A PROCEDURE FOR DETECTION OF DEFORMATIONS USING SURVEY CONTROL
NETWORKS
By
1
Ali H. Fagir and Mudathir O. Ahmed 2
1-Sudan University of Science & Technology –Faculty of Engineering,, P.O. Box 72 Khartoum, 2- Omdurman Islamic
University- Faculty of Engineering Sciences, e-mail- Modethirb@yahoo.com P.O. Box 382
ABSTRACT
Survey control networks are used for detecting deformations in a specific
area by measurements made at different epochs. The differences in coordinates
of stations, obtained from least squares adjustments, are compared in order to
assess if a deformation of a specified magnitude exists. Traditionally a global test
(the global congruency test) is carried out so as to detect if the area of the
network has undergone any changes in shape (uplift or subsidence) due to natural
or man made causes. As a next step, localization methods are used to determine
deformations at specific points in case there are changes in shape.
In this study, a procedure is developed to detect deformations at specific
points directly. The method is tested using a vertical control network measured
at various epochs. The results indicate that deformations greater than 0.03m can
be detected at a minimum significant level () of 0.05 (95% confidence level).
‫ازااحدة لمواقد‬
:‫الملخص‬
‫تستخدم شبكات الضبط في المساحة كواحددٍ ندت تياتدات اكتشدا‬
‫ بعد الحصول علي اليتاسات الاسبتة لهده الايداط علدي‬.‫الاياط المحددٍ على سطح األرض‬
‫فترات اناتة نختلفة تتم نيارنة ازحداثتات المحسوبة عت طريد نرريدة لقدل التربتعدات‬
.‫لتحديد نيدار ازااحة لو الحركة لاياط الشبكة األرضتة‬
‫ ) لجمتد‬Global Congruency Test ‫جرت العادٍ علي استخدام االختبار الشدانل‬
‫نيدداط الشددبكة لتحديددد نددا إذا كدداه ااالددا إااحددة كلتددة لمواق د نيدداط الشددبكة كاتددا للعوانددل‬
‫ َيتب ادا االختبار بأحد االختبارات الجزئتة لتحديد نيدار ازااحدة‬.‫الطبتعتة لو الحضارية‬
.‫عاد كل نيطة على حدٍ في حالة وجود تغترات في الشكل‬
‫ تدم‬.‫في اذا البحث قدنت طريية لتحديد نيدار ازااحةَ نباشرٍ ً للاياط بصفة فرديدة‬
‫تطبتد اددده الطرييددة علددي شددبكة راسددتة لخددذت لرصدداداا علددي فتددرات اناتددة نختلفددة ثددم‬
.ً‫قورنت الاتائج التي تم الحصول علتها ن واحدٍ نت الارريات المستخدنة سلفا‬
‫بعددد تطبت د اددده الطرييددة علددي الشددبكة نوضددود االختبددار ىوجددد له الحددد األدنددى‬
‫ نتر) عادنا تستخدم نسبة احتمال لوقود‬0.030 ‫لإلااحة المكتشفة حوالي ثالثة ساتمترات‬
.5% ‫) ال تيل عت‬ ‫خطأٍ نت الاود األول‬
INTRODUCTION
One important application of survey control networks is the detection of
expected deformations at a specified area. This is done by measurements made
at successive epochs and the most probable values of the coordinates are
obtained using the well known method of least squares. From the results of the
two epochs adjustment it is possible to calculate the displacement
(deformation) vector dxˆ and its associated variance covariance matrix C dxˆ from
(James, 1985).
dxˆ  xˆ 2  xˆ1
.......................................................
(1)
C dxˆ  C xˆ1  C xˆ2
(assuming that xˆ1 and xˆ 2 are uncorrelated).
However, the use of coordinates and, therefore, coordinate differences to
define deformations depends on the way the system of coordinates is defined.
The convenience of using coordinates as parameters to be estimated from
measurements and their associated weight matrix has a disadvantage in
deformation studies since such coordinates are datum dependent (Cross, 1983).
In constrained (fixed) networks the datum is usually defined by the
fixation of some points. The positions of such points should not be affected by
the adjustment process adopted i.e. the least squares corrections are assumed to
be zero.
The least squares estimates of parameters is given by the well known
equations (Cross, 1983):
…………………………….
(2)
xˆ  ( AT WA) 1 AT Wb
and the variance-covariance matrix of the parameters is readily obtained from:
C xˆ  ( AT WA) 1
………………………………..
(3)
where: A: is the design matrix, W: is the weight matrix, b: is a vector related to
the observations.
Conversely, the free network concept depends on the assumption that all
points are affected by the least squares process and the parameters are obtained
using the Moore-Penrose inverse instead of the usual Caiely inverse in equations
(2) and (3) (Serif Hekimogolu, Huseyn Demiret and Caneyt Aydin, 2002).
The credibility of using constrained or free networks depends on
whether the fixed points shall undergo any deformations expressed by changes
in their position. Therefore, a thorough understanding of the nature of the
expected deformations and their extent is of prime importance in the choice of
constrained or free datum networks (Chen, 1983).
In this research a constrained simulated vertical control network (Fig. 1) is
used to test for the smallest vertical deformation that can be detected in
a particular area. The network is simulated to represent different epochs and
a procedure is developed to test for expected vertical movements.
Review of Existing Procedures for the Detection of Deformations
The global congruency test is the most commonly methodology adopted
for the detection of general deformations in a given area i.e. an overall change
in shape (Wan Aziz, W.A. Othman, Z. and Nagib, H., 2001).
The displacement vector ( dxˆ ) from a two epochs adjustment and its
associated variance-covariance matrix are given by equations (1).
The null hypothesis of the global congruency test can be stated as
follows:
H o : E (dxˆ )  0
and the alternative hypothesis is given by:
H A : E(dxˆ)  0
where the letter E indicates expectation (expected value).
If the null hypothesis is accepted, then the points are assumed to be
stable, (i.e. the network is stable). Conversely, if the null hypothesis is rejected,
then the network has undergone a change in shape.
The test statistic known as the global congruency test is established as
follows (Caspary 1987, Pelzer, 1971):


 Fh,f (i.e.  has a Fisher (F) distribution with h and f
h o2
degrees of freedom.
where:
  dxˆ T C dxˆ1 dxˆ
h: is the rank of C dxˆ .
ˆ o2  ( f1ˆ o21  f 2ˆ o22 ) / f
f = f1 + f 2
f1 : is the degrees of freedom of x̂1 , f2 :is the degrees of freedom of x̂ 2 .
 o21 : is the unit variance of the first epoch,  o22 : is the unit variance of the
second epoch.
A suitable significance level is selected and the result of the test will
show if the two networks are congruent or not. Furthermore, localization
methods (Niemeir, 1985; Welsch et al., 2000) are used to identify the presence of
deformations at particular points in the area of the established network.
The success of the global congruency test depends on the following
factors (Caspary, 1987):
i- the use of one datum (free or constrained ) for both epochs.
ii- the influence of systematic errors must be accounted for by standard testing
procedures.
iii- the effect of the largest undetectable gross errors on the coordinates must be
known and accounted for.
An important drawback for the use of this method is that it gives an
overall picture of the expected deformations in the area. This implies that
deform-ations in particular places, which might be more important, can only be
detected by some other secondary steps.
A New Strategy for Deformation Detection
Following the ideas of Baarda concerning gross error detection e.g. (Baarda,
1968).
we can formulate the deformation problem as follows:
The displacement dxˆ i for any single point (i) obtained from a two epochs
solution is assumed to be normally distributed with zero mean and variance  d2x̂
i.e dxˆ i N (0,  d2xˆ ).
In this case no movement has taken place i.e. no deformation exists.
On the other hand, if the point is subjected to any movement, then the
displacement dxˆ i , will be normally distributed with a shifted mean  and
variance  d2xˆ i following the mean shift model i.e dxˆ i N (  ,  d2xˆ ).
Therefore we can set up the null and alternative hypotheses as follows:
Ho : E (dxˆ i )  0 , HA: E (dxˆ i )  0
After standardizing each element of the displacement vector, the testing
can be established as follows:
dxˆ
Ho: i = i  N (0,1)
 dxˆi
and
HA : i
=
dxˆ i
 dxˆ
 N (  ,1)
i
RESULTS
A levelling network consisting of seven stations is simulated such that
individual loop misclosuers shall not exceed 10 k millimeters where k is the
distance levelled in kilometers and station (1) is assumed to be fixed with zero
variance (Fig. 1).
The simulated network is used as a reference i.e. representing the first
epoch. (Table 1) shows the reference network i.e. before expected deformations
are inflicted. The observations are altered three times by multiples of their
standard errors while keeping the loop misclosures to acceptable limits.
Heights and their standard errors representing four different epochs are
obtained using equations (2) and (3) respectively.
The displacement vectors and their standard errors are shown in Table (2).
The procedures outlined in sections (2) and (3) are applied. A
significance level of 0.05 is used and results are outlined in (Tables 3) and (4)
respectively.
7
6
5
3
4
1
Fig. (1): A simulated vertical control network
2
Table (1): Simulated Observations of the First Epoch
From
point
1
1
1
2
3
3
4
4
5
5
6
To
point
2
3
4
3
4
6
5
6
6
7
7
Observation
(Height diff.) (m)
1.510
2.775
1.618
1.250
-1.175
1.322
1.800
2.480
0.700
1.876
1.200
Distance
(km)
2.000
2.500
2.549
2.061
1.118
2.828
1.118
3.354
2.549
4.031
2.121
Standard error
(m)
0.014
0.016
0.016
0.014
0.011
0.017
0.011
0.018
0.016
0.020
0.015
Table (2): Displacements and Their Standard Errors at the Three Epochs
Epoch (2)
Point No.
dxˆ i
2
3
4
5
6
7
(m)
-0.0140
-0.0160
-0.0160
-0.0270
-0.0330
-0.0470
Epoch (3)
 dxˆ
i
dxˆ i
( m)
0.0158
0.0147
0.0159
0.0202
0.0202
0.0254
(m)
-0.0280
-0.0320
-0.0320
-0.0540
-0.0660
-0.0940
Table (3): Global Tests Results
Epoch No.
   (h. o2 )
2
3
5.523
4
12.427
.
1.381
*
*
Where (*) is denoting the rejected value of the deformation test.
Epoch (4)
 dxˆ
dxˆ i
 dxˆ
( m)
0.0158
0.0147
0.0159
0.0202
0.0202
0.0254
(m)
-0.0420
-0.0480
-0.0480
-0.0810
-0.0990
-0.1410
(m)
0.0158
0.0147
0.0159
0.0202
0.0202
0.0254
i
i
Table (4): Results of the New Method
Epoch(2)
Point No.
i 
Epoch (3)
dxˆ i
 dx
2
0.886
3
1.091
4
i 
i
Epoch(4)
dxˆ i
 dx
1.772
i
i 
dxˆ i
 dx
2.658
*
2.183
*
3.274
*
1.006
2.012
*
3.018
*
5
1.339
2.678
*
4.017
*
6
1.635
3.270
*
4.905
*
7
1.853
3.706
*
5.559
*
i
Where (*) is denoting the rejected value of the deformation test.
DISCUSSTION
From the results of this study, the global test in (Table 3) show that all
deform-ations of the second epoch are accepted using the significance level of
(0.05) i.e. the value of the global test is found to be less than the tabulated one
of (3.22) from (Cross, 1983). Conversely, the method succeeded in detecting
deformations from displacements obtained from the third and fourth epochs. On
the other hand the results of the new method shown in (Table 4) give the
acceptance of all deformations at the second epoch using the same level of
significance i.e. the individual test values for displacements are accepted
compared with the critical value of (1.96) using (Cross, 1983). More than one
value of the test is found to be rejected for deformations of the third and fourth
epochs. This means that the two methods succeeded to detect the deformation
at the two last epochs.
The rejection of the global test needs to be followed by a localization
method to detect individual deformations. This is introduced by more than
study, such as (Caspary, 1987). Using the new procedure deformations can be
detected directly at specific points, this what is shown by results of the third
epoch in (Table 4) i.e. only the first point can be assumed as a stable point.
In (Table 4) it can be seen that the minimum size of detected deformation given at point (3) is about 0.03m. Then, it can be said that the new
strategy failed to detect deformations less than 0.03m. However, the minimum
size of deformation to be detected depends on the size of its standard error. In
this particular network the size of the standard error corresponding to
displacements of 0.03m must be less than 0.015. This size of deformation with
this standard error is accepted by (Wan Aziz, W.A. Othman, Z. and Nagib, H.,
2001) using other methods.
CONCLUSIONS
The new method detects deformations at specific points directly,
deformations greater than (0.03m) can easily be detected by the new method at
a significant level greater than 0.05 and the method gives the same results when
compared with the global method.
REFERENCES
1- Baarda, W. (1967). Statistical Concepts in Geodesy. Neth. Geod. Com. Publ.
on Geodesy. New Series 2, No. 4, Delft, Netherlands.
2- Caspary, W.F. (1987). Concepts of Networks and Deformation Analysis,
Monograph No.21, the University of New South Wales, School of
Surveying.
3- Chen, Y.Q. (1983). Analysis of Deformation Survey-A Generalized Method,
Ph.D. Dissertation, University of New Brunswick, Department of surveying
Engineering, T.R. 94.
4- Cross, P.A. (1983). Advanced Least Squares As Applied to Position-Fixing,
Working Paper No.6, University of London.
5- James, M. Secord, (1985). Implementation of A Generelized Method for the
Analysis of Deformation Surveys, Department of Geodesy and Geomatics
Engineering, Canada, T.R. No. 117.
6- Niemeier, W. (1985). Deformationanalyse, Geodaetische Netze in Landesund Ingeniuervemessung II Hearusg. Stuttgard, [559-623].
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Geod. Komm., Reihe C., Heft 164 Munchen.
8- Serif Hekimogolu, Huseyn Demiret and Caneyt Aydin, (2002). Reliability of the
Conventional Deformation Analysis Methods for Vertical Networks, Intern-
ational Net Published Paper, FIG. XXII International Congress, Washinton,
D.C. USA.
9-
(2001). Monitoring High-rise Building
Deformation Using Global Positioning System, International Net Published
Paper, Department of Geomatics, Faculty of Engineering and Geoinformation Science, University Technology-Malysia, Skudai Malaysia.
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Herbert.
Wan Aziz, W.A. Othman, Z. and Nagib, H.