DEPARTMENT OF GEOMATICS ENGINEERING

TOPOGRAPHY

2012 – 2013 SPRING TERM

WEEK 5

CLASS PRESENTATIONS FOR SURVEYING I COURSE BY E.TARI, H. KARAMAN

ISTANBUL TECHNICAL UNIVERSITY

1ITU DEPARTMENT OF GEOMATICS ENGINEERING

ERRORS IN OBSERVATION

Good measurements require a combination of human skill andmechanical equipment applied with the utmost judgment. Howeverno matter how carefully made, observation are never exact and willcontain errors. Surveyors, whose work must be performed toexacting standards, should therefore thoroughly understand thedifferent kinds of errors, their sources and expected magnitudesunder varying conditions, and their manner of propagation. Onlythen can they select instruments and procedures necessary toreduce error sizes within tolerable limits.

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ERRORS IN OBSERVATION

Consequently, it can be stated that :

no observation is exact

every observation contains error

the true value of an observation is never known

the exact error present is always unknown

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TYPES OF OBSERVATION

1. Direct Observation:

The quantity of observation can be determined by applying ameasuring instrument directly.

applying a tape to a line

turning an angle with a theodolite

2.Indirect Observation:

Observation is determined by its relationship to some other observedvalue or values.

the distance across a river can be determined by observing thelength of line on one side, the angle at each end of this line to apoint on the other side, and then computing the distance by one ofthe standard trigonometric formulas.

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ERROR

By definition an error is the difference between an observed valuefor a quantity and its true value,

εi = yi – μ

εi = the error an observation

yi = the observed value

μ = the true value

True value: a quantity’s theoretically correct or exact value.

(True value can never be determined!)

True value is the simply the population’s arithmetic mean if allrepeated measurements have equal precision.

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SOURCES OF ERRORS1.Natural errors: are caused by variations in wind, temperature,humidity, atmospheric pressure, atmospheric refraction, etc. Anexample is a steel tape whose length varies with the changes intemperature.

2.Instrumental errors: result from any imperfection in theconstruction or adjustment of instruments and from the movementof individual parts. For instance, the graduations on scale may notbe perfectly spaced or the scale may be warped.

3.Personal errors: arise principally from limitation of the humansenses. As an example; a small error occurs in the observed valueof a horizontal angle if the vertical crosshair in a theodolite is notaligned perfectly on the target.

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TYPES OF ERRORS

Errors in observation are of two types; systematic and random.

1.Systematic errors: result from factors that comprise the measuringsystem and include environment, instrument, and observer. So longas system conditions remain constant, the systematic errors willlikewise remain constant. If condition change, the magnitudes ofsystematic errors also changed.

Condition producing systematic errors conform to physical laws thatcan be observed, a correction can be computed and applied toobserved values. An error due to the effects of temperature on asteel tape is an example of systematic error. If temperature isknown, the shortening or lengthening effects on a steel tape can bedetermined.

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TYPES OF ERRORS

Before observations, systematic errors that result from instrumentalerror should be removed or minimize by controlling and adjustinginstruments.

Also, systematic errors should be remove by selecting suitablesurvey techniques.

2.Random errors: are those that remain in measured values aftermistakes and systematic errors have been eliminated. They arecaused by factors beyond the control of the observer, obey the lawsof probability, and are sometimes called accidental errors. They arepresent in all surveying observations.

There is no absolute way to compute or eliminate them, but theycan be estimated using adjustment procedures.

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TYPES OF ERRORS

This type of errors are not cumulative. The magnitude and algebraicsign of random errors are matter of chance. They can be (-) or (+)sign.

Errors caused by improper plumbing are random, they may makedistance either too long or short.

Random errors are present in every horizontal angle observation. Forexample; careless plumbing , poor focusing etc.

Random errors tend to partially cancel themselves in a series ofobservation.

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MISTAKES

These are blunders made by observer. Examples of mistakes includetransposing figures (recording a tape value of 68 as 86),miscounting the number of full tape length in a long measurement,measuring to or from the wrong point, and like that.

Mistakes must be discovered and eliminated. All measurements aresuspect until they have been verified. Verification may be as simpleas repeating the measurement, or verification can result fromgeometric or trigonometric analysis of related measurements. As arule, every measurement is immediately checked and repeat. Thisrepetition enables the surveyor to eliminate most mistakes and, thesame time, improve the precision of the measurement.

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PRECISION AND ACCURACY

Precision:

It refers to the degree of refinement or consistency of a group ofobservations and is evaluated on the basis of discrepancy size. Ifmultiple observations are made of the same quantity and smalldiscrepancies result, this indicates high precision. The degree ofprecision attainable is dependent on equipment sensitivity andobserver skill.

Accuracy:

It denotes the absolute nearness of observed quantities to their truevalues. In other words, accuracy is the relationship between thevalue of a measurement and the “true” value of the dimension beingmeasured.

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PRECISION AND ACCURACY

( Figure:1 Ghilani & Wolf, 2008)

(a) = precise but not accurate

(b) = neither precise nor accurate

(c) = both precise and accurate

In some case, more precise methods can result in less accurateanswers. For instance; if the steel tape had previously been brokenand then incorrectly repaired, the results would still be relativelyprecise but very inaccurate.

(a) (b) (c)

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ELIMINATING MISTAKES AND SYSTEMATIC ERROR

All field operations and office computations are governed by aconstant effort to eliminate mistakes and systematic errors.

Comparing several observations of the same quantity is one of thebest ways to identify mistakes. Assume that five observations of aline are recorded as; 67.91, 76.95, 67.89, 67.90, 67.89. The secondvalue disagrees with the others, apparently because of atransposition of figures in reading or recording.

When a mistake is detected, it is usually best to repeat theobservation. However, if sufficient number of other observations ofthe quantity are available, the widely divergent result may bediscarded.

Systematic errors can be calculated and proper corrections appliedto the observations. In some instances it may be possible to adopt afield procedure that automatically eliminates systematic errors.

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MOST PROBABLE VALUE ( y’ )

The true of any quantity is never known. However, its mostprobable value can be calculated if redundant observation havebeen made. Redundant observations are measurements in excessof minimum needed to determine a quantity.

Most probably value is derived from a sample set of data ratherthan the population, and simply the mean if the repeatedmeasurements have the same precision.

For a single unknown such as a line length that has been directlyand independently observed a number of times using the sameequipments and procedures, the most probable value in this casesimply the arithmetic mean;

y’ = (Σy / n)

Σy = the sum of the individual measurements y,

n = the total number of observations

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RESIDUALS

It is the differences between any individual measured quantity andthe most probable value for that quantity.

The mathematical expression for a residual is;

Ʋi = y’ – yi

Where Ʋi is the residual in any observation yi and y’ is the mostprobable value for the quantity.

Σ Ʋi = 0 [Ʋ] = 0 ;

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OCCURRENCE OF RANDOM ERRORS

It will be assumed that all mistakes and systematic errors have beeneliminated before random errors are considered.

To analyze the manner in which random errors occur, consider thedata of 100 repetitions of an angle observation made with a totalstation instrument. Assume these observations are free frommistakes and systematic errors.

It is first necessary to compute the most probable value for theobserved angle. Then residuals for all observed values arecomputed. And these values are listed and frequency, the number oftimes they occurred, is determined.

The range of observation or their residuals can be seen easily and aclass interval is calculated. Then gather observations or theirresiduals into groups which are determined by the help of classinterval.

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OCCURRENCE OF RANDOM ERRORS

To analyze the distribution of pattern of the observations or theirresiduals, a histogram, which is a simple bar graph showing thesizes of observation or their residuals versus their frequency ofoccurrence, can be prepared to give an immediate visual impressionof the distribution pattern of the observation or their residuals.

Figure:2 (Ghiliani & Wolf, 2008)

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GENERAL LAWS OF PROBABILITY

Small residuals (errors) occur more often than large ones; that is;they are more probable.

Large errors happen infrequently and are therefore less probable

Positive and negative errors of the same size happen with equalfrequency; that is; they are equally probable.

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BASIC DEFINITIONS

Variance (σ2):

It is a value by which the precision for a set of data is given.Population variance applies to a data set consisting of an entirepopulation. It is mean of the squares of the errors and given by;

= = is the sum of the square of the errors.

= number of measurements.

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BASIC DEFINITIONS

Sample variance applies to a sample set of data. It is unbiasedestimate for the population variance given above, and is calculatedas;

Degrees of freedom; is the number of observations that are inexcess of the number necessary to solve the unknowns. In otherwords, the number of degrees of freedom equals the number ofredundant observations.

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BASIC DEFINITIONS

Standard Deviation (S):

It is the square root of the sample variance.

Where S is the standard deviation, n-1 is the degrees of freedom,

and Σni=1 Ʋi

2 is the sum of squares of the residuals.

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BASIC DEFINITIONS

Standard Deviation of the Mean:

Because all measured values contains errors, the mean that iscomputed from a sample set of measured values will also containserrors.

S = standard deviation of a measured value

n = number of observations

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BASIC DEFINITIONS1. Average error (Average of absolute error)(t):

ε = errors

n = number of observation

ʋ = residuals

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BASIC DEFINITIONS2. Root Mean Square Error (m):

ε = errors

n = number of observations

ʋ = residuals

n-1 = redundant observations

If two or more unknowns are exists,

u = number of unknowns n= number of observations

(n-u) = redundant observations

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BASIC DEFINITIONS3. Relative Error (T):

mi = root mean square error

l = value of thing observed

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BASIC DEFINITIONS

4. Probable Error (r):

If residuals or errors are organized from smallest value to largestvalue in a set of numbers according to their absolute values, themiddle value of this set is described as probable value.

If n = odd number

If n = even number

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REFERENCES:

C.D. Ghilani, P.R. Wolf; Elementary Surveying , Pearson Education International Edition,

Twelfth Edition, 2008 .

Barry F. Kavanagh, Surveying Principles and Applications, Pearson Education International

Edition, Eighth Edition, 2009.

Ü.Öğün , Topografya Ders Notları ,

E.Tarı , M.Sahin , Surveying II Lecture Notes - Slides ,

U.Özerman, Topografya Ders Notları , Bahar 2010