Question 2 of 12: Measurement of Horizontal Directions by Reiteration, and its Checks

Question 2: Measurement of Horizontal Directions by Reiteration, and its Checks (14 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(1) The reiteration (directions) method. When several targets must be observed from one station, the directions method measures them all in a single coordinated set rather than as isolated angles. The procedure is:

1. Choose a reference object (RO). Level the instrument carefully and select a sharp, distant target as the zero or reference direction. Set the horizontal circle to (or near) $0^\circ00'00''$ on the RO.
2. Observe the half set on Face Left. With the telescope in Face Left (direct), point and read the RO, then turn only clockwise and point-and-read every target in turn — $2,3,\dots,n$ — recording the horizontal circle reading to each.
3. Close the horizon. Continue turning clockwise back onto the RO and re-read it. The change in the RO reading is the horizon closure.
4. Observe the second half set on Face Right. Reverse the telescope (transit), and observe the same targets in the reverse order (anticlockwise), reading each. Face-Left + Face-Right = one full set.
5. Reduce to directions. Mean each target’s two-face readings, then subtract the (meaned) RO reading so that every direction is referred to the RO as zero.
6. Repeat for additional full sets. For higher orders take several sets, each begun on a different part of the circle (circle re-orientation by $180^\circ/m$ for $m$ sets) to randomise graduation errors, and mean the sets.

Because all targets share one circle orientation, the reiteration method is more efficient and internally more consistent than measuring each angle separately, and it distributes the graduation and centring errors evenly among all the directions.

(2) The checks and their permissible values. Three nested checks apply at increasing scope:

• Within a half set — horizon closure (the “closing the horizon” check). The RO is read at the start and again at the end of the half set; the difference (the closure) reflects instrument drift, tribrach instability or pointing error accumulated during the half set. It must not exceed a specified tolerance (typically a small multiple of the pointing standard deviation, e.g. a few seconds for a 1″ instrument). If it fails, the half set is rejected and re-observed.
• Within a full set — the two-face difference (2C / collimation check). For each target, Face-Left minus (Face-Right ± $180^\circ$) gives twice the horizontal collimation ($2C$). $C$ should be sensibly constant for all targets in the set; a target whose $2C$ departs markedly from the others signals a blunder. The spread of $2C$ across targets must stay within tolerance.
• Among multiple full sets — the set-to-set spread of each reduced direction. Once reduced to the RO, each direction should reproduce from set to set within a permissible range (the range of the several set values, or their standard deviation, compared to a stated limit). A set whose value is an outlier is re-measured; the accepted sets are meaned to give the final direction.