Surveying II. The adjustment of the observations of a single quantity.

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!


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


.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


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;


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!


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:


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


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.


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


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


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.


• 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

Surveying II. The adjustment of the observations of a single quantity.

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