With the advent of total station instruments, survey data are being collected in three dimensions. Thus, it is advantageous to develop an adjustment model that works in three dimensions. Rigorous triangulation adjustment models date back to Bruns (1878). The main observational data consists of horizontal angles, vertical angles, azimuths, and slant distances. It is also possible to include differential leveling in the model. Since all data are collected on the Earth's surface, the local geodetic coordinate system provides a natural system in which to perform the adjustment.
As shown in Figure 23.1, the local geodetic system is oriented such that the n axis points along the meridian of the ellipse (local geodetic north), the u axis is aligned along the normal of the ellipsoid, and the e axis creates a right-handed coordinate system. The local geodetic coordinate system can be related to the geocentric coordinate system (see Section 17.5) through a series of three-dimensional rotations discussed in Section 18.7. To align the X axis with the e axis, the Z axis is rotated by an amount of λ − 180°. Then the Z axis is aligned with the u axis by a rotation of φ − 90° about the once rotated X axis. The resultant expression is