2.3.3 Variance-component estimation

In addition to the magnitudes of the measured values, their standard deviations also influence the adjustment results. Also, reliable recognition of observation errors is only possible if the of the residuals have been calculated correctly. These in return depend on the standard deviations of the measured values. If there are several groups of different observations, a global s₀

r = n - u

(2.3.3/1)

of the adjustment is no longer sufficient for verification of the weight functions. A rigorous computation of s₀ values for each observation group is time consuming. A good approximation, however, is obtained from

(2.3.3/2)

As independence of all observations is assumed here, the r_G can be computed from (2.3.1/3) and (2.2/6):

(2.3.3/3)

Here only the diagonal is required from the matrix product . Therefore those elements of must be known which differ from zero also in the normal equation matrix N. Thus the complete inversion of the normal equation matrix can be avoided, a task which is impossible for large data blocks.

The a posteriori standard deviation calculated according to (2.3.3/2) is compared to the a priori function for each group. If there is sufficient redundancy in this group the a priori is to be changed by the user until convergence has been achieved. Then observation errors can generally be traced without difficulty.

(2.3.3/4)

Although this is an approximate method, the resulting estimation of convergence (quotient = 1.0) is unbiased.