## WEIGHT MATRIX

**Description:
**

Different surveyors may make different observations with different types of instruments. Therefore the quality of the observations will vary and for a least squares solution to be rigorous the solution must take account of this variation of quality. Consider a grossly overdetermined network where an observation has an assumed value for its own standard error, and thus an expected value for the magnitude of its residual. If all the terms in the observation equation are divided by the assumed, or a priori, standard error of the observation, then the statistically expected value of the square of the residual will be 1.

If all the observation equations are likewise scaled by the assumed standard error of the observation, then the expected values of all the residuals squared will also be 1. In this case the expected value of the mean of the squares of the residuals must also be 1. The square root of this last statistic is commonly known as the ‘standard error of an observation of unit weight’, although the statistic might be better described as the square root of the ‘variance factor’ or the square root of the ‘reference variance’.

If all the terms in each observation equation are scaled by the inverse of the standard error of the observation then this leads to the solution:

whereW is a diagonal matrix and the terms on the leading diagonal, wii, are the inverse of the respective standard errors of the observations squared and all observations are uncorrelated. More strictly, the weight matrix is the inverse of the variance-covariance matrix of the ‘estimated observations’, that is the observed values of the observations. If:

where σ(b) is the variance-covariance matrix of the estimated observations, σ11 is the variance of the first observation and σ23 is the covariance between the 2nd and 3rd observations, etc., then

In most practical survey networks it is assumed that all the off-diagonal terms in σ(b) −1 are 0. In other words there are no covariances between observations and so all observations are independent of each other. This will usually be true. An exception is where a round of horizontal angles has been observed at a point. If there is an error in the horizontal pointing to one point and that pointing is used to compute more than one horizontal angle then the error in the pointing will be reflected in both computed horizontal angles in equal, Figure and possibly opposite, Figure amounts. In this case the observations will be correlated, positively and negatively, respectively.

The problem is often ignored in practice but a rigorous solution may still be achieved, if observation equations are formed for directions rather than angles. If there are no covariances, then σ(b) becomes: