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)