ODM principle, tacheometric field procedure

Principle of optical distance measurement (ODM)

Optical distance measurement finds the horizontal distance to a point from the angle subtended at the instrument by a known length (a staff intercept or a bar), using only angles and the geometry of similar triangles. No tape is used. Two types are common.

• Stadia (tacheometric) method: The diaphragm of the theodolite carries two extra horizontal hairs (stadia hairs) at a fixed spacing $i$. The rays through these hairs make a fixed angle at the instrument, so the staff intercept $s$ (top reading minus bottom reading) is proportional to the distance:

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

For an anallactic lens the additive constant is zero, so $D = k\,s$ with multiplying constant $k = f/i = 100$ and $c = 0$.

• Subtense method: a bar of known length $b$ is set perpendicular to the line of sight and the small angle $\theta$ it subtends is measured: $D = \frac{b}{2}\cot\frac{\theta}{2}$.

For a vertical staff and a line of sight inclined at angle $\theta$: $H = k s\cos^2\theta$ and $V = \frac{ks}{2}\sin 2\theta$.

Field procedure of tacheometric survey for a topographic map

1. Reconnaissance and control: Fix a traverse around the area with theodolite stations, and find their coordinates and RL (traverse computation plus levelling or trigonometric levelling from a BM).
2. Setting up: Set up the theodolite over a control station, level it and measure the height of the instrument (HI) with a tape.
3. Orientation: Sight a back station, set the horizontal circle to its known bearing.
4. Detail observation: Hold the staff vertically at each detail point (corners of buildings, road edges, streams, spot heights, break points of slope). For each point read:
   • the three hair readings (top, middle, bottom),
   • the horizontal circle (bearing or angle from the reference line),
   • the vertical circle (vertical or zenith angle) with the middle hair on the staff.
5. Booking: Enter the readings in the field book with a sketch of the detail and the point number.
6. Computation: Find $H$, $V$ and the RL of each point: $RL = RL_{station} + HI + V - r$ ($r$ = middle hair reading).
7. Plotting: Plot the control stations by coordinates, then plot each detail point by its bearing and horizontal distance to scale. Write the RL beside the point and interpolate contours between points.

Points are spaced closer where the ground changes slope suddenly. Check readings by taking a few points from two stations.