RELIABILITY OF THE OBSERVATIONS
Where there are many more observations than the strict minimum necessary needed to solve for the unknown coordinates then all the observations give a degree of independent check to each other. Where only the minimum number of observations has been made then there is no independent check upon those observations and they will all be unreliable in the sense that if one or more of them are grossly in error there is no way that the error can be detected. Hence any coordinates computed using a grossly erroneous observation will also be grossly in error. Figure 7.12 illustrates the problem where points A and B are fixed and points C and D are to be found.
In Figure the angles in the triangle ABC and the distances AC and AB have been measured with a low quality total station but the angle DAB and the distance AD have been measured with a high precision instrument. Therefore it would be expected that the uncertainty of the coordinates of C would be greater than those of D. However, the coordinates of C would be very reliable because point C is connected to the fixed points, A and B, by five observations. There are five observations to calculate the two coordinates of pointC, so three of the observations are ‘redundant’.
If one of those observations had a gross error, that error would be easy to detect and deal with. By contrast the coordinates of D would be very unreliable because that point is connected to the fixed points by only two observations. There are only two observations to calculate the two coordinates of point D and therefore a redundancy of zero. If one of those observations had a gross error the error would be undetectable.
The variance-covariance matrix of the estimated observations:
The estimated observations are the observations computed from the estimated coordinates. These contrast with the observations actually observed by the surveyor with survey instruments. By a derivation similar to that of the section on the variance-covariance matrix of the residuals above it can be shown that the variance-covariance matrix of the estimated observations, σ(l), is
Precision and reliability
It will be noted that this matrix has already been computed as part of the process needed to compute the variance-covariance matrix of the residuals and so very little extra work will be required. In this matrix the leading diagonal contains the squares of the standard errors of the estimated observations and the off-diagonal terms are the covariances between them. It is only the leading diagonal that is likely to be needed.
By comparing the standard error of an estimated observation with the standard error of the equivalent observed observation, it is possible to see what the estimation process has done to improve the quality of the observed observation to that of the estimated observation. If there is no improvement then that implies that, whatever the value of the observation, it has been accepted unmodified by the estimation process and therefore the observation is unreliable. Any coordinates that are computed from that observation would also be unreliable. It is likely therefore that if the unreliable observation is removed from the estimation process then the normal equations matrix will become singular and the parameter vector will need to be modified to remove the point that depended only upon the unreliable observation.
Although totally unreliable observations may be identified, all observations have different levels of reliability in the network solution. If the ratio of the variances of the computed and observed values of the observation is calculated then this ratio may be tested in an ‘F test’ to determine its statistical significance. This may be tedious for all observations, and a general rule of thumb is that the relationship:
σ(l) is the standard error of an estimated observation and σ(b) is the standard error of an observed observation. If this test is not satisfied then the observation is considered unreliable and further observations should be undertaken to improve the quality of the network in the suspect area.