4. Example

Adjust the level circuit of Chapter C shown in Figure D-2.

Figure D-2 Level Circuit

(1) Observation Equations

Use Equation D-4 as the initial observation equation format, then rearrage to place the unknowns on the left side, constants and residuals on the right.

Equation D-43

(2) Set up Matrices

(3) Solve Unknowns: [U] = [Q] x [C^TK]

[C^T] x [C]

[C^T] x [K]

Invert [C^TC] to get [Q]

[Q] x [C^TK]

These are the same elevations as in the Direct Minimization method.

(4) Adjustment Statistics

Residuals: [V] = [CU] - [K]

Compute S_o

Standard deviations for points Q and R elevations using Equation D-2

Adjusted observations

Adjusted observation uncertainties using Equation D-7.

Obs 1

First row of [C] and first column of [C^T].

Because rows 1-4 of the C matrix are the same, all four observations will have the same expected error.

Ditto for observations 5-7

Obs 5

Fifth row of [C] and fifth column of [C^T].

Obs 8

Eighth row of [C] and eighth column of [C^T].

Outside of having the same [C] coefficients, why do observations 1-5 have the same expected errors? Because each connects a benchmark to the adjusted point Q (that's why they have same rows in [C]). The only error affecting those observations is point Q's. Similarly, observations 5-7 are only affected by point R. Only the eighth observation has uncertainties at both ends.

(5) Adjustment Summary

Degrees of freedom:  DF = 8-2 = 6
Std Dev Unit Wt: So = ±0.028

Adjusted elevations

Adjusted observations