1. Adjustment of Trilateration

2. Introduction
For a brief period of time, trilateration was a highly effective method of establishing horizontal control coordinates.
This was after EDM technology matured but before GPS
Measuring distances-only avoided the more labor-intensive task of angle measurement.
This was a role-reversal, since previously, angle measurement was the easier operation

3. Intro - Continued
These trilateration surveys generally covered a large area
Correction for earth curvature needed to be applied
For some cases, state plane coordinate systems were used as a coordinate base
Often coordinates are large, so offsets may be subtracted prior to adjustment and then added after the adjustment

4. Distance Observation
What is the observation? What are the unknowns?

5. Taylor’s Series
The observation equation is non-linear so Taylor’s Series is applied.

6. Partial Derivatives

7. Distance Observation Equation

8. Example 13.2

9. Substitution for I and J in Equation
I J
A U
B U
C U
Note: In this example, the unknown station U is always in the J station position of the prototype line.

10. Matrix Form

11. Initial Approximations

12. Approximations - Continued

13. Approximations - Continued

14. Set Up Matrices

15. Set Up Normal Equations

16. Solve and Update

17. Second Iteration

18. Form Matrices

19. Update
Further iterations are negligible.

20. Post Adjustment Statistics

21. Statistics - Continued

22. Estimated Errors of Adjusted Observations

23. Solution Summary

24. More Complex Network
The same process applies to more complex networks. You just need to be careful with selection of I and J

25. Iteration Termination
Maximum number of iterations
Correction threshold
Convergence of estimate of standard deviation of unit weight