in Two Epochs
Adjustment of two epochs provides a posteriori reference variances cj'^ and CJi2 . It is therefore necessary to establish whether the measured values in both epochs have homogenous accuracy. For that purpose, we are establishing a zero (H^) and an alternative (H) hypothesis (Pelzer, 1971; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2001; Ašanin, 2003; Vrce, ^011):
H0 : E {^C ) = E ) = against H^ : E ) ^ E (cC^^ ) ^ .	(1)
Tests statistics is:
T	for > or T ~ ^ for >	(2) |
■■ 2	■	2	CTo,	" 2
where f^ and_/2 are number of freedom degrees in zero and control measurement epoch.	H
If the T < F^j,, zero hypothesis is not rejected and unique a posteriori variance is being calculated: ®
+ f -CT.
(3)
fi -f^o, + fi -f^o'
0f
where the / is a sum of numbers of freedom degrees in zero and control measurement epoch.
2.2	Testing the Network Congruence in Two Epochs
Network congruence is tested using mathematical statistics testing methods. For that purpose, the zero (H^) and the alternative (H) hypothesis is being set (Pelzer, 1971; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2001; Ašanin, 2003; Vrce, 2011):
Ho-. E (xT, ) = E (xX') against Ha : E (xX, E (xT^)	(4)
where x, and xl^ are vectors of zero and control measurement epoch coordinates.
Average non-fitting 6 2, which contains information on movements of points, is calculated using the formula:
e' = d - Q+- d, d = XT' - X,,	(5)
h
where: d - vector of coordinate differences, Qd - pseudo-inversion of coordinate differences cofactor matrix, and h - rank of the coordinate differences cofactor matrix Q^. The test statistics is as following:
T	.	(6)
f^o
When the T < Fh,f, zero hypothesis is not being rejected, i.e. so that the network is congruent in two epochs; then an alternative hypothesis will be accepted.
2.3	Testing Congruence of Basic Network Points
If unstable points' existence in the network is established, a hypothesis on basic network points' congruence is being tested. Hypotheses are being set as (Pelzer, 1971; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2001; Ašanin, 2003; Vrce, 2011):
Ho : E (XLs, ) = E (Xs2) against Ha : E (XE (T^^)	(7)
Where X^^ and xS^ are the vectors of basic points' coordinates in the zero and control measurement epoch.
Vector of the coordinate differences d and the pseudo-inversion of coordinate differences cofactor matrix Q + are decomposed as follows:
d =
Q+= Pd =
p	P
SS	'■SO
p	p
OS OO
(8)
where the S designates the basic network points and O designates the points representing the construction.
Average lack of coordination for basic network points is calculated using following formula:
d^ • P ■ d.
hs
where Pss = Pss - Pso " PoO " Pos . The test statistics reads:
T ~ F where the h^ is a rank of Pss .
(9)
(10)
When the T < F^ , zero hypothesis is not being rejected, i.e. basic network points are congruent in two epochs or else; when the (6) alternative hypothesis is accepted.
2.4 Localization of Unstable Basic Points
When the basic network points' congruence in two epochs has not been established, there is a need to localize unstable basic points, i.e. to determine which basic points are unstable. For that purpose, coordinate differences vector dS and coordinate differences cofactor matrix Pss are decomposed as follows (Pelzer, 1971; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2001; Ašanin, 2003; Vrce, 2011):
d s =
d F "	Pss =	Pff	Pf, "
.d, _		.P,F	P BB
(11)
where the F designates basic points being considered conditionally stable and the B designates basic points considered conditionally unstable.
For each basic network point, average gap is set as following:
d T • ^ • d.


where d ^ = d ^ + P,-,; • P^f
^,(J = 1,2, k )
• d F.
(12)
In the set of values , the maximum value is recognized, so that the point to which the maximum value refers to is considered unstable and rejected (left out) from the set of basic points, which are still considered to be conditionally stable points.
Average mismatch for the remaining k - 1 basic points is being determined as:
d^ • Pff • d F
d^T -
h.
(13)
where PpF - PpF - P^g • Pg^^ • P^^ and h^ is rank of P^ The test statistics reads:
e'
T--
- F
cr„
hp. f ■
(14)
If the T < F^.f .1-a , zero hypothesis is not rejected, i.e. it is considered that all k - 1 basic points are stable. If the T > F^.f .1-a , an alternative hypothesis is being accepted, i.e. there are still some unstable points among k - 1 basic points, when it is necessary to identify unstable points in the manner shown.
2.5 Testing Congruency of Construction Points
To test movements of construction points, the coordinate differences vector d and pseudo-inversion of coordinate differences cofactor matrix Q+ are decomposed as follows (Pelzer, 1971; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2001; Ašanin, 2003; Vrce, 2011):
d p
Q+- Pd -
P	P
'■FF	'■FO
P	P
OF OO
(15)
where the F designates basic points identified as stable, and O designates the basic points identified as unstable and construction points.
Average mismatch is determined using the formula: dO ■ POO • do
e'o -
h
(16)
where do - do + PoO ' PoF 'df and hO is rank of POO. The test statistics reads:
T-eL . F
cr„
ho. f .
(17)
If the T < F^.f .j-a , zero hypothesis is not rejected; or else an alternative hypothesis is being accepted. 3 KARLSRUHE METHOD
Karlsruhe method is based on independent adjustment of zero and the control epoch and also on their joint adjustment. In the first phase, measured values in the individual epochs are adjusted using the Least Squares Method (LSM). In the second phase, measured values in zero and control epoch are jointly adjusted. Joint adjustment of the two epochs is being done under the assumption that the basic points are congruent in two epochs and that the network scale is the same in both epochs.
In the first phase, adjustment is made independently for all measured values in each epoch, using indirect adjustment method:
v, = A • x, + f, , i = 1,2, k	(18)
where k is the number of epochs. The network may be adjusted usually or with the minimum trace.
From each individual adjustment, a square form Q. is being determined, and joint square form for all epochs is obtained by summing square forms of individual epochs' adjustments:
= = ^Tvf • P, • v, = v^ • P • v .	(19)
i=i i=i
Total number of the degrees of freedom b is obtained by summing up the degrees of freedom b. (b. = n. - «.) from individual epochs' adjustments.
During the second phase, measured values in zero and control measurement epoch are jointly adjusted. In joint adjustment of the two epochs, vector of unknown coordinates is divided into three sub vectors:
x^ = (z^, xf, x2)	(20)
where:
z	- sub vector of basic points is assumed to be stable in both epochs;
Xj, x2 - sub vectors of construction points or points assumed to be unstable.
From the joint adjustment, square form Q.^ is being determined, containing information on measurement errors and movements of unstable points. Joint adjustment square form Q^ gets deducted from square form Q0, which contains information about measurement errors only (Heck, 1983; Ninkov, 1985; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2004):
Q,	= v^ • P• v^ -vT • P• v .	(21)
The new square form Q^ contains only the information about unstable points' movement. Test statistics (Heck, 1983; Ninkov, 1985; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2004):
Q / f (vT • P • v - vT • P • v) b
P	-f-L .'L	(22)
Q0/ b	vT •P•v	f
where: f= (k - 1) • n ■ p0 — d, k - is number of epochs, n - is the geodetic network dimension,p0 - is the number of conditionally stable points, d - rank defect of matrix A.
If the F< 1—a, zero hypothesis is not rejected, i.e. all of the points from conditionally stable points are indeed stable points; or else an alternative hypothesis is being accepted.
3.1 Determining Unstable Points in the Set of Conditionally Stable Points
With F> Ff,b1—a, the set of conditionally stable points contains unstable points. It is necessary to determine such points. For that purpose, joint adjustments are repeated from which one conditionally stable point is being excluded successively. The adjustment providing the minimum value of the square form
Q . indicates that the point excluded from the adjustment is to be considered unstable. That point is definitely excluded from the set of conditionally stable points, and the entire procedure is repeated without it. The procedure is repeated iteratively, until the condition F <	is met, and the points
left in the set of conditionally stable points afterwards are considered stable.
3.2 Localization of Deformations
Deformations are being localized for each point. The zero hypothesis assumes that the point T did not move, while the alternative hypothesis assumes that the point T. did move. Zero (H0) and alternative (Ha) ^^ hypotheses are being set (Heck, 1983; Ninkov, 1985; Mihailovic and Aleksic, 1994, 2008; Ambrožič, 2004):
E (d ) = O
The test statistics reads:
d. • Q^1 • d
against
H
: E ( d.)
^ O.

cr„
- F
m -(7,,
where:
- Qx -B
qyyi
m, f
q-y'i
(23)
(24)
y, =BT-x=
.y2.
.■ - -y,

j "i.i 2
b -(702 + b2-(77. (7o2 = 1 0- / 02 .f =bi + b.,
f
m - geodetic network dimension.
If the F j <	zero hypothesis is not rejected, i.e. the point is stable; or else an alternative hypothesis
is being accepted.
4 MODIFIED KARLSRUHE METHOD
The method consists of free adjustment zero measurement epoch using the LSM with minimization of the part of the trace matching the assumed stable points. Adjustment of control measurement epoch is also being done using the LSM with minimization of the part of trace matching the assumed stable points; by accepting coordinates of adjusted zero measurement epoch as the approximate coordinates of the control measurement epoch (Ninkov, 1985).
Points' stability control is being done using the Unimodal Transformation Method (without changing the scale). After estimating transformation parameters under the LSM, the control of differences of the transformed control epoch coordinates and of zero epoch coordinates difference is under threshold of double coordinate standards from the control epoch adjustment. In the event of discovered unstable points, it is necessary to perform new adjustments by minimizing the part of a trace matching confirmed stable points.
After adji^^t^me^t^^ defoi-im£Lt^(^i^s	kca^izisd «^ach	^^ t^ ^^^^ ^^^^^^^
of points, zero {H0) and alternative (Ha) hypothesis are being set (Ninkov, 1985):
E (d) = 0
The test statistics rej^(is:
Cdn

against
- F
H
(d.)
^ 0.
(25)
(26)
l^f the F. < F^r^^, aiero h}^]3c^ttLw\s is no trej^ctfd; or else an alternative hypothesis is being accepted.
The pr«;s^nteid st^jitistical t£^st^ng of t^yp othesis may also be in terpretef geometrically. In the even t of zero hypothesis (H0), formula (26) may be written as follows:
iii\\f\fi;.^t^t2.cti.tftm_f.	t2n)
If we replace < with = in the formula (27), the expression becomes an ellipse equation, which is identical to the relative error ellipse between the points T^ . and increased for the factor .^2 • F^,^ . Axis
of the rektiw err or e^ pses are c^ cukted u^^^^ ^f f^^^^f
A. • 2 •	, (i = 1,2) .	(28)
The deformation analysis using relative error ellipses can be presented graphically with ease (Figure 1). Po^r^i movcmr^nt n^ectc^i^rino^i^ ir^cnes^s^d relat^^)); eorM eUipsns Mfc^i-^^n i^r^ to^e S2n^(£ j^k^et^li.T^^ moi^i^-
n^^r^t n^clfs d^- i^ta^t^ eoe j^o^ns T^ .	t^vef oir^t ^T^^ an^d l^l^^j^t^1^ T"^^1 ir tlFe c^enl^]. of ellij^sf.
Io th^^jr i^[^r^\:n2i. isi^\^t^^tde fl^t; ol^[^ipo l^	r2rol:^lypo	]^s^1no stf K^f^j^d 1 nerotmoti(^:ei ^r; ni^]f^t
T^ . are accepted with a certain probability.
Figure 1: Deformations analysis using relative errors ellipses.
Increments of approximate (adjusted) coordinates in the adjustment process of spatial movement components are calculated as follows (Ninkov, 1985):

(29)
Standards for determining spatial movements' components are approximately equal to the square root of the sum of coordinates' determination squares.
5 JAG3D METHOD
JAG3D is Open-Source Software developed for the geodetic networks adjustment and deformation analysis, by Michael Lösler (URL1). In the first phase, both the zero and control measurement epoch are independently adjusted using indirect adjustment method. Testing measurement accuracy homogeneity is being done in the manner explained in Pelzer Method. In the next step, congruency of the basic network points is being tested. If basic points' set contains unstable points, such unstable basic points are being localized. Zero (H0) and alternative (Ha) hypotheses are being set:
H„ : E (V,,^.) = 0 against H^ : E (V,,^.) ^ 0 . ^e test statistics is formed (URL2):
r..
VT •o
^ ,,. vv,, j
, ,j

- a priori test statistics
(30)
(31)
m •c
where:
Q
m o:
vr •o
^ ,, j vv,, j
V
, ,j
m •(jC

m, f -m
- a posteriori test statistics
(32)
VVÄ,j
^ •ovv,,j
v
=
R, j
-	is movement vector;
-	is cofactor matrix of movement estimate;
-	is geodetic network dimension;
-	is a priori variance;
-	is reduced a posteriori variance.
f — m
If the T . . < F or T . < F , , zero hypothesis is not rejected, i.e. the point is stable; or else an alterna-
pno, j m, M	post, j m, f - m	>	r
tive hypothesis is being accepted. Gauss-Markov model of the joint adjustment of both epochs is (URL2):
l1		Vi	
Ic	+		=
		V 2	
AO ,1 0
(33)
where:
11	- is the vector of zero measurement epoch measured values;
12	- is the vector of control measurement epoch measured values;
x^	- is the sub vector of basic points with stability confirmed (conditionally stable points);
x^1, x^2 - is the sub vector of points on construction or points considered conditionally unstable.
After joint adjustment, unstable points on construction are localized. In order to localize unstable points on construction, zero (H0) and alternative (Ha) hypothesis are being set:
Ho: E (d, ) = 0	against	H^ : E ( d, 0 .
(34)
or
^e test statistics reads (URL2):
dT • Q-1,k • d k
T
m
F
a priori test statistics
(35)
Tpost ,k
where:
^ = F •
Qdd,k = F
dl • Q-1,k • d k
'F„
m •(T„
■, f -m
■ a posteriori test statistics
(36)
QxO,1,xO,1	QxO,1,xO,2
Qxo,2,Xo,l Qxo,2,Xo,2
-	vector of movement;
-	movement estimate cofactor matrix;
F =
0 ^ -Ik ^ Ik ^ 0
If the^ . < F or T . < F , , zero hypothesis is not rejected, i.e. the point is stable; otherwise an
prio, k m, M	post, k m, f — m	^ ^	'	^
alternative hypothesis is being accepted.
6 CALCULATION EXAMPLE
For the purposes of applying Deformation Analysis Methods described in the paper, the geodetic control GNSS 2D network was simulated, consisting of 9 points. Basic network points are points 1, 2, 3 and 4, while points 5, 6, 7, 8 and 9 interpret the construction (Figure 2). All observations and deformations in the network are simulated. ^e simulated vectors are linearly independent, and the standard deviation of two-dimensional component of GNSS baseline vector is 5 mm + 0.5 ppm. Approximate points' coordinates and simulated deformations are shown in Table 1, with the data on two epoch vector observations being given in Table 2, while the network sketch is shown in Figure 2.
Table 1:
Approximate points' coordinates and simulated deformations.
Point number	F [m]	X [m]	d. [mm]	Vi [°]
1	1320	1400	--	--
2	1370	1270	--	--
3	1650	1125	--	--
4	1670	1310	--	--
5	1785	1250	--	--
6	1740	1400	15	218
7	1625	1530	40	225
8	1470	1585	10	197
9	1325	1570	5	182
^ or
Table 2: Two epoch vector observation data.
Vectors		Zero	epoch	Control	epoch
From	To	AY [m]	AZ [m]	AY [m]	AZ [m]
	2	50.00290	-129.9970	49.9975	-129.9990
	3	330.0092	-275.0010	330.0024	-275.0010
	5	465.0007	-149.9990	465.0031	-150.0020
	4	349.9914	-89.9941	350.0097	-89.9971
	6	420.0068	-0.0091	419.9920	-0.0065
	7	305.0051	129.9935	304.9722	129.9799
	8	149.9977	185.0017	149.9947	184.9907
	9	5.0009	169.9974	4.9991	169.9957
2	3	280.0031	-145.0010	279.9998	-145.0025
2	5	415.0015	-20.0060	414.9930	-19.9936
2	4	300.0028	39.9964	299.9990	39.9997
2	6	369.9995	129.9984	369.9938	129.9863
2	7	254.9998	259.9971	254.9697	259.9802
2	8	100.0004	314.9965	99.9990	314.9937
2	9	-45.0032	299.9983	-44.9962	299.9979
2	1	-49.9943	129.9962	-49.9957	129.9960
3	5	135.0000	125.0054	135.0031	125.0017
3	6	90.0027	274.996	89.9818	274.9902
3	4	19.9966	185.0073	19.9977	184.9956
3	7	-25.0043	404.9985	-25.0244	404.9705
3	8	-180.0000	459.9942	-180.0014	459.9888
3	9	-325.001	444.9975	-325.0001	444.9971
3	1	-329.995	275.0037	-329.9983	274.9960
3	2	-279.998	145.0042	-279.9955	145.0044
4	3	-19.9978	-184.999	-19.9994	-184.9929
4	5	114.9942	-59.9989	114.9970	-60.0012
4	6	69.9961	90.0025	69.9901	89.9881
4	7	-44.9992	219.9942	-45.0288	219.9735
4	8	-200.0010	274.9977	-199.9994	274.9953
4	9	-344.9950	259.9927	-344.9985	259.9971
4	1	-349.9950	89.9982	-349.9978	90.0016
4	2	-300.0020	-39.9962	-299.9994	-40.0019
Zero and control measurement epoch were adjusted, along with joint adjustment of both measurement epochs. Datum was defined by minimum trace to the basic network points. Adjustments were performed using the JAG3D and MatGeo (developed at the Faculty of Technical Sciences in Novi Sad, using Matlab environment) software. Deformation analysis by means of the JAG3D Method was performed
using the JAG3D software, while deformation analysis using other methods was implemented in the MatGeo program. The significance level used in calculations is a = 0.05.
Figure 2: Network sketch in the scale of 1:5000.
^e value a'0 = 1 was accepted as a priori standard deviation. Following values were obtained from the adjustment of zero and control measurement epochs: cr^ -1.095, cr^ -1.023, - 51.521,
Qj - 50.212,^0 = 48 and f^ = 48. Following values were obtained from adjustment of both epochs:
CT^ - 1.059, Qz - 114.381 andf^ = 102. Adjusted coordinates ofzero and control epoch measurements
are shown in Table 3.
Table 3: Adjusted points' coordinates from zero and control measurement epoch.
	Zero epoch		Control epoch	
Point number	F [m]	X [m]	F [m]	X [m]
1	1320.0001	1399.9995	1319.9999	1399.9992
2	1369.9995	1270.0017	1370.0000	1270.0001
3	1650.0011	1124.9983	1650.0000	1125.0011
4	1669.9993	1310.0004	1670.0002	1309.9996
5	1784.9991	1250.0004	1784.9991	1250.0013
6	1740.0012	1399.9970	1739.9893	1399.9895
7	1625.0003	1529.9958	1624.9722	1529.9761
8	1469.9991	1584.9976	1469.9982	1584.9921
9	1325.0004	1569.9965	1325.0011	1569.9969
6.1 Pelzer Method
In the first step, homogeneity of measurement accuracy in two epochs is being tested. For that purpose, the test statistics is being formed according to the formula (2), while the test statistics T = 1.146 is lower than critical value F48 48 0975 = 1.773, so that the values measured in two epochs have homogeneous accuracy. Unified degrees of freedom from two epochs are f = 96. Unified standard deviation from two epochs is calculated using the formula (3) being cr0 = 1.059.
In the second step, the network congruency in two epochs is being tested. According to the formula (6), test statistics is thus calculated: T = 12.400, so that the value obtained is greater than the critical value F16 96 0 95 = 1.7 5 0, with the conclusion that network points' coordinates from two epochs are not congruent.
In this step, congruency of the basic network points is being tested. ^e test statistics T = 0.987, being calculated using formula (10), is lower than critical value F696095 = 2.195, thus the conclusion can be made that the basic network points from two epochs are congruent.
After congruency of basic network points had been established, stability of construction points is being tested. Test statistics T = 19.248, determined using formula (17), is greater than critical value F1096095 = 1.931, thus the conclusion can be made that the construction points' coordinates from two epochs are not congruent.
When non-congruency of construction points is established, unstable construction points are being localized next. For each point, average mismatch is being calculated using formula (12) (Table 4).
Localization procedure is done through iterations. In the first iteration, point 7 is identified as unstable (Table 4), and is being excluded from the set of construction points.
Table 4: Localization of unstable construction points by Pelzer Method.
	Iteration 1	Iteration 2	Movement	Slope
Point number	00	00	d. [mm]	V, [°]
5	0.066	0.066	0.919	0.623
6	15.088	15.088	14.029	238.224
7	90.543	--	34.313	235.004
8	2.264	2.264	5.487	189.325
9	0.048	0.048	0.794	54.713
In order to test the stability of the remaining construction points, the test statistics is being calculated using the formula (14). In the set of remaining points, some are unstable since the test statistics T= 3.891 is greater than critical value F8,96,0.95 = 2.036. In the second iteration, the point number 6 has the greatest value of average mismatch values and is thus considered as unstable. It is necessary to test the stability of the remaining construction points. ^e set of remaining points does not contain unstable points, since the test statistics reads T = 0.706, being lower than critical value F696095 = 2.195. Discovered movements of points are shown in Table 4. Points 6 and 7 are identified as unstable, while the remaining points are identified as stable.
l 549 |
6.2 Karlsruhe Method
The set of conditionally stable points are the basic network of points. In order to test the stability of conditionally stable points, the test statistics is calculated using the formula (22). Test statistics of the F = 0.987 is lower than the critical value ^^„„„^ = 2.195, thus the conclusion can be made that the set
6,96,0.95
of conditionally stable points contains no unstable points. Localization of conditionally unstable points is being performed by calculating test statistics for each point using the formula (24). Table 5 shows values of test statistics F and critical values F for each point. If test statistics F is lower than the critical value F, the point is stable; otherwise it is unstable. Using Karlsruhe Method, points 6 and 7 are identified as unstable. Points' movements are shown in Table 5.
Table 5: Localization of unstable points by Karlsruhe Method.
Point number	F	F 2,96,0.95	d. [mm]	v, [°]
5	0.059	3.091	0.919	0.623
6	13.454	3.091	14.029	238.224
7	80.738	3.091	34.313	235.004
8	2.018	3.091	5.487	189.325
9	0.043	3.091	0.794	54.713
6.3 Modified Karlsruhe Method
Basic network points' congruency has been verified using the Unimodal transformation. In the next step, localization of unstable points is being performed. For each point, value of F test statistics is calculated using formula (26), and the values obtained are shown in Table 6. If tested F statistics is lower than F critical value, the point is stable; otherwise the point is unstable. Using this method, points 6 and 7 were identified as unstable. Points' movements discovered are shown in Table 6.
Table 6: Localization of unstable points by modified Karlsruhe Method.
Point number	F	F 2,48,0.95	d. [mm]	v, [°]
1	0.055	3.191	0.431	218.454
2	0.849	3.191	1.684	163.821
3	2.625	3.191	2.987	339.197
4	0.427	3.191	1.197	133.983
5	0.058	3.191	0.909	0.630
6	13.469	3.191	14.042	238.155
7	80.746	3.191	34.329	234.977
8	2.033	3.191	5.510	189.291
9	0.042	3.191	0.782	56.514
6.4 JAG3D Method
After independent adjustment of zero and of control measurement epochs, the testing of basic network points' congruency in two epochs has been performed. Basic network points' congruency in two epochs has been determined. In the second step, the joint adjustment of zero and of the control measurement
epoch has been performed, with basic network points being included in the set of conditionally stable points. After joint adjustment of both epochs, localization of unstable construction points has been performed. For each point, test statistics for the T has been calculated using formula (35) and (36).
Table 7: Localization of unstable points, JAG3D Method.
Point number	T .	T	T 2,^,0.95	T 2,100,0.95	d. [mm]	^i- [°]
5	0.066	0.059	2.996	3.087	0.919	0.623
6	15.088	13.454	2.996	3.087	14.029	238.202
7	90.543	80.738	2.996	3.087	34.313	235.004
8	2.264	2.018	2.996	3.087	5.487	189.342
9	0.048	0.043	2.996	3.087	0.794	54.713
Method			Basic network	points			Construction		points	
	Points	1	2	3	4	5	6	7	8	9
"Ö	dy [mm]	0	0	0	0	0	-9.21	-28.01	-3.03	-0.15
te	dx [mm]	0	0	0	0	0	-11.84	-28.56	-9.53	-5
imu	d [mm]	0	0	0	0	0	15	40	10	5
CO	v[°]	--	--	--	--	--	218	225	197	182
Values of test statistics for each point are shown in Table 7. If the test statistics for the T is lower than the F critical value, the point is stable; otherwise the point is unstable. Using the method implemented in the JAG3D software, points 6 and 7 were identified as unstable. ^e discoverent points' movements are shown in Table 7.
7 analysis of comparative results
^e paper describes some of the current methods of the deformation analysis. For the purpose of Deformation Analysis Methods application, a geodetic control network was conceptualized, with simulations of all deformations. By the application of methods by Pelzer, by Karlsruhe, by the modified Karlsruhe method and the one being implemented in the JAG3D, the Open-Source Software has identified unstable points giving movements of determined points. Points 6 and 7 were identified as unstable, while the remaining points were identified as stable. Nevertheless for other points no significant movements have been discovered, with simulated movements being on the measurement accuracy threshold. Information about stability, simulated and determined points' movements are shown in Table 8. Movements detected through application of the above Deformation Analysis Methods are nearly identical and comparable to the simulated movements. The greatest difference has the magnitude order of several millimeters, referring to the point 7. Reliability of the Deformation Analysis Methods is dependent on stable datum framework, along with the accuracy of measurement, indicating the lower threshold of movement that can be discovered "with certainty", which is verified by the results obtained in the practical part of this paper (points with simulated smaller movements were not identified as unstable).
Table 8: Simulated and identified movements of points.
	Method	Points		Basic network points				Construction		points	
			1	2	3	4	5	6	7	8	9
		dy [mm]	0	0	0	0	0.01	-11.93	-28.11	-0.89	0.65
S3	r	dx [mm]	0	0	0	0	0.92	-7.39	-19.68	-5.42	0.46
Ü	el	d [mm]	0	0	0	0	0.92	14.03	34.31	5.49	0.80
11	P	v[°]	--	--	--	--	0.62	238.22	235.00	189.33	54.71
		Stable	Yes	Yes	Yes	Yes	Yes	No	No	Yes	Yes
		dy [mm]	0	0	0	0	0.01	-11.93	-28.11	-0.89	0.65
	e h	dx [mm]	0	0	0	0	0.92	-7.39	-19.68	-5.42	0.46
	rl	d [mm]	0	0	0	0	0.92	14.03	34.31	5.49	0.80
		v[°]	--	--	--	--	0.62	238.22	235.00	189.33	54.71
		Stable	Yes	Yes	Yes	Yes	Yes	No	No	Yes	Yes
		dy [mm]	-0.27	0.47	-1.06	0.86	0.01	-11.93	-28.11	-0.89	0.65
	d e eh	dx [mm]	-0.34	-1.62	2.79	-0.83	0.91	-7.41	-19.70	-5.44	0.43
		d [mm]	0.43	1.69	2.98	1.2	0.91	14.04	34.33	5.51	0.78
	S	v[°]	218.45	163.82	339.20	133.98	0.63	238.15	234.98	189.29	56.51
		Stable	Yes	Yes	Yes	Yes	Yes	No	No	Yes	Yes
		dy [mm]	0	0	0	0	0.01	-11.92	-28.11	-0.89	0.65
	Q	dx [mm]	0	0	0	0	0.92	-7.39	-19.68	-5.41	0.46
	3	d [mm]	0	0	0	0	0.92	14.02	34.31	5.48	0.80
		v[°]	--	--	--	--	0.62	238.20	235.00	189.34	54.71
		Stable	Yes	Yes	Yes	Yes	Yes	No	No	Yes	Yes
8 conclusions
It is known that certain methods of deformation analysis provide different results - the conclusion being made by the FIG Commission, consisting ofesteemed scientists from five university centers (Hanover, Delft, Fredericton, Karlsruhe and Munich). Of course, conclusions on reliability of Deformation Analysis Methods cannot be made based on the research performed on a single deformation model. Examining reliability of individual deformation analysis method needs to be implemented over several different deformation models.
Pursuant to the results obtained in the paper, the conclusion can be drawn that modified Karlsruhe Method and method implemented in the JAG3D Open-Source Software provide satisfying results regarding the discovery of unstable points, against the most commonly applied Deformation Analysis Methods-by Pelzer and Karlsruhe.
Precondition for the efficient application of various Deformation Analysis Methods refers to the setting as many basic points on a geologically stable terrain as possible. Apart from reliability of information on ground and construction movements, an important factor refers to the least movement intensity determination, which may certainly be determined with chosen probability and the test power.
If a minimum of four datum points are situated on geologically stable terrain, outside zone of expected deformations, having not changed their positions between two epochs of measurement, reliability of
applying models noted in the paper will significantly increase. ^at is also impacted by application of the GNSS measurements using static or rapid static method, especially when significant movements in horizontal plane are being quantified.
alysis. Kensington: A Comparison of
References:
Ambrožič, T (2001). Deformacijska analiza po postopku Hannover. Geodetski vestnik. 45(1-2), 38-53.
Ambrožič, T. (2004). Deformacijska analiza po postopku Karlsruhe. Geodetski vestnik. 48 (3), 315-331.
Ašanin, S. (2003). Inženjerska geodezija. Belgrade: Ageo.
Caspary, W. F. (1987). Concepts of Network and Deformation The University of New South Wales, School of Surveying
Chrzanowski, A., Members of the "ad hoc" Committee (198
Different Approaches into the Analysis of Deformation Measurements. FIG XVI Congress, Montreux, Switzerland, Paper No. 602.3, Montreux 1981.
Heck, B. (1983). Das Analyseverfahren des geodätishen Instituts der Universität Karlsruhe Stand 1983. Deformationsanalysen ,83. Geometrische Analyse und Interpretation von Deformationen Geodätischer Netze. Mönchen: Hochschule der Bundeswehr. Heft 9.
Heck, B., Kok, J. J., Welsch, W., M., Baumer, R., Chrzanowski, A., Chen, Y. Q., Secord, J. M. (1982). Report of the FIG-working group on the analysis of deformation measurements. In: I. Joo in A. Detreköi (Eds.), 3rd International Symposium on Deformation Measurements by Geodetic Methods (pp. 337-415). Budapest: Akademiai Kiadd
Jäger, R., Kälber, S., Oswald, M. (2006). GNSS/GPS/LPS based Online Control and Alarm System (GOCA) - Mathematical Models and Technical Realisation of a System for Natural and Geotechnical Deformation Monitoring and Analysis, GEOS 2006.
Marjetič, A., Zemljak, M., Ambrožič, T. (2012). Deformacijska analiza po postopku Delft. Geodetski vestnik, 56 (1), 9-26. DOI: http://dx.doi.org/10.15292/ geodetski-vestnik.2012.01.009-026
Mihailovic', K., Aleksic', I. (1994). Deformaciona analiza geodetskih mreža. Belgrade: Gradjevinski fakultet, Institut za geodeziju.
Mihailovic', K., Aleksic', I. (2008). Koncepti mreža u geodetskom premeru. Belgrade: Geokarta.
Ninkov, T. (1985). Deformaciona analiza i njena praktična primena. Geodetski list, 39 /62 (7-9), 167-178.
Pelzer, H. (1971). Zur Analyse geodätischer Deformationsmessungen.Mönchen: Deutsche Geodätische Kommission, Reihe C, No. 164.
Setan, H., Singh, R. (2001). Deformation analysis of a geodetic monitoring network. Geomatica, 55 (3), 333-346.
Vrce, E. (2011). Deformacijska analiza mikrotriangulacijske mreže. Geodetski glasnik. 45 (40), 14-27.
Welsch,W. (1981). Description of homogenous horizontal strains and some remarks to their analysis. International symposium on geodetic network and computations of the International Association of Geodesy, Munich.
URL1. (2015). http://javagraticule3d.sourceforge.net, accessed 2. 2. 2015.
URL2. (2015). http://wiki.derletztekick.com/javagraticule3d/least-squares-adjustment/deformationanalysis, accessed 2. 2. 2015.
Sušic' Z., Batilovic' M., Ninkov T., Aleksic I., Bulatovic' V. (2015). Identification of movements using different geodetic methods of deformation analysis.
Geodetski vestnik, 59 (3): 10.15292/geodetski-vestnik.2015.03.537-553
Assist. Prof. Zoran Sušič, Ph.D.
University of Novi Sad, Faculty of Technical Sciences Trg Dositeja Obradovica 6,2i000 Novi Sad, Serbia e-mail: susic_zoran@yahoo.com
Mehmed Batilovič, Univ. Grad. Eng. of Geod.
University of Novi Sad, Faculty of Technical Sciences Trg Dositeja Obradovica 6,2i 000 Novi Sad, Serbia e-mail: batilovicm@gmail.com
Prof. Ivan Aleksič, Ph.D.
University of Belgrade, Faculty of Civil Engineering Bulevar kralja Aleksandra 73, 11 000 Belgrade, Serbia e-mail: aleksic@grfbg.acrs
Prof. Toša Ninkov, Ph.D.
University of Novi Sad, Faculty of Technical Sciences Trg Dositeja Obradovica 6,21 000 Novi Sad, Serbia e-mail: ninkov.tosa@gmail.com
Assist. Prof. Vladimir Bulatovič, Ph.D.
University of Novi Sad, Faculty of Technical Sciences Trg Dositeja Obradovica 6,21000 Novi Sad, Serbia e-mail: vbulat2003@gmail.com