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:
 i    Li
The total error (i) can be subdivided into two parts, the
systematic error (di), and the random error (xi):
 i    Li    M ( L)  M ( L)  L  d  x
system. error
random err.
A short revision
The probability density function
d
 f xdx  P(c  x  d )
c
The PDF of a normally distributed probabilistic variable:

 f x dx  1

A short revision
The „3-sigma” rule:
P(a  3  x  a  3 ) 99,73%
where:
a – is the expected value of
the x probabilistic variable
A short revision
The mean error of the variable can be computed using the
error (Gauss):
n
1
m  lim   i2
n n
i 1
.
In order to compute i the true value should be known.
Thus the mean error must be estimated from the observations:
1
2
L  Li 
m

n  1 i1
n
Number of redundant observations
(degree of freedom)
A short revision
The relationship between the weight and the mean error:
pi 
,

2
2
i
m
Accuracy vs. Precision
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)
Adjusting the observations of a single quantity
„A single observation is not an observation…”
In case of more observations, discrepancies are experienced:
L1  L2  L3  ...  Ln
Let’s assume, that
• the observations are statistically independent,
• the observations are free of systematic error :
M ( Li )  
• the mean error of the individual observations are known:
m1 , m2 , m3 ,..., mn
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:
L1  v1  L2  v2  L3  v3  ....  Ln  vn  Lˆ
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”
n
 p v  min ,
i 1
2
i i
vi  Lˆ  Li .
Let’s combine the two equations:
   pi Lˆ  Li   min.
n
i 1
2
Adjusting the observations of a single quantity
   pi Lˆ  Li   min.
n
2
i 1
Let’s find the minimum of the function:
d n
  2 pi Lˆ  Li   0,
dLˆ i 1
thus:
n
n
n
d n
  2 pi Lˆ    2 pi Li   2Lˆ  pi  2 pi Li  0.
dLˆ i 1
i 1
i 1
i 1
Adjusting the observations of a single quantity
n
n
i 1
i 1
2Lˆ  pi  2 pi Li  0,
By reordering this equation:
 pi Li
Note that this formula is
the formula of the
weighted mean:
 pi
- an undistorted
estimation;
- the most efficient one;
n
Lˆ 
i 1
n
i 1
Adjusting the observations of a single quantity
Let’s check the computations:
The corrections are computed for each observation:
vi  Lˆ  Li
n
Let’s compute the following sum:
Lˆ  pi
n
 p i vi  ?
i 1
i 1
 pi vi   pi Lˆ  Li   Lˆ  pi   pi Li
n
n
n
n
i 1
i 1
i 1
i 1
n
 p i vi  0
i 1
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
Lˆ 
 pi Li
pi
Lˆ
 n
Li
 pi
i 1
n
 pi
i 1
i 1
Thus the mean error of the adjusted value is:
2
n
n
2
2
i
Lˆ
i 1
i 1
i
 Lˆ
m   
 L
2

pi
2
 m  
mi
2
n 

  pi 
 i 1 
Adjusting the observations of a single quantity
The mean error of the adjusted value:
m 
2
Lˆ
2
i
p
n
i 1
n 
  pi 
 i 1 
2
2
i
m
Introducing the relationship between the weight and the mean error:
pi 

2
2
i
m
pi mi2   2
Adjusting the observations of a single quantity
Thus the mean error of the adjusted value can be computed by:
n
mL2ˆ  
i 1
pi  2


  pi 
 i 1 
n
2

2


  pi 
 i 1 
n
n
2
 pi 
i 1
2
n
 pi
i 1
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:
n

pv
2
i i
i 1
f
where f is the number of
redundant observations (degree
of freedom)
In case of n observations of a single quantity f=n-1:
n

2
p
v
 i i
i 1
n 1
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:
2
2
2
2
Lˆ
i
n
m 

 pi
i 1
and
m 

pi
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:
2
2
n


p Lˆ  2 
  pi
2
m Lˆ

i 1
n
 pi
i 1
Thus the weight of the adjusted value equals to the sum of the weights of the
observations.
The ‘a priori’ and the ‘a posteriori’ mean error
• 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 process of adjustment
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:
pi 
2
mi2
n
Compute the adjusted value:
Lˆ 
 pi Li
i 1
n
 pi
i 1
The process of adjustment
Compute the corrections:
vi  Lˆ  Li
n
Check the adjustment:
 p i vi  0
i 1
n
Compute the ‘a posteriori’ mean error of unit
weight:

 pi vi2
i 1
n 1
n
p Lˆ   pi
Compute the weight of the adjusted value:
i 1
Compute the ‘a posteriori’ mean
error of the observations and the
adjusted value:
mi 

pi
, and mLˆ 

n
 pi
i 1
When the ‘a priori’ mean error of observations are
equal
In this case:
m1  m2  m3  ...  mn
pi 
2
mi2
p1  p2  p3  ...  pn
Thus the observations have a unit weight!
n
The adjusted value:
L
 pi Li
i 1
n
 pi
i 1
The corrections:
vi  L  Li
n

 Li
i 1
n
When the ‘a priori’ mean error of observations are
equal
The ‘a posteriori’ mean error of unit weight:
n

pv
i 1
2
i i
n 1
n

 vi2
i 1
n 1
The weight of the adjusted value:
n
p L   pi  n
i 1
The ‘a posteriori’ mean error of the observations and the adjusted values:
mi 

pi
  and mL 

n
 pi
i 1


n
A simple example
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
mi
mi2
Pi
p iL i
vi
+5
10
100
1
+5
+4
+9
5
25
4
+36
+11
5
25
4
+8
10
100
+6
10
100
Li
Vi2
Pivi2
+4
16
16
0
0
0
0
+44
-2
-8
4
16
1
+8
+1
+1
1
1
1
+6
+3
+3
9
9
11
+99
Pivi
[mm]
S
0
Let’s choose an ‘a priori’ mean error of u.w.: =5
The adjusted value:
42
pi 
2
mi2
99
Lˆ  120,110m  mm  120,119m
11
n
The ‘a posteriori’ mean error of u.w.:

 pi vi2
i 1
n 1

42
 3,24
4