3140601_SURVEYING_GTU_Study_Material_civil engineering

Introduction and details of surveying for fourth semester civil engineering students

1. Ujjval J. Solanki Civil Engg . Deptt , u [email protected] Railway Bridge and Tunnel Engineeing (2160603) Darshan Institute of Engineering & Technology Surveying-3140601 Chapter - 8-Theory of Errors

2. Topics to be covered Introduction Types of Errors Definitions Laws of Accidental Errors Probable Error Theory of Least Square Laws of Weights Rules For Giving Weights and Distribution of Errors To The Field Observations Determination Of The Most Probable Values (MPV) of Quantiti es Chapter weightage as per University Syllabus :10% ( @ 7 marks)

3. INTRODUCTION Survey measurements may involve several operations like Setting up Centering Levelling Bisecting Reading ….. so it is impossible to determine the true value of error Hence survey measurements always contain some error ( માણસ માત્ર ભૂલ ને પાત્ર )….. bhul koni thay !!!.. kam kare eni Will focus on reduction and adjustment of field measurements to provide Most Probable Value (MPV) of field observation

4. INTRODUCTION Errors may occur….. Reasons Faulty instrument Climatic condition Human mistake or carelessness Civil engineer must know about errors developed and its elimination

5. TYPES OF ERROR Main three types of errors. Mistakes . OR Gross error Accidental Errors. Systematic errors Details …..in next slide

6. TYPES OF ERROR- Main three types of errors 1. Mistakes. OR Gross error REASONS Inattention Inexperience Carelessness Poor judgment/confusion in mind If undetected produce serious effect. Quite random both in occurrence an magnitude SOLUTION : Cross checking 2. Systematic errors CHARACTERISTICS Under same condition always be the same size and sign . Always follow some mathematical /physical law. Correction can be determined and applied. Effect is + ve / - ve , Cumulative effect. Constant in character + ve OR – ve , Results too great/too small. If undetected systematic errors are very serious REASONS (i) Shrinkage of topographical map. (ii) Wrong length of chain. (iii) Wrong scale of map. (iv) Atmospheric refraction. SOLUTION OF SYSTEMATIC ERRORS (i) Instrument design- errors should auto elimination. (ii) Find out the relationship of error 3. Accidental Errors. CHARACTERISTICS Remains after mistakes and systematic errors have been eliminated. Beyond the ability of observer to control. Represent the limitation of precision. Obey the laws of chance Must handle as per laws of probability. Lack of perfection in human eye. Observation in cm. tape 10mm,9mm,11mm May compensate each other- compensating error. Smaller error –better precisio n

7. TYPES OF ERROR: COMPARISON Mistakes Systematic errors Accidental Error Occur due to carelessness, in experience Same size, sign, under certain condition. Remain after elimination of all the errors Gross error Cumulative error Random error Do not follow any fixed pattern Follow mathematical law Follow law of probability Random in nature Cumulative error Compensating error Observer read 181° in place of 179° Temperature correction in steel tape Tape marked in cm. observer try to read in mm One observer estimate-4 mm Second observer estimate5 mm. This cause accidental error

8. DEFINITIONS Observation Observed value of quantity Weight of an Observation True value of a Quantity. Most probable value: (MPV) True Error Most probable Error. Residual Error or variation ( v ) Observation equation Condition equation Normal Equation

9. DEFINITIONS (GTU-2016) Observation Type-1 Direct observation If the value of a quantity is measured directly. (i.e. Angle A= 45˚ 20' 15“) Type-2 Indirect observation Value of the quantity calculated indirectly from direct observation. (e.g.-Angle computed at true station from satellite station) (e.g.- angle calculated from sine rule )

10. DEFINITIONS (GTU-2016) Observed value of quantity : Corrected value -> after correction of all the known errors.(mistakes and systematic errors) Type-1 Independent quantity Whose value is independent of the value of other quantity ( i.e. R.L. of B.M.) Type-2 Conditioned or Dependent quantity Whose value is dependent upon the value of one or more quantity. ( R. L. of other points other than B.M.)

11. DEFINITIONS Weight of an Observation A number giving an indication of its precision and trustworthiness (The trait of deserving trust and confidence) when making comparison between several quantities of different worth . ( e.g.- Weight-4- Four times much reliable as an observation of weight 1 ) Weights are assigned to observation as direct proportion to the number of observations

12. DEFINITIONS True value of a Quantity The value which is absolutely free from all the errors (hypothetical quantity) 5) Most probable value: (MPV) Is the one which has more chance of being true than has any other. MPV = Mean value ± Em (Probable error of mean) 6) True Error Difference between the true value and observed value. True error = True value - Observed value

13. DEFINITIONS Most probable Error (GTU 2011) Which is added to and subtracted from most probable value fixes the limits of true value. Limits of True value = MPV ± MPE 8) Residual Error or variation (v) (GTU 2011) = Observed value – Most probable value 9) Observation equation : ( GTU 2014) Relation between observed quantity and its numerical value

14. DEFINITIONS (GTU 2011,2013) Conditioned equation : Which is expressing the relation between several dependent quantities. (A+ B+ C = 180˚ ) 11) Normal Equation - Equation with unknown quantity - can be solved simultaneously - Using normal equation MPV evaluated - No of normal equation= No of unknown

15. 4 Laws of Accidental Errors Accidental error/ Probable error/ Standard error Accidental error follws law of probability Occurance of errors.  Expressed by equation Equation used to find probable error/ precision. Features of accidental error Small error occur more frequent than large= MPE + ve and –ve error of same size happen with equal frequency = Equally probable Large error occurs seldom = can not probable and may be impossible 15

16. 4 Laws of Accidental Errors Probability curve 16

17. Probable Error Probable Error It is calculated from the probability curve of errors It is a measure of accuracy of observation only with regards to accidental errors Purpose : 1) Measure of precision of any series of observation 2) Means of assigning weights to two or more quantities. From the probability curve need to discuss following six aspects 17 1 Probable error of Single measurement 2 Probable error of an average 3 Probable error of Sum of observations 4 Mean Square Error 5 Average Error 6 Probable error of Single measurement Detail discussion in Next Slide

18. Probable Error from probability curve 18 1 Probable error of Single measurement -   Probable error of an average-   = Difference between single observation and mean of the series ( variance) = No of observations in the series  

19. Probable Error 19 3 Probable error of Sum of observations-   = Difference between single observation and mean of the series ( variance) = No of observations in the series = Probable error of single measurement =Probable error of Sum of observations   4 Mean Square Error Is equal to the square root arithmetic mean of the square of the individual error =  

20. Probable Error 20 5 Average Error It is arithmatic mean of seperate errors of equal weight = Difference between single observation and mean of the series ( variance) = No of observations in the series = Probable error of single measurement =Probable error of Sum of observations   6 Standard Deviation - It is known as root mean square (RMS) error of measurement Indicate the amount of variation about central value  

21. 5 Probable error Significance of Standard deviation- It assess the precision of set of observation. Indicates the amount of variation about central value. Establish the limits of error bound within which 68.3% values of set lie. Smaller the value of ‘ σ ’ greater the precision  

22. Example 1 In carrying a line of level across a river, the following eight readings were taken with level under identical condition . ( GTU-June-2010) 2.322,2.346,2.352,2.306,2.312,2.300,2.306, 2.326 Calculate 1) the probable error of single observation - Es 2) the probable error of mean - Em 3) Most probable value of staff reading- MPV 4) Standard deviation-  

23. Solution Sr. No . Staff reading s Mean (2-3) ² 1 2 3 4 5 1 2.322 2.321 0.001 0.0000001 2 2.346 2.321 0.025 0.000625 3 2.352 2.321 0.031 0.000961 4 2.306 2.321 -0.015 0.000225 5 2.312 2.321 -0.009 0.000081 6 2.3 2.321 -0.021 0.000441 7 2.306 2.321 -0.015 0.000225 8 2.326 2.321 0.005 0.000025 Total 7 ∑ ² 0.0025831 Sr. No . Staff reading s Mean 1 2 3 4 5 1 2.322 2.321 0.001 0.0000001 2 2.346 2.321 0.025 0.000625 3 2.352 2.321 0.031 0.000961 4 2.306 2.321 -0.015 0.000225 5 2.312 2.321 -0.009 0.000081 6 2.3 2.321 -0.021 0.000441 7 2.306 2.321 -0.015 0.000225 8 2.326 2.321 0.005 0.000025 Total 0.0025831

24. Solution 1 - Probable error of single observation = 0.001295797   2 - Probable error of mean =0.00458133   3 Most probable value of the staff reading MPV = Mean = 2.321 ± 0.00458133 = 2.325 or 2.316   4 Standard deviation = 0.0019209744  

25. Determination of probable Error- Following 5 Cases Case-1 Direct observation of equal weight Probable error of single observation of unit weight Probable error of single observation of weight w Probable error of single arithmetic mean Case-2 Direct observation of Un-Equal weights Probable error of single observation of unit weight Probable error of single observation of weight w Probable error of weighted arithmetic mean Case-3 Indirect observation of Independent quantity Case-4 Indirect observation involving conditional Equations Case-5 Computed Quantities Department of Civil Engineering, Darshan Institute of Engineering and Technology-RAJKOT 25 a b c a b c

26. Determination of probable Error- Following 5 Cases Case-1 Direct observation of equal weight (a)Probable error of single observation of unit weight   (b)Probable error of single observation of weight w   (c) Probable error of single arithmetic mean   Case-2 Direct observation of Un-Equal weights (a) Probable error of single observation of unit weight -   (b) Probable error of single observation of weight w   (c )Probable error of weighted arithmetic mean  

27. Determination of probable Error- Following 5 Cases Case-3 Indirect observation of independent quantities Probable error of an observation of unit weight q= No of unknown quantities   The probable error of an observation of weight w   Case-4 Indirect observations involving conditional Equations   The probable error of an observation of weight w   Case-5 Computed Quantities The probable error of computed quantities may be calculated form the laws….. Later will discuss

28. 5 Determination of probable Error Example on :Case-2 -Direct observation of Un-Equal weights ( Equal weight same problem – as shown in Example-1) Example :2 The observed values of an angle are given below Find Probable error of single observation of unit weight Probable error of weighted arithmetic mean Probable error of single observation of weight -3 Angle Weight 85ᵒ 40’ 20” 2 85ᵒ 40’ 18” 2 85ᵒ 40’ 19” 3

29. 5 Determination of probable Error Solution : Example-2 Errors are in seconds only –Inputs are seconds 1 2 3= 1*2 4 = Mean of 3 -1 5=4*4 6= 2*5 n=3 1 2 3= 1*2 4 = Mean of 3 -1 5=4*4 6= 2*5 n=3

30. 5 Determination of probable Error Solution : Example-2 Probable error of single observation of unit weight , Ans-1 (b)Probable error of weighted arithmetic mean , Ans-2 (c) Probable error of single observation of weight 3 Ans-3  

31. 6 Theory of least squares Principle of distributing errors by the method of least squares is of great help to find the MPV of a quantity which has measured several times. Fundamental principle of method of least square The most probable value of an observed quantity available from a given set of observation is the one for which the sum of the squares of the residual error is minimum = minimum… If measurement of equal weight = minimum.. if unequal weight  

32. 7 LAWS OF WEIGHTS Concept of law of weight Weight : The weight of a quantity is trust worthiness of a quantity. The relative precision and trustworthiness of an observation as compared to the precision of other quantities is known as weight of the observation The weights are always expressed in numbers. Higher number indicate higher precision and trust as compared to lesser numbers Law of weight 1 Law of weight 2 Law of weight 3 Law of weight 4 Law of weight 5 Law of weight 6 Law of weight 7

33. 7 LAWS OF WEIGHTS Law of weight -1 The weight of the arithmetic mean of the measurements of unit weight is equal to the number of observation Example-1 Calculate the weight of the arithmetic mean of the following observation of an angle of unit weight Solution : Arithmetic mean = 65° 30’ 15” Hence the weight of the arithmetic mean 65° 30’ 15” is 3 Angle Weight A= 65° 30’ 10” 1 A= 65° 30’ 15” 1 A= 65° 30’ 20” 1

34. 7 LAWS OF WEIGHTS Law of weight -2 : The weight of the weighted arithmetic mean of a number of observations is equal to the sum of individual weights of observations Example-1 An angle A was measured three times as given below with their respective weights. What is the weight of the weighted arithmetic mean (WAM) Solution : Weighted Arithmetic Mean (WAM) of A = = 60° 15’ 12.33 ” So the 60 ° 15’ 12.33” has weight - 6 (1+2+3 )   Angle Weight A= 60° 15’ 10” 1 A= 60° 15’ 14” 2 A= 60° 15’ 12” 3

35. 7 LAWS OF WEIGHTS Law of weight -3 : The weight of algebraic sum of two or more quantities is equal to the reciprocal of the sum of reciprocals of individual weights Example-1 Calculate the weight of (A+B) and (A-B) if the measured values and weights of A and B respectively are:A = 60° 50’ 30” weight 3 and B= 50° 40’ 20” weight 4 Solution: A + B = 60° 50’ 30” + 50° 40’ 20” = 111° 30’ 50” A - B = 60° 50’ 30” - 50° 40’ 20” = 10° 10’ 10” As per law-3 the weight of (A + B) & (A-B ) = Hence A + B = 111° 30’ 50” weight 12/7 A -B = 10° 10’ 10” weight 12/7  

36. 7 LAWS OF WEIGHTS Law of weight -4 : The weight of product of any quantity multiplied by a constant, is equal to the weight of that quantity divided by the square of that constant Example-1 What is the weight of 3A if A = 2 0° 25’ 40” and its weight is 3. Solution : A is multiplied by 3, the constant of multiplication C is 3 and weight w of A is 3 . From law of weight 4 the weight of 3A is ( C is constant, w = weight) Weight of (3A) = 3 X 2 0° 25’ 40” = 61 ° 17’ 0 ” = Hence Weight of 61 ° 17’ 0” =  

37. 7 LAWS OF WEIGHTS Law of weight -5 : The weight of the quotient of any quantity divided by a constant is equal to the weight of that quantity multiplied by the square of that constant Example-1 Compute the weight of if A = 36 ° 20’ 40” of weight is 3 Solution : Solution: The constant of division is 4 and weight of the observation is 3. From law (5), the weight of is Weight of Hence Weight of =  

38. 7 LAWS OF WEIGHTS Law of weight -6- : The weight of the equation remains unchanged if all the sign of equation are changed or if the equation is added to or subtracted from a constant Example-1 If the weight of A+B =76 ° 20’ 30”is 3 , what is weight of –(A+B) or 180 –(A+B) ? From law of 6, the weight –(A+B) will remain the same . Weight of –(A+B) = 76 ° 20’ 30” weight 3 or Weight of 180 ° –(A+B) = 180 ° – 76 ° 20’ 30” = 103 ° 39’ 30” is equal to weight=3

39. 7 LAWS OF WEIGHTS Law of weight -7 : If an equation is multiplied by its own weight, the weight of resulting equation is equal to the reciprocal of the weight of that equation Example-1 Calculate the weight of the equation if weight of (A + B) is .The observed value of (A+B) is =120 ° 20’ 40” Solution: As the equation is being multiplied by its own weight, from Law of weight (7), the weight of {w(A+B)} will be if the w is the weight of (A+B). Weight of [ (A+B) = * 120 ° 20’ 40”= 90° 15’ 30 ” = Weight of 90° 15’ 30” =  

40. 8 RULES FOR GIVING WEIGHTS AND DISTRIBUTION OF ERRORS TO THE FIELD OBSERVATION 40

41. 8 RULES FOR GIVING WEIGHTS Following rules are applied in assigning the weights to the field. Weight of an angle varies directly to the number of observation made on the angle = Weight of the level lines vary inversely as the length of their routes = Weight of any angle measured large number of times is inversely proportional to square of probable error = Correction applied to various observed quantities, are inversely proportional to their weight  

42. 8 Distribution of Errors To The Field Observation In field any observation always some error (accidental error). Closing error ( Bowditch rule) FOLLOWING RULES OBSERVED FOR DISTRIBUTION OF ERROR Correction applied to observations are inversely proportional to their weight Correction applied to an observation is directly proportional to square of probable error In level line correction proportional to length In all observation of equal weight , the error is distributed is equally , so corrections are equal

43. Examples on Distribution of Errors Example:3 ( GTU-2012/2016) The following are the three angles A,B and C observed at a station O closing the horizon , along with their probable errors of measurement. Determine their corrected values Solution: Sum of three angles A = 82⁰ 15’ 20” ± 2” B = 128⁰ 26’ 10” ± 4” C = 149⁰ 18’ 15” ± 3” A + B + C = 82⁰ 15’ 20” + 128⁰ 26’ 10” + 149⁰ 18’ 15” =359⁰ 59’ 45” Discrepancy = 360⁰ 0’ 0” - 359⁰ 59’ 45” = 15” Hence each angle is to be increased , and the error of 15” is to be distributed in proportionate to the square of the probable error ( see Slide no-43, Rules no-2) Summation shall be 360⁰

44. Examples on Distribution of Errors Let be the correction to be applied to angles A,B and C   ……(1) ……………………………(2)   From Equation (1 ),   and in Equation (2) +4 + =15” (1 +4+ =15” ( 5 + =15” (5+ = 15”   ”= 4.65” Check :   Hence Corrected angles are A 82⁰ 15’ 20” + 2.07” 82⁰ 15’ 22.07 ” B 128⁰ 26’ 10” + 8.28” 128⁰ 26’ 18.28” C 149⁰ 18’ 15” + 4.65” 149⁰ 18’ 19.65” TOTAL 360⁰ 00’ 00”

45. Examples on Distribution of Errors Example:4 –GTU-Winter-2014 Adjust the following angles closing the horizon Solution : Sum of four angles A + B + C +D = 112 ⁰ 20’ 47” + 90 ⁰ 30’ 15” + 58 ⁰ 12’ 05” + 98 ⁰ 57’ 01” = 360⁰ 00’ 08” Discrepancy = 360⁰ 0’ 0” - 360⁰ 00’ 08” = - 08” Hence each angle is to be decreased, and the error of 8” is to be distributed in to the angles in an inverse proportionate to their weights ( see Slide no-43, Rules no-1) A = 112 ⁰ 20’ 47” Weight 2 B = 90 ⁰ 30’ 15” Weight 3 C = 58 ⁰ 12’ 05” Weight 1 D = 98 ⁰ 57’ 00” Weight 4

46. Examples on Distribution of Errors Let be the correction to be applied to angles A,B,C and D   ……………………..(2)   From Equation (1 ) We have   Hence Corrected angles are A 112 ⁰ 20’ 47” - 0.96” 112 ⁰ 20’ 45.04” B 90 ⁰ 30’ 15” - 0.64” 90⁰ 30’ 13.36” C 58 ⁰ 12’ 05” - 1.92” 58⁰ 12’ 2.08” D 98 ⁰ 57’ 01” - 0.48” 98 ⁰ 56’ 59.92 ” TOTAL 360⁰ 00 ’ 00” ? * 4     of , in Equation (2) we get   ( + )= 4”   + + = 4”   = 4 * = =0.96”   = =0.64”   = =1.92”   = =0.48”  

47. 9. DETERMINATION OF MOST PROBABLE VALUES MPV : The most probable value of a quantity is the value which has more chances of being true than any other value. ( close to the true value) Determined form Principle of least square. If systematic errors are eliminated from the observations, the arithmetic mean will be the most probable value of the quantity being observed

48. 9. DETERMINATION OF MOST PROBABLE VALUES METHODS Direct observation of equal weights Direct observation of unequal weight. Indirect observation involving unknown of equal weights Indirect observation involving unknown of unequal weights Observation equations accompanied by condition equation. The Normal equation The method of differences The method of correlates.

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50. 9. DETERMINATION OF MOST PROBABLE VALUES CASE-1 Direct observation of equal weight The MPV of direct observed quantity of observation of equal weight is the arithmetic mean of observations. X 1, X 2, X 3, X 4, …… X n , X is Most probable value. n = No. of observation

51. 9. DETERMINATION OF MOST PROBABLE VALUES Problem on CASE-1 Direct observation of equal weight Following direct observations of base line were taken: 2523.32 m; 2523.25m; 2523.17m; 2523.38m; 2523.47m; 2523.68m;.Calculate Most probable value of the length of base line. Solution = MPV = Arithmetic mean M = 2523.378 mt. ; MPV of base line =2523.378 m.  

52. 9. DETERMINATION OF MOST PROBABLE VALUES CASE-2 Direct observation of unequal weight The MPV of direct observed quantity of observation of unequal weigh is weighted arithmetic mean of observations. X 1, X 2, X 3, X 4, …… X n , X is Most probable value. n = No. of observation W = weight of observation w₁, w₂, w₃….

53. 9. DETERMINATION OF MOST PROBABLE VALUES Problem on CASE-2 Direct observation of unequal weight Ex-1 Find the MPV of the angle from the following observations. Angle A = 76ᵒ 35’ 00” weight 1 Angle A = 76ᵒ 33’ 40” weight 2 Solution: Most probable value (MPV) = = 76ᵒ 33’ 6.67”

54. 9. DETERMINATION OF MOST PROBABLE VALUES CASE-3 Indirect observation involving unknown of equal weights Most probable value found by method of Normal Equations. To form normal equation, ( steps) 1 Multiply equations by their coefficients of unknowns and add the results.

55. 9. DETERMINATION OF MOST PROBABLE VALUES Problem on CASE-3 Indirect observation involving unknown of equal weights Ex-1 Find the MPV of the angle A from the following observation equation .(GTU-2012) Angle A = 30ᵒ 28’ 40” weight 2 Angle 3A = 91ᵒ 25’ 55” weight 3 Solution: Most probable value (MPV) = The normal equation of A Multiply first eq n by 2 = (1 * 2) and second eq n by 9=(3 *3) 2 *A = 2* 30ᵒ 28’ 40”,  2A = 60ᵒ 57’ 20” ____(1) 9*3A = 9* 91ᵒ 25’ 55” 27A= 822ᵒ53’15”____(2) Sum (1) + (2) = 29A = 883ᵒ50’35” So Angle A = 883ᵒ50’35”/29 = 30ᵒ28’38.45 ”

56. 9. DETERMINATION OF MOST PROBABLE VALUES CASE-4 Indirect observation involving unknown of unequal weights Most probable value found by method of Normal Equations. To form normal equation, (steps) multiply equations by their coefficients of unknowns and the weight of the that observation and Add 2 the equations thus formed.

57. 9. DETERMINATION OF MOST PROBABLE VALUES CASE-5 Observation equations accompanied by condition equation One or more conditional equations are available. Methods to compute MPV The Normal equations. ( 2) The Methods of differences. ( 3) The Method of correlates.

58. Thank you