Description:

Most of this chapter, so far, has been concerned with applying least squares principles to estimation in two dimensions. The principles and processes are exactly the same in three dimensions except that everything is 50% bigger. The x vector will contain additional terms for the heights of the points and theA matrix will contain observations for difference height by levelling, slope distances as opposed to horizontal distances in two dimensions and vertical angles.

The A matrix:
The A matrix and the observed minus computed vector are constructed in exactly the same way in three dimensions as they are in two dimensions. The only major difference is that in three dimensions there are three parameters, δEi, δNi and δHi, associated with each point. The horizontal angle observation equation remains unchanged except to add 0s as the coefficients of the δHs.

Slope distance equation:
The slope distance equation is derived by applying Pythagoras’ theorem in three dimensions so the observation equation is

Upon linearizing and putting into matrix form this becomes:

where aij is the bearing of j from i and vij is the vertical angle to j from i and is found from provisional values of coordinates:

Vertical angle equation:

The vertical angle distance equation is:

Upon linearizing and putting into matrix form this becomes: