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Errors in Total Station Measurements

Lately I have seen heated discussion on LinkedIn about staking out in construction (typically hold down bolts) with pictures of residuals at 0.000m and sometimes great looking accessories to help with the prism stability.

I’d like to share some thoughts on a more fundamental level about measuring to tight tolerances using total stations and maybe in future posts extend into my recommendations for high accuracy staking out and total station calibration in particular.

The measurement of a total station is typically made up of two components:
1. The angular measurement
2. The distance measurement

The angular measurement, split into two components, the vertical angle and the
horizontal angle are governed by very fine angle encoders and typically the accuracy is quoted in seconds. For example it is typical in a construction situation to procure a ’5 second’ instrument. This means that the angle measurement fed into the software of a modern TPS resolves in 5 second increments. 0, 5, 10, 15 and so on.

This is a simplification of what actually happens, different manufacturers have
different approaches, but let’s go on with this an analogy.

In real terms this means that assuming the distance measurement is perfect (we’ll come to this later) that a granularity of 1mm can be achieved at a distance less than or equal to 41m. If a survey manager was to specify that there should be no more than 150m between control points on a job site, then assume that the instrument is no more than 75m approx. from the controls, then the position can be resolved to about 2mm if the distance is perfect. This is more than likely a suitable tolerance for most construction projects, but let’s pay the upcharge to get to a 2 second instrument for critical layout, and a 1 second instrument for a licensed surveyor.
Then we can say the following:
5” instrument = 1mm in 41m
2” instrument = 1mm in 104m
1” instrument = 1 mm in 206m

Leica Geosystems sells a TS60 instrument with 0.5” angular reading for those who need even more precision.

In my opinion, the distance measurement component is more critical and less
understood.

To understand this in a broad sense I will discuss two types of errors commonly
encountered (there are much more, but beyond the scope of this article). These are:
1. Zero error
2. Cyclic error

Zero error or index error is a combination of the effect of the zero point of the
instrument being elsewhere than over the mark which the instrument has been
centered on, together with the reflector’s position being other than centered above the mark at the other end of the line being measured. The offset will be constant for a given model of reflector so this is usually accounted for in the reflector constant set on the instrument.

Let’s say the erroneous zero position of a TPS is 5mm behind the centre of the
rotation. A point is measured facing west, this is 5mm longer that is should be, now the TPS turns east and another point is measured, this is also 5mm long, but the tie distance between the measured points is incorrect by 10mm! This is without any dependence on distance, so a user could easily be shamed by someone with a measuring tape.

In my career the worst TPS offender I came across had a zero error of -14mm.

Figure 1: Zero error, exaggerated!

Short periodic error or cyclic errors may occur in EDM instruments that measure
using the phase measuring method. These are periodic errors with wavelengths equal to the fine measuring unit length of the instrument and also can occur in the harmonics of these wavelengths. These errors can be caused by electrical and / or optical crosstalk or a systematic error in the phase measuring system.

Typically instruments with a shorter measuring wavelength will have a less
pronounced error. Instruments measuring with a pulse measurement are not affected by this by design.

Figure 2: Measurement wave

You can see that the error goes from +a to -a over the length of the measurement wave.
In practice, this means an error can occur at a certain distance increment, at 10m it may be a zero additional error, but at 10.2m it would mean an additional 1mm error, and back to zero at 10.4m.

Putting it together

Not accounting for any errors in the control network and the setup of the total
station, let’s say our user has a 2” instrument and is staking out at a distance of 50m.
The angular errors should be mitigated, but what about the distance errors?
A surveyor who quantifies and accounts for these errors will reduce the risk they are exposed to in the course of their day to day duties.

Consider for an example the very typical case of an engineering surveyor tasked with the setting out of holding down bolts to be cast into concrete for the construction of a structural steel building, with a typical spacing of 200 mm between bolts and a construction tolerance of 5 mm. Equipped with (Redacted)*, any measurements taken would be subject to a -3.5 mm zero error and a cyclic error of ± 0.9 mm. With the cyclic error going from a maximum positive to a maximum negative over the distance of half a wavelength (185 mm) the error is setting these bolts could be as much as – 4.4 mm to -2.6 mm solely from the distance measuring errors of the EDM. This is before any errors are accrued from angular readings, set up errors, errors of centering of prisms, errors in the control network, meteorological conditions etc. With the likelihood that the setup of the instrument being incorrect by the fact that distances measured to resection points or known backsights are incorrect by -3.5 mm (± cyclic errors) is becomes apparent that achieving an accuracy of position of these bolts to 5 mm is highly unlikely.

However if the surveyor knows that the zero error exists and adjust the prism
constant by this amount, then the magnitude of cyclic error present is small enough that the construction tolerances can still be achieved. It can be argued that the three or four hours required to perform a calibration once per year represent significant value for money when weighed against the cost of any possible rework to rectify poorly set bolts.

What calibration I can hear the comments say, well I’d like to write about that in a separate post, but Leica Geosystems has you covered with the Gold CCP package, where the instrument will be calibrated over the baseline** in the Rhine valley before delivery.

* I have results of analysis of various total station makes and models but I will not share the model here
** Baseline calibration is the best way to determine the zero error and the cyclic error in my opinion

Brendan Gilligan

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