Surveying Measurement Accuracy and Theodolite Error Analysis

Surveying Measurement Accuracy and Error Propagation

Tolerance should be imposed only by the requirements of the research. It should be conditional on the surveying instrument used for this tolerance; it would be naive to expect very great accuracy using a blunt instrument.

Transmission Error (Mean Square Error of a Sum)

The mean square error (MSE) of a sum is equal to the square root of the sum of the squares of the mean square errors of the summands, provided the measurements are made with different precision. If measurements are performed with equal precision, it can be shown that the mean square error is the same for all addends.

Exercises

1. Six measures of the length of a building's facade were made using a steel tape (listed in the table below). Calculate:

1. The most probable value of the facade's length.
2. The error of each measure.
3. The relative error of measure 4.
4. Tolerance in the following two cases:
   • Equal to 2.5 times the mean square error.
   • Same as the mean square error.

Instrumental Errors in Surveying

Errors in Theodolites

The accidental errors often committed when using theodolites include:

• Failure of Verticality (Verticality Error)
• Address Error (Steering Error)
• Pointing Error (Aim Error)
• Reading Error (Angle Reading Error)

Verticality Error

This occurs when the main axis is not perfectly vertical. It depends on the degree of sensitivity of the level, which must be specified by the manufacturer. It affects observations of both zenith angles and azimuth.

Address Error (Steering Error)

This occurs because the instrument's plummet does not exactly coincide with the center of the signal in the field, or because the pole is not exactly perpendicular to a target point. These two aspects cause the observed azimuth to differ from the true azimuth, resulting in the highly significant error called steering error. It is only relevant in observations of azimuthal angles. Since the steering error is inversely proportional to the distance between the instrument and the prism, it decreases as the distance between points increases. This error is the most important of all mistakes made with theodolites, representing nearly 99% of the total error. The steering error can be reduced by pointing to the spot on the floor where the prism is placed. This manages to reduce the position error of the prism on the point to zero.

Error of Aim (Pointing Error)

This occurs when the crosshairs do not exactly match the center of the prism.

Collimation Error (Colima-HZ)

The collimation error (C) is the deflection angle formed by the trunnion axis and the line of sight. It is eliminated by making measurements at two positions of the telescope.

Vertical Index Error (Index-V)

If the line of sight is horizontal, the vertical circle reading should be exactly 90° (100 g). Deviation from this value is called the vertical index error (i).

Angle Reading Error

This error arises in the assessment of verniers and micrometers when reading both zenith angles and azimuth. This error is not committed when using total stations and digital tachometers.

Maximum Total Error

The maximum total error is the sum of all previous errors. Arcsin Depp Ed Epi