Evaluation of the Cyclic Error in Topcon DL-101C Precision Digital Levels

Summary of 4th Year Thesis
by

H. C. Lok
School of Geomatic Engineering
University of New South Wales

Supervised by
Assoc. Prof. J. M. Rüeger

Aim
In 1998, the School of Geomatic Engineering of the University of New South Wales purchased ten Topcon precision digital levels DL-101C and the associated invar bar code staffs.  The main purpose of this thesis was to find out whether these Topcon DL-101C precision digital levels suffer from the cyclic errors found in earlier Topcon DL-101.  Only the cyclic error was tested and considered.  Special techniques were used in order to eliminate the other errors inherent in digital levels.  The project included field experiments, least squares adjustments for the processing of data and the estimation of the cyclic errors.

Field Experiment
Only four Topcon digital levels DL-101C were used to measure to an invar staff on twelve bolts. The bolts were placed at a height step of 30 mm giving a total difference in height of 330 mm.  Measurements were also made with a Zeiss Ni1 automatic precision level with a three-metre Wild line scale invar staff in order to obtain reference data.

Each Digital level was set-up at sighting distances of 2.0 m, 3.0 m, 5.0 m, 8.0 m, 15.0 m, and 20.0 m from the line of bolts.  At each sighting distance, measurements were made to the staff on the very first bolt and then to the staff on all other bolts.  After that, the bolts were immediately measured to in reverse order.  The complete sequence of measurement was:  1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 10 -> 11 -> 12 -> 12 -> 11 -> 10 -> 9 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 2 -> 1.  Before and after measuring all bolts with each Topcon Digital Level Dl-101C, the Zeiss Ni1 precise level was set-up, at a sighting distance of 2.0 m, to measure the bolts in the same sequence.  The above procedures were repeated for all four of the Topcon digital levels DL-101C tested.
 

Figure 2 ~ Instrument set-up

The figure above shows the instruement and the invar staff at the test site.  The bolts (obscured) were on the ramp behind the hand rail.  The instrument set-ups were on a perpendicular to the row of bolts, intersecting the row of bolts between Bolts 6 and 7.

Least Square Modelling

Topcon Measurements

Image36.gif - 1346 Bytes

Where
          HCorr ~   Corrected height of collimation (mm)
          Ti      ~   Staff reading at i-th bolt with a Topcon Digital Level (mm) from instrument station j
          a       ~   Coefficient of the sine term of the cyclic error (mm)
          b       ~   Coefficient of the cosine term of the cyclic errors (mm)
          Hi     ~    Height of the it-h bolt (mm)
          Zki      ~   Staff reading at ith bolt in scale k using Zeiss automatic Levels (mm)
          Lambda      ~   Wavelength of fitted cyclic error (taken as 315 mm)
 

Plot of "a priori" Standard Deviation versus Distance (Outliers Removed)
The "a priori" standard deviations were computed from the differences between the forward and reverse measurements to the 12 bolts from each set-up.  Less that 30 minutes elapsed between the forward and return measurement to a bolt.  Each diamond in the diagram represents the precision pooled over a maximum of eights values from four instruments.

Pooled Standard Deviation Vs Distance
 

Cyclic Errors Found
Below are the results of eight least squares adjustments that returned significant cyclic errors.  The uncertainties are given at 95 percent confidence level.  The cyclic errors in the remaining 16 least squares adjustments were not significant.
 
Instrument
Distance
(m)
a
(mm)
Ua (95% C.I.)
(mm)
b
(mm)
Ub (95% C.I.)
(mm)
NJ0095
2, 3
-9.6
± 8.4
-6.2
± 8.6
NJ0109
2, 3
-26.6
± 10.4
+3.7
± 9.8
NJ0109
5, 8
-19.4
± 9.6
-15.8
± 9.2
NJ0109
2, 3
-18.5
± 10.2
+9.1
± 9.8
NJ0109
15, 20
-4.1
± 23.9
-38.8
± 22.9
NJ0105
2, 3
-0.8
± 8.6
-11.2
± 8.4
NJ0105
5, 8
+17.1
± 9.4
-11.2
± 9.2
NJ0081
2, 3
-12.2
± 8.8
+2.5
± 8.8
 

Conclusion
The largest statistically significant cyclic error found in four of the School's ten Topcon DL-101C digital levels was 0.039 mm.  It should be noted that in most surveying applications, a cyclic error of 0.039 mm in amplitude is not important, as it averages out over consecutive set-ups, for example in a first order levelling network.
The measuring precision under field conditions is about ±0.02 mm at sighting distances below 8 m and then increases with the square of the distance.




Further Information
For more information, please contact:

A/Prof. J.M. Rüeger
Email:      J.Rueger@unsw.edu.au
Mail:        School of Surveying & Spatial Information Systems
               University of New South Wales
               Sydney NSW 2052
               Australia
Phone:   +61-2-9385-4173
Fax:       +61-2-9313-7493

Mr. H.C. Lok
Email:    HC_LOK@mail.com

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