1. Weights of Observations

2. Introduction
Weights can be assigned to observations according to their relative quality
Example: Interior angles of a traverse are measured – half of them by an inexperienced operator and the other half by the best instrument person. Relative weight should be applied.
Weight is inversely proportional to variance

3. Relation to Covariance Matrix
With correlated observations, weights are related to the inverse of the covariance matrix, Σ.
For convenience, we introduce the concept of a cofactor. The cofactor is related to its associated covariance element by a scale factor which is the inverse of the reference variance.

4. Recall, Covariance Matrix
For independent observations, the off-diagonal terms are all zero.

5. Cofactor Matrix
We can also define a cofactor matrix which is related to the covariance matrix.
The weight matrix is then:

6. Weight Matrix for Independent Observations
Covariance matrix is diagonal
Inverse is also diagonal, where each diagonal term is the reciprocal of the corresponding variance element
Therefore, the weight for observation i is:
If the weight, wi = 1, then
is the variance of an observation of unit weight (reference variance)

7. Reference Variance It is an arbitrary scale factor (a priori)
A convenient value is 1 (one)
In that case the weight of an independent observation is the reciprocal of its variance

8. Simple Weighted Mean Example
A distance is measured three times, giving values of 151.9, 152.5, and Compute the mean.
Same answer by weighted mean. The value appears twice so it can be given a relative weight of 2.

9. Weighted Mean Formula

10. Weighted Mean – Example 2
A line was measured twice, using two different total stations. The distance observations are listed below along with the computed standard deviations based on the instrument specifications. Compute the weighted mean.
D1 = m σ1 = m D2 = m σ2 = m
Solution: First, compute the weights.

11. Example - Continued Now, compute the weighted mean.
Notice that the value is much closer to the more precise observation.

12. Standard Deviations – Weighted Case
When computing a weighted mean, you want an indication of standard deviation of observations.
Since there are different weights, there will be different standard deviations
A single representative value is the standard deviation of an observation of unit weight
We can also compute standard deviation for a particular observation
And compute the standard deviation of the weighted mean

13. Standard Deviation Formulas
Standard deviation of unit weight
Standard deviation of observation, i
Standard deviation of the weighted mean

14. Weights for Angles and Leveling
If all other conditions are equal, angle weights are directly proportional to the number of turns
For differential leveling it is conventional to consider entire lines of levels rather than individual setups. Weights are:
Inversely proportional to line length
Inversely proportional to number of setups

15. Angle Example 9.2
This example asks for an “adjustment” and uses the concept of a correction factor which has not been described at this point. We will skip this type of problem until we get to the topic of least squares adjustment.

16. Differential Leveling Example
Four different routes were taken to determine the elevation difference between two benchmarks (see table). Computed the weighted mean elevation difference.

17. Example - Continued
Weights: (note that weights are multiplied by 12 to produce integers, but this is not necessary)
Compute weighted mean:
What about significant figures?

18. Example - Continued Compute residuals
Compute standard deviation of unit weight
Compute standard deviation of the mean

19. Example - Continued
Standard deviations of weighted observations:

20. Summary
Weighting allows us to consider different precisions of individual observations
So far, the examples have been with simple means
Soon, we will look at least squares adjustment with weights
In adjustments involving observations of different types (e.g. angles and distances) it is essential to use weights