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Weights
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Weights (W) of observations
Weight is a positive number assigned to an observation that indicates the relative accuracy to other observations
The smaller an observation error (variance) is, the more precise an observation and consequently the bigger weight the observation should
have
Weights are inversely proportional to variances W =
Weights are used to control the sizes of corrections applied to measurements in an adjustment
The bigger an observation weight is, the smaller the size of correction (residual) applied in the adjustment of observations
2
1
Weights of observations
Instrument
Measured
length AB Weight
Linen tape 625.79 1
Steel tape 625.71 2
EDM 625.69 4
Adjust the measurement of length AB by computing the MPV of the
distance (weighted mean) and its standard deviation
w
wdZ
Most Probable
Value
71.6257
97.4379
421
)69.625(4)71.625(2)79.625(1
Z
Weighted mean
Standard deviation of weighted mean
1
2
nw
wvS
Z
idZv 3n
08.079.62571.6251 v
00.071.62571.6252 v
02.069.62571.6253 v 0016.0)02.0(4
0000.0)00.0(2
0064.0)08.0(1
22
33
22
22
22
11
vw
vw
vw
0080.02wv 024.0)2(7
0080.0
ZS
Route
elevation
difference
between point
P and Q weight
1 25.35 18
2 25.41 9
3 25.38 6
4 25.30 3
Compute:
1. Weighted mean (MPV) of elevation difference
2. Std dev of weighted mean
3. Std dev of weighted observations
4. Std dev of unit weight
Weighted mean (MPV) of elevation difference
366.2536918
)30.25(3)38.25(6)41.25(9)35.25(18
Std dev of weighted mean
018.0
1
2
nw
wvS
Z
w
wdZ
Std dev of weighted observations
)1(
2
nw
wvS
n
n
037.0)3(9
0363.02 S026.0
)3(18
0363.01 S
045.0)3(6
0363.03 S 063.0
)3(3
0363.04 S
Std dev of unit weight
1
2
n
wvS
110.0)3(
0363.0S
No. Steel Tape
(m)
Total Station
(m)
1 85.984 85.316
2 85.031 85.002
3 85.442 85.652
4 85.883 85.121
5 85.344 85.422
N 5 5
Mean 85.537 85.303
Std 0.394 0.255
A length AB was measured five times by using a steel tape and
Total Station instrument. The readings are tabulated as follows;
Calculate the most probable value of the length AB and its
standard deviation.
m1 85.537 m2 85.303
std1 0.394 std2 0.255
w1 6.4 w2 15.4
Wt Mean 85.372
Inst
Horizontal angle
(deg-min-sec) Std dev (")
A 49-27-20 15
B 49-27-24 6
C 49-27-27 2
Calculate the weighted mean of the angle and its standard
deviation
Weights are used to control the amount of error to be
distributed or adjusted in the observed values so that
the adjusted values would conform with the related
geometric condition of the problem
COMPARISON BETWEEN
WEIGHTED AND UNWEIGHTED OBSERVATIONS
Example: Horizon Angles 1
The following angles were observed at the horizon;
X = 420 12 22 +/- 10 Y = 590 56 24 +/- 10 Z = 2570 51 44 +/- 10
Assuming equal weight observations, compute the most probable
values of X,Y and Z and the residuals of the observations
X
Z
Y
Adjusted Observed Correction
() Std (")
X 42 12 12.0 42 12 22 -10 10
Y 59 56 14.0 59 56 24 -10 10
Z 257 51 34.0 257 51 44 -10 10
360 0 0 360 0 30 -30
Example: Horizon Angles 2
The following angles were observed at the horizon;
X = 420 12 22 +/- 10 Y = 590 56 24 +/- 20 Z = 2570 51 44 +/- 30
Given the standard deviation of the observations, compute the
most probable values of X,Y and Z and the residual of the
observations
X
Z
Y
Adjusted Observed Correction
() Std (")
X 42 12 19.5 42 12 22 -2.5 10
Y 59 56 15.8 59 56 24 -8.2 20
Z 257 51 24.7 257 51 44 -19.3 30
360 0 0 360 0 30 -30
Adjustment of elevation differences for the
determination of elevation or height above MSL
Line
Measured
Elevation
Differences Distances
1 5.10 5
2 2.34 5
3 -1.25 5
4 -6.13 5
5 -0.68 5
6 -3.00 5
7 1.70 5
Find elevations of A, B and C using least squares method
Note: Weight for elevation differences = 1/(distance)
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