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 [C^{T}WK]
Multiply [C^{T} ] x [W] and [C^{T}W] x [C]
Multiply [C^{T}W] x [K]
Invert [C^{T}WC]
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.
d. Adjustment Statistics
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 S_{o} increased, it's a better overall indicator of the mixed quality observations.