### 4. Example

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

 Figure D-2Level Circuit
 Obs Line dElev Obs Line dElev 1 BMA-Q +8.91 5 BMA-R -3.56 2 BMB-Q -2.92 6 BMC-R -17.12 3 BMC-Q -4.67 7 BMD-R -21.10 4 BMD-Q -8.66 8 Q-R -12.47
##### (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

##### (3) Solve Unknowns: [U] = [Q] x [CTK]

[CT] x [C]

[CT] x [K]

Invert [CTC] to get [Q]

[Q] x [CTK]

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

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

Compute So

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

Adjusted observation uncertainties using Equation D-7.

Obs 1

First row of [C] and first column of [CT].

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 [CT].

Obs 8

Eighth row of [C] and eighth column of [CT].

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.

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

 Point Elevation Std Dev Q 815.418 ±0.013 R 802.962 ±0.014