Triangulation; EDM principle for trilateration sides

Triangulation is a method of horizontal control survey in which control stations are fixed at the vertices of a chain or net of triangles. One side (the base line) is measured very accurately, all the angles of the triangles are measured, and the lengths of the other sides are computed with the sine rule.

Trilateration is the method in which the lengths of all the sides of the triangles are measured directly with an EDM (electromagnetic distance measuring) instrument, and the angles are computed from the sides using the cosine rule.

Principle of EDM for measuring a side

An EDM instrument sends out a modulated electromagnetic wave (infrared, light or microwave) from one end of the line to a reflector (prism) at the other end. The wave returns to the instrument, and the distance is found from the phase difference between the transmitted and the received signals.

If the wavelength of the modulated wave is $\lambda$, the double path (to the reflector and back) contains $n$ whole waves and a part wave $\Delta\lambda$:

$$2D = n\lambda + \Delta\lambda \quad\Rightarrow\quad D = \frac{1}{2}\left(n\lambda + \frac{\phi}{2\pi}\,\lambda\right)$$

where $\phi$ is the measured phase difference (angle). The instrument measures $\phi$ accurately, so $\Delta\lambda$ is known, while the integer $n$ is resolved by using several modulation frequencies (a coarse frequency to give approximate distance and finer ones for precise value). The velocity of the wave depends on temperature, pressure and humidity, so meteorological corrections are applied. Alternatively, in a pulse instrument, the time $t$ taken by a pulse for the double path gives $D = \frac{c\,t}{2}$.

   EDM  ====== modulated wave =====>  reflector (prism)
   unit <=====   reflected wave ====
   phase difference -> distance D

Typical accuracy is $\pm (3\text{ to }5\ \text{mm} + 1\text{ to }2\ \text{ppm})$. The sides of the trilateration figure are measured this way, and the angles are computed with the cosine rule.