Measuring with Electronic Total Stations

Alojzy Dzierzega, René Scherrer

1 SUMMARY

Since the introduction of the electronic total station, the art of measuring seems to have been rele-

gated to the backseat. However, optimal results can still only be achieved by applying the “Rules of

the art of measuring“. These cover the various remaining sources of error in total stations, in particular

electronic total stations, as well as the methods that can minimize or eliminate influences on meas-

urements.

2 INTRODUCTION

Nowadays automation in surveying is not only taken for granted for office based procedures but has

also become standard expectation for field applications. Automation in the field is being brought about

mostly by electronic total stations. The push of a button opens up almost any desired function: target

tracking, precision targeting, calculation of the calibration values for instrument errors, standing axis tilt,

data storage, etc. This however only facilitates the surveyors workload and reduces the need for a

steady hand at precision targeting. The art of measuring must still be mastered, particularly for work

requiring a high degree of precision.

Fundamental skills in surveying include among other things the ability to understand:

- the accuracy specification of an instrument and how to determine it

- the difference between single and dual face measurements

- the influence of instrument errors and how to determine them

- the influence of ambient temperature on the results of measurements

The following is based primarily on experience gained using Leica instruments. In general, the accu-

racy considerations are similar for all electronic total stations. To achieve optimal results the following

should be observed.

3 SOURCES OF ERRORS

All so-called instrument errors are residual errors that come about during the production phase of the

instrument and are unavoidable. The influence on the measurement results can be eliminated by se-

lecting suitable measuring methods or reduced to a minimum by applying calibration values. For in-

formation on how to determine these instrument errors and how to take calibration values into account

please consult the relevant technical literature and/or the appropriate user manuals.

3.1 AXIAL ERRORS

These are caused by the fact that certain instrument axes are not perpendicular to each other. Spatial

orientation plays no part.

A collimation error (line-of-sight error) occurs when the line-of-sight is not perpendicular to the tilt-

ing axis. It’s influence on the Hz- reading depends on the zenith angle.

(1.1)

P

c

Q

C

H

TC

Z

Z‘

Hz(c)

( )

sin( )

c

Hz c

z

( ) = influence of collimation error

on Hz-reading

= collimation error

= zenith angle

Hz c

c

z

At the horizon the influence of the collimation error on the Hz-reading corresponds to the value of the

error itself. It increases towards the zenith and the nadir and at these points themselves it becomes

singular.

A tilting axis error occurs when the tilting axis is not perpendicular to the standing axis. It’s influence

on the horizontal circle reading is also dependant on the zenith angle.

(1.2)

For measurements on the horizon the error has no influence on the Hz-reading, it is equal to zero. It

increases towards the zenith and the nadir and at these points themselves it becomes singular.

The influence of these two axis errors on the result of the Hz-reading can be eliminated by dual face

measurements (averaging the measurements in face I and face II). These can be seen in the symmet-

ric curves of diagram 1.

Diagram 1: Influence of collimation and tilting axis errors on Hz-readings

( )

tan( )

i

Hz i

z

C

Z

Q

i

Z

i

P

H

T

Hz(i)

() = influence of tilting axis error

on Hz-reading

= tilting axis error

= zenith angle

Hz i

i

z

In the diagram, so-called corrective limits are displayed. They show the angle limits of the zenith angle

up to which the corrections are calculated. These angle limits each form a section of +/- 20 [gon]

around the zenith and theoretically also around the nadir, in which for practical reasons no measure-

ments can be made. For telescopic positions with a zenith angle within these sectors no actual correc-

tions are calculated but the corresponding corrections for these angles are applied.

This is done for two reasons. For one the singularity problem is avoided. The second but far more

important reason is the problem of accuracy based on error propagation. At a zenith angle of +/- 20

[gon] there is an amplification factor of three for the calibration values of the collimation and tilting axis

itself as well as for it’s standard deviation.

3.2 ZERO POINT ERRORS

This error occurs when the zero point of a scale is not in correct reference (correct position) to the

defining factor.

A (mechanical) vertical-index error occurs when the zero point of the vertical scale reading is not in

(or parallel to) the standing axis of the instrument.

A compensator-index error occurs when the zero point of the compensator is not in the plumb-line.

With a dual axis compensator the index error of the compensator is divided into two components, one

alongside and the other crosswise to the telescope. In order to simplify the user interface but at the

cost of transparency, certain instruments (e.g. the TPS 700 / TPS 300 series by Leica) determine the

mechanical vertical–index error together with the compensator-index error in one step. Only the verti-

cal-index error is displayed; the compensator-index error is stored in temporary memory and used to

correct the measured standing axis components.

The influence of these two zero point errors mentioned above on the results of angle measurements

can also be eliminated by forming averages of face I and face II measurements.

3.3 ATR (AUTOMATIC TARGET RECOGNITION) CALIBRATION

A further zero point error is the deviation of the center of the CCD-array from the line-of-sight. The

influence of this error is determined in the so called ATR calibration process and the angle readings

are corrected automatically by the corresponding values.

3.4 SET UP ERRORS

The so-called standing axis error is often listed as an axis error. It is actually not an axis error in the

sense of the word but an instrument set up error. It occurs when the standing axis of the instrument is

not aligned with the plumb-line.

It’s influence on the results of angle measurements can only be determined mathematically by meas-

uring the tilt of the standing axis in the direction of the telescope and crosswise to it e.g. with the help

of a dual axis compensator. It can not be eliminated by dual face measurements and therefore it

should be given more consideration.

3.5 ERRORS DUE TO TEMPERATURE DIFFERENCES

Temperature differences between the instrument and the environment it is in, will result in changes to

the characteristics of the instrument, in particular of the compensator over time. This can mostly but

not totally be eliminated by measuring the same target in two faces. It is important to let the instrument

adjust to the ambient temperature before starting measurements.

Diagram 2: Typical compensator behavior during temperature adjustment

For most compensator instruments the following rules of thumb apply:

- Time for temperature adjustment is approx. 2 Min / C of temperature difference

- Ascending curve is significantly steeper than the descending curve

- Amplitude not only depends on the components but also on the instrument

4 SINGLE FACE VERSUS DUAL FACE MEASUREMENTS

The manufacturer’s specification of angle measurement accuracy is based on the standard deviation

of a dual face measurement if nothing else is specified. If there is an indication of a DIN or ISO stan-

dard then it is an additional reference to the method used in determining this standard deviation. The

reference to the standard deviation of a dual face measurement stems from the fact that with this

method the influences of most systematic errors, the magnitudes of which are generally unknown, can

be eliminated and the best possible results can be achieved. Furthermore it is assumed that the in-

strument has been calibrated.

To save time a single face measurement would be preferable to a dual face one. A properly calibrated

instrument is required for a single face measurement. Unlike earlier optical-mechanical instruments,

with the electronic instruments the user can quickly carry this out by himself.

With the optical-mechanical instruments mechanical adjustments had to be made, e.g. moving the

reticle. Certain adjustments, such as e.g. to the tilting axis, could only be done in service centers and

were not available to the user.

Electronic total stations are software calibrated. The error (calibration value) is determined and its

influence is mathematically added to the raw angle reading. A steady and skilled hand is no longer

required. At times it would be a fatal error to try to carry out mechanical adjustments. For example,

moving the reticle because of a collimation error would miss-align the optical and EDM axis.

The degree to which a (Leica) instrument can be calibrated differs. With the less sophisticated instru-

ments of the TPS 300 / 700 series e.g. the tilting axis error cannot be calibrated by the user. He has to

live with the built-in tolerances or have it corrected at a service center. As an alternative there is also

the possibility of dual face measurements.

A statement on the necessary frequency of calibration by the user can hardly be made. It should al-

ways be done when there is the suspicion that or danger of the settings having changed. The user

manual lists concrete examples: after long storage periods, rough transport, extreme temperature

differences, etc. It is important to remember that only instruments that have adjusted to the ambient

temperatures should be calibrated, otherwise the values determined will not be correct.

The standard deviation of a single face measurement is of course less precise than that of a dual face

measurement. To evaluate this, not only half the number of measurements compared to dual face

measurements have to be taken into consideration, but also the standard deviation of the calibration

values and their error propagation in steep sightings.

For e.g. a single face measurement pointing at a zenith angle of 20 [gon], an approx. three time worse

standard deviation compared to the manufacturer’s specifications (two face measurement) would re-

sult. This is assuming manufacturer’s specifications of 0.5 [mgon], and a standard deviation of the

collimation and tilting axis error of 0.3 [mgon] each.

This example shows two things. First of all the reason why from a certain zenith angle onwards the

corrective values of the instrument errors are “frozen”. Their values become too large and the results

would be too inaccurate. Secondly, the fact that even under the assumption of having correct calibra-

tion values the standard deviation for single measurements can be up to three times worse than that of

the dual face measurements due purely to statistical error propagation.

Under which circumstances a dual face measurement is to be favored over a single face one mostly

depends on the situation and the accuracy requirements. It is always recommended when:

- highest degree of accuracy is required (making full use of the manufacturer’s specifications)

- targeting within +/- 20 gon of the zenith

- high temperature differences occur within a short period

- calibration values are not determined or for some reason they do not seem correct.

5 EXAMPLES

The following examples should lend some practical meaning to the explanations given so far.

5.1 INFLUENCE OF TEMPERATURE ADJUSTMENT AND MEASURING METHODS

Based on the examples of vertical angle measurements, which best reflect the sensitivity of the com-

pensator, the influence of temperature adjustment of the instrument to ambient temperature and the

influence of the measuring method should be made clear. Four targets spread out over a zenith angle

range of 41 [gon] were observed from a distance of approx. 24m vertical angle. Four series of meas-

urements of three sets each were made in both faces (according to the new ISO standard 17123-3).

Four different sets of measurements were done with the same instrument (see table 1).

Measurement

Instrument

temperature

Ambient

temperature

Adjustment to

ambient tem-

perature

Measuring sequence

1. Sample

22 C 17 C

no 1

st

half set in 1

st

face, then

2

nd

half set in 2

nd

retro face

2. Sample

22 C 17 C

no Each target measured in 1

st

and

2

nd

face at once

3. Sample

22 C 16 C

yes

1

st

half set in 1

st

face, then

2

nd

half set in 2

nd

retro face

4.Sample

22 C 16 C

yes

Each target measured in 1

st

and

2

nd

face at once

Table 1: Temperature adjustment and measuring method of the experimental measurements

Diagrams 3 and 4 essentially state the same thing; one indicates the achieved standard deviation per

series, the other shows the largest differences of the measured values per target point.

Diagram 3: Standard deviation per series and overall total of all series

Diagram 4: Maximum differences in V-angle per target

Diagram 5 essentially reflects in form of the vertical index error the compensator characteristics as

shown in diagram 2. The curve characteristics in the individual examples not only depend on the ad-

justment or non-adjustment of instruments to ambient temperatures but is also strongly dependant on

the sequence (method) of measurements taken.

Diagram 5: Vertical-index error dependant on the method of measurement

Max. differences in V-angle

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Pkt 1 Pkt 2 Pkt 3 Pkt 4

Target points

difference [mgon]

1.Sample

2.Sample

3.Sample

4.Sample

Standard deviation for V-angle

0.00

0.10

0.20

0.30

0.40

0.50

I II III IV over all

Series

standard deviation

[mgon]

1.Sample

2.Sample

3.Sample

4.Sample

V-Index error

-1

-0.5

0

0.5

1

1.5

2

1

4

7

10

13

16

19

22

25

28

31

34

37

40

43

46

Measurements in chronological order

V-index error [mgon]

1.Sample

2.Sample

3.sample

4.Sample

The result clearly show that the instruments must be adjusted to the ambient temperature before tak-

ing measurements. If this cannot be done optimally then sequential dual face measurements must be

made to the targets within a short period.

5.2 EXTENDING A STRAIGHT LINE

The straight line AB is to be extended to C. A classic job with a classic solution. The instrument is set

up in B, A in face I is targeted, telescope is plunged to face II and at C the image of the reticle C1 is

marked. Then A in face II is targeted, the telescope is plunged to face I and on C the image of the

reticle C2 is marked. The middle of the two marks (C1+C2)/2 then forms the extended line.

Diagram 6: Extending a straight line

This is an elegant method. By forming the average the influence of instrument error is eliminated and

as no angles have to be set out the problem of possible circle graduation errors (this influence how-

ever can be neglected with today’s electronic total stations) does not occur. This method does how-

ever have a drawback. The standing axis tilt (caused by set up error) is not considered, as it cannot be

eliminated by forming the average of the two face measurements (the instrument is turned around the

standing axis and not around the plumb line).

As long as these straight lines are to be extended horizontally the standing axis tilt plays no part as its

influence in the horizon is zero (just like the influence of the tilting axis error). Mostly at construction

sites extending a straight line into a ditch or to an upper floor is the requirement, though due to the

steep sightings the standing axis tilt has to be considered here.

Most electronic total stations have a dual axis compensator that measures the components of the

standing axis tilt and adds the corresponding calculated corrections to V and Hz. These corrections do

not come about by merely inverting the telescope, a measurement has to take place setting out an

angle of 200 gon.

Upon these reflections a slightly modified procedure emerges: In face I target point A, turn off 200 gon

and mark direction C1 in point C. Then target point A in face II, turn the telescope by 200 gon and

mark direction C2 in point C. Forming the average of both faces results in the correct straight line ex-

tension even for steep sightings (both directions C1 and C2 should practically coincide if the instru-

ment has been properly calibrated).

5.3 SUBSEQUENT DETERMINATION OF COLLIMATION, TILTING AXIS AND VERTICAL INDEX ERROR

Let’s assume a large amount of points were measured in single face from one station, some of them

however in dual face mode. The points measured in the two telescope faces were additionally dis-

persed over a height range of e.g. +/- 20 gon.

A subsequent check of the instrument shows that for some reason the stored corrective values for the

collimation, tilting axis and vertical index error were not correct.

For the points measured in two faces this make no differences as forming the average brings about a

correct result. The points measured in one face are prone to errors. Thanks to the dual face meas-

urement of some points it is possible to subsequently determine the instrument error and to apply the

correct calibration values.

The individual vertical index error can be calculated from zenith angle measurements in face I and

face II:

(1.3)

400[ ]

2

I II

z z gon

v

A

B

C

C1

C2

The calculated arithmetic average of the vertical index error is introduced as the average vertical index

error of the points measured from this station. This includes the mechanical share as well as the share

of a compensator generated zero point error.

Determining the collimation and tilting axis error is more sophisticated as the two are coupled together.

(1.4)

A least squares adjustment is done with “c“ and “i“ as the unknowns. Observations are half the differ-

ences of the Hz- directions of face I and face II.

In this example measurements were made to collimators dispersed over a range of 40 – 130 gon.

Out of this collection of data, measurements were selected where only a certain range of zenith angles

were included in the evaluation. In the same way the amount of measurements taken in this range

were varied. Table 2 provides an overview of the adjustments and the results, while the subsequent

diagrams 7–10 group and display the results graphically according to ranges of zenith angles in func-

tion of the amount of measured points.

Range of vertical

angle

[gon]

Amount

of points

Collimation error c

[mgon]

(c)

[mgon]

Tilting axis error i

[mgon]

(i)

[mgon]

40 – 130

12

-0.15 0.04 -0.34 0.07

10

-0.16 0.04 -0.35 0.08

8

-0.16 0.04 -0.30 0.07

6

-0.18 0.04 -0.24 0.08

4

-0.14 0.03 -0.25 0.04

60 – 130

10

-0.16 0.03 -0.46 0.09

8

-0.18 0.03 -0.54 0.08

6

-0.19 0.03 -0.51 0.08

4

-0.19 0.06 -0.49 0.12

70 – 120

8

-0.14 0.03 -0.37 0.11

6

-0.15 0.03 -0.52 0.11

4

-0.16 0.04 -0.47 0.12

80 – 120

6

-0.18 0.02 -0.61 0.09

4

-0.17 0.03 -0.68 0.11

100 – 130

4

-0.10 0.05 -0.29 0.16

70 – 100

4

-0.20 0.07 -0.36 0.22

Table 2: Axis error and its standard deviation in function of the amount of measured points

and their distribution over the range of the zenith angles.

( 200[ ])

sin( ) tan( ) 2

I II

Hz Hz gon

c i

z z

= vertical index error

= zenith angle in face I

= zenith angle in face II

I

II

v

z

z

= collimation error

= tilting axis error

= zenith angle

= Hz-reading in face I

= Hz-reading in face II

I

II

c

i

z

Hz

Hz

Diagram 7: Collimation error in dependence of the range of zenith angles

Diagram 8: Tilting axis error in dependence of the range of zenith angles

The diagrams show that the tilting axis error is subjected to more variations than the collimation error.

Both seem to converge with the amount of measured points, independent of the selected range of

zenith angles.

Collimation Error c in Function of Range of V-angle

-0.25

-0.20

-0.15

-0.10

2 4 6 8 10 12 14

number of points

collimation errorr c

[mgon]

V=40-130 [gon]

V=60-130 [gon]

V=70-120 [gon]

V=80-120 [gon]

V=100-130 [gon]

V=70-100 [gon]

Tilting Axis Error in Function of Range of V-angle

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

2 4 6 8 10 12 14

number of points

tilting axis error i [mgon]

V=40-130 [gon]

V=60-130 [gon]

V=70-120 [gon]

V=80-120 [gon]

V=100-130 [gon]

V=70-100 [gon]

Diagram 9: Standard deviation of the collimation error dependant on the range of zenith angles

Diagram 10: Standard deviation of the tilting axis error in dependence of the range of zenith angles

From the standard deviation it is evident that the tilting axis error cannot be determined as accurately

as the collimation error. The least amount of measured points to reasonably determine either error

seems to be 6 target points.

It can be said that for subsequent determination of either axis errors from dual face measurements, at

least 6 target points dispersed over a range of zenith angles of 40 gon are required to provide accept-

able results. The 6 target points may be selected at random and need not be connected to the actual

surveying job.

6 CLOSING REMARKS

It was not intended to treat instrument errors in this article. There is enough specific literature on the

subject. Moreover, the idea was to show some of the complexities behind these modern, easy to use

instruments. Modern electronic total stations greatly facilitate surveying, but to achieve optimal results

some surveying knowledge is still paramount.

Authors address: Dr. Alojzy Dzierzega,

René Scherrer

Leica-Geosystems AG

Standard Deviation of Collimation Error c

0.02

0.03

0.04

0.05

0.06

0.07

2 4 6 8 10 12 14

number of points

standard deviation [mgon]

V=40-130 [gon]

V=60-130 [gon]

V=70-120 [gon]

V=80-120 [gon]

V=100-130 [gon]

V=70-100 [gon]

Standard Deviation of Tilting Axis Error i

0.00

0.05

0.10

0.15

0.20

0.25

2 4 6 8 10 12 14

number of points

standard deviation [mgon]

V=40-130 [gon]

V=60-130 [gon]

V=70-120 [gon]

V=80-120 [gon]

V=100-130 [gon]

V=70-100 [gon]

Heinrich Wild Strasse

CH-9435 Heerbrugg (Switzerland)

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