Inaccessible object, stations in different vertical plane

When the base of the object (tower, chimney, hill top) is inaccessible and the two instrument stations are not in the same vertical plane as the object, the horizontal distance to the object is found by the triangulation (base line) method and the height by trigonometric levelling.

Field work

1. Select two stations X and Y a convenient distance $b$ apart (base line) so that P is visible from both and the triangle XYP is well conditioned (angles at X and Y between 30° and 120°).
2. Measure the base line XY and level the instrument at X. Take a back-sight on a BM of known RL to get the RL of the instrument axis; do the same at Y.
3. At X measure the horizontal angle $\alpha$ = $\angle$YXP (face left and right) and the vertical angle $\theta_X$ to P. At Y measure $\beta$ = $\angle$XYP and the vertical angle $\theta_Y$.

                 P (top)
                /  \
               /    \
          XP  /      \ YP
             /        \
            /α        β\
           X ---- b ---- Y
         (plan view)

Computation

In the plan triangle XYP, $\angle$XPY $= 180^\circ - (\alpha + \beta)$. By the sine rule:

$$\begin{aligned} XP &= \frac{b\sin\beta}{\sin(\alpha+\beta)} \\ YP &= \frac{b\sin\alpha}{\sin(\alpha+\beta)} \end{aligned}$$

These are horizontal distances. The height of P above each axis:

$$h_X = XP\tan\theta_X, \qquad h_Y = YP\tan\theta_Y$$

(for zenith angle $z$, use $\theta = 90^\circ - z$.)

RL of P from each station:

$$\begin{aligned} RL_P &= RL_{BM} + BS_X + h_X \quad (\text{from X}) \\ RL_P &= RL_{BM} + BS_Y + h_Y \quad (\text{from Y}) \end{aligned}$$

where $BS$ is the back-sight on the BM (axis RL = RL of BM + BS). The mean of the two values is taken as the RL of P. If the two stations are on very different levels, the axis RLs are found by BM sights at each station, as above. If the angle of elevation is given from the station peg, add the height of instrument $h_i$ to the RL of the peg.

The two values should agree closely; a large difference shows an error in the angles or the base line.