We know intuitively that some measurements are more reliable than others.

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2 Some measurements are more equal than others. We know intuitively that some measurements are more reliable than others. Measurements that are more reliable should have greater weight or influence in our surveys. 2

3 Weights can be assigned according to a number of different strategies: Assigning weights according to judgment: You might assign less weight to an angle that was measured from an unstable set-up Assigning weights according to an Index: The sum of the interior angles = (n-2)*180, the angles in a closed polygon must conform to this index More weight (or influence) is given to values that are measured more times Higher precision measurements are given more weight A distance measured with a +/-2mm + 2ppm EDM should receive more weight than a distance measured with an EDM rated at +/- 5mm + 5ppm 3

4 It would be improper to calculate a simple mean of these two measurements. To do so would give the value derived from 5 observations the same weight (or influence) as the value that is derived from 50 measurements. 4

5 In this example, measurement B should receive10 times the weight of measurement A, because it was measured 50 times instead of 5 times. 5

6 This is how a weighed mean based on the number of observations is computed: Weighted mean = the sum of (each measurement times the number each measurements observations) divided by (the sum of the number of observations) This idea finds practical application in surveying, particularly when it becomes necessary to compute a weighted mean bearing 6

7 In the example above, the bearings are first converted to azimuths in decimal degrees. Next, the azimuth of each line segment is multiplied by its length. These are summed, and the result is divided by the sum of the lengths. The result is an azimuth in decimal degrees, which is converted to a bearing in degrees-minutes and seconds. It is critical that the computations be performed on azimuths, because using bearings will result in erroneous answers if the bearings of the lines are in different quadrants! 7

8 Equal weights are assigned to values measured under the same conditions For example, you might assign equal weights three interior angles you d measure in a triangle. The more times a quantity is measured the more weight it should receive On the other hand, if one of the interior angles was measured 10 times, and the others only once, the two angles should get a weight of 1, and one should get a weight of 10 Weights derived from standard errors are set proportional to the inverse square of the Standard Error (smaller SE = more weight) If two of the angles were measured with a 1-second theodolite and one with a 1-minute transit, the angles measured with the theodolite should receive more weight since the standard errors of these angles would be smaller. 8

9 In this illustration, both measurements are assigned equal weights. The mean is calculated by dividing the sum of the observations by the sum of the weights. This is just a simple mean. 9

10 In this instance, our observations must meet a geometric constraint (the sum of the angles in triangles ABC must sum to 180 degrees). The observed angles must then be adjusted by small amounts to satisfy this index. Each angle was measured using the same instrument under similar conditions, so equal weights are assigned to each angle. The correction to each angle is computed by subtracting the (total error)/(total weight) from each observation, thus: Adjusted Angle = Observed angle [(total error)/(total weight)] (note that in this example, the total error is considered to be negative 2.7 ) 10

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13 Remember we are looking at Standard Errors here, and weights are proportional to the inverse square of the Standard Error (smaller SE = more weight) Therefore, Set B has the most weight, while Set C has the least. 13

14 1. Weight Ratio = Inverse Square of Standard Error 2. Assign a Weight of 1 to observation with largest Standard Error 3. Assign weights to remaining observations proportionately: (Observation s Weight Ratio) / (Weight Ratio of observation with a weight of 1) 4. Multiply Observations by their Weights, and find the sum 5. Divide the sum by the sum of the weights to get a weighted mean 14

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16 The sum of three interior angles should sum to 180 degrees, but in this example, the three angles sum to less than that there is 2.7 of error in the observational data. For whatever reason, we ve decided to assign different weights to the observations. Observation B has ¼ the weight of A and Observation B has ½ the weight of Observation C. Adjustment to an observation will be made in inverse proportion to its weight so the observation with the least weight will receive the most adjustment. 16

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19 In this example, the measurement having the largest Standard Error (angle BOC) should receive the largest adjustment. 19

20 In this example, adjustments are made in direct proportion to the square of an observation s Standard Error, so the observation with the smallest Standard Error will be adjusted the least, and the observation with the most Standard Error will be adjusted the most. 20

21 A complaint that one often hears about Least Squares is that you can always get good statistical results by inflating the centering errors. While this is true, there is no free lunch: If you inflate the centering errors, you also inflate the error ellipses! Remember, for a meaningful adjustment, you must REMOVE ALL blunders and systematic errors from the data, and you must use Standard Errors that are representative of the observing conditions and the equipment being used. 21

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25 Standardized Residual = Residual / Standard Error 25

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28 Clean data should result in Standardized Residuals that are 1 or less. 28

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