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GEODETSKI VESTNIK | letn. / Vol. 64 | št. / No. 1 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
KLJUČNE BESEDE KEY WORDS
ABSTRACT IZVLEČEK 
Caspary approach, geodetic network, deformation analysis, 
observations, numerical example
postopek Caspary, geodetska mreža, deformacijska analiza, 
meritve, računski primer
UDK : 528.23  
Klasifikacija prispevka po COBISS.SI: 1.02
Prispelo: 10. 1. 2020
Sprejeto: 22. 2. 2020
DOI: 10.15292/geodetski-vestnik.2020.01.68-88
REvIEW ARTICLE
Received: 10. 1. 2020
Accepted: 22. 2. 2020
Veton Hamza, Bojan Stopar, Tomaž Ambrožič
Deformacijska analiza po 
postopku c aspary
Deformation analysis: the 
c aspary approach
In this paper, the theoretical background of the Caspary 
method of geodetic deformation analysis is described and 
implemented in a simulated geodetic network in which two 
epochs of measurements are used. The Caspary approach foresees 
congruence testing of the geodetic network, the determination 
of the stable points between two analysed epochs, the 
transformation of the geodetic network using S-transformation, 
the calculation of displacements, and, in the last step, the 
graphical presentation of displacement vectors. Results obtained 
from the presented example are similar to those presented in 
the Hannover, Karlsruhe, Delft, Fredericton, München, and 
robust methods.
V članku je opisan postopek Caspary, ki je eden izmed postopkov 
deformacijske analize. Značilnosti tega postopka so testiranje 
skladnosti geodetske mreže, določitev stabilnosti točk med 
dvema terminskima izmerama, transformacija geodetske mreže 
s transformacijo S, izračun premikov in grafična predstavitev 
vektorjev premikov. V članku je najprej podano teoretično 
ozadje postopka, nato je postopek uporabljen na primeru 
simuliranih meritev dveh terminskih izmer. Rezultati postopka 
Caspary na obravnavanem primeru ne odstopajo bistveno od 
rezultatov, dobljenih s postopki Hannover, Karlsruhe, Delft, 
Fredericton, München in z robustnimi metodami.
Veton Hamza, Bojan Stopar, Tomaž Ambrožič | DEFORMACIJSKA ANALIZA PO POSTOPKU CASPARY | DEFORMATION ANAL YSIS: THE CASPARY APPROACH | 68-88 |
SI  |  EN

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1 introDuction
The Caspary approach was developed at the University of New South Wales in Australia by Caspary 
(1988). This methodology foresees to use geodetic observations that are carried out in two epochs; it also 
includes geological and geophysical analyses in order to have more information about the stability of the 
reference points (Mihailović and Aleksić, 1994). These analyses can obtain information’s for the point’s 
stability, but statistical testing does the verification of point’s stability between two epochs of observations 
that are analysed. Caspary approach has similarities with Hannover method in defining stable points 
while it foresees to transform the geodetic datum by S-transformation and use as datum points only those 
points that have remained stable and are confirmed by statistical tests (Mihailović and Aleksić, 1994).
In general, methods that are using geodetic observations for estimating displacements contain four steps 
that are also implemented in the Caspary approach (Caspary, 1988; Mihailović and Aleksić, 1994):
1. Firstly, the geodetic network needs to be established, after the accuracy that needs to be achieved, 
and the observation plan has to be defined.
2. In the second step, the network adjustment is carried out for each epoch separately, outliers are 
removed, and point coordinates are calculated.
3. Points from the reference block that have been shifted between two epochs are confirmed in the 
third step; these points are not used as datum points in the upcoming computations.
4. In the last step, after S-transformation displacements are estimated for both object and reference 
points, error ellipses and displacement vectors are shown graphically.
2 theoretical Back GrounDs
2.1 establishment of the geodetic network and definition of the observation plan 
The geometry of the geodetic network depends on the terrain configuration, type and size of the object 
that will be monitored and the ability of the surveyor to establish such geodetic network in certain ter-
rain conditions that will be used after to monitor the object stability regarding the projected accuracy 
(Mihailović and Aleksić, 2008). Observation plan needs to be determined in such a way that it can be 
realized taking into account the terrain configuration. Aiming to define the geometry of the geodetic 
network and the observation plan, the optimisation of the first order is used while the weights of planned 
measurements are defined in the optimisation of the second order.
2.2 Geodetic network adjustment and outlier detection
Caspary approach is classified in the group of methods that analyse two epochs of observations, and 
the coordinate differences between two epochs are presented as displacements. Firstly, the accuracy of 
measurements should be tested; homogenous accuracy is achieved in case that the a posteriori variance  
is statistically equal in both of epochs, which is tested with the following hypothesis (Ambrožič, 2001; 
Caspary, 1988; Mihailović and Aleksić, 1994):
 
2 22
0 01 02 0
ˆˆ
:( )( ) , HEσσσ = = (1)
 
2 22
01 02 0
ˆˆ
:( )( ) .
a
HEσσσ ≠≠ (2)

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Verification of null hypothesis is done by the following T-test that belongs to the F-distribution:
 
12
2
01
1 ,, 2
02
ˆ
,
ˆ
ff
TF
α
σ
σ
−
= ≤ (3)
 f
1
  n
1
  u
1
  d
1
, (4)
 f
2
  n
2
  u
2
  d
2
, (5)
α – risk level,
f
i
 – redundancy number (degrees of freedom) for certain epoch,
n
i
 – number of observations for certain epoch,
u
i
 – number of unknowns for certain epoch,
d
i
 – datum defect for certain epoch.
In case that the null hypothesis isn’t rejected, then the common variance need to be computed (Caspary, 
1988; Mihailović and Aleksić, 1994):
 
22
2 1 01 2 02
12
ˆˆ
.
q ff
s
ff f
σσ +
= =
+
 (6)
The Caspary approach foresees to use inner constrained datum definition, during adjustment computa-
tion of the geodetic network; the following conditions need to be fulfilled (Kuang, 1996; Caspary, 1988):
 
T
.,
i ii
min = v Pv
 (7)
 
T
0
.  and  ˆ ,
ii i i i
min = = + xx x x x
 (8)
v
i
 – vector of residuals for certain epoch,
P
i
 – weight matrix for certain epoch,
x
0i
 – vector of approximate coordinates for certain epoch,
x
i
 – vector of parameter corrections for certain epoch,
ˆ
i
x – vector of adjusted coordinates for certain epoch.
The geodetic network is adjusted as free network in both epochs, the orientation unknowns (and the 
potential unknown of the scale factor) need to be eliminated by using the Gauss or other methods 
(Mihailović and Aleksić, 1994; Mihailović, 1981). Baarda’s Data Snooping, Pope’s T au Method or Dan-
ish Method can be implemented to detect outliers (Caspary, 1988; Grigillo and Stopar, 2003) and the 
accuracy of measured angles and distances need to be harmonized before the adjustment of the geodetic 
network (Ambrožič, 2004).
2.3  Detection of unstable point by c aspary approach
Caspary approach has similarity with some of the other methods used in deformation analyses in the 

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process of determining stable points from the reference block, especially with the method of Hannover 
(Mihailović and Aleksić, 1994). Firstly, after adjusting the geodetic network as free network the vector 
∆ of adjusted coordinate differences and the pseudo inverse matrix of normal equations Q

∆∆ 
 are defined 
as follows (Caspary, 1988):
 
21
ˆˆ , ∆= − xx (9)
 
TT
1 2 111 22 ˆˆ 2 ˆˆ
,
−++
∆∆
=+=+
xx xx
Q Q Q BP B BPB
 (10)
B
i
 – coefficient matrix of observations for certain epoch.
The vector ∆ (9) need to be decomposed in two sub vectors ∆
T
 = (∆
T
n  
∆
T
p
) where ∆
T
n 
presents points from 
the reference block that will be tested for their stability and ∆
T
p
 presents object points that are treated as 
unstable points in all steps of the upcoming computations:
 
.
n
p
∆

∆=

∆

 (11)
The same logic is followed for decomposing matrix  (Caspary, 1988):
 
.
nn np
pn pp
−
∆∆

= ∆∆ =


PP
QP
PP
 (12)
In order to define if points have been shifted between two epochs, the following hypothesis are set 
(Caspary, 1988):
 
( ) ( )
012
ˆˆ : ˆ ,
n nn
HE = = xxx
 (13)
 
( ) ( )
12
ˆˆ : ˆ .
an nn
HE ≠≠ xxx (14)
Not rejection of null hypothesis means that all tested points have remained stable while in the opposite case, 
there are some points that have changed their position between two epochs. Hypothesis defined in (13) and 
(14) belongs to the F-distribution, and they are verified by the following congruency test (Caspary, 1988):
 
( ) ( )
(1 , , ) 2
//
 
/
, ~ 
nn
af f
qfqf
TF
qf S
∆
∆∆∆∆
−
= = (15)
whereas:
 
( )
T
,
n nn n
n
q
−
∆
= ∆∆ Q
 (16)
 f
∆
  2m  d, (17)
m – number of points that are tested for their stability.
The matrix Q

nn
 is estimated as follows (Caspary, 1988):
 
1
.
nn nn np pp pn
−−
= − Q P PPP (18)
In case that of all points of the reference network are tested, and the network is not divided into reference 

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block and object block then the matrix Q

nn
 (18) is equal to the matrix Q

∆∆
 (12). The value of the T-test 
defined by the equation (15) need to be compared with the corresponding value F of the F-distribution 
F
(1a, f
∆
, f )
, if T  F
(1a, f
∆
, f )
 then the null hypothesis is not rejected, in case that T  F
(1a, f
∆
, f )
 the null 
hypothesis is rejected. Rejection of the null hypothesis means that there are reference points than have 
been shifted, and further analyses need to take place in order to detect unstable points.
The process of detecting the unstable points is executed iteratively. Another decomposition of the vector 
∆ and matrix Q

∆∆
 is performed following the same logic as in the equations (11) and (12). The sub-vector 
∆
n
 represents all points from the reference block that are tested for stability, for which the assumption that 
they are stable is considered in the process of defining stable points while sub-vector ∆
p
 represents only 
one point from the reference block in each of the iterations (Caspary, 1988). The same rule is followed 
for the decomposition of the matrix Q

∆∆
, the P
nn
 block belongs to all points from the reference block 
while the P
pp
 block belongs to one point. The quadratic form (q
∆
)
n
 (16) need to be decomposed in two 
subforms after transformation of the vector ∆ that is carried out as follows (Caspary, 1988):
 
I0
PP I
1
.
n
pp pn p
−
 ∆ 
= ∆ 
∆
 
 
 (19)
The sub-vector 
p
∆ is referring to tested point and sub-vector 
n
∆ is referring to all remain points from 
reference block, in order to define the value of (q
∆
)
p
 for each point separately in all iterations, this de-
composition is needed. As an unstable point will be declared the point that will have a greater value of 
(q
∆
)
p
 in each of iterations, the value of (q
∆
)
p
 is estimated by the following expression (Caspary, 1988):
 
( ) .
p pp p
p
q
∆
= ∆∆
T
P
 (20)
In the next iteration, the already defined unstable point will be in the same group with the object points, 
the decomposition of the vector ∆ (11) and the matrix Q

∆∆
 (12) will continue iteratively until all the 
shifted points are detected. Congruency test defined in (15) is repeated until the null hypothesis (13) is 
not rejected, which means that remained points are stable.
2.4 estimation of displacements
Displacements are estimated after all shifted points are detected, the vector ∆ and the matrix Q

∆∆
 are 
partitioned, as is shown (Caspary, 1988):
 
,
r
o
∆

∆=

∆

 (21)
 
.
rr ro
or oo
−
∆∆ ∆∆

= =


PP
QP
PP
 (22)
The sub-vector ∆
r
 represents stable points while the sub-vector ∆
o
 represents all shifted points; for the 
decomposition of Q

∆∆
 matrix the same logic is followed, coefficients from P
rr
 block belongs to stable 
points while P
oo
 block belongs to object points and unstable reference points. In the last transformation, 
points that have remained stable will define the geodetic datum. S-transformation is implemented to 

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transform the geodetic datum in stable points for each epoch separately in order to not carried out another 
adjustment. Vector ∆ and matrix Q
∆∆
 need to be transformed into the new geodetic datum defined by 
stable points with the following equations (Stopar and Marjetič, 2007):
 ∆
stab
  S
stab
∆, (23)
 Q
∆∆stab
  S
stab 
Q
∆∆ 
S
T
stab
, (24)
 S
stab
  I  H(HE
stab 
H)
-1
H
T
E
stab
. (25)
In the matrix E
stab
 all elements are equal to null except diagonal elements of the stable points that are 
equal to one. I matrix is the identity matrix, and the H matrix is defined according to the datum defect 
of the geodetic network (Krüger, 1980). Displacements of the object points (shifted points) are estimated 
by the following equation (Caspary, 1988):
 
1
.
o oo or r o
−
∆ = ∆ +∆ PP (26)
The Caspary approach foresees to show displacements also graphically, vectors of displacements and 
error ellipses  (confidence ellipses/deformation ellipses) with a significance level of 95% need to be 
estimated (Caspary, 1988). Coordinates of the first (null) epoch are defining the origin of error ellipses 
while coordinate differences between first and second epoch are presenting the length of displacements 
vectors. In case that the vector will be out of the error ellipse that point will be reconfirmed as unstable, 
in stable point the vector of displacement need to be inside the error ellipse. Equations that are used to 
estimate the error ellipses from the matrix Q
∆∆stab
  Q
∆∆stab1
  Q
∆∆stab2
 are define as follows (Caspary, 1988):
 
1 1 ,2,
2,
f
as F
α
λ
−
=
 (27)
 
2 1 ,2,
2,
f
bs F
α
λ
−
=
 (28)
 
ˆˆ
ˆˆ ˆˆ
2
tan2 ,
xy
xx yy
q
qq
Θ=
−
 (29)
 
( ) ˆ 1 ˆ ˆˆ
1
,
2
xx yy
qqz λ= ++
 (30)
 
( ) ˆ 2 ˆ ˆˆ
1
,
2
xx yy
qqz λ= +−
 (31)
 
( )
2
ˆˆ ˆˆ
2
ˆˆ ˆˆ
. 4
xx yy xy xy
z q q qq =−+
 (32)
Graphic interpretation of the obtained results is the last step of the Caspary approach that is carried out 
after estimation of the error ellipses defined by the presented equations (27–32).
3 practical eXample
The Caspary method was tested in simulated geodetic network consisted of 7 points, 24 horizontal 
directions and 24 distances. The a priori variance for the directions is σ
si
  1" and the a priori vari-
ance for the distances is equal to σ
di
  5 mm. The plan of observations is the same in both epochs 
while previously the geodetic network was tested by different researchers with other methods used 
in deformation analyses such are:

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 – Hannover (Ambrožič, 2001),
 – Karlsruhe (Ambrožič, 2004),
 – Delft (Marjetič, Zemljak and Ambrožič, 2013),
 – Fredericton (Vrečko and Ambrožič, 2013),
 – München (Soldo and Ambrožič, 2018),
 – Robust methods (Ambrožič et al., 2019).
Caspary approach foresees to fulfil the conditions defined in equation (7) and (8) during the adjustment 
of the network, regarding this the cofactor matrix of unknowns is calculated by the following expression 
(Krüger, 1980):
 
T T1
ˆ
T
ˆ
. ()
−
=+−
xx
Q B PB HH HH
 (33)
The datum matrix H
T
 matrix depends on the observations that are carried out, and it is defined accord-
ing to the defect of datum equal to three in both epochs. More information for datum definition can be 
founded in (Krüger, 1980). Adjustment computation of the geodetic network was carried out for each 
epoch separately, and results are presented in Table 1.
Table 1: Results from the adjustment computation.
First epoch Second epoch
i  1 i  2
2
0
ˆ
i
σ 0.941 1.337
n
i
48 48
u
i
14 + 7 14 + 7
d
i
3 3
f
i
30 30
s
2 
; equation (6) 1.337
f ; equation (6) 60
From the results presented in Table 1, it can be concluded that homogenous accuracy has been reached 
in both of epochs, the value of common variance s
2
 was used in further calculations. The vector ∆ and 
the matrix Q

∆∆
 were estimated regarding equations (9) and (10), respectively. Taking into account that 
the geodetic network is not divided into reference block and object block the matrix Q

nn
 is equal to the 
matrix P
nn
. The next step was to test points stability between two epochs regarding hypothesis defined 
in (13) and (14) while to check hypothesis the congruency test presented in equation (15) was used. 
The statistical test has shown that the null hypothesis (13) is rejected and results from this calculation 
are presented in Table 2.
Table 2: Results from the congruency test.
Parameter m f
∆
(q
∆
)
n
T F
(0.975,11,60)
Values 7 11 1772.21 141.48 1.95
The congruency test has confirmed that some points from the geodetic network have been shifted between 
two epochs. Results from the computations regarding the explanations from section 2.3 are summarized in 
T able 3 while results from the statistical congruency test (15) in each of the iterations are shown in T able 4.

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Table 3: Localization of unstable points.
Point
1
st
 iteration 2
nd
 iteration 3
rd
 iteration 4
th
 iteration
(q
∆
)
p
(q
∆
)
p
(q
∆
)
p
(q
∆
)
p
1 755.51
2 562.59 321.26 505.70
3 415.29 347.99 394.94 146.15
4 94.53 99.03 52.00 75.72
5 67.62 75.56 17.13 3.87
6 8.99 95.81 52.00 0.51
7 664.76 363.61
Table 4: Results from the statistical test (15) and critical values.
Parameter 1
st
 iteration 2
nd
 iteration 3
rd
 iteration 4
th
 iteration 5
th
 iteration
m 7 6 5 4 3
f
∆
11 9 7 5 3
T 141.48 99.20 81.93 25.89 0.36
F 1.95 2.04 2.17 2.37 2.76
From the presented results, it was concluded that point 1, 2, 3 and 7 had been shifted while point 4, 5 
and 6 have remained stable. In the fifth iteration, the congruency test has confirmed that the null hy-
pothesis (13) isn’t rejected and in the following step the defined stable points have been used as datum 
points to carried out the S- transformations presented with equations (23), (24) and (25). Displace-
ments are calculated after the transformation of the geodetic datum, error ellipses and displacement 
values through Y and X-axis are presented in Table 5 while error ellipses and displacements vectors are 
graphically shown in Figure 1.
Table 5: Displacements and error ellipses.
Point
Displacements Error ellipses
d
y
 [mm] d
x
 [mm] a [mm] b [mm] Θ [º]
1 – 19.2 – 37.9 11.1 9.2 104
2 – 38.4 49.4 13.2 10.4 62
3 20.8 – 43.9 11.9 9.1 2
4 – – 6.4 3.7 74
5 – – 6.2 5.4 159
6 – – 6.5 2.9 55
7 23.9 43.1 7.2 6.7 151

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Figure 1: Error ellipses and displacements.
4 comparison of results from caspary methoD With other methoDs
Results from the Caspary method are compared with the results from the method of Hannover, Karlsruhe, 
Delft, Fredericton and München approach and presented in Table 6. Point 4, 5 and 6 are defined as 
stable points while point 1, 2, 3 and 7 are confirmed as unstable points in all of the methods. Slight 
difference in results between methods was noticed, but the estimated values of displacements are very 
similar to simulated displacements.
Table 6: Simulated displacements and results from deformation analyses by Hannover, Karlsruhe, Delft, Fredericton, München 
and Caspary approach.
Point 1 2 3 4 5 6 7
Simulated
 d
y
 [mm] – 20.0 – 30.0 25.0 0.0 0.0 0.0 25.0
 d
x 
[mm] – 34.6 52.0 – 43.3 0.0 0.0 0.0 43.3
 d
 
[mm] 40.0 60.0 50.0 0.0 0.0 0.0 50.0
υ [º] 210 330 150 – – – 30
Hannover
d
y
 [mm] – 19.6 – 38.7 20.6 – 4.0 – 6.4 3.3 23.6
d
x
 [mm] – 38.0 49.0 – 44.3 5.1 – 7.1 – 10.6 42.9
d [mm] 42.8 62.4 48.9 6.5 10.0 11.1 49.0
υ [º] 207 322 155 322 222 163 29
Movement yes yes yes no no no yes
Karlsruhe
d
y
 [mm] – 19.7 – 38.8 20.6 – – – 23.6
d
x
 [mm] – 38.0 49.0 – 44.4 – – – 42.9
d [mm] 42.8 62.5 48,9 – – – 49,0
υ [º] 207 322 155 – – – 29
Movement yes yes yes no no no yes

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Point 1 2 3 4 5 6 7
Delft
d
y
 [mm] – 19.4 – 38.1 21.4 0.7 – 0.8 0.0 24.0
d
x
 [mm] – 37.5 49.5 – 43.5 1.0 – 2.3 1.3 42.9
d [mm] 42.2 62.5 48.5 1.2 2.4 1.3 49.2
υ [º] 207 322 154 35 199 0 29
Movement yes yes yes no no no yes
Fredericton
d
y
 [mm] – 19.6 – 38.7 20.6 – – – 23.6
d
x 
[mm] – 38.0 49.0 – 44.3 – – – 42.9
d [mm] 42.8 62.5 48.9 – – – 48.9
υ [º] 207 322 155 – – – 29
Movement yes yes yes no no no yes
München
d
y
 [mm] – 19.5 – 38.2 21.4 0.7 – 0.8 0.0 24.0
d
x
 [mm] – 37.6 49.5 – 43.6 1.0 – 2.2 1.4 42.9
d [mm] 42.4 62.5 48.6 1.2 2.3 1.4 49.2
υ [º] 207 322 154 35 200 0 29
Movement yes yes yes no no no yes
Caspary
d
y
 [mm] – 19.2 – 38.4 20.8 – – – 23.9
d
x
 [mm] – 37.9 49.4 – 43.9 – – – 43.1
d [mm] 42.5 62.5 48.6 – – – 49.3
υ [º] 207 322 154 – – – 29
Movement yes yes yes no no no yes
5 conclusion
Deformation analyses by the Caspary approach is the seventh method described an implemented in the 
same simulated geodetic network. This methodology is developed by W. F . Caspary, and it is presented 
on his monograph and in other literature (Caspary, 1988; Mihailović and Aleksić, 1994). Caspary 
method has many similarities with the method of Hannover, but it includes additionally geological and 
geophysical analysis to define which points need to be treated as conditionally stable (Mihailović and 
Aleksić, 1994). These points are analysed additionally for their stability by the explained methodology, 
moved points in the further analyses will belong in the same group with the object points.
Stable points need to define the geodetic datum in both epochs. T o transform the geodetic datum of the 
first and second epoch into stable points was used S-transformation. After the datum transformations, 
coordinate differences between the second and first epoch are defining point’s displacements. Caspary 
approach foresees to show displacements also graphically, unstable points from reference block are con-
firmed by error ellipses additionally while the displacement vector needs to be out of the error ellipse 
which origin is defined from coordinates of the first epoch.
Results presented by Caspary method have minor differences with results obtained by other methods of de-
formation analyses published in the previous articles of one of the authors of this article, common points are 
confirmed as unstable by all methods, and estimated displacements are very close to simulated displacements.

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literature and references:
Ambrožič, T . (2001). Deformacijska analiza po postopku Hannover . Geodetski vestnik, 
45 (1&2), 38–53. http://www.geodetski-vestnik.com/45/gv45-12.pdf, 
accessed on 10. 2. 2020.
Ambrožič, T . (2004). Deformacijska analiza po postopku Karlsruhe. Geodetski vestnik, 
48 (3), 315–331. http://www.geodetski-vestnik.com/56/1/gv56-1_009-026.
pdf, accessed on 10. 2. 2020.
Ambrožič, T., Mulahusić, A., Tuno, N., Topoljak, J., Hajdar, A., Kogoj, D. (2019). 
Deformation analysis with robust methods in geodetic nets. Geodetski 
vestnik, 63 (2), 163–178. DOI: https://doi.org/10.15292/geodetski-
vestnik.2019.02.163-178
Caspary, W . F . (1988). Concepts of Network and Deformation Analysis. Kensington: The 
University of New South Wales, School of Surveing, Australia.
Grigillo, D., Stopar, B. (2003). Methods of Gross Error Detection in Geodetic 
Observations. Geodetski vestnik, 47 (4), 387–403. http://www.geodetski-
vestnik.com/47/4/gv47-4_387-403.pdf, accessed on 10. 2. 2020.
Krüger , J. (1980). Numerische Behandlung von Datums- und Konfigurationsdefekten. 
In H. Pelzer (Ed.), Geodätische Netze in Landes- und Ingenieurvermessung, 
Vorträge des Kontaktstudiums. Hannover, Stuttgart: Konrad Wittwer.
Kuang, S. (1996). Geodetic Network Analysis and Optimal Design: Concepts and 
Applications. Chelsea: Ann Arbor Press, Inc.
Marjetič, A., Stopar , B. (2007). Geodetski datum in S-transformacija. Geodetski vestnik, 
51 (3), 549-564. http://www.geodetski-vestnik.com/51/3/gv51-3_549-564.
pdf, accessed on 10. 2. 2020.
Marjetič, A., Zemljak, M., Ambrožič, T. (2013). Deformacijska analiza po postopku 
Delft. Geodetski vestnik, 57 (3), 479–497. DOI: https://doi.org/10.15292/
geodetski-vestnik.2013.03.479-497
Mihailović, K. (1981). Geodezija II. Beograd: Građevinski fakultet Univerziteta u 
Beogradu.
Mihailović, K., Aleksić, I. (1994). Deformaciona analiza geodetskih mreža. Beograd: 
Građevinski fakultet Univerziteta u Beogradu.
Mihailović, K., Aleksić, I. R. (2008). Koncepti mreža u geodetskom premeru. Beograd: 
Privredno društvo za kartografiju Geokarta d.o.o.
Soldo, J., Ambrožič, T. (2018). Deformacijska analiza po postopku München. 
Geodetski vestnik, 62 (3), 392–414. DOI: https://doi.org/10.15292/geodetski-
vestnik.2018.03.392-414
Vrečko, A., Ambrožič, T. (2013). Deformacijska analiza po postopku Fredericton. 
Geodetski vestnik, 57 (3), 479–497. DOI: https://doi.org/10.15292/geodetski-
vestnik.2013.03.479-497
Hamza V ., Stopar B., Ambrožič T.  (2020). Deformation Analysis: the Caspary Approach. Geodetski vestnik, 64 (1), 68-88. 
DOI: https://doi.org/10.15292/geodetski-vestnik.2020.01.68-88
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Deformacijska analiza po postopku c aspary
OSNOVNE INFORMACIJE O ČLANKU:
GLEJ STRAN 68
1 uVoD
Postopek Caspary je razvil W . F . Caspary na Univerzi Novi južni Wales v Sydneyju v Avstraliji (Caspary, 
1988). V postopku uporabimo geodetske meritve, ki jih izvedemo v dveh neodvisnih terminskih izmerah, 
vključuje pa tudi geološke in geofizikalne analize (Mihailović in Aleksić, 1994). S temi analizami lahko 
dobimo podatke o stabilnosti točk, vendar določitev stabilnih točk med dvema terminskima izmerama 
opravimo s statističnim testiranjem premikov točk. Postopek Caspary je podoben postopku Hannover, 
razlika med njima je v tem, da se pri prvem na koncu predvideva uporaba transformacije S v datum tistih 
referenčnih točk, ki jih s predhodno opravljenimi statističnimi testi določimo kot stabilne (Mihailović 
in Aleksić, 1994).
Postopke, pri katerih se za izračun premikov uporabljajo geodetske meritve, v splošnem delimo na štiri 
korake. Podobno lahko delimo postopek Caspary (Caspary, 1988; Mihailović in Aleksić, 1994):
1. Najprej vzpostavimo geodetsko mrežo, določimo plan izmere, po njem izvedemo izmero in dobimo 
meritve zahtevane natančnosti.
2. V drugem koraku lociramo in odstranimo morebitne grobe pogreške med meritvami, izvedemo 
izravnavo in izračunamo izravnane koordinate točk.
3. V tretjem koraku ugotovimo, katere referenčne točke so se med dvema terminskima izmerama 
premaknile. V nadaljnjih izračunih teh točk ne smemo uporabiti kot datumske točke.
4. V zadnjem koraku naredimo transformacijo S ter grafično prikažemo premike točk na objektu in 
tistih referenčnih točk, za katere smo ugotovili, da so se premaknile.
2 teoretiČno ozaDje
2.1 Vzpostavitev geodetske mreže in določitev plana izmere
Geometrija geodetske mreže je odvisna od konfiguracije terena ter vrste in velikosti objekta, ki ga želimo 
spremljati. V danih terenskih razmerah moramo za spremljanje stabilnosti objekta vzpostaviti takšno 
geodetsko mrežo, da bomo dosegli projektirano natančnost meritev (Mihailović in Aleksić, 2008). Plan 
meritev moramo sestaviti tako, da ga bo ob upoštevanju konfiguracije terena mogoče realizirati. Za 
določitev geometrije geodetske mreže in plana izmere uporabimo optimizacijo prvega reda, za določitev 
uteži načrtovanih meritev pa optimizacijo drugega reda.
2.2 izravnava meritev in ugotovitev morebitno grobo pogrešenih meritev
Postopek Caspary je razvrščen v skupino metod, s katerimi analiziramo meritve dveh terminskih izmer. 
Koordinatne razlike identičnih točk med dvema terminskima izmerama predstavimo kot premike. Najprej 
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moramo testirati homogenost natančnosti meritev, torej če sta a-posteriori referenčni varianci 
2
0
ˆ
i
σ obeh 
terminskih izmer statistično enaki. Test opravimo s preizkusom ničelne H
0
 hipoteze (Ambrožič, 2001; 
Caspary, 1988; Mihailović in Aleksić, 1994):
 
( )
2 22
0 01 02 0
ˆˆ
: () , HEσσσ = =
 (1)
 
( )
2 22
01 02 0
ˆˆ
: () .
a
HEσσσ ≠≠
 (2)
Preverjanje ničelne hipoteze opravimo s testno statistiko T, ki se porazdeljuje po porazdelitvi F:
 
12
2
01
1 ,, 2
02
ˆ
,
ˆ
ff
TF
α
σ
σ
−
= ≤
 (3)
 f
1
  n
1
  u
1
  d
1
, (4)
 f
2
  n
2
  u
2
  d
2
, (5)
α – izbrana stopnja značilnosti testa,
f
i
 – število nadštevilnih meritev (prostostnih stopenj) v posamezni terminski izmeri,
n
i
 – število meritev v posamezni terminski izmeri,
u
i
 – število neznank v posamezni terminski izmeri,
d
i
 – defekt datuma (defekt ranga matrike normalnih enačb) v posamezni terminski izmeri.
Če ničelne hipoteze ne moremo zavrniti, izračunamo skupno a-posteriori referenčno varianco (Caspary, 
1988; Mihailović in Aleksić, 1994):
 
22
2 1 01 2 02
12
ˆˆ
.
q ff
s
ff f
σσ +
= =
+
 (6)
Postopek Caspary predvideva, da je datum geodetske mreže določen z notranjimi vezmi, zato morajo 
biti v izravnavi izpolnjeni naslednji pogoji (Kuang, 1996; Caspary, 1988):
 
T
.,
i ii
min = v Pv
 (7)
 
T
0
.  in  ˆ ,
ii i i i
min = = + xx x x x
 (8)
v
i
 – vektor popravkov meritev v posamezni terminski izmeri,
P
i
 – matrika uteži v posamezni terminski izmeri,
x
0i
 – vektor približnih vrednosti koordinat v posamezni terminski izmeri,
x
i
 – vektor popravkov približnih vrednosti koordinat v posamezni terminski izmeri,
ˆ
i
x – vektor izravnanih koordinat v posamezni terminski izmeri.
Geodetsko mrežo v posamezni terminski izmeri izravnamo kot prosto mrežo, orientacijske neznanke 
(in morebitno neznanko merila mreže) moramo odstraniti z Gaußovo metodo eliminacije ali drugimi 
metodami (Mihailović in Aleksić, 1994; Mihailović, 1981). Za odkrivanje grobo pogrešenih meritev 
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lahko uporabimo Baardovo metodo Data Snooping, Popeovo metodo Tau ali dansko metodo (Caspary, 
1988; Grigillo in Stopar, 2003). Pred izravnavo moramo uskladiti tudi natančnosti kotnih in dolžinskih 
meritev (Ambrožič, 2004).
2.3 testiranje nestabilnih točk po postopku c aspary
Postopek Caspary je, v postopku določitve stabilnih referenčnih točk, podoben nekaterim drugim me-
todam deformacijske analize, še posebej postopku Hannover (Mihailović in Aleksić, 1994). Po izravnavi 
geodetske mreže kot proste mreže sta vektor izravnanih koordinatnih razlik ∆ in psevdoinverzna matrika 
kofaktorjev izravnanih koordinatnih razlik Q

∆∆
 definirana kot (Caspary, 1988):
 
21
ˆˆ , ∆= − xx
 (9)
 
TT
1 2 111 22 ˆˆ 2 ˆˆ
,
−++
∆∆
=+=+
xx xx
Q Q Q BP B BPB
 (10)
B
i
 – matrika koeficientov enačb popravkov meritev v posamezni terminski izmeri.
Vektor ∆ (9) moramo razstaviti na podvektorja ∆
T
 = (∆
T
n  
∆
T
p
). V podvektorju ∆
T
n
 so koordinatne razlike 
referenčnih točk, ki jih bomo v nadaljevanju testirali glede stabilnosti, v podvektorju ∆
T
p
 pa so koordinatne 
razlike točk na objektu, ki jih bomo v nadaljnjih korakih obravnavali kot nestabilne točke:
 
.
n
p
∆

∆=

∆

 (11)
Podobno razstavimo tudi pripadajočo matriko Q

∆∆
 (Caspary, 1988):
 
.
nn np
pn pp
−
∆∆ ∆∆

= =


PP
QP
PP
 (12)
Stabilnost referenčnih točk med dvema terminskima izmerama določimo s testiranjem naslednje hipoteze 
(Caspary, 1988):
 
( ) ( )
012
ˆˆ : ˆ ,
n nn
HE = = xxx
 (13)
 
( ) ( )
12
ˆˆ : ˆ .
an nn
HE ≠≠ xxx
 (14)
Če ničelne hipoteze ne moremo zavrniti, pomeni, da so vse v testiranje vključene referenčne točke v 
dveh terminskih izmerah ostale stabilne, v nasprotnem primeru je med dvema terminskima izmerama 
nekaj referenčnih točk spremenilo svoje koordinate. Testiranje ničelne hipoteze (13) opravimo s testno 
statistiko T, ki se porazdeljuje po porazdelitvi F (Caspary, 1988):
 
( ) ( )
(1 , , ) 2
//
 
/
, ~ 
nn
af f
qfqf
TF
qf S
∆
∆∆∆∆
−
= =
 (15)
kjer je:
 
( )
T
,
n nn n
n
q
−
∆
= ∆∆ Q
 (16)
 f
∆
  2m  d, (17)
m – število referenčnih točk, ki so vključene v testiranje.
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Matriko Q

nn
 izračunamo z naslednjo enačbo (Caspary, 1988):
 
1
.
nn nn np pp pn
−−
= − Q P PPP (18)
Če točke geodetske mreže niso razdeljene na referenčne točke in točke na objektu, torej obravnavamo vse 
točke mreže kot referenčne, potem je matrika Q

nn
 (18) enaka matriki Q

∆∆
 (12). Vrednost testne statistike 
T izračunano z enačbo (15), primerjamo z ustrezno kritično vrednostjo F
(1a, f
∆
, f )
. Če je T   F
(1a, f
∆
, f )
, 
potem ničelne hipoteze ne moremo zavrniti, če je T   F
(1a, f
∆
, f )
, pa ničelno hipotezo zavrnemo. Zavrnitev 
ničelne hipoteze pomeni, da imamo med referenčnimi točkami tudi točke, ki so se dvema terminskima 
izmerama premaknile. T ako moramo opraviti dodatne analize, da odkrijemo nestabilne referenčne točke.
Postopek določitve nestabilnih referenčnih točk izvedemo iterativno. Razstavljanje vektorja ∆ in matrike 
Q

∆∆
 naredimo podobno, kot smo zapisali v enačbah (11) in (12). V posamezni iteraciji so v podvektorju 
∆
n
 koordinatne razlike vseh referenčnih točk razen ene, za katere smo predpostavili, da so stabilne in jih 
bomo v nadaljevanju testirali, ali so res stabilne. V podvektorju ∆
p
 sta le koordinatni razliki (za 2D-mre-
žo) samo ene referenčne točke (tiste, ki je nismo vključili v podvektor ∆
n
) (Caspary, 1988). Pripadajočo 
matriko Q

∆∆
 razstavimo podobno: podmatrika P
nn
 vsebuje elemente vseh referenčnih točk razen ene, 
podmatrika P
pp
 pa elemente le ene točke (tiste, ki je ni v podmatriki P
nn
). Kvadratno formo (q
∆
)
n
, enačba 
(16), lahko razstavimo na dva dela. To naredimo po transformaciji vektorja ∆ na podvektorja naslednje 
oblike (Caspary, 1988):
 
pp pn

= ∆ 
 
  
I0
PP I
 (19)
Podvektor 
p
∆ se nanaša na točko, ki jo testiramo, podvektor 
n
∆ pa se nanaša na vse preostale referenčne 
točke. Razstavljanje vektorja ∆ in matrike Q

∆∆
 ter izračun (q
∆
)
p
 naredimo za vsako točko posebej. Iteracij 
je toliko, kolikor imamo referenčnih točk, za katere smo predpostavili, da so stabilne. Kot nestabilno 
točko določimo točko, za katero izračunamo po vsaki iteraciji največjo vrednost (q
∆
)
p
 (Caspary, 1988):
 
( ) .
p pp p
p
q
∆
= ∆∆
T
P
 (20)
V naslednji iteraciji prej določeno nestabilno referenčno točko prestavimo v skupino točk na objektu. 
Razstavljanje vektorja ∆ (11) in matrike Q

∆∆
 (12) nadaljujemo iterativno, dokler ne odkrijemo vseh 
nestabilnih referenčnih točk. Testno statistiko, določeno v (15), ponavljamo tolikokrat, dokler ničelne 
hipoteze (13) ne moremo zavrniti, kar pomeni, da so vse preostale referenčne točke stabilne.
2.4 izračun premikov
Ko odkrijemo vse točke, ki niso stabilne, izračunamo premike točk tako, da vektor ∆ in pripadajočo 
matriko Q

∆∆
 razstavimo na (Caspary, 1988):
 
,
r
o
∆

∆=

∆

 (21)
 .
rr ro
or oo
−
∆∆ ∆∆

= =


PP
QP
PP
 (22)
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V podvektorju ∆
r
 so koordinatne razlike stabilnih točk, medtem ko so v podvektorju ∆
o
 koordinatne 
razlike točk, ki so se premaknile. Za razstavitev matrike Q

∆∆
 velja podobno: elementi podmatrike P
rr
 se 
nanašajo na stabilne točke, medtem ko se elementi podmatrike P
oo
 nanašajo na točke, ki so se premaknile, 
torej gre za točke na objektu in nestabilne referenčne točke. V zadnji transformaciji, ki jo izvedemo, 
bodo točke, ki so stabilne, določale nov datum geodetske mreže. Za transformacijo geodetske mreže 
uporabimo transformacijo S za vsako terminsko izmero posebej, datum transformirane mreže je določen 
s stabilnimi točkami. Vektor ∆ in matriko Q
∆∆
 transformiramo v nov geodetski datum stabilnih točk z 
naslednjimi enačbami (Stopar in Marjetič, 2007):
 ∆
stab
  S
stab
∆, (23)
 Q
∆∆stab
  S
stab 
Q
∆∆ 
S
T
stab
, (24)
 S
stab
  I  H(HE
stab 
H)
-1
H
T
E
stab
. (25)
V matriki E
stab
 so vsi elementi enaki nič, le diagonalni elementi imajo vrednosti enake ena na mestih, ki 
pripadajo koordinatni komponenti stabilne referenčne točke. Matrika I je enotska matrika, matrika H 
je matrika geodetskega datuma, določenega z notranjimi vezmi (Krüger, 1980). Premike točk na objektu 
in nestabilnih referenčnih točk izračunamo z naslednjo enačbo (Caspary, 1988):
 
1
.
o oo or r o
−
∆ = ∆ +∆ PP (26)
Postopek Caspary predvideva tudi grafično predstavitev vektorjev premikov in elips zaupanja. Elipse 
zaupanja koordinat točk so določene s stopnjo zaupanja 95 % (Caspary, 1988). V grafični predstavitvi 
določajo koordinate točk prve (ničelne) terminske izmere središča elips zaupanja, razlike koordinat med 
terminskima izmerama pa določajo dolžino vektorjev premikov točk. Če se konec vektorja premika 
nahaja izven elipse zaupanja, lahko ponovno potrdimo, da se je točka premaknila, če je konec vektorja 
premika znotraj elipse zaupanja, pa lahko trdimo, da se točka ni premaknila. Iz elementov matrike 
Q
∆∆stab
  Q
∆∆stab1
  Q
∆∆stab2
 izračunamo elemente elips zaupanja na naslednji način (Caspary, 1988):
 
1 1 ,2,
2,
f
as F
α
λ
−
=
 (27)
 
2 1 ,2,
2,
f
bs F
α
λ
−
=
 (28)
 
ˆˆ
ˆˆ ˆˆ
2
tan2 ,
xy
xx yy
q
qq
Θ=
−
 (29)
 
( ) ˆ 1 ˆ ˆˆ
1
,
2
xx yy
qqz λ= ++
 (30)
 
( )
λ= +−
ˆ 2 ˆ ˆˆ
1
,
2
xx yy
qqz
 (31)
 
( )
2
ˆˆ ˆˆ
2
ˆˆ ˆˆ
. 4
xx yy xy xy
z q q qq =−+
 (32)
Grafična predstavitev dobljenih rezultatov je zadnji korak pristopa Caspary in jo izvedemo po izračunu 
elementov elips zaupanja, predstavljenih z enačbami od (27) do (32).
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3 ra Čunski primer
Uporabnost postopka Caspary želimo testirati na simulirani geodetski mreži, sestavljeni iz 7 točk ter 
merjenih 24 horizontalnih smeri in 24 dolžin. Za vrednost a-priori variance za smeri izberemo σ
si
  
1", za vrednost a-priori variance za dolžine pa σ
di
  5 mm. Za obe terminski izmeri je plan meritev 
enak. Isto geodetsko mrežo so uporabili tudi drugi raziskovalci, ko so testirali druge postopke defor-
macijske analize:
 – Hannover (Ambrožič, 2001),
 – Karlsruhe (Ambrožič, 2004),
 – Delft (Marjetič, Zemljak in Ambrožič, 2013),
 – Fredericton (Vrečko in Ambrožič, 2013),
 – München (Soldo in Ambrožič, 2018),
 – robustne metode (Ambrožič et al., 2019).
Postopek Caspary predvideva, da so v izravnavi mreže izpolnjeni pogoji, določeni v enačbah (7) in (8). 
Matriko kofaktorjev neznank izračunamo z naslednjim izrazom (Krüger, 1980):
 
T T1
ˆ
T
ˆ
. ()
−
=+−
xx
Q B PB HH HH (33)
Datumska matrika H
T
 je odvisna od vrste izvedenih meritev v geodetski mreži in je določena glede na 
defekt datuma geodetske mreže, ki je v našem primeru v obeh terminskih izmerah enak tri (Krüger, 
1980). Vsako terminsko izmero smo izravnali, rezultate prikazujemo v preglednici 1.
Preglednica 1: Rezultati izravnave.
Prva terminska izmera Druga terminska izmera
i  1 i  2
2
0
ˆ
i
σ
0,941 1,337
n
i
48 48
u
i
14 + 7 14 + 7
d
i
3 3
f
i
30 30
s
2
; enačba (6) 1,337
f ; enačba (6) 60
Iz rezultatov, predstavljenih v preglednici 1, ugotovimo, da smo dosegli homogeno natančnost meritev 
v obeh terminskih izmerah. Vrednost skupne a-posteriori referenčne variance s
2
 uporabimo v nadaljnjih 
izračunih. Vektor ∆ in matriko Q

∆∆
 izračunamo po enačbah (9) in (10). Ker geodetske mreže nismo 
razdelili na referenčne točke in točke na objektu, sta matriki Q

nn
 in P
nn
 enaki. V naslednjem koraku te-
stiramo stabilnost točk med dvema terminskima izmerama z ničelno hipotezo (13). Izračunamo vrednost 
testne statistike po enačbi (15) in ugotovimo, da moramo ničelno hipotezo (13) zavrniti. Rezultate tega 
izračuna predstavljamo v preglednici 2.
Preglednica 2: Rezultati testiranja skladnosti.
Parameter m f
∆
(q
∆
)
n
T F
(0.975,11,60)
Vrednost 7 11 1772,21 141,48 1,95
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Test skladnosti je potrdil, da so se nekatere točke geodetske mreže med dvema terminskima izmerama 
premaknile. Zato postopek nadaljujemo z izračuni, opisanimi v podpoglavju 2.3. Rezultate teh izračunov 
prikazujemo v preglednici 3, rezultate testne statistike (15) in kritične vrednosti po vsaki iteraciji pa v 
preglednici 4.
Preglednica 3: Določitev nestabilnih referenčnih točk.
Točka
1. iteracija 2. iteracija 3. iteracija 4. iteracija
(q
∆
)
p
(q
∆
)
p
(q
∆
)
p
(q
∆
)
p
1 755,51
2 562,59 321,26 505,70
3 415,29 347,99 394,94 146,15
4 94,53 99,03 52,00 75,72
5 67,62 75,56 17,13 3,87
6 8,99 95,81 52,00 0,51
7 664,76 363,61
Preglednica 4: Rezultati testne statistike (15) in kritične vrednosti.
Parameter 1. iteracija 2. iteracija 3. iteracija 4. iteracija 5. iteracija
m 7 6 5 4 3
f
∆
11 9 7 5 3
T 141,48 99,20 81,93 25,89 0,36
F 1,95 2,04 2,17 2,37 2,76
Iz predstavljenih rezultatov sklepamo, da so se točke 1, 2, 3 in 7 premaknile, točke 4, 5 in 6 pa so stabilne. 
V peti iteraciji ugotovimo, da s testom skladnosti ne moremo zavrniti ničelne hipoteze (13). V naslednjem 
koraku potrjeno stabilne točke uporabimo kot točke, ki definirajo datum mreže v transformaciji S, ki 
je predstavljena z enačbami (23), (24) in (25). Po transformaciji S, s katero v obeh terminskih izmerah 
zagotovimo identični datum geodetske mreže, izračunamo premike točk in elemente elips zaupanja, ki 
jih prikazujemo v preglednici 5. Elipse zaupanja in ugotovljene vektorje premikov grafično prikazujemo 
na sliki 1.
Preglednica 5: Premiki in elementi elips zaupanja.
Točka
Premiki Elipse zaupanja
d
y
 [mm] d
x
 [mm] a [mm] b [mm] Θ [º]
1 – 19,2 – 37,9 11,1 9,2 104
2 – 38,4 49,4 13,2 10,4 62
3 20,8 – 43,9 11,9 9,1 2
4 – – 6,4 3,7 74
5 – – 6,2 5,4 159
6 – – 6,5 2,9 55
7 23,9 43,1 7,2 6,7 151 
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Slika 1: Elipse zaupanja in premiki točk.
4 primerja Va z rezultati DruGih postopko V
Primerjavo rezultatov postopka Caspary in rezultatov postopkov Hannover, Karlsruhe, Delft, 
Fredericton in München predstavljamo v preglednici 6. Z uporabo vseh postopkov trdimo, da 
so točke 4, 5 in 6 stabilne, točke 1, 2, 3 in 7 pa nestabilne. Opazimo lahko le majhno razliko v 
rezultatih izračunanih premikov, vendar so izračunane vrednosti premikov zelo podobne simuli-
ranim premikom.
Preglednica 6: Simulirani premiki točk mreže in rezultati deformacijske analize po postopkih Hannover, Karlsruhe, Delft, 
Fredericton, München in Caspary.
Točka 1 2 3 4 5 6 7
Simulirano
 d
y
 [mm] – 20,0 – 30,0 25,0 0,0 0,0 0,0 25,0
d
x
 [mm] – 34,6 52,0 – 43,3 0,0 0,0 0,0 43,3
d [mm] 40,0 60,0 50,0 0,0 0,0 0,0 50,0
υ [º] 210 330 150 – – – 30
Hannover
d
y
 [mm] – 19,6 – 38,7 20,6 – 4,0 – 6,4 3,3 23,6
d
x
 [mm] – 38,0 49,0 – 44,3 5,1 – 7,1 – 10,6 42,9
d [mm] 42,8 62,4 48,9 6,5 10,0 11,1 49,0
υ [º] 207 322 155 322 222 163 29
Premik da da da ne ne ne da
Karlsruhe
d
y
 [mm] – 19,7 – 38,8 20,6 – – – 23,6
d
x
 [mm] – 38,0 49,0 – 44,4 – – – 42,9
d [mm] 42,8 62,5 48,9 – – – 49,0
υ [º] 207 322 155 – – – 29
Premik da da da ne ne ne da
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Točka 1 2 3 4 5 6 7
Delft
d
y
 [mm] – 19,4 – 38,1 21,4 0,7 – 0,8 0,0 24,0
d
x
 [mm] – 37,5 49,5 – 43,5 1,0 – 2,3 1,3 42,9
d [mm] 42,2 62,5 48,5 1,2 2,4 1,3 49,2
υ [º] 207 322 154 35 199 0 29
Premik da da da ne ne ne da
Fredericton
d
y
 [mm] – 19,6 – 38,7 20,6 – – – 23,6
d
x
 [mm] – 38,0 49,0 – 44,3 – – – 42,9
d [mm] 42,8 62,5 48,9 – – – 48,9
υ [º] 207 322 155 – – – 29
Premik da da da ne ne ne da
München
d
y
 [mm] – 19,5 – 38,2 21,4 0,7 – 0,8 0,0 24,0
d
x
 [mm] – 37,6 49,5 – 43,6 1,0 – 2,2 1,4 42,9
d [mm] 42,4 62,5 48,6 1,2 2,3 1,4 49,2
υ [º] 207 322 154 35 200 0 29
Premik da da da ne ne ne da
Caspary
d
y
 [mm] – 19,2 – 38,4 20,8 – – – 23,9
d
x
 [mm] – 37,9 49,4 – 43,9 – – – 43,1
d [mm] 42,5 62,5 48,6 – – – 49,2
υ [º] 207 322 154 – – – 9
Premik da da da ne ne ne da
5 sklep
Deformacijska analiza po postopku Caspary je sedma opisana metoda, izvedena na isti simulirani geo-
detski mreži. Postopek je razvil W . F . Caspary, predstavil ga je v monografiji (Caspary, 1988), obravnavali 
pa so ga tudi drugi avtorji (Mihailović in Aleksić, 1994). Postopek Caspary je delno podoben postopku 
Hannover, vključuje pa dodatne geološke in geofizikalne analize, ki jih uporabimo za odločitev, katere 
referenčne točke lahko obravnavamo kot pogojno stabilne (Mihailović in Aleksić, 1994). Stabilnost teh 
točk dodatno analiziramo s predstavljeno metodologijo. Referenčne točke, za katere smo ugotovili, da 
niso stabilne, pa v nadaljnjih analizah obravnavamo kot točke na objektu.
Stabilne referenčne točke morajo določati geodetski datum v obeh terminskih izmerah. Za transfor-
macijo geodetske mreže prve in druge terminske izmere v datum stabilnih referenčnih točk uporabimo 
transformacijo S. Po transformaciji mreže prve in druge terminske izmere v geodetski datum stabilnih 
referenčnih točk izračunamo koordinatne razlike točk med drugo in prvo terminsko izmero. Postopek 
Caspary predvideva grafično predstavitev vektorjev premikov in elips zaupanja. V ektor premika nestabil-
nih referenčnih točk, katerega začetek določajo koordinate točke iz prve terminske izmere in ima konec 
izven elipse zaupanja, potrdi, da se je referenčna točka premaknila.
Rezultati, ki jih dobimo s postopkom Caspary, se le malo razlikujejo od rezultatov, ki jih dobimo z 
drugimi postopki deformacijske analize in so objavljeni v predhodnih člankih enega od avtorjev tega 
članka. Z vsemi pristopi deformacijske analize smo potrdili iste nestabilne točke, izračunani premiki pa 
so zelo podobni simuliranim premikom.
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Literatura in viri:
Glej literaturo na strani 78.
Veton Hamza, mag. inž. geod. in geoinf.
Univerza v Ljubljani, Fakulteta za gradbeništvo in geodezijo
Jamova cesta 2, SI-1000 Ljubljana
e-naslov: veton.hamza@fgg.uni-lj.si
prof. dr. Bojan Stopar, univ. dipl. inž. geod.
Univerza v Ljubljani, Fakulteta za gradbeništvo in geodezijo
Jamova cesta 2, SI-1000 Ljubljana
e-naslov: bojan.stopar@fgg.uni-lj.si
izr. prof. dr. Tomaž Ambrozič, univ. dipl. inž. geod., 
univ.dipl. inž. rud.
Univerza v Ljubljani, Fakulteta za gradbeništvo in geodezijo
Jamova cesta 2, SI-1000 Ljubljana
e-naslov: tomaz.ambrozic@fgg.uni-lj.si
Hamza V ., Stopar B., Ambrožič T. (2020). Deformacijska analiza po postopku Caspary. Deformation Analysis: the Caspary Approach 
Geodetski vestnik, 64 (1), 68-88.  DOI: https://doi.org/10.15292/geodetski-vestnik.2020.01.68-88
Veton Hamza, Bojan Stopar, Tomaž Ambrožič | DEFORMACIJSKA ANALIZA PO POSTOPKU CASPARY | DEFORMATION ANAL YSIS: THE CASPARY APPROACH | 68-88 |