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

jj

jj

ii

ii

jjiijjii

dyyFdx

xFdy

yFdx

xF

yxyxFyxyxF

0000

),,,(),,,(0000

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