Lecture 5 - errors.pdf
- 1. 1 ERROR THEORY • Scope – Depending on their respective purposes, measurement have to be performed to a certain accuracy and reliability. – Due to shortcoming of measuring instruments and human senses it is impossible to obtain truly error free measurements.
- 2. 2 • As a rule, measurements are repeated several times, and if possible supported by additional measurements. • Example – Measuring hypotenuse in addition to the sides of right angled triangle, or – Observing the third angle of a triangle in addition to the two that are needed. • The evaluation of the measurements leads to the following tasks:
- 3. 3 – To derive the most probable mean value of the desired unknown quantity. – To provide a figure for the accuracy or the dispersion of a single measurement. – To estimate the accuracy or dispersion of the mean and its confidence region.
- 4. 4 • Surveying measurements are subjected to three types of errors – Gross Errors/mistakes – Systematic Errors – Random Errors
- 5. 5 • Gross Errors – Often called mistakes or blunders – Usually much larger than other categories – Usually result due to inexperience of the observer who is not familiar with equipment and methods used – Gross errors are due carelessness or incompetence of the observer – Can be spotted by check measurements and then eliminated.
- 6. 6 • Examples – displacement of arrows or station marks – miscounting tape lengths – misreading the tape – wrong booking
- 7. 7 • Systematic Errors – Systematic Errors are those which follow some mathematical law and they will have the same magnitude and sign in a series of measurements when repeated under the same condition. – They are cumulative in nature – Can be eliminated by applying some mathematical corrections. – Can also be removed by calibrating the observing equipment and quantifying the the errors
- 8. 8 – Proper selection of measuring procedure • Examples – Wrong length of tape – Poor ranging – poor straightening – slope – sag – temperature variation – wrong tensioning
- 9. 9 • Random Errors – Random errors are errors that remain after all gross and systematic errors are removed – Random errors cannot be removed from measurement but methods can be adopted to ensure that they are kept within acceptable limits. – Random errors are inherent in all types of measurement, the magnitude and the sign of which are not constant. – Random errors can either be positive or negative and hence they tend to compensate each other.
- 10. 10 • In surveying the true value of a quantity is usually never known • The exact error in a measurement or observation can never be known. • Random errors follow general laws of probability and these are: – Small errors occur more frequently and therefore are more frequent than large ones – Large errors happen infrequently and are therefore less probable, very large errors may be mistakes and not random errors – Positive and negative errors of the same size are equally probable and happen with equal frequency.
- 11. 11 • Examples – holding and marking – variation in tension 0 -10 10 Magnitude of Error Error frequency
- 12. 12 In general, the distance measurement obtained in the field will be in error. Errors in the distance measurement can arise from a number of sources: 1) Instrument errors. A tape may be faulty due to a defect in its manufacturing or from kinking. 2) Natural errors. The actual horizontal distance between the ends of the tape can vary due to the effects of: temperature, elongation due to tension, and sagging. 3) Personal errors. Errors will arise from carelessness by the survey crew: poor alignment tape not horizontal
- 13. 13 • Redundancy – Redundant observations are additional measurements taken for a quantity in order to evaluate standard errors and establish probabilities. – Redundant observation are also used to detect mistakes in the field work • Precision – Pertain to the closeness to one another of a set of repeated observations in a random variable
- 14. 14 – If such observation are clustered together, then they are said to have been obtained with a high precision Accuracy – Refers to the closeness between measurements and their true values. – The further the measurement is from the true value, the less accurate it is. – Quite often words like the most probable value, expected value and mean are used as the true value can never be achieved
- 15. 15 • Reliability – In all surveying measurements, attempts are made to detect and eliminate mistakes in fieldwork and computations and the degree to which a survey is able to do this is a measure of its reliability. – Unreliable observations are those which may contain gross errors without the observer knowing – Reliable observations are unlikely to contain undetected mistakes.
- 16. 16 • Mean – The single li, which would be obtained if a quantity were to be determined by an infinite number of equal precise independent measurements (n ) with random errors, would oscillate around a mean value . – Mean is called the mathematical expectation or the “true” value of the quantity.
- 17. 17 – Usually only a limited number of measurements is available (a random sample of size n) the sample mean (arithmetic mean is used to estimate the true value. ẋ =1/n(l1+l2+…+ ln) = 1/nli – The residuals or correction which remain after the formation of arithmetic mean v1 = ẋ -l1; v2 = ẋ-l2; vn = ẋ-ln are such that the sum of the square (v) is minimum
- 18. 18 – This the basic requirement for the method called least square • Measure of spread/dispersion – To obtain a measure of spread of a single measurement, the deviations of the observation li from the true value are considered. – For n this leads to the so called “true errors”. 1= 1-l1; 2= 1-l2; …; n= 1-ln
- 19. 19 – The measure of accuracy of a single observation li is defined by the “theoretical standard deviation”. = ±(2/n)1/2 for n – Usually and i are unknown, sample mean x and residuals vi are used.
- 20. 20 – Based on the estimation the sample standard error of the observation is obtained as an approximation or estimation of as: s = ± (v2/n-1)1/2 – The sample standard error of the mean of n observations becomes sẋ =±s/(n)1/2
- 21. 21 x f(x) measurements High precision small Low precision high 2 2 1 1
- 22. 22 • A small deviation (1) indicates a small spread amongst results • A large deviation (2) indicates a large spread • In other words the set of measurements with small standard deviation (1) has a higher precision than the other set with larger standard deviation (2) • For statistical reasons standard deviation should be derived from a large number of observation.