surveying ii. the adjustment of the observations of a single quantity
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Surveying II.
The adjustment of the observations
of a single quantity
A short revision
Total error: the difference between the true value and the observation:
ii LThe total error (i) can be subdivided into two parts, the systematic error (i), and the random error (i):
LLMLMLii )()(system. error random err.
The PDF of a normally distributed probabilistic variable:
1dxxf
The probability density function
d
c
dxcPdxxf )(
A short revision
The „3-sigma” rule:
%73,99)33( aaPwhere:a – is the expected value of the probabilistic variable
A short revision
The mean error of the variable can be computed using the error (Gauss):
n
ii
n nm
1
21lim
.
n
iiLL
nm
1
2
1
1
In order to compute i the true value should be known. Thus the mean error must be estimated from the observations:
Number of redundant observations(degree of freedom)
A short revision
The relationship between the weight and the mean error:
,
2
2
ii m
p
A short revision
The result is accurate but not precise.
The result is precise but not accurate.
Precision: The value is close to the true value (total error is low)
Accuracy: The approach gives similar results under changes conditions, too. (the mean error computed from the observations is low)
Accuracy vs. Precision
„A single observation is not an observation…”
In case of more observations, discrepancies are experienced:
nLLLL ...321
Let’s assume, that
• the observations are statistically independent,
• the observations are free of systematic error :
• the mean error of the individual observations are known:
)( iLM
nmmmm ,...,,, 321
Adjusting the observations of a single quantity
Task: to remove the discrepancies from the observations and to compute the most likely value of the quantity.
Corrections are applied to each observation:
LvLvLvLvL nnˆ....332211
The adjusted value of the observations (no system. error)
Question: How could the correction values be determined (infinite number of possible correction sets)
Let’s minimize the corrections!
Adjusting the observations of a single quantity
Usually the weighted square of the corrections are minimized:
„least squares adjustment”
.ˆ
,min1
2
ii
n
iii
LLv
vp
Let’s combine the two equations:
.minˆ1
2
n
iii LLp
Adjusting the observations of a single quantity
.minˆ1
2
n
iii LLp
Let’s find the minimum of the function:
n
iii LLp
Ld
d
1
,0ˆ2ˆ
thus:
n
i
n
i
n
iiiiii
n
ii LppLLpLp
Ld
d
1 1 11
.02ˆ22ˆ2ˆ
Adjusting the observations of a single quantity
n
i
n
iiii LppL
1 1
,02ˆ2
By reordering this equation:
n
ii
n
iii
p
LpL
1
1ˆ
Note that this formula is the formula of the weighted mean:
- an undistorted estimation;- the most efficient one;
Adjusting the observations of a single quantity
Let’s check the computations:
ii LLv ˆThe corrections are computed for each observation:
Let’s compute the following sum:
?1
n
iiivp
n
i
n
i
n
iiiiii
n
iii LppLLLpvp
1 1 11
ˆˆ
n
iipL
1
ˆ
01
n
iiivp
Check!
Adjusting the observations of a single quantity
Task: The computation of the mean error of the adjusted value.
The law of error propagation can be used:
n
ii
n
iii
p
LpL
1
1ˆ
n
ii
i
i p
p
L
L
1
ˆ
n
ii
n
ii
in
ii
iL m
p
pm
L
Lm
1
22
1
2
1
2
2
2ˆ
ˆThus the mean error of the adjusted value is:
Adjusting the observations of a single quantity
n
ii
n
ii
iL m
p
pm
1
22
1
22ˆ
The mean error of the adjusted value:
2
2
ii m
p
Introducing the relationship between the weight and the mean error:
22 iimp
Adjusting the observations of a single quantity
n
in
ii
n
ii
n
ii
n
ii
iL
pp
pp
pm
1
1
2
12
1
2
2
1
22ˆ
Thus the mean error of the adjusted value can be computed by:
Please note:
It is necessary to know the mean error of the unit weight before the computation of the mean error (‘a priori’ mean error of unit weight)
In this case the computed mean error values are based on variables, which are known before the adjustment process. These are the ‘a priori’ mean error values. They can be used for planning the observations.
Adjusting the observations of a single quantity
After the adjustment, the mean error of the unit weight can be estimated using the following equation:
f
vpn
iii
1
2
where f is the number of redundant observations (degree of freedom)
In case of n observations of a single quantity f=n-1:
11
2
n
vpn
iii
This equation uses quantities available after the adjustment process.
‘a posteriori’ mean error of unit weight
Adjusting the observations of a single quantity
Using the ‘a posteriori’ (after the adjustment) mean error of unit weights, the mean error of the observations as well as the adjusted value can be computed:
i
in
ii
Lp
mandp
m2
2
1
22ˆ
Adjusting the observations of a single quantity
Question: How much is the weight of the adjusted value?
Since the mean error of the unit weight is known, using the relationship between the weight and the mean error, the weight of the adjusted value can be computed:
n
ii
n
ii
LL p
p
mp
1
1
2
2
2ˆ
2
ˆ
Thus the weight of the adjusted value equals to the sum of the weights of the observations.
Adjusting the observations of a single quantity
• the ‘a priori’ mean error reflect our knowledge before the observations (instrument specifications, prior experiences);
• the ‘a posteriori’ mean error can be computed after the adjustment of the observations (experienced mean error);
• in both cases the mean error of the unit weight can be computed;
• when the two values are close to each other, then:• the ‘a priori’ mean error values are realistic (our observations are accurate enough)• our observations are not affected by blunders
The ‘a priori’ and the ‘a posteriori’ mean error
Given observations and mean err.: L1, L2, L3, …, Ln and m1, m2, m3, …, mn
Let’s choose an ‘a priori’ mean error of the unit weight:
Define the weights of the observations: 2
2
ii m
p
Compute the adjusted value:
n
ii
n
iii
p
LpL
1
1ˆ
The process of adjustment
Compute the corrections: ii LLv ˆ
Check the adjustment: 01
n
iiivp
Compute the ‘a posteriori’ mean error of unit weight:
11
2
n
vpn
iii
Compute the weight of the adjusted value:
n
iiL pp
1ˆ
Compute the ‘a posteriori’ mean error of the observations and the adjusted value:
n
ii
Li
i
p
mandp
m
1
ˆ,
The process of adjustment
In this case:
2
2
ii m
p
nmmmm ...321
npppp ...321
Thus the observations have a unit weight!
n
L
p
LpL
n
ii
n
ii
n
iii
1
1
1
ii LLv
The adjusted value:
The corrections:
When the ‘a priori’ mean error of observations are equal
111
2
1
2
n
v
n
vpn
ii
n
iii
The ‘a posteriori’ mean error of unit weight:
nppn
iiL
1
The weight of the adjusted value:
The ‘a posteriori’ mean error of the observations and the adjusted values:
np
mandp
mn
ii
Li
i
1
When the ‘a priori’ mean error of observations are equal
The results of the distance observations between two points are given with their mean error values:
L1 = 121.115m ± 10mm
L2 = 121.119m ± 5mm
L3 = 121.121m ± 5mm
L4 = 121.118m ± 10mm
L5 = 121.116m ± 10mm
A simple example
L0=120,110
Li
[mm]
mi mi2 Pi piLi vi Pivi Vi
2 Pivi2
+5 10 100 1 +5 +4 +4 16 16
+9 5 25 4 +36 0 0 0 0
+11 5 25 4 +44 -2 -8 4 16
+8 10 100 1 +8 +1 +1 1 1
+6 10 100 1 +6 +3 +3 9 9
11 +99 0 42
Let’s choose an ‘a priori’ mean error of u.w.: =52
2
ii m
p
The adjusted value: mmmmL 119,12011
99110,120ˆ
The ‘a posteriori’ mean error of u.w.: 24,34
42
11
2
n
vpn
iii
A simple example
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