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: NJ, EJ, NK, EK.  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 [CTWK]

Rather than detail the complete solution process, the matrices and partial products are shown for each iteration. Just remeber that after each iteration, the entire [C] and [K] matrices must be recomputed. 

(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 So 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 So , then So 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 [CT].


Obs 6: Angle BKJ

Use sixth row of [C] and sixth column of [CT].


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  SN SE
5359.086' 2494.852' ±0.126  ±0.210 
5097.253' 3313.197' ±0.167  ±0.076 




  Adj Obs
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"