Measuring Inaccessible Points in Land Surveying and ...
38
Measuring Inaccessible Points in Land Surveying and Analysis of their Uncertainty Yue Zhuo June 2012 Supervisor: Stig-Göran Mårtensson Examiner: Mohammad Bagherbandi Examensarbete, kandidatnivå, 15 hp Geomatik Degree Project for a Bachelor of Science/Technology in Geomatics
Transcript of Measuring Inaccessible Points in Land Surveying and ...
Measuring Inaccessible Points in Land Surveying and
Analysis of their Uncertainty
Yue Zhuo
June 2012
Supervisor: Stig-Göran Mårtensson
Examiner: Mohammad Bagherbandi
Examensarbete, kandidatnivå, 15 hp Geomatik
Degree Project for a Bachelor of Science/Technology in Geomatics
I
Preface This thesis work is a summary and finalization of my bachelor´s degree study at the
University of Gävle. I would like to say thanks to all the teachers who taught me during
my study these years; especially my supervisor, Stig-Göran Mårtensson, he guided me
patiently, helped me to correct my poor English, give beneficial suggestion and
constructive comments on my report; I would like to express my appreciation to him for
all his help at my thesis work. Furthermore, I am grateful to my friends, for assisting on
my land surveying work. Finally, thanks to my family, for their unselfish supports and
infinite love.
June 2012, Gävle
Yue Zhuo
II
Abstract When surveying inaccessible points with a total station or a GNSS, some special indirect methods might be required. The objective of this report is to find suitable special indirect methods for specific surveying cases; furthermore, some recommendations for the methods are given. This report covers the remote elevation measurement method (REM), the double survey station method, the double-side survey method, the two prism method and the bearing and distance method. These five indirect methods are carried out either with a total station or with a GNSS. The theory behind each method is given and their measurement uncertainty is analyzed from a numerical and a practical point of view. The findings are: the REM method should be preferred for remote objects, the two prisms method or the bearing and distance method for close objects where the ratio between the two participating distances should not exceed 2, a derived and simplified formula for finding uncertainties of bearing calculations is recommended.
Key words: total station, GNSS, inaccessible points, land surveying, uncertainty.
III
Table of Contents Preface ........................................................................................................... I Abstract ........................................................................................................ II Table of Contents ...................................................................................... III 1. Introduction ........................................................................................... 1
1.1 Previous studies ................................................................................................ 1 2. Methods .................................................................................................. 2
2.1 The REM method .............................................................................................. 2 2.2 The double survey station method ................................................................ 4 2.3 The double-side survey method ..................................................................... 6 2.4 The two prisms method................................................................................... 8
2.4.1 Plane coordinates ..................................................................................................... 8 2.4.2 Height coordinates ................................................................................................. 10
2.5 Bearing and distance ..................................................................................... 12 2.6 Statistics ........................................................................................................... 13 2.7 Materials ........................................................................................................... 13
3. Results .................................................................................................. 14 3.1 Numerical study .............................................................................................. 14
3.3.1 Analysis of the measurement uncertainty of the REM method ....................... 15 3.3.2 Analysis of the measurement uncertainty of the double survey station method ............................................................................................................................... 15 3.3.3 Analysis of the measurement uncertainty of the double-side survey method ............................................................................................................................................ 16 3.3.4 Analysis of the bearing uncertainty ..................................................................... 17 3.3.5 Analysis of the measurement uncertainty of the two prisms method ............ 17 3.3.6 Analysis of the measurement uncertainty of bearing and distance ................ 19
3.4 Practical study ................................................................................................. 20 3.4.1 Survey with the REM method ............................................................................... 20 3.4.2 Survey with the double survey station method ................................................. 20 3.4.3 Survey with the double-side survey method ...................................................... 21 3.4.4 Survey with the two prisms method .................................................................... 22 3.4.5 Survey with bearing and distance method ......................................................... 23 3.4.5 Numerical study vs. practical study ..................................................................... 24
4. Discussion ............................................................................................. 25 4.1 Numerical study .............................................................................................. 25 4.2 Practical study ................................................................................................. 26
5. Conclusions .......................................................................................... 27 6. References ............................................................................................ 28 Appendix ..................................................................................................... 29
1
1. Introduction With the development of science and technology, total stations and GNSSs’ used for land surveying work become more and more intelligent. Total stations and GNSSs’ can provide a lot of surveying methods, such as: the remote elevation measurement (REM), offset measurement, triangulation method, area computation, bearing and distance method and so on (Kavanagh, 2003 a); methods that apply to different surveying situations. Using these methods flexibly may increase the efficiency of surveying work. Choosing a right and suitable method according to the surveying case is of critical importance for a surveyor (Feng et al., 2001; Hu et al., 2005). Particularly, one such case is when it is difficult to place a prism when needed on the point to be measured, like when measuring a high-tension cable or a pipeline (or some large or high buildings); such points are termed inaccessible points. To solve this kind of problem, some indirect methods by total stations or GNSSs’ are needed. The aim of this thesis project is to evaluate some suitable indirect special methods that can be used by total stations or GNSSs’ to measures inaccessible points, and to numerically and practically analyze the measurement uncertainty of each method. The basic idea was to particularly focus on methods for finding heights, but since one such method also deliver plane coordinates, an additional method based on the same theory but mainly used for finding plane coordinates is also included. The REM method, together with four other indirect methods, are such methods introduced in this thesis project. The methods are (apart from REM): 1) the double survey station method, 2) the double-side survey method, 3) the two prisms method and 4) the bearing and distance method. The first four methods are carried out with a total station and the last one with a GNSS. The REM method and 1) and 2) are used for finding heights above ground of unknown points, method 3) is used for finding both heights and plane coordinates of unknown points, and 4) for finding plane coordinates only. The theory of these methods is presented and the theoretical uncertainty of each method is derived and numerically analyzed. Then the performance by experiments, including usability and measurement uncertainty, is evaluated. Furthermore, some recommendations for the methods are given. With the help of these indirect methods, heights above ground and plane coordinates of inaccessible points may be measured with less risk and less effort, which will eventually result in time-saving and efficiency-enhancing.
1.1 Previous studies
Since all methods and most theory and formulas are well known, they were known by the author, or have been collected from recognized textbooks like Kahmen & Faig (1988) available at most university libraries. One adequate paper concerning uncertainty of inaccessible points is Cederholm & Jensen (2009). They discussed the uncertainty of the inaccessible point found by the bearing and distance method. The uncertainty formulas are in their presentation divided into two components; one along the line formed according to the method, and one across. Several studies that were found concerning finding inaccessible points were referring to the usage of reflectorless total stations, e.g. Aiquan (2008), but since such methods are not applicable in this thesis, they are not mentioned.
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2. Methods
2.1 The REM method
The REM method is used to find the height above ground where a prism cannot be placed directly on the target (Wei and Cheng, 2006). Surveyors can determine the ground of inaccessible points (like tall buildings, bridges, etc.) with this method (Kavanagh, 2003 b). The principle of the REM method is simple and the observation process is convenient. With this method, a prism is positioned under the unknown points, the height above ground of the prism is measured with a tape measure; heights above ground of the remote targets are easily measured with a total station (Duggal, 2004). The principle of the REM method is shown in Figure 1, where a prism B is positioned vertically below an unknown point A, the height h above ground of the prism is measured and the slope distances to the prism from a remote position like F is determined by a total station D together with the two zenith distances z1 and z2.
Figure 1: A principle sketch of the REM method, where S is the slope distance to point B. The zenith distances z1 and z2 are measured by a total station at D.
With the reference to Figure 1, the height H above ground is found by trigonometry:
hH AB ; where 212 coscotsinAB zSzzS . Thus, the height H above ground of the inaccessible unknown point A is:
hzSzzSH 212 coscotsin (1) The standard uncertainty of the REM method of the height H above ground is found according to JCGM 100 (2008) by the formula for combined standard uncertainty:
ii xucHu 222 (2)
This is the law of propagation of uncertainties (Zhang and wang, 2007), where ic is the
sensitivity coefficient of the input estimate ix . ic is defined as:i
i x
f
c .
3
The sensitivity coefficients of Formula 1 will then become:
2121 coscotsin zzzS
Hc
12
2
12 sin
sin
z
zS
z
Hc
2122
3 sincotcos zSzzSz
Hc
14
h
Hc
Applying the combined standard uncertainty Formula 2, the standard uncertainty of the height H of the unknown point A is:
huzuzzzS
zuz
zSSuzzzHu
22
22212
2
12
14
222
22212
2
sincotcos
sin
sincoscotsin
(3)
Formula 3 can be simplified introducing the following assumptions:
- At “normal” surveying work the zenith distances are > 50 gon, thus we do not have to “fear” any exaggerated contributions from z1
- 22 1 2(sin cot cos ) 1z z z
- 4sin
sin
14
22
z
z
- 4)sincos(cot 2221 zzz
- vuzuzu 21
- duhuSu
Then:
vuSduHu 2222 82 (4)
It is important to note the zenith distance z1 in the first three terms of Formula 3; the smaller z1 is, the worse will the standard uncertainty of the height be. The theory of the REM method is simple, but the observation process is a little bit more complex since when positioning the prism under the unknown point, the prism should be placed very close to the plumb line of the unknown point.
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2.2 The double survey station method
If the prism cannot be placed directly under the unknown point, contrary to the REM method, then the double survey station method may be used to measure the height above ground of an inaccessible point. In this method, a prism can be placed near the inaccessible point, but not necessarily below it. The principle of the double survey station method is shown in Figure 2. The heights H1 and H2 above ground are measured with the REM method. Like in Figure 2, where a prism is positioned at point F and a total station is positioned at point O1, the zenith distance z1 to the unknown point A is measured with a total station at O1. The total station can be moved to point O2 where point O2 is closer to the unknown point A, the zenith distance z2 to the unknown point A is measured by a total station at point O2. H3 is the height difference between point E and F can be measured with a levelling instrument. The height difference H3 can be measured with an application of a total station as well (Lee and Rho, 2001). Finally calculate the height H above ground by trigonometry.
Figure 2: A principle sketch of the double survey station method, where the zenith distances z1 and z2 are measured by a total station at point O1 and O2 respectively; H1 and H2 are the heights above ground of point C and D respectively.
Notice that, the value (H1 – H2) can either be positive or negative; when negative, the unknown point A is lower than the prism at F, then point C is also lower than point D and thus the value (H1 – H2) is negative. With the reference to Figure 2, the height H above ground is found: 31 BC HHH ;
where )(sin
sincosBC
21
2121 zz
zzHH
.
5
Thus, the height above ground H of the inaccessible unknown point A is:
321
21211 )sin(
sincosH
zz
zzHHHH
(5)
The sensitivity coefficients of Formula 5 are:
212
2221
11 -sin
sincos
zz
zzHH
z
Hc
212
1121
22 -sin
sincos
zz
zzHH
z
Hc
)sin(
sincos1
21
21
13 zz
zz
H
Hc
)sin(
sincos
21
21
24 zz
zz
H
Hc
13
5
H
Hc
The standard uncertainty of the height H above ground of the inaccessible unknown point A is found by the combined standard uncertainty Formula 2:
32
22
212
22
12
12
2
21
21
22
12
12
12
22
22
214
2212
)(sin
sincos)(
)sin(
sincos1
)(sincos)(sincos)(sin
HuHuzz
zzHu
zz
zz
zuzzzuzzzz
HHHu
(6)
The standard uncertainty formula gives a clear indication of the unfavorable situation which might occur in cases when the difference between the two measured zenith distances is small. A disadvantage of this method is that it is needed to place a total station at two positions, the process then becomes a little bit more complex than for the REM method. An advantage when using the method is that a prism can be positioned at any place, not necessarily under the plumb line of the unknown points; the method could be widely used in land surveying work.
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2.3 The double-side survey method
If the unknown point is above a water area or some other inaccessible area, where the prism cannot be placed directly under the unknown point, but on stable grounds in front of, the double-side survey method can be used to solve the problem. With this method the REM method is not used any more, like it was in the double survey station method; thus the double-side survey method is an independent survey method. The principle of this method is shown in Figure 3, where a total station is placed at point M and its height h1 above ground is measured with a tape measure; observe the unknown point A by measuring the zenith distance z1. A prism D exchanges the total station at point M; the prism's height is adjusted so it is equal to h1. The total station is moved to point K which is closer to the inaccessible area, observe the unknown point A again by measuring the zenith distance z2. Then, place another prism N at point J. The zenith distances z3 and z4 are measured by the total station at G.
Figure 3: A principle sketch of the double-side survey method, where the zenith distance z1 is measured by the total station D; SDG and SGN are slope distances measured by the total station at G as are the zenith distances z2, z3 and z4. h2 is the height above ground of prism N and h1 is the height above ground of prism D (h1
is also the height above ground of the total station at D). With the reference to Figure 3, the height H above ground is found:
2EPBEAB hHHHH ; where )sin(
)sin(cos
21
421DGAB zz
zzzSH
, 4DGBE cos zSH and
3GNEP cos zSH .
Thus, the height H above ground of the inaccessible unknown point A is:
23GN4DG21
421DG coscos)sin(
)sin(coshzSzS
zz
zzzSH
(7)
7
This height is the distance from the unknown point A to the e.g. water plane. The sensitivity coefficients of Formula 7 are:
)(sin
cos)cos()sin(sin)sin(
212
12121142DG
11 zz
zzzzzzzzS
z
Hc
212
411DG
22 sin
)sin(cos
zz
zzzS
z
Hc
3GN3
3 sin zSz
Hc
421
421DG
44 sin
sin
coscosz
zz
zzzS
z
Hc
4
21
421
DG5 cos
sin
sincosz
zz
zzz
S
Hc
3GN
6 coszS
Hc
12
7
h
Hc
Applying the combined standard uncertainty Formula 2 to calculate the standard uncertainty of the double-side survey method. The standard uncertainty of the height H above ground of the inaccessible unknown point A is:
22
GN2
32
DG2
212
422
12
42
2
421
4212DG
32
322
GN22
214
422
122
DG
12
214
212121142
22DG2
cos)(sin
)(sincos
sin)sin(
)cos(cos
sin)(sin
)(sincos
)(sin
cos)cos()sin(sin)(sin
huSuzSuzz
zzz
zuzzz
zzzS
zuzSzuzz
zzzS
zuzz
zzzzzzzzSHu
(8)
The double-side method can be used to find heights of objects that are above e.g. water areas; compared with the REM method and the double survey station method, this method is more flexible when choosing the position of the prisms. The method can be used for land measurements as well, not just water areas. The disadvantage of this method is that the calculation process is a little bit more complex.
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2.4 The two prisms method
The two prisms method is a special surveying method for finding coordinates of inaccessible points. This method is carried out with a total station. When using the method, two prisms on a straight pole are used. The principle of the two prisms method is shown in Figure 4, where point M is a total station. Points B and C are the two observed prisms near the unknown inaccessible point A.
Figure 4: A principle sketch of the two prisms method, where points A, B and C are on a straight pole; point A is the inaccessible point.
2.4.1 Plane coordinates
The plane coordinates of the inaccessible point A, provided the distance dAB is horizontal, is calculated by the polar method (Kahmen & Faig, 1988), here x-direction has been chosen for northing and y-direction for easting:
A B AB BA
A B AB BA
cos
sin
x x d
y y d
(9)
As point A, B and C are located on a straight line, where C and B have been observed, the bearing BAφ , which is equal to CBφ , can be found by:
B CCB BA
B C
arctany y
φ φx x
Derived, by using Formula 2 applied on Formula 9, the standard uncertainty formula of a polar measurement is found to be:
BA22
ABAB2
B2
A2 uddururu (10)
Where )( Aru and )( Bru are the radial uncertainties of points A and B respectively, ( )u d and ( )u φ are the standard uncertainties of the distance and the bearing in use.
M
A
B
C dBC
dAB
SB
SC
9
Points C and B are measured by a total station at M. With the distance d0 and the bearing 0φ from M to B, the radial standard uncertainty of point B is found by:
0
2200
22B
2 M udduuru (11)
Where Mu is the combined positional and directional uncertainty of the total station at point M. Inserting the above formula in Formula 9, the radial standard uncertainty of the inaccessible unknown point A will be:
BA22
ABAB2
022
0022
A2 M udduudduuru
Where the standard uncertainty of the bearing, defined according to Formula 2, is:
2
2BC
2BC
BCC
2
2
2BC
2BC
BCB
2
2
2BC
2BC
BCC
2
2
2BC
2BC
BCB
2BA
2
yyxx
xxyu
yyxx
xxyu
yyxx
yyxu
yyxx
yyxuu
C2
B2
2BC
CB2
C2
B2
2BC
CB2 cossin
yuyud
xuxud
(12)
Thus, the radial standard uncertainty of the inaccessible point A is:
CBCB
A
yuyud
dyuyu
d
d
duudduuru
22
2
BC
ABCB
222
2
BC
ABCB
2
AB2
022
00222
coscos
M
(13)
The bearing uncertainty Formula 12 and the radial uncertainty Formula 13 can both be simplified and generalized by introducing the following assumptions:
- dududu AB0
- xuxuxu CB
- yuyuyu CB
- 2 2 2( ) ( ) ( )u x u y u r
- CB2sin and CB
2cos are both 1
- the radial uncertainty of Formula 13 is relative to the total station M
Then the simplified bearing uncertainty will be:
2
BCBA
2 2
d
ruu (14)
10
And the simplified radial standard uncertainty of the inaccessible point A is:
rud
dudduru 2
2
BC
AB0
220
2A
2 22
(15)
It is important to note the dependence of the ratio between dAB and dBC in Formula 15 for the radial uncertainty of the unknown point A; the longer dAB is with respect to dBC, the worse is the radial uncertainty.
2.4.2 Height coordinate
The two prisms method could be used to find vertical positions as well. Measure the relative heights with respect to the total station of point B and C to find the height of the inaccessible unknown point A. The heights of the two prisms (HB and HC), with respect to the height of the total station, are obtained by the formula:
CCC
BBB
cos
cos
zSH
zSH
Then the height H of the unknown point A is:
CCBC
AB
BC
ABBBC
BC
AB
BC
ABB cos1cos1 zS
d
d
d
dzSH
d
d
d
dHH
(16)
Where SB and SC are slope distances between the total station and point B and C respectively, and zB and zC are the zenith distances. The sensitivity coefficients of Formula 16 are:
BC
ABB
B1 1cos
d
dz
S
Hc
BC
ABBB
B2 1sin
d
dzS
z
Hc
BC
ABC
C3 cos
d
dz
S
Hc
BC
ABCC
C4 sin
d
dzS
z
Hc
2BC
ABCCBB
BC5 coscos
d
dzSzS
d
Hc
BC
CCBBAB
6
1coscos
dzSzS
d
Hc
11
The relative standard uncertainty of the height H of the unknown point A with respect to the total station is calculated according to the combined standard uncertainty Formula 2:
AB2
2BC
2CCBB
BC2
4BC
2AB2
CCBB
C2
2BC
2AB
C22
CC2
2BC
2AB
C2
B2
2
BC
ABB
22BB
2
2
BC
ABB
22
1coscos
coscos
sincos
1sin1cos
dud
zSzS
dud
dzSzS
zud
dzSSu
d
dz
zud
dzSSu
d
dzHu
(17)
The standard uncertainty Formula 17 can be simplified introducing the following assumptions:
- 2Bsin z , 2
Csin z , 2Bcos z and 2
Ccos z are all 1
- )()()()()( ABBCCB dududuSuSu
- vuzuzu CB
- SSS CB
Then:
vuSdud
dHu 222
2
BC
AB2 12
(18)
The ratio between dAB and dBC occurs here as well as in Formula 15, indicating that the uncertainty of the height H will be affected such that the longer dAB is with respect to dBC, the worse is the uncertainty of the height.
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2.5 Bearing and distance
The method bearing and distance, here described as an outdoor method, is used to measure plane coordinates of inaccessible points like building corners or other hidden points. The method is preferably carried out by a GNSS because it is far easier to operate and to handle than a total station. The principle of bearing and distance is shown in Figure 5, where point A is an unknown point in an area which does not allow observations by a GNSS, for instance it cannot be physically reached, or it is situated such that satellite signals are obstructed from reaching the GNSS. Aim at point A to establish a straight line on the ground, measure the coordinates of point B and C by GNSS on the line. The process to find the coordinates of A is now equal to the process of finding them with the two prisms method. The distances AB, or both distance AB and BC, can be measured as horizontal on ground with a tape measure.
Figure 5: A principle sketch of the bearing and distance method where the coordinates of C and B are obtained with a GNSS.
The mathematical process to find coordinates as well as the analysis of uncertainty is identical to the two prisms method, as well as the notations used in Figure 5. The major difference is the uncertainties of the observables; they are anticipated to be larger with this method. To find the coordinates, Formula 9 can be used, and to find the measurement uncertainty Formula 10 integrated by the simplified bearing formula (Formula 14):
rud
dduru
d
ddururu 2
2
BC
ABAB
22
2
BC
ABAB
22A
2 212
(19)
Where ABdu is the standard uncertainty of the distance AB along the line to the
inaccessible point, ru is the radial standard uncertainty of either of the two measured points B and C.
dBC
dAB
13
2.6 Statistics
To find the standard uncertainties of the experiments done during the field tests, the formula recommended by JCGM 100 (2008) is used:
11
2
n
xxxu
n
ii
(20) Where xi is an individually measured value, x is the mean value and n is the number of
measurements. The mean value is defined as n
xx
n
ii
1 .
2.7 Materials
Data for finding the numerical measurement uncertainty of the presented methods is collected from a Swedish regulation termed SIS-TS 21143:2009. In the regulation total stations and GNSSs’ have been classified according to anticipated standard uncertainties, standard uncertainties that will be used during calculations (see Tables 1 and 2).
Table 1: Classification of total stations according to SIS-TS 21143:2009.
Class Utilities Standard uncertainty
in direction u dir (one set)
Standard uncertainty in
distance u d
T1
For industrial application and surveillance. Control measurements of structures with particularly high demands.
0,8 mgon 1 mm + 2 ppm
T2
Geodetic survey for road and railway projects. Detail and control measurement of railways, bridges and tunnel constructions.
2,4 mgon 3 mm + 3 ppm
T3
Geodetic survey in general. Detailed measurement of roads and other engineering structures. Control measurements of other facilities and structures. Detailed measurements within physically planned areas.
4 mgon 3 mm + 3 ppm
T4 Other detail measurements. 8 mgon 5 mm + 5 ppm
2 ppm indicates an additional 2 mm/km on the measured distance
14
Table 2: Classification of GNSSs’ according to SIS-TS 21143:2009.
Class Method Utilities Standard
uncertainty
(plane) u r
Standard uncertainty
(height) u H
St 1 Static
measurement 2nd order networks.
5 mm + 1 ppm Two-frequency
receivers -
St 2 Static
measurement
3rd order networks for infrastructure. Photogrammetric tie points.
5–10 mm + 2ppm One-frequency
receivers -
RTK1 RTK-
measurement Detail measurements in plane and height.
10 mm + 1 ppm 20 mm+ 1ppm
1 ppm concerning RTK is an additional 1 mm/km based on the baseline length receiver to rover Instruments used for the practical study was a total station Lecia Viva TS 15I (belongs to class T2 according to Table 1) and a GNSS Lecia GS 15 (belongs to any class in Table 2, but here used as belonging to RTK1) with an antenna AS 15.
3. Results
3.1 Numerical study
Theoretical measurement uncertainties are calculated for each method using one and each of the total stations classified in Table 1 and GNSS according to RTK-measurements in Table 2. Full uncertainty formulas are used and where appropriate also the simplified formulas for comparison. Where total stations have been used, two distances to the inaccessible point have been considered; one short to be compared with the experimental work presented in section 3.2, and one long to resemble what is believed to be the longest practiced. The standard uncertainties in direction diru in Table 1 are valid for one full set of directions (two-face observation), but surveys in this presentation are all detail measurements (one-face observation), i.e. they are assumed to be done with half sets
only, thus the standard uncertainty used is 2 u dir .
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3.3.1 Analysis of the measurement uncertainty of the REM method
The full uncertainty formula (Formula 3) of the REM method is simplified with the following assumptions and supposed values:
- duhuSu
- vuzuzu 21
- the zenith distances are z1 = 80 gon and z2 = 90 gon
du and vu are the standard uncertainties of the used total station in distance and
direction, respectively. Then:
vuSduHu 2222 2709,20271,1
Suppose the shortest distance between the total station and the inaccessible point S is 20 m and the longest distance S is 200 m, then the standard uncertainty results of the formula and the simplified formula will be as in Table 3.
Table 3: Measurement uncertainty of the REM method by class of total station and distances.
Class
Measurement uncertainty Hu [mm]
Full formula Simplified formula
S = 20 m S = 200 m S = 20 m S = 200 m
T1 1,1 5,5 1,7 10
T2 3,4 16 5,2 30
T3 4,1 27 6,6 50
T4 7,4 54 12 100
3.3.2 Analysis of the measurement uncertainty of the double survey station
method
The measurement uncertainty of the double survey station method (Formula 6) is dependent on the heights H1 and H2 and the two zenith distances. H1 and H2 are obtained by the REM method, as well as their uncertainties. The full formula can be simplified with the following assumptions and supposed values:
- duHuHuHu 321
- vuzuzu 21
- the zenith distances are z1 = 80 gon and z2 = 70 gon
du and vu are the standard uncertainties of the used total station in distance and
direction respectively. Then:
vuHduHu 2222 4178579,6
16
Suppose the shortest distance between the closest total station and the inaccessible point first is 20 m and then 200 m, then the results will be as in Table 4.
Table 4: Measurement uncertainty of the double survey station method by class of total station and distances.
Class Standard uncertainty Hu [mm]
S = 20 m S = 200 m
T1 3,2 18
T2 9,6 55
T3 12 91
T4 22 180
3.3.3 Analysis of the measurement uncertainty of the double-side survey
method
The measurement uncertainty of the double-side survey method (Formula 8) is in particular dependent on two measured slope distances and two zenith distances. The full formula can be simplified with the following assumptions and supposed values:
- SSS GNDG
- duSuSu GNDG
- vuzuzuzuzu 4321
- the zenith distances are z1 = 80 gon, z2 = 70 gon, z3 = 110 gon and z4 = 80 gon
du and vu are the standard uncertainties of the used total station in distance and
direction, respectively. Then:
vuSduHu 2222 6812356,5
Suppose the shortest distance between the closest total station and the inaccessible point first is 20 m and then 200 m, then the standard uncertainty results will be as in Table 5.
Table 5: Measurement uncertainty of the double-side survey method
by class of total station and distances.
Class Standard uncertainty Hu [mm]
S = 20 m S = 200 m
T1 9,6 46
T2 29 140
T3 47 230
T4 93 460
17
3.3.4 Analysis of the bearing uncertainty
Simplify the full bearing uncertainty formula (Formula 12) and the simplified formula (Formula 14) with the following assumptions:
- xuxuxu CB = 5 mm
- yuyuyu CB = 5 mm
Then the full formula will be:
2BC
2BC
CB2
CB2
2 00005,0cossin00005,0
ddu
And the simplified:
2BC
2BC
222 0001,0
2dd
yuxuu
If the distance dBC is varying from 0,5 to 100,0 m, then the uncertainty results of the full formula and the simplified formula will be as in Table 6.
Table 6: Numerical measurement uncertainties of bearing calculation dependence of distance.
dBC [m] 0,5 1,0 5,0 10,0 50,0 100,0
Bearing uncertainty u [gon]
Full formula
0,90 0,45 0,090 0,045 0,0090 0,0045
Simplified formula
1,3 0,64 0,13 0,064 0,013 0,0064
3.3.5 Analysis of the measurement uncertainty of the two prisms method
By simplifying the full uncertainty formula (Formula 13) as was the case when deriving Formula 15, and additionally assume that:
- 2u d u x u y u r u d
then the full formula will be:
2
2 2 2 2ABA 0 0
BC
2 1d
u r u d d ud
(21)
du and u are the standard uncertainties of the used total station in distance and direction, respectively.
18
For the simplified formula it is assumed that the ratio BC
AB
d
d is 0,5. Then the simplified
formula will be:
022
02
A2 5,2 udduru
Suppose the shortest distance between the closest total station and the inaccessible point first is 5 m and then 20 m, then the standard uncertainty results of the full formula and the simplified formula will be as in Table 7.
Table 7: Measurement uncertainties of the plane coordinate of the two prisms method by class of total station.
Class Standard uncertainty Aru [mm]
d0 = 5 m d0 = 20 m
T1 1,6 1,6
T2 4,8 4,9
T3 4,8 5,1
T4 8,0 8,7
The measurement uncertainty of the height coordinates of the two prisms method (Formula 17 and Formula 18) can simplified with the following assumptions and supposed values:
- SSS CB
- dududuSuSu ABBCCB
- vuzuzu CB
- The zenith distances are zB = 80 gon, zC = 70 gon, - The distances are dBC = 1,0 m and dAB = 0,5 m
du and vu are the standard uncertainties of the used total station in distance and
direction, respectively. Then:
vuSduHu 2222 3626,22716,1
And the simplified:
vuSduHu 2222 66
Suppose the shortest distance between the closet total station and the inaccessible point first is 5 m and then 20 m, then the standard uncertainty results of the full and the simplified formula will be as in Table 8.
19
Table 8: Measurement uncertainty of the height coordinate of the two prisms method by class of total station and distances.
Class
Standard uncertainty Hu [mm]
Full formula Simplified formula
S = 5 m S = 20 m S = 5 m S = 20 m
T1 1,1 1,3 2,5 2,6
T2 3,4 3,8 7,4 7,8
T3 3,5 4,3 7,4 8,5
T4 5,8 7,8 12 15
3.3.6 Analysis of the measurement uncertainty of the bearing and distance method
Formula 19 has been used for finding the measurement uncertainty of this method. Since it is highly sensitive to the ratio between dAB and dBC, five such ratios believing to be reasonable to use in practice, have been used. The radial standard uncertainty ru has been chosen from Table 2, a value most likely valid for single baseline RTK since it is associated with a distance parameter. It is, however, also valid for network RTK in Sweden based on a recent publication by Mårtensson et al. (2012). 2du , the standard uncertainty of the closest distance along the line to the inaccessible
point, has for simplicity been set to 10 mm. With the above data and assumptions, the measurement uncertainty of the inaccessible point will be as in Figure 6.
Figure 6: Increase of measurement uncertainty of the bearing and distance method with the increase of the ratio between dAB and dBC.
20
3.4 Practical study
To test the five special surveying methods for finding inaccessible points, convenient test sites were chosen close to, or indoor, the geodetic laboratory at the campus of the university. For the first three methods described in section 2, an “inaccessible” point was marked on top of a high port (Figure 7). An instrument set-up resembling Figures 1, and 3 was chosen, an set-up which was shifted ten times to obtain sufficient numbers of observations in order to achieve reliable measurement uncertainties of the inaccessible point. All the observations and results of each method are listed in tables in appendix.
3.4.1 Survey with the REM method The REM method set-up is shown in Figure 7.
Figure 7: The test set-up with the REM method. Notations used are the same as in Figure 1.
Observations and results are shown in Table 1 in the appendix. The mean value of the heights with this method is 3,401 m, which is regarded to be the most probable value of the correct height of the inaccessible point. The standard uncertainty according to Formula 20 is 2,6 mm.
3.4.2 Survey with the double survey station method
The close set-up of the double survey station method is shown in Figure 8. H3 in Figure 2 is zero here. Heights H1 and H2 are obtained by the REM method.
D B
A
S h
21
Figure 8: The test set-up with the double survey station method where the total station is closest to the inaccessible point. Notations used are the same as in Figure 2.
Observations and results are shown in Table 2 in appendix. The mean value of the heights with this method is 3,404 m, which is regarded to be the most probable value of the correct height if the inaccessible point. The standard uncertainty according to Formula 20 is 5,2 mm.
3.4.3 Survey with the double-side survey method
The double-side survey method set-up is shown in Figure 9. Both positions of the total station (D and G) are shown; here the total station is at its closest position.
Figure 9: The test set-up with the double-side survey method. Notations used are the same as in Figure 3.
Observations and results are shown in Table 3 in appendix. The mean value of the heights with this method is 3,403 m, which is regarded to be the most probable value of the correct height if the inaccessible point by this method. The standard uncertainty according to Formula 20 is 6,9 mm.
A
F O2
H2
G D
A
N
SDG
SGN
h1 h2
22
3.4.4 Survey with the two prisms method
The two prisms method was tested indoor for two different set-ups of poles and prisms; one can be regarded as ordinary equipment – two ranging poles that can be attached to each other and two prisms meant for mounting on ranging poles. The second equipment was a specially made equipment for hidden-point observations – it was a set of basic series mini prisms supplemented by a set of hidden point poles (GMP111 + GMP112 from Leica, Figure 10). To keep the poles steady during observations, they were supported by a vertical pole raised from the floor and locked by hand to the measuring pole. The local coordinate system used during the tests is aligned such that the x-axis (northing) is pointing in the same direction as the measuring pole when held at right angle with respect to the observations made by the total station. The direction (bearing) of the measuring pole is thus 0 gon. Perpendicular to the x-axis, forming a left-handed system, is the y-axis (easting). With such an arrangement, uncertainties along and across the measuring pole are easily obtained.
Figure 10: A test set-up to illustrate the two prisms method, here the ratio between dAB and dBC is 1. Notations used are the same as in Figure 4.
For the ordinary equipment, the distance between the prisms was 1,0 m throughout the test, observations and results are in Tables 4, 5 and 6 in appendix. The distance between the total station and the inaccessible point was approximately 20 m.
Table 9: Results for the two prisms method with respect to the ratio between the distances dAB and dBC.
Ratio values
Mean values [m] Uncertainty [mm]
Height Coordinates Height Coordinates Radial
0,5 9,513 (11,653; 30,915) 1,9 (8,3; 5,0) 9,6
1,0 9,658 (11,660; 30,905) 14 (8,3; 5,8) 10
2,0 9,597 (11,669; 30,878) 13 (19; 44) 49
C
B
A
dBC
dAB
SB
SC
23
For the special equipment, the distance between the prisms was 0,70 m throughout the test, observations and results are in Tables 7, 8 and 9 in appendix. The distance between the total station and the inaccessible point was approximately 13 m.
Table 10: Results for the two prisms method with respect to the ratio between distances dAB and dBC.
Ratio values
Mean values [m] Uncertainty [mm]
Height Coordinates Height Coordinates Radial
0,5 9,724 (13,061; 22,489) 15 (1,4; 14) 14
1,0 9,720 (13,108; 22,483) 12 (1,9; 14) 14
1,857 9,689 (13,108; 22,487) 22 (12; 38) 40
3.4.5 Survey with the bearing and distance method
At these observations a GNSS with the antenna on a two metre pole was used, the pole was held by free-hand to resemble a realistic surveying situation. The test site was chosen outdoor on a flat ground (Figure 11). The observations and results are shown in Tables 10, 11 and 12 in appendix.
Figure 11: Observations for the bearing and distance method. The “inaccessible” point is A and the start-point C is marked by a wooden pole.
Table 11: Results for the bearing and distance method with respect to the ratio between the distances dAB and dBC.
Ratio values
Mean values [m] Uncertainty [mm]
Coordinates Radial
0,5 (1,604; 13,239) (15; 10) 18
1,0 (1,592; 13,248) (14; 13) 19
2,0 (1,582; 13,245) (18; 11) 21
C
B
A dBC
dAB
24
3.4.5 Numerical study vs. practical study
The numerical height measurement uncertainty results are compared with corresponding practical uncertainty results. The distance between the total station at its furthest position and the inaccessible point for numerical study in Table 12 is 20 m, but for the practical study it is 3 m.
Table 12: Comparison for a total station class T2 of height uncertainty between the numerical and the practical study for the first three methods.
Method Height uncertainty [mm]
Numerical study Practical study
The REM method 3,4 2,6
The double survey station method 9,6 5,2
The double-side survey method 29 6,9
For the two prisms method the numerical height and radial measurement uncertainty results are compared with corresponding practical uncertainty results. The distance between the total station and the measuring pole is 20 m for the numerical study, but for the practical study the distances are 20 m for the ordinary equipment and 13 m for the special equipment (Table 13). The bearing and distance method is in Table 13 only compared for radial uncertainties and three selected ratios.
Table 13: Comparison of height and radial uncertainty between the numerical and the practical study for the last two methods.
Methods Ratio values
Height uncertainty [mm] Radial uncertainty [mm]
Numerical study
Practical study Numerical
study
Practical study
Ordinary Special Ordinary Special
The two prisms method
(Class T2)
0,5 3,8 1,9 15 4,9 9,6 14
1,0 4,7 14 12 6,1 10 14
2,0 1,857 5,3 13 22 9,6 49 40
Bearing and distance
(Class RTK1)
0,5
-
16 18
1,0 20 19
2,0 32 21
25
4. Discussion
4.1 Numerical study
Measurement uncertainties are compared between different methods, particularly concerning heights, but also, where appropriate, concerning plane coordinates. Also, when appropriate, full formulas are compared with simplified formulas. It is found that the REM method and the two prisms method are comparable in height measurement uncertainties at short distances (<20 m). But it is believed that the two prisms method is more sensible to use at such distances. Both methods show larger uncertainties using the simplified formula compared to the full formula. This is, of course, expected, but the differences are not very large. For the REM method the simplified formula gives a value slightly smaller than two times the full-formula-value, whilst for the two prisms method it is slightly larger than two times. Comparing the first three methods, methods that are entirely meant for height observations, the uncertainty results differ considerable, from the REM method being the best, via the double survey station method, to the double-side survey method being the worse. In fact there is a factor of approximately three involved between each method; like for total station class T2 the uncertainty of the REM method is 3,4 mm (at 20 m) and 16 mm (at 200 m), for the double survey station method it is 9,6 mm and 55 mm at the same distances, and finally for the double-side survey method the comparable values are 29 mm and 140 mm. The REM method is highly sensitive to small zenith distances observed at the furthest position of the total station (Formula 3). But since “small” zenith distances hardly occur at ordinary surveying, we can rely on the small uncertainties numerically found. Both methods where the total station during observations is moved from one place to another are very sensitive to the difference in zenith distances obtained at each place. The values for the double-side survey method in Table 5 are obtained under the assumption that the difference is 10 gon, which is very much where we have a fairly flat ground. But even so, the measurement uncertainties are much larger than for the other methods, an indication of the sensitivity of small zenith distance differences. The last two methods, the two prisms method and the bearing and distance method, can be used for finding planar coordinates. They both have an identical calculation process, but the difference is that when using the two prisms method the involved distances on the measuring pole are not necessarily horizontal during observations, but have to be during calculations. The two prisms method shows smaller measurement uncertainty results for several reasons, one of which is that the distances involved (dAB and dBC) for finding the inaccessible point are generally smaller using the two prisms method compared to the bearing and distance method. Another reason is the equipment used; a total station gives a smaller measurement uncertainty than a GNSS. Finally, the uncertainty for both methods is dependent on the ratio between the distances dAB and dBC, something that is for the bearing and distance method clearly seen in Figure 6.
26
4.2 Practical study
To verify, or examine, the numerical measurement uncertainties, a practical study was performed. All five methods were experimentally tested, the first three with fairly short distances to the inaccessible points (3–20 m) due to limited space where control could be maintained. In reality those methods are anticipated to be used over distances ranging up to several hundred metres. Zenith distances were varying from 47 to 115 gon, also a range not very realistic. The last two methods were, however, tested under realistic conditions. Even though short distances were used, the internal difference in uncertainty between the REM method, the double survey station method and the double-side survey method is kept compared to the numerical study. The measurement uncertainty of the REM method is 2,6 mm, for the double survey station method it is 5,2 mm and for the double-side survey method it is 6,9 mm. Compared to the numerical study for a class T2 total station, the uncertainty of the REM method is very close to the corresponding numerical result (3,4 mm), whilst the other two show smaller values at the test compared to the numerical values (9,6 mm and 29 mm). The remarkable difference shown by the double-side survey method is hard to explain, but most likely it has to do with the very short distances used during the test since distances are influencing almost all terms in the uncertainty Formula 8. The fear of the sensitivity of small zenith distance differences discussed in the previous section is not revealed in the test; only one difference is smaller than 10 gon, the others much larger. When comparing heights and height uncertainties for the two prism methods, it is seen that the heights change from 9,513 m at ratio 0,5, via 9,658 m at ratio 1, to 9,597 m at ratio 2 for the ordinary equipment. The larger heights at ratios 1 and 2 are due to the support by an extra pole during observations held at prism B bending the unstable measuring pole set-up such that the direction of the prisms is pointing above the inaccessible point. This phenomena is not as clear for the special equipment where the heights do not differ too much at different ratios; 9,724 m (ratio 0,5), 9,720 m (ratio 1) and 9,689 m (ratio 1,9). Even if the measuring pole was thinner for the special equipment, it felt more stable at ratios 0,5 and 1, but as unstable as for the ordinary equipment at ratio 2. It is surprising to find that the ordinary equipment generally has a better result than the special when comparing uncertainties. The feeling at the experiment was that the instability of the ordinary equipment should influence the result to the worse compared to the special equipment, particularly at ratios 2 and 1. The spread among heights, 145 mm for the ordinary equipment and 35 mm for the special, is probably more relevant for a comparison; it gives an indication of an advantage to the special equipment even if the uncertainty is worse. The two prisms method was also tested for finding planar coordinates and their uncertainties. When comparing the ordinary and the special equipment used and different ratios for the two prisms method, it is found that the observations made by the ordinary equipment do not differ significantly nor for coordinates neither for uncertainties at ratios 0,5 and 1, but for ratio 2. In fact the same statement can be made for the special equipment, but with two exemptions; 1) the x-coordinate uncertainty (along the measuring pole) is very small compared to the y-coordinate uncertainty (across the measuring pole), 2) there is a systematic difference between the x-coordinates obtained with ratios 0,5 and 1. The difference is 47 mm and it is highly significant since the uncertainties at each observation are in the range 1–2 mm. A convincing explanation cannot be found to explain this. The radial uncertainty for the ordinary equipment and for the ratios 0,5 and 1 is approximately 10 mm, for the special
27
equipment a comparable figure is 14 mm. For both equipment and for the ratios 2 and 1,9 respectively, the uncertainties are much larger than for the other ratios, a result of the unstable behavior of the measuring poles when the part closest to the inaccessible point is two times the length of the distance between the prisms. The bearing and distance method is showing a minor increase in uncertainty as the ratio is increasing from 0,5 to 2; 18 mm, 19 mm and 21 mm respectively. In fact not too far from the values presented in Table 13 which are 16 mm, 20 mm and 32 mm. Comparing the x- and y-coordinate uncertainties, coordinates which now are in a national coordinate system, there is a small tendency that the uncertainty of the y-coordinate is smaller than the uncertainty of the x-coordinate. Since the test line is in the y-direction (east-west direction), this finding is in line with the uncertainty formulas in Cederholm & Jensen (2009).
5. Conclusions - The REM method should be favoured among the methods in this thesis report
that can be used for long distances, provided a prism can be placed under the object of interest. The reason is its numerical simplicity and that it is showing small uncertainties both in the numerical study as well as in the practical study.
- The double survey station method could be used if a prism cannot be placed under the object of interest.
- Do avoid the double-side survey method if possible. Mainly because it has a very unfavourable uncertainty propagation.
- The two prisms method should be chosen for short distances, preferably with ratios not larger than 1 between the distances dAB and dBC on the measuring pole.
- The simplified bearing formula can be used for almost all bearing uncertainty calculations.
- Avoid ratios between the distances dAB and dBC larger than 2 when using the bearing and distance method.
28
6. References Aiquan M., Xingchu M. & Jian Y., (2008). Application of non-Reflect Total Station in Cadastration, Urban Geotechnical Investigation & Surveying. doi: CNKI:SUN:CSKC.0.2008-04-039, vol.04, no.P271, pp123-137. Cederholm P. & Jensen K., (2009). GPS Measurement of Inaccessible Detail points. Surveying Review, 41. 352 - 363. DOI 10.1179/003962609X41591. Duggal S. K., (2004). Remote Elevation Measurement (REM). Surveying (second edition, pp.593). New Delhi, Tata McGraw-Hill Publishing Company Limited,. Feng Q., Sjogren P., Stephansson O. & Jing L., (2001). Measuring fracture orientation at exposed rock faces by using a non-reflector total station. Engineering Geology 59, 133-146. Hu H., Zheng C. & Yang Q., (2005). Discussion about how to use total station in special condition, Kunming Metallurgy News Papper, issue NO.1, pp.37 – 52. JCGM 100 (2008). Evaluation of measurement data – Guide to the expression of uncertainty in measurement. Joint Committee for Guides in Metrology (JCGM/WG 1). Kahmen H. & Faig W., (1988). Polar survey of object points. In Tutte GmbH, Hubler R., Luderitz R. & Bauer G. (Eds.). Surveying (pp 250 -255). Germany, Berlin: Walter de Gruyter & Co. Kavanagh B. F., (2003 a). Geomatics. In University of Michigan (Eds.), United States of America, USA: Upper Saddle River HJ: Prentice Hall. Kavanagh B. F., (2003 b). Remote object elevation calculation. In University of Michigan (Eds.), Geomatics (pp.553). United States of America, USA: Upper Saddle River HJ: Prentice Hall. Lee J. & Rho R., (2001). Application to leveling using total station. Denmark: published by the Interactional Federation of Surveyors (FIG). Mårtensson S-G., Reshetyuk Y. & Jivall L., (2012). Measurement uncertainty in network RTK GNSS-based positioning of a terrestrial laser scanner. Journal of Applied Geodesy. doi: 10.1515/jag-2011-0013. SIS-TS 21143:2009. Byggmätning – Geodetisk mätning, beräkning och redovisning vid långsträckta objekt. Stockholm: SIS Förlag AB. Wei Z. & Cheng M., (2006). Discussion about REM of Total Station, Engineering surveying, vol.37, pp.17 – 20. Zhang Y. & Wang B., (2007). A new method of triangular elevation by total station and accuracy estimation. Engineering of Surveying and Mapping. doi: CNKI:SUN:CHGC.0.2007-06-013, Kunming, China.
29
Appendix Table 1: Observations and results for the REM method. Notations used according to Formula 1, zenith distances in gon and linear distances in metre.
NO z1 z2 S h H
1 69,8710 77,8130 3,835 1,652 3,403
2 77,5616 85,0416 4,019 1,645 3,401
3 66,0617 81,1987 3,520 1,653 3,398
4 66,6425 77,6745 4,937 1,636 3,400
5 81,9369 89,0119 4,831 1,707 3,404
6 59,1121 67,8911 5,361 1,731 3,400
7 76,6868 87,0488 6,241 1,688 3,399
8 63,3824 72,8924 5,477 1,704 3,402
9 67,6962 78,9172 5,252 1,547 3,397
10 60,5973 69,7733 5,288 1,562 3,405
Table 2: Observations and results for the double survey station method. Notations used according to Formula 5, zenith distances in gon and heights in metre.
NO z1 z2 H1 H2 H
1 68,7812 64,4034 2,136 1,919 3,397
2 72,1565 67,0096 2,375 2,149 3,405
3 71,2736 67,1987 1,778 1,503 3,409
4 78,3954 74,3415 2,279 2,044 3,410
5 74,2678 66,5579 2,484 2,159 3,399
6 73,6120 66,3871 2,333 1,982 3,411
7 79,9819 66,2350 3,153 2,955 3,400
8 76,4667 66,6798 2,706 2,365 3,403
9 67,8360 64,1172 1,653 1,404 3,398
10 79,0160 65,3213 3,010 2,704 3,407
30
Table 3: Observations and the results obtained by the double-side survey method. Notations used according to Formula 7, zenith distances in gon and linear distances in metre.
NO z1 z2 z3 z4 SDG SGN h2 H
1 95,6555 68,9885 52,2892 110,4624 3,852 2,793 1,978 3,399
2 82,6051 65,5927 60,4802 110,6892 3,923 2,835 1,355 3,403
3 81,9099 62,0113 54,1406 105,2446 1,839 1,630 1,752 3,407
4 86,7322 55,4594 48,3799 103,3505 2,375 1,964 1,592 3,410
5 78,7910 54,9780 62,7353 114,8019 3,510 2,894 1,421 3,395
6 87,0198 65,6266 64,4607 111,5763 4,947 3,471 1,713 3,404
7 79,4777 63,3909 61,0760 107,1350 2,792 1,988 1,454 3,408
8 82,9944 75,2358 58,3660 106,0081 2,591 1,776 1,455 3,411
9 85,8723 60,1391 47,8058 112,0178 3,807 2,901 1,304 3,398
10 84,1656 55,3472 65,5967 114,9117 4,935 3,877 1,558 3,390
Table 4: Observations and results obtained with the ordinary equipment by the two prisms method (with ratio equal to 0,5). Notations used according to Formula 9 and Formula 16, zenith distances and bearing in gon and linear distances in metre.
NO SB SC B C zB zC HA (xA; yA)
1 20,880 20,791 96,4815 99,5059 101,4055 101,2597 9,513 (11,654; 30,905)
2 20,779 20,464 96,3790 99,2822 101,6083 101,8612 9,511 (11,661; 30,917)
3 20,713 20,262 96,2125 98,8426 101,8791 102,7093 9,514 (11,667; 30,916)
4 20,768 20,425 96,3453 99,1799 101,2091 100,6635 9,514 (11,660, 30,920)
5 20,902 20,853 96,4033 99,2463 100,9578 99,9061 9,512 (11,651; 30,909)
6 21,019 21,184 96,4718 99,3563 101,1419 100,4881 9,515 (11,644; 30,919)
7 21,060 21,316 96,4935 99,3842 101,4980 101,5398 9,514 (11,640; 30,915)
8 20,984 21,076 96,4879 99,4308 101,7564 102,3274 9,516 (11,646; 30,920)
9 20,890 20,807 96,4893 99,5180 101,4789 101,4645 9,511 (11,652; 30,914)
10 20,809 20,554 96,4171 99,3634 101,3412 101,0490 9,511 (11,657; 30,918)
31
Table 5: Observations and results obtained with the ordinary equipment by the two prisms method (with ratio equal to 1). Notations used according to Formula 9 and Formula 16, zenith distances and bearing in gon and linear distances in metre.
NO SB SC B C zB zC HA (xA; yA)
1 20,768 20,641 97,9616 101,0118 101,4395 101,3416 9,687 (11,685; 30,905)
2 20,604 20,304 97,7998 100,7620 101,7671 102,0082 9,657 (11,667; 30,910)
3 20,473 20,047 97,6085 100,4361 101,2385 100,9355 9,661 (11,676; 30,900)
4 20,645 20,399 97,7383 100,6084 100,6541 99,7450 9,657 (11,661; 30,897)
5 20,905 20,911 97,8882 100,8064 100,6130 99,7370 9,647 (11,651; 30,910)
6 21,032 21,172 98,0059 100,9939 101,3199 101,1500 9,633 (11,648; 30,906)
7 20,996 21,090 97,9772 100,9624 101,9584 102,3794 9,653 (11,653; 30,912)
8 20,810 20,723 97,8904 100,8663 102,2216 102,9240 9,664 (11,660; 30,908)
9 20,792 20,687 97,9733 101,0269 101,4540 101,3940 9,661 (11,658; 30,908)
10 20,755 20,625 97,6624 100,4341 100,1960 98,9183 9,658 (11,665; 30,896) Table 6: Observations and results obtained with the ordinary equipment by the two prisms method (with ratio equal to 2). Notations used according to Formula 9 and Formula 16, zenith distances and bearing in gon and linear distances in metre.
NO SB SC B C zB zC HA (xA; yA)
1 20,646 20,592 100,9912 104,0544 100,8966 100,8966 9,605 (11,660; 30,863)
2 20,379 20,204 100,8297 103,9038 102,0541 102,0541 9,609 (11,682; 30,832)
3 20,080 19,737 100,5227 103,5214 101,5826 101,5826 9,578 (11,691; 30,852)
4 19,985 19,596 100,3072 103,2214 100,6678 100,6678 9,608 (11,699; 30,847)
5 20,140 19,817 100,2517 103,0929 99,5308 99,5308 9,615 (11,688; 30,876)
6 20,743 20,748 100,9332 103,9421 100,3045 100,3045 9,603 (11,657; 30,842)
7 21,140 21,318 100,9956 103,9348 100,9665 100,9665 9,586 (11,642; 30,888)
8 21,030 21,122 101,0364 104,0328 101,8770 101,8770 9,597 (11,652; 30,951)
9 20,690 20,607 100,8894 103,8973 102,7936 102,7936 9,585 (11,655; 30,961)
10 20,654 20,602 101,0190 104,0940 101,6550 101,6550 9,585 (11,661; 30,863)
32
Table 7: Observations and results obtained with the special equipment by the two prisms method (with ratio equal to 0,5). Notations used according to Formula 9 and Formula 16, zenith distances and bearing in gon and linear distances in metre.
NO SB SC B C zB zC HA (xA; yA)
1 12,736 12,554 86,3161 89,6275 101,7796 102,5357 9,703 (13,064; 22,485)
2 12,687 12,404 86,1478 89,1925 102,0078 103,1251 9,729 (13,063; 22,466)
3 12,667 12,381 86,2249 89,4541 101,2286 100,9841 9,744 (13,061; 22,479)
4 12,710 12,487 86,2092 89,3458 100,6995 99,5331 9,748 (13,062; 22,498)
5 12,855 12,888 86,2500 89,3271 100,4655 98,9049 9,732 (13,060; 22,508)
6 12,899 13,001 86,4332 89,8134 101,1590 100,8199 9,713 (13,063; 22,509)
7 12,852 12,855 86,4092 89,7965 101,7695 102,4742 9,710 (13,060; 22,496)
8 12,782 12,671 86,3550 89,6953 101,8609 102,7227 9,718 (13,061; 22,482)
9 12,695 12,436 86,2797 89,5569 101,5416 101,8392 9,731 (13,060; 22,480)
10 12,670 12,362 86,2137 89,3646 101,0793 100,5430 9,717 (13,060; 22,485) Table 8: Observations and results obtained with the special equipment by the two prisms method (with ratio equal to 1). Notations used according to Formula 9 and Formula 16, zenith distances and bearing in gon and linear distances in metre.
NO SB SC B C zB zC HA (xA; yA)
1 12,905 12,995 87,9215 91,3274 101,2520 101,0755 9,712 (13,107; 22,486)
2 12,513 12,208 87,4941 90,6818 100,9076 100,3419 9,708 (13,107; 22,481)
3 12,528 12,254 87,4090 90,5119 100,3252 99,2494 9,727 (13,108; 22,466)
4 12,716 12,611 87,6286 90,8201 100,0324 98,6796 9,726 (13,104; 22,493)
5 12,864 12,909 87,8254 91,1557 100,4684 99,5795 9,725 (13,106; 22,493)
6 12,815 12,792 87,9379 91,4083 101,2762 101,2361 9,735 (13,109; 22,510)
7 12,726 12,632 87,8235 91,2453 102,0337 102,6336 9,709 (13,109; 22,491)
8 12,620 12,432 87,6658 90,9901 102,1394 102,9483 9,726 (13,110; 22,476)
9 12,791 12,782 87,8799 91,3322 101,9360 102,3743 9,699 (13,110; 22,469)
10 12,659 12,520 87,7112 91,0673 102,1782 103,0618 9,735 (13,109; 22,466)
33
Table 9: Observations and results obtained with the special equipment by the two prisms method (with ratio equal to 1,857). Notations used according to Formula 9 and Formula 16, zenith distances and bearing in gon and linear distances in metre.
NO SB SC B C zB zC HA (xA; yA)
1 12,756 12,763 90,9124 94,3892 101,2691 101,0590 9,667 (13,102; 22,478)
2 12,587 12,503 90,7813 94,2877 102,1737 102,4461 9,665 (13,112; 22,475)
3 12,229 12,236 90,1616 93,4833 102,4369 102,1232 9,662 (13,110; 22,467)
4 12,194 11,905 90,1155 93,4370 100,6784 100,1681 9,686 (13,115; 22,442)
5 12,338 12,097 90,1741 93,3760 99,3909 98,2335 9,713 (13,089; 22,511)
6 12,631 12,552 90,6244 93,9920 99,5979 98,6206 9,723 (13,104; 22,513)
7 12,802 12,843 90,8629 94,2816 100,5108 99,9934 9,704 (13,097; 22,460)
8 13,046 13,206 90,7942 94,0846 101,5734 101,6295 9,705 (13,098; 22,477)
9 13,015 13,119 90,6950 94,0078 102,6133 103,1956 9,695 (13,127; 22,548)
10 12,636 12,611 90,5333 93,9311 103,3973 104,3514 9,672 (13,122; 22,413)
Table 10: Results obtained by the bearing and distance method (ratio equal to 0,5).
NO (xB; yB) (xC; yC) (xA; yA)
1 (1,528; 8,737) (1,627; 8,847) (1,594; 13,237)
2 (1,541; 8,742) (1,648; 8,853) (1,604; 13,242)
3 (1,544; 8,754) (1,642; 8,842) (1,601; 13,254)
4 (1,537; 8,735) (1,633; 8,844) (1,604; 13,235)
5 (1,525; 8,748) (1,624; 8,841) (1,584; 13,248)
6 (1,532; 8,731) (1,635; 8,838) (1,595; 13,231)
7 (1,531; 8,755) (1,632; 8,855) (1,592; 13,255)
8 (1,546; 8,739) (1,657; 8,852) (1,608; 13,239)
9 (1,553; 8,730) (1,651; 8,840) (1,619; 13,230)
10 (1,568; 8,728) (1,662; 8,836) (1,635; 13,227)
34
Table 11: Results obtained by the bearing and distance method (ratio equal to 1,0).
NO (xB; yB) (xC; yC) (xA; yA)
1 (1,545; 8,732) (1,591; 8,783) (1,611; 13,232)
2 (1,537; 8,759) (1,584; 8,799) (1,592; 13,259)
3 (1,529; 8,761) (1,583; 8,816) (1,591; 13,261)
4 (1,541; 8,757) (1,595; 8,803) (1,596; 13,257)
5 (1,518; 8,768) (1,560; 8,815) (1,584; 13,268)
6 (1,532; 8,732) (1,587; 8,784) (1,592; 13,232)
7 (1,527; 8,733) (1,572; 8,787) (1,596; 13,232)
8 (1,523; 8,750) (1,585; 8,804) (1,579; 13,250)
9 (1,539; 8,744) (1,572; 8,792) (1,615; 13,243)
10 (1,514; 8,747) (1,575; 8,795) (1,566; 13,247)
Table 12: Results obtained by the bearing and distance method (ratio equal to 2,0).
NO (xB; yB) (xC; yC) (xA; yA)
1 (1,519; 8,746) (1,524; 8,757) (1,609; 13,245)
2 (1,537; 8,762) (1,548; 8,774) (1,602; 13,262)
3 (1,539; 8,733) (1,552; 8,747) (1,604; 13,233)
4 (1,498; 8,755) (1,503; 8,752) (1,573; 13,244)
5 (1,521; 8,755) (1,536; 8,763) (1,559; 13,255)
6 (1,517; 8,739) (1,525; 8,748) (1,583; 13,239)
7 (1,496; 8,740) (1,515; 8,759) (1,558; 13,240)
8 (1,535; 8,761) (1,547; 8,769) (1,581; 13,261)
9 (1,522; 8,748) (1,532; 8,755) (1,570; 13,248)
10 (1,531; 8,729) (1,540; 8,736) (1,582; 13,229)
