Least Squares - Positional Tolerance

Positional tolerance is a standard of accuracy for survey points. A computed position either meets the standard or it does not. The standard defines the radius of a circle about a theoretical point based on its distance to the nearest controlling station. As long as a computed position lies within the positional tolerance of the theoretical point, it meets the standard. The higher the standard the smaller the radius about the theoretical point and thus the smaller the positional tolerance.

A published positional tolerance can include two parts – a distance multiplier and a not to exceed value. The distance multiplier computes a tolerance based on the distance from the nearest controlling corner. The not to exceed value puts a cap or limit on the value computed from the distance multiplier. An example might be a distance multiplier of +/-0.0008 and a not to exceed value of +/- 0.25 feet. A 500 foot observation would allow an error of 0.0008X 500 or 0.04 feet. A 5,000 foot observation would allow an error of 0.4 feet, so the not to exceed limit of 0.25 feet would apply instead.

Control Stations

Positional tolerances are derived from a Least Squares solution that gives consideration to stations that control the positions of dependent stations. Generally, these controlling stations are monuments of published legal position and the dependent stations are other monuments that are being tied or set.

Using Least Squares to Compute Positional Tolerance

In a Least Squares solution, the precision of the control stations is fixed, forcing the Least Squares solution to hold their positions. By editing the point records in the Least Squares analysis and ‘fixing’ the control stations, TPC can compute valid positional tolerances as part of the Least Squares solution.

The positional tolerances are reported in the Coordinate Error Ellipse section of the report. Just turn on Ellipse Data in the Report page and TPC will include this information in the report.

On the Report tab of the Least Squares dialog,  turn on the [x] Positional Tolerance toggle.

What is Reported

TPC reported the diameter of the positional tolerance circle about the adjusted point (station). It uses one-half the semi-major axis reported in the Error Ellipse section and further reduces this value by the Ellipse Scale Factor so that you end up with a value in actual feet or meters.

In a network, there is no specified path along with the positional tolerance is computed. The network computes the position of the station independent of a specific path. So the distance portion of the 'traditional' or 'traversed' Positional Tolerance is not relevant in a network solution. Instead, we simply interpret this value as the 'not to exceed' portion of the Positional Tolerance.

Also see Relative Positional Accuracy.

Related Topics

Least Squares
Least Squares Overview
Least Squares Blunder Detection


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