The principle of least squares adjustment with condition equations can be summarised as below (Gotthardt 1978; Cui et al. 1982):
1. The linearised observation equation system can be represented by Eq. 7.1 (cf. Sect. 7.2).
2. The corresponding condition equation system can be written as
C : coefficient matrix of dimension r x n,
W: constant vector of dimension r, and
r : number of conditions. 3. The least squares criterion for solving the observation equations with condition equations is well-known as
where K is a gain vector (of dimension r) to be determined.
The function F reaches minimum value if the partial differentiation of F with respect to X equals zero, i.e.,
then one has
5. Combining Eqs. 7.43 and 7.39 together, one has
7. The precisions of the solutions are then
This can be obtained by substituting Eq. 7.1 into VTPV and using the relations of Eqs. 7.39 and 7.42.
Up to now the complete formulas of conditional least squares adjustment have been derived.
Sequential Application of Conditional Least Squares Adjustment
Recalling the least squares adjustment discussed in Sect. 7.2, the linearised observation equation system
has the solution
The precisions of the solutions can be obtained by
and VTPV can be calculated by using
Similarly, in the conditional least squares adjustment discussed in Sect. 7.4, the linearised observation equation system and conditional equations read
the solution follows
where K is the gain, and
The precision vector of the solution vector can be obtained by using Eqs. 7.48-7.52. Using the notations obtained in least squares solution, one has
Equation 7.62 indicates that the conditional least squares problem can be solved first without the conditions, and then through the gain K to compute a modification’s term. The change of the solution is caused by the conditions. For computing the weighted squares of the residuals, Eq. 7.63 can be used (by adding two modification’s terms to the weighted squares of residuals of the least squares solution). This property is very important for many practical applications such as ambiguity fixing or coordinates fixing. For example, after the least squares solution and fixing the ambiguity values, one needs to compute the ambiguity fixed solution. Of course, one can put the fixed ambiguities as known parameters and go back to solve the problem once again. However, using the above formulas, one can use the fixed ambiguities as conditions to compute the gain and the modification’s terms to get the ambiguity fixed solution directly. Similarly, this property can be also used for solutions with some fixed station coordinates.