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Surveying II. Lecture 1.

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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. - PowerPoint PPT Presentation

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Page 1: Surveying II. Lecture 1

Surveying II.

Lecture 1.

Page 2: 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.

Page 3: Surveying II. Lecture 1

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.

Page 4: Surveying II. Lecture 1

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.

Page 5: Surveying II. Lecture 1

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)

Page 6: Surveying II. Lecture 1

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.

Page 7: Surveying II. Lecture 1

Probability Distribution Function

Frequ

ency

per

unit g

roupi

ng

0

2

4

6

10

8

90°4’17.0” 18.0” 19.0” 20.0” 21.0” 22.0” 23.0” 24.0” 25.0”

Proba

bility

0.0

0.1

0.3

0.2

90°4’17.0” 18.0” 19.0” 20.0” 21.0” 22.0” 23.0” 24.0” 25.0”

Proportional Frequency curve

Probability Distribution curve

Frequency curve

Page 8: Surveying II. Lecture 1

Proba

bility

0.0

0.1

0.3

0.2

90°4’17.0” 18.0” 19.0” 20.0” 21.0” 22.0” 23.0” 24.0” 25.0”

Probability Distribution Function

Properties of PDF:

1dxxf

d

c

dxxf

The probability, that the PV is within the interval (c,d)

Page 9: Surveying II. Lecture 1

The Normal Distribution

Proba

bility

0.0

0.1

0.3

0.2

90°4’17.0” 18.0” 19.0” 20.0” 21.0” 22.0” 23.0” 24.0” 25.0”

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

2

21

x

exf

Page 10: Surveying II. Lecture 1

The Bell-curve

Proba

bility

0.0

0.1

0.3

0.2

90°4’17.0” 18.0” 19.0” 20.0” 21.0” 22.0” 23.0” 24.0” 25.0”

Change in the mean value

Change in the standard deviation

22

2

21

x

exf

Page 11: Surveying II. Lecture 1

The standard normal distribution

f(x)

0

x

Instead of PV the standardized PV could also be used for computations:

1

0

Page 12: Surveying II. Lecture 1

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).

Page 13: Surveying II. Lecture 1

Important quantities

dxxfxM

The Mean Value

The Variance

22

22

MM

or

dxxfx

The Standard Deviation

2

2

MM

or

dxxfx

Page 14: Surveying II. Lecture 1

Observation errors

ii

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

Page 15: Surveying II. Lecture 1

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

Page 16: Surveying II. Lecture 1

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

Page 17: Surveying II. Lecture 1

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

Page 18: Surveying II. Lecture 1

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

Page 19: Surveying II. Lecture 1

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).

Page 20: Surveying II. Lecture 1

Error propagation

x

y

x

y

y=x*g

y=G(x)

where g = G’(x)

Page 21: Surveying II. Lecture 1

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.

n

i LxiiiiG

iixggwhereg

1

Page 22: Surveying II. Lecture 1

Error propagation

Propagation of mean errors:

n

i LxiiiiG

iixggwheremgm

1

222

Page 23: Surveying II. Lecture 1

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

22

1

2

1

2121

22

2121

mmthenmmifmmm

ggLLG

GLLLLG

221

222

122121

21 LLG mLmLm

LgésLgLLG

nmm

nmm

ngggL

nLG

n

iLLG

n

n

ii

i

1

22

211

1

1...1