## 4. Example

Adjust the same level circuit from Chapter D, Figure F-5.

 Figure F-5 Level Circuit

Additional information are the number of instrument setups on each line.

 Obs Line dElev Setups Obs Line dElev Setups 1 BMA-Q +8.91 3 5 BMA-R -3.56 3 2 BMB-Q -2.92 2 6 BMC-R -17.12 4 3 BMC-Q -4.67 1 7 BMD-R -21.10 2 4 BMD-Q -8.66 3 8 Q-R -12.47 3

### a. Observation equations

The observation equations are the same as in the Chapter D example.

### b. Set up matrices

The [C], [U], [K], and [V] matrices are also the same.

Compute weights from number of setups. Weights are inversely proportional to number of setups.

 Obs Line Setups Weight Obs Line Setups Weight 1 BMA-Q 3 1/3 5 BMA-R 3 1/3 2 BMB-Q 2 1/2 6 BMC-R 4 1/4 3 BMC-Q 1 1/1 7 BMD-R 2 1/2 4 BMD-Q 3 1/3 8 Q-R 3 1/3

The weights can be multiplied by 12 to make them integers. The weight matrix is:

### c. Solve Unknowns: [U] = [Q] x [CTWK]

Multiply [CT ] x [W] and [CTW] x [C]

Multiply [CTW] x [K]

Invert [CTWC]

Since this is a 2x2 matrix, it can be quickly inverted using its the determinant.

Compute the elevations

Carry enough significant figures to avoid rounding errors.

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

### e. Comparison of Unweighted and Weighted Adjustment

 Point Unweighted, So=±0.028 Weighted; So=±0.067 Q 815.418 ±0.013 815.424 ±0.012 R 802.962 ±0.014 802.958 ±0.016

Weighing observations changed the elevations changed slightly. Although So increased, it's a better overall indicator of the mixed quality observations.