Surveying II. Lecture 1.
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Transcript of Surveying II. Lecture 1.
Surveying II.
Lecture 1.
Types of errors
There are several types of error that can occur, with different characteristics.
Mistakes
Such as miscounting the number of tape lengths when measuring long distances or transposing numbers when booking.
Can occur during the whole surveying process, including observing, bboking, computing or plotting.
Solution:Creating suitable procedures, and checking the measurements.
Probability theory
The effect of a blunder is much larger than the acceptable error of the applied measurement technique.
Types of errors
Systematic errors
Systematic errors arise from sources that act in a similar manner on observations.
Examples: • expansion of steel tapes due to temperature changes• frequency changes in electromagnetic distance measurements
These errors are dangerous, when we have to add observations, because they act in the same direction. Hence the total effect is the sum of each error.
Solution:Calibrating the instruments - comparing the observations with other observations made by other instruments.
Types of errors
Random errors
All the discrepancies remaining once the mistakes and systematic errors have been eliminated. Even when a quantity is measured many times with the same technology and instrumentation, it is highly unlikely that the results would be identical.
Although these errors are called random, they have the following charachteristics:• small errors occur more frequently than large ones• positive and negative errors are equally likely• very large errors occur rarely
Due to this, the normal statistical distribution can be assumed.
Solution:Repetitions of observations.
The aim of processing the observations
Questions:
How can the variability of the observations be described numerically. (Error theory)
How can we describe the variability of functions of observations (area, volume, etc.)? (Error propagation laws)
How can we remove the discrepancies from the observations? (Computational adjustment)
Basics of error theory
Probabilistic Variables (PV): quantities on which random processes has an effect.
Discrete PV: the variable can have a unique number of values.
Continuous PV: the variable can have infinite number of values.
Probability Distribution Function
Frequ
ency
per
unit g
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Proba
bility
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Proportional Frequency curve
Probability Distribution curve
Frequency curve
Proba
bility
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Probability Distribution Function
Properties of PDF:
1dxxf
d
c
dxxf
The probability, that the PV is within the interval (c,d)
The Normal Distribution
Proba
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If the value of PV depends on a large number of independent and random factors, and their effects are small than the PV usually follows the normal distribution.
22
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x
exf
The Bell-curve
Proba
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Change in the mean value
Change in the standard deviation
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x
exf
The standard normal distribution
f(x)
0
x
Instead of PV the standardized PV could also be used for computations:
1
0
The 3 rule
f(x)
0
9973,033 P
The probability, that the PV is within the interval +/-3 around its mean value (), is 99,73% (almost sure).
Important quantities
dxxfxM
The Mean Value
The Variance
22
22
MM
or
dxxfx
The Standard Deviation
2
2
MM
or
dxxfx
Observation errors
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i
L
errornobservatiovalueltheoreticansobservatioL
Let’s denote the difference of the theoretical value and the mean value ()
LM
Let’s denote the difference of the i-th observation and the mean value ()
LLM
Then
LLMLML )()(
Total Error = systematic error + random error
The mean error
Gauss:
n
iin n
m1
21lim Recall the definition of standard deviation
2 MM
If we separate the systematic and the random errors:
22 mm
Where - mean systematic errorm - mean random error
The correlation and the covariance
In case of two PVs may arise the question: Are they independent? Do they depend on each other?
Correlation:
MMMcr ,,
Covariance:
MMMc ,
Is there a linear relationship between and ?
If r = +1 or -1 -> linear relationship between the two quantities,if r = 0 -> it is necessary , but not suitable criteria for the independence
EstimationsPlease note that up to now, all PVs were continuous PVs.
BUT. We do not know the probability distribution of the PVs. Therefore it should be estimated from a number of samples (observations).
Undistorted estimations: If the mean value of the estimation equals to the estimated quantity.
Efficiency of the estimation: If two estimations are undistorted, the more efficient is the one with the lower variance.
aaM ˆ
Estimations
n
iiLn
L1
1
The Mean Value - the arithmetic mean
The mean error
n
iin
m1
21
Since the observation errors are not known (i), we could use the difference from the arithmetic mean instead.
n
iiLL
nm
1
21
The estimation above is distorted, therefore we use the corrigated standard deviation:
n
iiLL
nm
1
2
11
Error propagationIf the observations are PVs, then their functions are PVs, too. That’s the law of propagation.
We assume that the observations are independent.
Let’s have n observations (L1, L2, …, Ln), and their function G = g L1, L2, …, Ln)
Questions:
• how big is the error of the value of G (G), when the error of Li are known (i). • what is the standard deviation of G (G), when the std. dev of Li are known.
Let’s suppose that the function G is linear (if not, it should be reformatted as Taylor series).
Error propagation
x
y
x
y
y=x*g
y=G(x)
where g = G’(x)
Error propagation
Propagation of observation errors:
n
i LxiiiiG
iixggwhereg
1
The propagation of observation error is linear.
If we still have some systematic errors in the observations, then their effect
propagates linearly.
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i LxiiiiG
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1
Error propagation
Propagation of mean errors:
n
i LxiiiiG
iixggwheremgm
1
222
Error propagationSimple cases:
• observation multiplied with a constant (G=cL)
• sum of two quantities
• product of two quantities
• mean value of the samples
LLG cmmcm
cg
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2121
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GLLLLG
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21 LLG mLmLm
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