Measuring Inaccesible Points in Land Surveying

7/27/2019 Measuring Inaccesible Points in Land Surveying

1/38

Measuring Inaccessible Points in

Land Surveying andAnalysis of their Uncertainty

Yue Zhuo

June 2012

Supervisor: Stig-Gran Mrtensson

Examiner: Mohammad Bagherbandi

Examensarbete, kandidatniv, 15 hp

Geomatik

Degree Project for a Bachelor of Science/Technology in Geomatics


2/38

I

Preface

This thesis work is a summary and finalization of my bachelors degree study at the

University of Gvle. I would like to say thanks to all the teachers who taught me during

my study these years; especially my supervisor, Stig-Gran Mrtensson, he guided me

patiently, helped me to correct my poor English, give beneficial suggestion andconstructive 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, Gvle

Yue Zhuo


3/38

II

Abstract

When surveying inaccessible points with a total station or a GNSS, some specialindirect methods might be required. The objective of this report is to find suitablespecial indirect methods for specific surveying cases; furthermore, somerecommendations for the methods are given. This report covers the remote elevationmeasurement method (REM), the double survey station method, the double-side survey

method, the two prism method and the bearing and distance method. These five indirectmethods are carried out either with a total station or with a GNSS. The theory behindeach method is given and their measurement uncertainty is analyzed from a numericaland a practical point of view. The findings are: the REM method should be preferred forremote objects, the two prisms method or the bearing and distance method for closeobjects where the ratio between the two participating distances should not exceed 2, aderived and simplified formula for finding uncertainties of bearing calculations isrecommended.

Key words: total station, GNSS, inaccessible points, land surveying, uncertainty.


4/38

III

Table of ContentsPreface ........................................................................................................... IAbstract........................................................................................................ IITable of Contents ...................................................................................... III1. Introduction ........................................................................................... 1

1.1 Previous studies ................................................................................................ 12. Methods .................................................................................................. 22.1 The REM method .............................................................................................. 22.2 The double survey station method ................................................................ 42.3 The double-side survey method..................................................................... 62.4 The two prisms method................................................................................... 8

2.4.1 Plane coordinates ..................................................................................................... 82.4.2 Height coordinates ................................................................................................. 10

2.5 Bearing and distance ..................................................................................... 122.6 Statistics........................................................................................................... 132.7 Materials........................................................................................................... 13

3. Results .................................................................................................. 143.1 Numerical study .............................................................................................. 14

3.3.1 Analysis of the measurement uncertainty of the REM method ....................... 153.3.2 Analysis of the measurement uncertainty of the double survey stationmethod ............................................................................................................................... 153.3.3 Analysis of the measurement uncertainty of the double-side survey method............................................................................................................................................ 163.3.4 Analysis of the bearing uncertainty ..................................................................... 173.3.5 Analysis of the measurement uncertainty of the two prisms method ............ 173.3.6 Analysis of the measurement uncertainty of bearing and distance ................ 19

3.4 Practical study ................................................................................................. 203.4.1 Survey with the REM method ............................................................................... 203.4.2 Survey with the double survey station method ................................................. 203.4.3 Survey with the double-side survey method ...................................................... 213.4.4 Survey with the two prisms method .................................................................... 223.4.5 Survey with bearing and distance method ......................................................... 233.4.5 Numerical study vs. practical study ..................................................................... 24

4. Discussion............................................................................................. 254.1 Numerical study .............................................................................................. 254.2 Practical study ................................................................................................. 26

5. Conclusions .......................................................................................... 276. References............................................................................................ 28Appendix..................................................................................................... 29


5/38

1

1. IntroductionWith the development of science and technology, total stations and GNSSs used forland 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 distancemethod and so on (Kavanagh, 2003 a); methods that apply to different surveyingsituations. Using these methods flexibly may increase the efficiency of surveying work.Choosing a right and suitable method according to the surveying case is of criticalimportance 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 thepoint to be measured, like when measuring a high-tension cable or a pipeline (or somelarge 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 tonumerically and practically analyze the measurement uncertainty of each method.The basic idea was to particularly focus on methods for finding heights, but since onesuch method also deliver plane coordinates, an additional method based on the sametheory but mainly used for finding plane coordinates is also included.

The REM method, together with four other indirect methods, are such methodsintroduced in this thesis project. The methods are (apart from REM): 1) the doublesurvey station method, 2) the double-side survey method, 3) the two prisms method and4) the bearing and distance method. The first four methods are carried out with a totalstation 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 bothheights and plane coordinates of unknown points, and 4) for finding plane coordinatesonly. The theory of these methods is presented and the theoretical uncertainty of eachmethod is derived and numerically analyzed. Then the performance by experiments,including usability and measurement uncertainty, is evaluated. Furthermore, somerecommendations for the methods are given.

With the help of these indirect methods, heights above ground and plane coordinates ofinaccessible points may be measured with less risk and less effort, which will eventuallyresult in time-saving and efficiency-enhancing.

1.1 Previous studies

Since all methods and most theory and formulas are well known, they were known bythe author, or have been collected from recognized textbooks like Kahmen & Faig(1988) available at most university libraries. One adequate paper concerning uncertaintyof inaccessible points is Cederholm & Jensen (2009). They discussed the uncertainty ofthe inaccessible point found by the bearing and distance method. The uncertaintyformulas are in their presentation divided into two components; one along the lineformed according to the method, and one across. Several studies that were foundconcerning 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.


6/38

2

2. Methods2.1 The REM method

The REM method is used to find the height above ground where a prism cannot beplaced directly on the target (Wei and Cheng, 2006). Surveyors can determine theground of inaccessible points (like tall buildings, bridges, etc.) with this method

(Kavanagh, 2003 b). The principle of the REM method is simple and the observationprocess 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 aboveground 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 positionedvertically below an unknown point A, the height h above ground of the prism ismeasured and the slope distances to the prism from a remote position like F isdetermined by a total station D together with the two zenith distancesz1 andz2.

Figure 1: A principle sketch of the REM method, where Sis the slopedistance to point B. The zenith distances z1 andz2 are measured by atotal station at D.

With the reference to Figure 1, the heightHabove ground is found by trigonometry:

hH AB ; where 212 coscotsinAB zSzzS .

Thus, the heightHabove ground of the inaccessible unknown point A is:

hzSzzSH 212 coscotsin (1)

The standard uncertainty of the REM method of the height Habove ground is foundaccording 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

ix

f

c .


7/38

3

The sensitivity coefficients of Formula 1 will then become:

2121 coscotsin zzzS

Hc

1

2

2

1

2sin

sin

z

zS

z

Hc

212

2

3 sincotcos zSzzSzHc

14

h

Hc

Applying the combined standard uncertainty Formula 2, the standard uncertainty of theheightHof the unknown point A is:

huzuzzzS

zuz

zSSuzzzHu

2

2

22

212

2

1

2

1

4

2

2222

212

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 nothave to fear any exaggerated contributions fromz1

- 22 1 2(sin cot cos ) 1z z z - 4sinsin 14 2

2

zz

- 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; thesmallerz1 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 morecomplex since when positioning the prism under the unknown point, the prism should

be placed very close to the plumb line of the unknown point.


8/38

4

2.2 The double survey station method

If the prism cannot be placed directly under the unknown point, contrary to the REMmethod, then the double survey station method may be used to measure the heightabove ground of an inaccessible point. In this method, a prism can be placed near theinaccessible point, but not necessarily below it.

The principle of the double survey station method is shown in Figure 2. The heightsH1andH2 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 zenithdistance z1 to the unknown point A is measured with a total station at O1. The totalstation can be moved to point O2where point O2 is closer to the unknown point A, thezenith distancez2to the unknown point A is measured by a total station at point O2.H3 isthe height difference between point E and F can be measured with a levelling instrument.The height differenceH3 can be measured with an application of a total station as well(Lee and Rho, 2001). Finally calculate the heightHabove ground by trigonometry.

Figure 2: A principle sketch of the double survey station method, where thezenith distancesz1 andz2 are measured by a total station at point O1 and O2respectively; H1 andH2 are the heights above ground of point C and Drespectively.

Notice that, the value (H1 H2) can either be positive or negative; when negative, theunknown point A is lower than the prism at F, then point C is also lower than point Dand thus the value (H1 H2) is negative.

With the reference to Figure 2, the heightHabove ground is found: 31 BC HHH ;

where )(sin

sincosBC

21

2121

zz

zzHH

.


9/38

5

Thus, the height above groundHof the inaccessible unknown point A is:

321

21211

)sin(

sincosH

zz

zzHHHH

(5)

The sensitivity coefficients of Formula 5 are:

21

2

2221

1

1-sin

sincos

zz

zzHH

z

Hc

21

2

1121

2

2-sin

sincos

zz

zzHH

z

Hc

)sin(

sincos1

21

21

1

3zz

zz

H

Hc

)sin(

sincos

21

21

2

4zz

zz

H

Hc

13

5

H

Hc

The standard uncertainty of the height H above ground of the inaccessible unknownpoint A is found by the combined standard uncertainty Formula 2:

32

2

2

21

2

2

2

1

2

1

2

2

21

21

2

2

1

2

1

2

1

2

2

2

2

2

21

4

2

212

)(sin

sincos)(

)sin(

sincos1

)(sincos)(sincos)(sin

HuHu

zz

zzHu

zz

zz

zuzzzuzzzz

HHHu

(6)

The standard uncertainty formula gives a clear indication of the unfavorable situationwhich might occur in cases when the difference between the two measured zenithdistances 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. Anadvantage when using the method is that a prism can be positioned at any place, notnecessarily under the plumb line of the unknown points; the method could be widelyused in land surveying work.


10/38

6

2.3 The double-side survey method

If the unknown point is above a water area or some other inaccessible area, where theprism cannot be placed directly under the unknown point, but on stable grounds in frontof, the double-side survey method can be used to solve the problem. With this methodthe 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 atpoint M and its height h1 above ground is measured with a tape measure; observe theunknown point A by measuring the zenith distance z1. A prism D exchanges the totalstation at point M; the prism's height is adjusted so it is equal to h1. The total station ismoved to point K which is closer to the inaccessible area, observe the unknown point Aagain by measuring the zenith distance z2. Then, place another prism N at point J. Thezenith distancesz3 andz4 are measured by the total station at G.

Figure 3: A principle sketch of the double-side survey method, where the zenithdistance z1is measured by the total station D; SDGandSGN are slope distancesmeasured by the total station at G as are the zenith distancesz2,z3andz4. h2 is theheight above ground of prism N andh1 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 heightHabove ground is found:

2EPBEAB hHHHH ; where)sin(

)sin(cos

21

421DGAB

zz

zzzSH

, 4DGBE coszSH and

3GNEP coszSH .

Thus, the heightHabove ground of the inaccessible unknown point A is:

23GN4DG

21

421DG coscos)sin(

)sin(coshzSzSzz

zzzSH

(7)


11/38

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(

21

2

12121142DG

1

1zz

zzzzzzzzS

z

Hc

212

411DG

2

2sin

)sin(coszzzzzS

zHc

3GN

3

3 sinzSz

Hc

4

21

421DG

4

4 sinsin

coscosz

zz

zzzS

z

Hc

421

421

DG

5 cossin

sincosz

zz

zzz

S

Hc

3

GN

6 coszS

Hc

12

7

h

Hc

Applying the combined standard uncertainty Formula 2 to calculate the standarduncertainty of the double-side survey method. The standard uncertainty of the heightHabove ground of the inaccessible unknown point A is:

22

GN

2

3

2

DG

2

212

42

2

1

2

4

2

2

4

21

4212

DG

3

2

3

22

GN2

2

21

4

42

2

1

22

DG

1

2

21

4

2

12121142

22

DG2

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. waterareas; compared with the REM method and the double survey station method, thismethod is more flexible when choosing the position of the prisms. The method can beused for land measurements as well, not just water areas. The disadvantage of thismethod is that the calculation process is a little bit more complex.


12/38

8

2.4 The two prisms method

The two prisms method is a special surveying method for finding coordinates ofinaccessible points. This method is carried out with a total station. When using themethod, 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 inaccessiblepoint A.

Figure 4: A principle sketch of the two prisms method, where points A, Band 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 ishorizontal, is calculated by the polar method (Kahmen & Faig, 1988), here x-directionhas been chosen for northing andy-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 BA , which is equal to CB , can be found by:

B CCB BA

B C

arctany y

x x

Derived, by using Formula 2 applied on Formula 9, the standard uncertainty formula ofa polar measurement is found to be:

BA22

ABAB

2

B

2

A

2 uddururu (10)

Where )( Aru and )( Bru are the radial uncertainties of points A and B respectively, ( )u d

and ( )u are the standard uncertainties of the distance and the bearing in use.

M

A

B

CdBC

dAB

SB

SC


13/38

9

Points C and B are measured by a total station at M. With the distance d0 and the

bearing 0 from M to B, the radial standard uncertainty of point B is found by:

022

00

22

B

2 M udduuru (11)

Where Mu is the combined positional and directional uncertainty of the total stationat point M.

Inserting the above formula in Formula 9, the radial standard uncertainty of theinaccessible unknown point A will be:

BA22

ABAB

2

0

22

00

22

A

2 M udduudduuru

Where the standard uncertainty of the bearing, defined according to Formula 2, is:

2

2

BC

2

BC

BCC

2

2

2

BC

2

BC

BCB

2

2

2

BC

2

BC

BCC

2

2

2

BC

2

BC

BCB

2

BA

2

yyxx

xxyu

yyxx

xxyu

yyxx

yyxuyyxx

yyxuu

C2B22BC

CB

2

C

2

B

2

2

BC

CB

2 cossinyuyu

dxuxu

d

(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

AB

2

0

22

00

222

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 CB2cos are both 1- the radial uncertainty of Formula 13 is relative to the total station M

Then the simplified bearing uncertainty will be:

2

BC

BA2 2

d

ruu (14)


14/38

10

And the simplifiedradial standard uncertainty of the inaccessible point A is:

rud

dudduru 2

2

BC

AB0

22

0

2

A

2 22

(15)

It is important to note the dependence of the ratio between dAB anddBCin Formula 15 forthe radial uncertainty of the unknown point A; the longer dABis with respect to dBC, theworse is the radial uncertainty.

2.4.2 Height coordinate

The two prisms method could be used to find vertical positions as well. Measure therelative heights with respect to the total station of point B and C to find the height of theinaccessible unknown point A.

The heights of the two prisms (HB andHC), with respect to the height of the total station,are obtained by the formula:

CCC

BBB

cos

cos

zSH

zSH

Then the heightHof the unknown point A is:

CC

BC

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 Crespectively, andzBandzCare the zenith distances.

The sensitivity coefficients of Formula 16 are:

BC

ABB

B

1 1cosd

dz

S

Hc

BC

ABBB

B

2 1sind

dzS

z

Hc

BC

ABC

C

3 cosd

dz

S

Hc

BC

ABCC

C

4 sind

dzS

z

Hc

2

BC

ABCCBB

BC

5 coscosd

dzSzS

d

Hc

BC

CCBB

AB

6

1coscos

dzSzS

d

Hc


15/38

11

The relative standard uncertainty of the heightHof the unknown point A with respect tothe total station is calculated according to the combined standard uncertainty Formula 2:

AB2

2

BC

2

CCBB

BC

2

4

BC

2

AB2

CCBB

C

2

2

BC

2AB

C

22

CC

2

2

BC

2AB

C

2

B

2

2

BC

ABB

22

BB

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:

- 2 Bsin z , 2 Csin z , 2 Bcos z and 2 Ccos z are all 1- )()()()()( ABBCCB dududuSuSu - vuzuzu CB - SSS CB

Then:

vuSduddHu222

2

BC

AB2 12

(18)

The ratio between dAB anddBCoccurs here as well as in Formula 15, indicating that theuncertainty of the heightHwill be affected such that the longerdABis with respect to dBC,the worse is the uncertainty of the height.


16/38

12

2.5 Bearing and distance

The method bearing and distance, here described as an outdoor method, is used tomeasure 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 tooperate and to handle than a total station. The principle of bearing and distance is shownin 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 situatedsuch that satellite signals are obstructed from reaching the GNSS.

Aim at point A to establish a straight line on the ground, measure the coordinates ofpoint B and C by GNSS on the line. The process to find the coordinates of A is nowequal 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 methodwhere the coordinates of C and B are obtained with a GNSS.

The mathematical process to find coordinates as well as the analysis of uncertainty isidentical to the two prisms method, as well as the notations used in Figure 5. The majordifference is the uncertainties of the observables; they are anticipated to be larger withthis method.

To find the coordinates, Formula 9 can be used, and to find the measurement uncertaintyFormula 10 integrated by the simplified bearing formula (Formula 14):

rud

dduru

d

ddururu 2

2

BC

ABAB

22

2

BC

ABAB

22

A

2 212

(19)

Where ABdu is the standard uncertainty of the distance AB along the line to theinaccessible point, ru is the radial standard uncertainty of either of the two measured

points B and C.

dBC

dAB


17/38

13

2.6 Statistics

To find the standard uncertainties of the experiments done during the field tests, theformula recommended by JCGM 100 (2008) is used:

11

2

n

xx

xu

n

i

i

(20)

Wherexi is an individually measured value, is the mean value andn is the number of

measurements. The mean value is defined asn

x

x

n

i

i 1 .

2.7 Materials

Data for finding the numerical measurement uncertainty of the presented methods iscollected from a Swedish regulation termed SIS-TS 21143:2009. In the regulation totalstations 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 uncertaintyin direction u dir (one set)

Standarduncertainty in

distance u d

T1

For industrial application and

surveillance.Control measurements ofstructures with particularlyhigh demands.

0,8 mgon 1 mm + 2 ppm

T2

Geodetic survey for road andrailway projects.Detail and controlmeasurement of railways,

bridges and tunnelconstructions.

2,4 mgon 3 mm + 3 ppm

T3

Geodetic survey in general.Detailed measurement ofroads and other engineeringstructures.Control measurements ofother 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


18/38

14

Table 2: Classification of GNSSs according to SIS-TS 21143:2009.

Class Method Utilities

Standarduncertainty

(plane) u r Standard

uncertainty

(height) u H

St 1 Staticmeasurement

2ndorder networks. 5 mm + 1 ppmTwo-frequencyreceivers

-

St 2Static

measurement

3rdorder networks forinfrastructure.Photogrammetric tie

points.

510 mm + 2ppmOne-frequency

receivers-

RTK1RTK-

measurementDetail measurementsin 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. Results3.1 Numerical study

Theoretical measurement uncertainties are calculated for each method using one andeach 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 alsothe simplified formulas for comparison.

Where total stations have been used, two distances to the inaccessible point have beenconsidered; one short to be compared with the experimental work presented in section3.2, 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 ofdirections (two-face observation), but surveys in this presentation are all detailmeasurements (one-face observation), i.e. they are assumed to be done with half sets

only, thus the standard uncertainty used is 2 u dir .


19/38

15

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 thefollowing assumptions and supposed values:

- duhuSu - vuzuzu

21

- the zenith distances arez1 = 80 gon andz2 = 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 theformula and the simplified formula will be as in Table 3.

Table 3: Measurement uncertainty of the REM methodby 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 heightsH1 andH2 and the two zenith distances.H1 andH2 are obtainedby 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 arez1 = 80 gon andz2 = 70 gon

du and vu are the standard uncertainties of the used total station in distance and

direction respectively. Then:

vuHduHu 2222 4178579,6


20/38

16

Suppose the shortest distance between the closest total station and the inaccessible pointfirst 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 methodby 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 surveymethod

The measurement uncertainty of the double-side survey method (Formula 8) is inparticular dependent on two measured slope distances and two zenith distances. The fullformula can be simplified with the following assumptions and supposed values:

- SSS GNDG - duSuSu GNDG - vuzuzuzuzu 4321 - the zenith distances arez1 = 80 gon,z2 = 70 gon,z3 = 110 gon andz4 = 80 gon



vuSduHu 2222 6812356,5

Suppose the shortest distance between the closest total station and the inaccessible pointfirst 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 methodby class of total station and distances.

ClassStandard uncertainty Hu [mm]

S = 20 m S = 200 m

T1 9,6 46

T2 29 140

T3 47 230

T4 93 460


21/38

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:

2

BC

2

BC

CB

2

CB

22 00005,0cossin00005,0

ddu

And the simplified:

2

BC

2

BC

222 0001,02

dd

yuxuu

If the distance dBC is varying from 0,5 to 100,0 m, then the uncertainty results of the fullformula and the simplified formula will be as in Table 6.

Table 6: Numerical measurement uncertainties of bearingcalculation dependence of distance.

dBC [m] 0,5 1,0 5,0 10,0 50,0 100,0

Bearinguncertainty

u [gon]

Full

formula0,90 0,45 0,090 0,045 0,0090 0,0045

Simplifiedformula

1,3 0,64 0,13 0,064 0,013 0,0064

3.3.5 Analysis of the measurement uncertainty of the two prisms methodBy simplifying the full uncertainty formula (Formula 13) as was the case when derivingFormula 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 anddirection, respectively.


22/38

18

For the simplified formula it is assumed that the ratioBC

AB

d

dis 0,5. Then the simplified

formula will be:

022

0

2

A

2 5,2 udduru

Suppose the shortest distance between the closest total station and the inaccessible pointfirst is 5 m and then 20 m, then the standard uncertainty results of the full formula andthe simplified formula will be as in Table 7.

Table 7: Measurement uncertainties of the plane coordinate ofthe two prisms method by class of total station.

ClassStandard 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 andsupposed values:

- SSS CB - dududuSuSu ABBCCB - vuzuzu CB - The zenith distances arezB= 80 gon,zC= 70 gon,- The distances are dBC= 1,0 m anddAB= 0,5 m



vuSduHu 2222 3626,22716,1

And the simplified:

vuSduHu 2222 66

Suppose the shortest distance between the closet total station and the inaccessible pointfirst is 5 m and then 20 m, then the standard uncertainty results of the full and thesimplified formula will be as in Table 8.


23/38

19

Table 8: Measurement uncertainty of the height coordinateof 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 thebearing 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 anddBC, 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 itis associated with a distance parameter. It is, however, also valid for network RTK inSweden based on a recent publication by Mrtensson et al. (2012).

2du , the standard uncertainty of the closest distance along the line to the inaccessiblepoint, 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 anddistance method with the increase of the ratio between dAB anddBC.


24/38

20

3.4 Practical study

To test the five special surveying methods for finding inaccessible points, convenienttest sites were chosen close to, or indoor, the geodetic laboratory at the campus of theuniversity. For the first three methods described in section 2, an inaccessible pointwas 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 inaccessiblepoint. 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 arethe same as in Figure 1.

Observations and results are shown in Table 1 in the appendix. The mean value of theheights with this method is 3,401 m, which is regarded to be the most probable value ofthe correct height of the inaccessible point. The standard uncertainty according toFormula 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 Figure2 is zero here. HeightsH1 andH2 are obtained by the REM method.

DB

A

Sh


25/38

21

Figure 8: The test set-up with the double survey station method wherethe 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 theheights with this method is 3,404 m, which is regarded to be the most probable value ofthe correct height if the inaccessible point. The standard uncertainty according toFormula 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. Notationsused are the same as in Figure 3.

Observations and results are shown in Table 3 in appendix. The mean value of theheights with this method is 3,403 m, which is regarded to be the most probable value ofthe correct height if the inaccessible point by this method. The standard uncertaintyaccording to Formula 20 is6,9 mm.

A

F

O2H2

G D

A

N

SDG

SGN

h1

h2


26/38

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 toeach other and two prisms meant for mounting on ranging poles. The second equipmentwas a specially made equipment for hidden-point observations it was a set of basicseries mini prisms supplemented by a set of hidden point poles (GMP111 + GMP112

from Leica, Figure 10). To keep the poles steady during observations, they weresupported 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 rightangle 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-handedsystem, is the y-axis (easting). With such an arrangement, uncertainties along and acrossthe measuring pole are easily obtained.

Figure 10: A test set-up to illustrate the two prisms method, here theratio between dAB anddBC is 1. Notations used are the same as inFigure 4.

For the ordinary equipment, the distance between the prisms was 1,0 m throughout thetest, observations and results are in Tables 4, 5 and 6 in appendix. The distance betweenthe total station and the inaccessible point was approximately 20 m.

Table 9: Results for the two prisms method with respectto the ratio between the distances dAB anddBC.

Ratiovalues

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


27/38

23

For the special equipment, the distance between the prisms was 0,70 m throughout thetest, observations and results are in Tables 7, 8 and 9 in appendix. The distance betweenthe total station and the inaccessible point was approximately 13 m.

Table 10: Results for the two prisms method with respectto the ratio between distances dAB anddBC.

Ratiovalues

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 polewas held by free-hand to resemble a realistic surveying situation. The test site waschosen outdoor on a flat ground (Figure 11). The observations and results are shown inTables 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 tothe ratio between the distances dAB anddBC.

Ratiovalues

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

AdBC

dAB


28/38

24

3.4.5 Numerical study vs. practical study

The numerical height measurement uncertainty results are compared with correspondingpractical uncertainty results. The distance between the total station at its furthestposition and the inaccessible point for numerical study in Table 12 is 20 m, but for thepractical study it is 3 m.

Table 12: Comparison for a total station class T2 of height uncertaintybetween the numerical and the practical study for the first three methods.

MethodHeight 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 uncertaintyresults 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 forthe practical study the distances are 20 m for the ordinary equipment and 13 m for thespecial equipment (Table 13).

The bearing and distance method is in Table 13 only compared for radial uncertaintiesand three selected ratios.

Table 13: Comparison of height and radial uncertainty between thenumerical and the practical study for the last two methods.

MethodsRatiovalues

Height uncertainty [mm] Radial uncertainty [mm]

Numericalstudy

Practical studyNumerical

study

Practical study

Ordinary Special Ordinary Special

The twoprisms method

(Class T2)

0,5 3,8 1,9 15 4,9 9,6 14

1,0 4,7 14 126,1

10 14

2,0 1,857 5,3 13 22 9,6 49 40

Bearing anddistance

(Class RTK1)

0,5

-

16 18

1,0 20 19

2,0 32 21


29/38

25

4. Discussion4.1 Numerical study

Measurement uncertainties are compared between different methods, particularlyconcerning 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 heightmeasurement uncertainties at short distances (


30/38

26

4.2 Practical study

To verify, or examine, the numerical measurement uncertainties, a practical study wasperformed. All five methods were experimentally tested, the first three with fairly shortdistances to the inaccessible points (320 m) due to limited space where control could

be maintained. In reality those methods are anticipated to be used over distances rangingup 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 realisticconditions.

Even though short distances were used, the internal difference in uncertainty betweenthe REM method, the double survey station method and the double-side survey methodis kept compared to the numerical study. The measurement uncertainty of the REMmethod 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 totalstation, the uncertainty of the REM method is very close to the corresponding numericalresult (3,4 mm), whilst the other two show smaller values at the test compared to thenumerical 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 veryshort distances used during the test since distances are influencing almost all terms inthe uncertainty Formula 8. The fear of the sensitivity of small zenith distancedifferences discussed in the previous section is not revealed in the test; only onedifference is smaller than 10 gon, the others much larger.

When comparing heights and height uncertainties for the two prism methods, it is seenthat the heights change from 9,513 m at ratio 0,5, via 9,658 m at ratio 1, to 9,597 m atratio 2 for the ordinary equipment. The larger heights at ratios 1 and 2 are due to thesupport by an extra pole during observations held at prism B bending the unstablemeasuring pole set-up such that the direction of the prisms is pointing above theinaccessible point. This phenomena is not as clear for the special equipment where theheights 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 specialequipment, it felt more stable at ratios 0,5 and 1, but as unstable as for the ordinaryequipment 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 theexperiment was that the instability of the ordinary equipment should influence the resultto the worse compared to the special equipment, particularly at ratios 2 and 1. Thespread 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 theiruncertainties. When comparing the ordinary and the special equipment used anddifferent ratios for the two prisms method, it is found that the observations made by theordinary equipment do not differ significantly nor for coordinates neither foruncertainties at ratios 0,5 and 1, but for ratio 2. In fact the same statement can be madefor 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 12 mm. Aconvincing explanation cannot be found to explain this. The radial uncertainty for theordinary equipment and for the ratios 0,5 and 1 is approximately 10 mm, for the special


31/38

27

equipment a comparable figure is 14 mm. For both equipment and for the ratios 2 and1,9 respectively, the uncertainties are much larger than for the other ratios, a result ofthe 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 ratiois 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. Comparingthe x- and y-coordinate uncertainties, coordinates which now are in a nationalcoordinate system, there is a small tendency that the uncertainty of the y-coordinate issmaller 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 reportthat 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 showingsmall 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 placedunder the object of interest.

- Do avoid the double-side survey method if possible. Mainly because it has avery unfavourable uncertainty propagation.

- The two prisms method should be chosen for short distances, preferably withratios not larger than 1 between the distances dAB anddBC on the measuring pole.

- The simplified bearing formula can be used for almost all bearing uncertaintycalculations.

- Avoid ratios between the distances dAB and dBC larger than 2 when using thebearing and distance method.


32/38

28

6. ReferencesAiquan 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(secondedition, pp.593). New Delhi,Tata McGraw-Hill Publishing CompanyLimited,.

Feng Q., Sjogren P., Stephansson O. & Jing L., (2001). Measuring fracture orientationat exposed rock faces by using a non-reflector total station. EngineeringGeology 59, 133-146.

Hu H., Zheng C. & Yang Q., (2005).Discussion about how to use total station inspecial condition, Kunming Metallurgy News Papper, issue NO.1, pp.37 52.

JCGM 100 (2008).Evaluation of measurement data Guide to the expression ofuncertainty in measurement. Joint Committee for Guides in Metrology(JCGM/WG 1).

Kahmen H. & Faig W., (1988). Polar survey of object points. In Tutte GmbH, HublerR., 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.),UnitedStates of America,USA: Upper Saddle River HJ: Prentice Hall.

Kavanagh B. F., (2003 b). Remote object elevation calculation. In University ofMichigan (Eds.), Geomatics (pp.553).United States of America,USA: UpperSaddle River HJ: Prentice Hall.

Lee J. & Rho R., (2001).Application to leveling using total station. Denmark: publishedby the Interactional Federation of Surveyors (FIG).

Mrtensson S-G., Reshetyuk Y. & Jivall L., (2012). Measurement uncertainty in

network RTK GNSS-based positioning of a terrestrial laser scanner.Journal ofApplied Geodesy. doi: 10.1515/jag-2011-0013.

SIS-TS 21143:2009.Byggmtning Geodetisk mtning, berkning och redovisning vidlngstrckta objekt. Stockholm: SIS Frlag AB.

Wei Z. & Cheng M., (2006).Discussion about REM of Total Station, Engineeringsurveying, vol.37, pp.17 20.

Zhang Y. & Wang B., (2007).A new method of triangular elevation by total station andaccuracy estimation. Engineering of Surveying and Mapping. doi:

CNKI:SUN:CHGC.0.2007-06-013, Kunming, China.


33/38

29

Appendix

Table 1: Observations and results for the REM method. Notations used according toFormula 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 usedaccording 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 3,3972 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


34/38

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 inmetre.

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 twoprisms method (with ratio equal to 0,5). Notations used according to Formula 9 andFormula 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)


35/38

31

Table 5: Observations and results obtained with the ordinary equipment by the twoprisms method (with ratio equal to 1). Notations used according to Formula 9 andFormula 16, zenith distances and bearing in gon and linear distances in metre.


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 twoprisms method (with ratio equal to 2). Notations used according to Formula 9 andFormula 16, zenith distances and bearing in gon and linear distances in metre.


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)


36/38

32

Table 7: Observations and results obtained with the special equipment by the two prismsmethod (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.


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 prismsmethod (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.


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)


37/38

33

Table 9: Observations and results obtained with the special equipment by the two prismsmethod (with ratio equal to 1,857). Notations used according to Formula 9 and Formula16, zenith distances and bearing in gon and linear distances in metre.


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)


38/38



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)



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)

Measuring Inaccesible Points in Land Surveying

Documents

Transcript of Measuring Inaccesible Points in Land Surveying