Conditional Least Squares Adjustment (Adjustment and Filtering Methods) (GPS)

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

tmpD-387_thumb

where

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

tmpD-388_thumb

wheretmpD-389_thumbis the transpose of the related vector V. 4. To solve X and compute V, a function F can be formed as


tmpD-391_thumb

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.,

tmpD-392_thumb

then one has

tmpD-393_thumb


or

tmpD-394_thumb

wheretmpD-395_thumbare transpose matrices of A and C, respectively.

5. Combining Eqs. 7.43 and 7.39 together, one has

tmpD-397_thumb

6. For simplification, lettmpD-399_thumbwhere superscript 1 is an inverse operator. The solutions of Eqs. 7.44 and 7.45 are then

tmpD-401_thumb

7. The precisions of the solutions are then

tmpD-402_thumb

where i is the element index of a vector or a matrix,tmpD-403_thumbis the so-called standard deviation (or sigma),tmpD-404_thumbelement of the precision vector,tmpD-405_thumbis the tmpD-406_thumbdiagonal element of the quadratic matrixtmpD-407_thumband

tmpD-413_thumb

8. For convenience of sequential computation,tmpD-414_thumbcan be calculated by using

tmpD-416_thumb

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

tmpD-418_thumb

has the solution

tmpD-419_thumb

The precisions of the solutions can be obtained by

tmpD-420_thumb

where

tmpD-421_thumb

and VTPV can be calculated by using

tmpD-422_thumb

For convenience, the least squares solution vector is denoted bytmpD-423_thumband weighted residuals square bytmpD-424_thumb

Similarly, in the conditional least squares adjustment discussed in Sect. 7.4, the linearised observation equation system and conditional equations read

tmpD-427_thumb

the solution follows

tmpD-428_thumb

where K is the gain, and

tmpD-429_thumb

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

tmpD-430_thumb

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.

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