Adjustment Computations Adjustment Computations: Spatial Data Analysis

CHAPTER 10

WEIGHTS OF OBSERVATIONS

Example 10.1 -- Weighted Mean

As a simple example of computing a weighted mean using Equation (10.13), suppose that a distance d is measured three times with the following results: 92.61 with weight 3, 92.60 with weight 2, and 92.62 with weight 1. Calculate the weighted mean.

Data Entry

SOLUTION

Example 10.2 -- Weights in Angles

Suppose the angles in an equilateral triangle ABC were measured by the same operator using the same instrument, but the number of repetitions for each angle varied. The results were: A = 45°15'25", n = 4; B = 83°37'22", n = 8; and 51°07'39", n = 6. Adjust the angles.

Data Entry

SOLUTION

Correction Factor

Corrected angles

Example 10.3 -- Weighting in Differential Leveling

In the leveling network of Figure 10.1, recall that the lengths of lines 1, 2, and 3 were 2, 3, and 4 miles, respectively. If the observed elevation differences in lines 1, 2 and 3 were +21.20 ft, +21.23 ft and +21.29 ft, respectively, find the weighted mean for the elevation difference, and the adjusted elevation of BMX (Note: All level lines were run from BMA to BMX.)

Data Entry

SOLUTION

Example 10.4 -- Weights in Taped Distances

A distance is measured as 625.79 ft using a cloth tape and given weight of 1; it is measured again as 625.71 ft using a steel tape and assigned a weight of 2; and finally, it is measured a third time as 625.69 ft with an EDM instrument and given a weight of 4. Calculate the most probable value of the length (weighted mean), and find the standard deviation of the weighted mean.

Data Entry

SOLUTION

Example 10.5 -- Weights in Differential Leveling

In leveling from benchmark A to benchmark B, four different routes of varying length are taken. The data of Table 10.2 are obtained. Calculate the most probable elevation difference (weighted mean), the standard deviation of unit weight, the standard deviation of the weighted mean, and the standard deviations of the weighted observations.

Data Entry

SOLUTION

 

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