6. Example Adjustment
Given the network and data in Figure F-5, determine:
- adjusted coordinates of points J and K.
- their standard errors
- adjusted observations
- their standard errors
Figure F-5 Combined Observations |
There are seven observations and four unknowns: N_{J}, E_{J}, N_{K}, E_{K}. The network has 7-4 = 3 DF.
a. Matrix structures
Matrix structures are:
b. Initial approximations for points K and J
The computations are summarized and results shown:
- Distance and direction for line BA by inverse
- Solve angle ABK using Law of Cosines: 28°28'09"
- Compute direction of BK using direction BA and angle ABK: 66°33'27"
- Use direction BK and 1309.94' to compute coordinates of K: 5097.77' N, 3312.73' E
- Solve angle JBA using Law of Cosines: 11°56'48"
- Compute direction of BJ using direction BA and angle JBA: 26°08'30"
- Use direction BJ and 871.35' to compute coordinates of J: 5358.86' N, 2494.82' E
c. Compute weight matrix
d. Distances
The distances are computed from the fixed coordinates of A and B and the initial approximations of J and K.
e. Build [C] and [K] matrices
Note: it is extremely important to minimize rounding errors by carrying enough digits in computations. Not doing so can either increase the number of iterations needed for a solution or could cause the solution to diverge.
(1) Distance observations
Obs 1: Line KB
Obs 2: Line KJ
Obs 3: Line KA
Obs 4: Line JA
Obs 5: Line JB
(2) Angle observations
Obs 6: Angle BKJ
Obs 7: Angle JKA
(3) Assemble [C] and [K] matrices
f. Solve U=[Q] x [C^{T}WK]
Rather than detail the complete solution process, the mtrices and partial products are shown for each iteration.
(1) First Iteration
The corrections aren't small enough so update the coordinates and repeat solution.
Updated coordinates:
Repeat as corrections are not small enough.
(2) Second Iteration
Using the updated coordinates, go back to Step d, recompute distances and matrices, and solve for [U].
The corrections still aren't small enough so update the coordinates and repeat solution.
Updated coordinates:
(3) Third Iteration
Corrections are acceptably small enough. Update coordinates one last time.
g. Compute statistics
(1) Compute S_{o} and adjusted point uncertainties
Using last [C], [K] and [U] matricies, determine residuals from [V] = [C] x [U] - [K]
Use [V] and [W] to compute S_{o} , then S_{o} and [Q] to compute standard deviations of the adjusted coordinates.
(2) Adjusted observations
Add residuals to the original observations
Compute standard errors for the adjusted observations. Example comps are shown for distance KB and angle BKJ.
Obs 1: distance KB
Use first row of [C] and first column of [C^{T}].
Obs 6: Angle BKJ
Use sixth row of [C] and sixth column of [C^{T}].
Remaining standard errors are included in the following section.
h. Adjustment Summary
Degrees of freedom: DF = 7-4 = 3
Std Dev Unit Wt: So = ±2.136'
Point | North | East | S_{N} | S_{E} |
J | 5359.086' | 2494.852' | ±0.126 | ±0.210 |
K | 5097.253' | 3313.197' | ±0.167 | ±0.076 |
Adj Obs | S | |
Dist KB | 1310.671' | ±0.037' |
Dist KJ | 858.663' | ± 0.023' |
Dist KA | 938.422' | ± 0.033' |
Dis JA | 1019.338' | ± 0.023' |
Dist JB | 866.611' | ± 0.022' |
Ang BKJ | 40°47'23.4" | ±09.2" |
Ang JKA | 68°56'54.8" | ±13.0" |