If an observation contains a gross error and it is included with a set of otherwise good observations then the least squares process will accommodate it by distorting the network to make it according to the normal least squares criteria. A quick scan down a list of residuals for the largest may identify the erroneous observation but if there are several gross errors in the set of observations then there will be many large residuals. However, by computing the statistic of residual divided by its own standard error it is possible to identify the most significant gross error.
The gross error may be dealt with either by correcting the error if it can be traced or by eliminating the rogue observation from the set. Now the next worst gross error may be traced and so on until all gross errors have been removed from the set of observations. The standard error of a residual may be found as the square root of the element of the leading diagonal of the variance-covariance matrix of the residuals.
The variance-covariance matrix of the estimated residuals:
The original least squares problem:
was solved for the parameters as:
and so, on putting the estimated value of x back into the original equation, the estimated residuals become:
Using the Gauss propagation of error law:
which, when terms, preceded by their own inverse, are accounted for, simplifies to:
The matrixA(ATWA)−1AT will be a large one if the number of observations is large. It is unlikely, however, that the covariances between derived observations will be required. If so, then only the leading diagonal of the matrix will be of interest. By considering only non-zero products of the terms of A(ATWA)−1AT it can be shown that the variance of an observation may be computed from:
where a1, a2, a3, a4, a5 and a6 are the non-zero terms of the row of theA matrix relating to the observation concerned and the variance-covariance matrix is a sub-set of the already computed (ATWA)−1 where all the rows and columns containing variances and covariances of parameters not represented in a1 to a6 have been removed. The above example might be for a derived angle as there are six terms involved. If the variance for a derived distance was required then the matrix product would look like this.
The variance of a derived observation that was not observed may also be found in exactly the same way. The terms a1 to a4 are computed as if an observation had actually been made and the variance is computed exactly as above.
The test to be applied is to compare the estimated residual with its own standard error. The idea is that if the residual is a significant multiple of its own standard error, then, in a large network, there will be cause for concern that the error in the observation is more than random and so the observation will need to be investigated. Although there are strict statistical tests they do not take account of the fact that the residuals are usually correlated. Therefore the simple rule of thumb often used is that if the residual divided by its own standard error is within the range of plus or minus 4, then the observation will be accepted ascontaining no significant non-random error, i.e. that: