## 5. Examples

### a. Example 1

A TSI is used to measure an angle twice direct and reverse with a resulting average angle of 69°35'47". The BS mark is a tripod mounted target with a centering error of ±0.005 ft. The FS mark is a handheld prism pole. The bottom half of the pole is not visible because of the terrain so a centering error of ±0.02 ft will be used. The TSI has a 01" display and its centering error is ±0.003 ft. From the instrument manual, the TSI's stated ISO 17123:3 angle uncertainty is ±06". Approximate horizontal distances to site marks are shown in Figure E-12.

Figure E-12 Example 1 |

What is the expected error in the measured angle?

*TSI Pointing and Reading error *

*TSI Centering error*

To determine this error we need the distance between the targets. This can be computed from the Law of Cosines using the measured angle and distances to the target:

then

*Target Centering error*

*Combined*

The expected error in the angle is ±0°00'19"

*Analysis*

What's the largest source of error? Target centering, the prism pole in particular, is the largest error contributor.

If we replace the pole with a mark similar to that of the BS, the target centering error drops to ±07.23" and expected angular error to ±00°00'10."

### b. Example 2

The angle shown in Fig 13 will be measured with a TSI having an ISO 17123:3 angle uncertainty of ±05" and centering error of ±0.003 ft. Both BS and FS marks are tripod mounted, also with a centering error of ±0.003 ft.

Figure E-13 Example 2 |

How many D/R sets must be turned to achieve an expected angular error no greater than ±09"?

The only error affected by the number of angles turned is TSI Pointing and Reading, Eqn (E-3). After substituting TSI Centering and Target Centering errors in Eqn (E-6), it should be solved for the TSI Pointing and Pointing error. Equation (E-3) can then be solved for n.

*Compute TSI Centering error, E _{tsi}*

Compute distance between targets

compute the error

*Compute Target Centering error, E _{t}*

*Set up Equation (E-6) and solve for n*

Square both sides, sort terms, and replace E_{pr} with Eqn (E-3)

solve for n:

6.49 is the minimum value for n to achieve an expected accuracy of ±09"

n is the number of times the angle is measured, direct or reverse. Since we always measure direct and reverse sets, n must be an even integer. Since it's not possible to measure 6.49/2 = 3.245 D/R sets, n must be 6 (3 D/R) or 8 (4 D/R).

For n=6, the angular error, E_{ang}, is 09.1" ±juuuust outside the criterion.

For n=8 the angular error is ±08.8".

To meet the ±09" criteria, measure 4 D/R