In control surveying, making measurements and computing coordinates of points from those measurements is only one half of the surveyor’s business. The other half is error management, which is concerned with assessing the quality of the work and drawing the appropriate conclusions. In the following sections various quality indicators are considered.
The first set of statistics that are available from the least squares computation are the residuals of each of the observations. A residual gives an indication of how well a particular observation fits with the coordinates computed from that, and all the other observations.Aquick glance at a set of residuals will give an indication if there are any observations that have a gross error. One gross error will distort the whole network but its worst effect will be in the residual associated with the erroneous observation.
The fact that all residuals are large does not necessarily indicate that there is more than one gross error. In this case, however, the observation with the largest residual, ignoring weights, will probably be the one that is in error. The residuals are computed by putting the final computed values of the parameters, with the observed value of the observation into the original observation equation. In matrix terms:
where A and b are computed using the final values of x. A check can be applied to the computation at this stage. Pre-multiplying both sides of the above equation by ATW gives:
but in the derivation of the least squares formulae it was shown that:
which are the normal equations, and therefore,
How close ATWv is to a null vector will give an indication of the arithmetic correctness of the solution and of its completeness, i.e. have there been sufficient iterations