Stadia principle and field procedure

Principle of stadia tacheometry: the telescope has two extra horizontal hairs (stadia hairs), one above and one below the central cross-hair. The rays through these hairs make a fixed small angle at the focus, so the staff intercept $s$ between them is proportional to the distance of the staff from the instrument.

 Diaphragm     Objective              Staff
   a |           |   \                  A  top
     | i         |     \  ...           |
  ---+-----------+-------F--------------+---- axis
     |           |      /               |
   b |           |   /                  B  bottom
     |<-- f --->|<--- D1 ---->|

Horizontal sight, staff vertical

From the similar triangles formed at the external focus $F$ (stadia hair interval $i$, focal length $f$, intercept $s$):

$$\frac{D_1}{f}=\frac{s}{i}\ \Rightarrow\ D_1=\frac{f}{i}\,s$$

Adding the distance of the focus from the instrument axis, $(f+d)$, where $d$ is the distance from the objective to the vertical axis:

$$D=\frac{f}{i}\,s+(f+d)=Ks+C$$

$K=f/i$ is the multiplying constant (usually 100) and $C=f+d$ is the additive constant (about 0.3 m for an external focusing telescope, and zero for an internal focusing or anallatic telescope).

A horizontal sight gives $D=Ks+C$. For an inclined sight with a vertical staff, $D=Ks\cos^2\theta+C\cos\theta$ and $V=\tfrac12Ks\sin2\theta+C\sin\theta$.

Field procedure of a tacheometric survey for a topographic map

1. Reconnaissance and control: establish control stations (a traverse) over the area, and find the RL of a benchmark or station.
2. Set up the tacheometer on a station, then centre and level it. Measure the height of the instrument (HI) with a tape.
3. Orient: with the horizontal circle at zero (or at a known bearing), sight a known reference station or magnetic north.
4. Back-sight to a benchmark or known point with the staff to find the RL of the instrument axis.
5. Detail points: the staff-man holds the staff vertical at each ground feature (corners, break lines, spot heights). For each point:
   • bisect the staff and read the horizontal circle (bearing),
   • read the top, middle and bottom hairs (staff readings),
   • read the vertical angle, setting the middle hair at the same reading as the HI where possible (this simplifies the calculation) or at any convenient reading,
   • book the readings with a description of the point.
6. Check that the mean of the top and bottom hair readings equals the middle reading (within 0.005 m), and close on a known point at the end of the day.
7. Computation: $s=$ top $-$ bottom, $D=Ks\cos^2\theta$, $V=\frac{1}{2}Ks\sin2\theta$, RL $=$ RL of axis $\pm V-h$.
8. Plotting: plot each point by its bearing and horizontal distance (protractor and scale) or by coordinates, write its RL beside it, and interpolate contours for the topographic map.

Field book (typical)