When surveying data are collected, they usually must conform to a given set of geometric conditions, and when they do not, the observations are adjusted to force that geometric closure. For a set of uncorrelated observations, a measurement with a high precision, as indicated by a small variance, implies a good observation, and in the adjustment it should receive a relatively small portion of the overall correction. Conversely, a measurement with a lower precision as indicated by a larger variance implies an observation with a larger error, and thus it should receive a larger portion of the correction.
The weight of an observation is a measure of an observation's relative worth compared to other observations. Weights are used to control the sizes of corrections applied to observations in an adjustment. The more precise an observation is, the higher its weight should be; in other words, the smaller the variance, the higher the weight. From this analysis, it can be stated intuitively that weights are inversely proportional to variances. Thus, it also follows that correction sizes should be inversely proportional to weights.
In situations where observations are correlated, weights are related to the inverse of the covariance matrix, ∑. As discussed in Chapter 6, the elements of this matrix are variances and covariances. Since weights ...