## 1. General

Since the MPV is the value which meets the least squares criterion, why not compute every possible value of the each unknown and then average them? Depending on the number of observations and unknowns, this can involve a lot of computations. We can refer to this approach as the Brute Force method of LS adjustment.Will it work in all situations?

## 2. Level Circuit LS Adjustment

For the leveling circuit in Figure B-1

 Figure B-1 Level Circuit

and these observed elevation differences

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

what are the elevations of points Q and R?

A traditional simple adjustment is difficult to apply in this situation. Because point Q is connected to four benchmarks and point R to three, any simple adjustment attempted will result in multiple elevations for each point.

With a small network like this, a Brute Force approach might be feasible. Using the observed elevation differences, compute all possible elevations for points Q and R, then simply average them.

Point Q elevations and their average are:

Similarly, for point R:

Compute So

Compute residuals using the adjusted elevations and the individually computed elevations used for the averages.
 v = MPVQ-ElevQ v2 sums -0.015 0.000225 +0.015 0.000225 -0.035 0.001225 +0.035 0.001225 0.002900 v = MPVR-ElevR v2 +0.007 0.000049 -0.033 0.001089 +0.027 0.000729 0.001867 0.004767
There are seven observations (m) and two unknown elevations (n).

## 3. Will Brute Force Work in All Situations?

Add an observation, dElev(8)=-12.47, between points Q and R, Figure B-2.

A fourth elevation for point R must be computed from point Q using the eighth observation. But which elevation of point Q should be used? Any of them? All of them? Its average?

The issue is more fundamental. When two unknowns are connected with a measurement, both are affected; after all, it's a network. Even though the observation is from point Q to point R, point Q's elevation is also affected. So point R's elevation depends on point Q's and point Q's elevation depends on point R's....

Adding another observation in the opposite direction from point R to point Q, Figure B-3, further complicates it.