Transcript Adjustment of Trilateration

Adjustment of Trilateration
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
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
Distance Observation
What is the observation? What are the unknowns?
Taylor’s Series
The observation equation is non-linear so Taylor’s Series is
applied.
F ( xi , yi , x j , y j )  F ( xi0 , yi0 , x j0 , y j0 )
 F 
 F 
 F 
 F 
   dxi    dyi    dx j    dy j
 xi  0
 yi  0
 x j  0
 y j  0
Partial Derivatives
Distance Observation Equation
Example 13.2
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.
Matrix Form
Initial Approximations
Approximations - Continued
Approximations - Continued
Set Up Matrices
Set Up Normal Equations
Solve and Update
Second Iteration
Form Matrices
Update
Further iterations are negligible.
Post Adjustment Statistics
Statistics - Continued
Estimated Errors of Adjusted
Observations
Solution Summary
More Complex Network
The same process applies to more complex networks. You just
need to be careful with selection of I and J
Iteration Termination
• Maximum number of iterations
• Correction threshold
• Convergence of estimate of standard deviation
of unit weight