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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