1. Errors in survey measurements
Lecture 2

2. Previous lecture Definition of surveying Plane and geodetic surveying
Different types of surveying
Principles in surveying
Control
Economy of accuracy
consistency

3. contents Measurements Errors Source of errors Types of errors
Quality of measurements
Significant figures
Error propagation

4. Measurements Definition of measurements
The process of estimating the magnitude of some attribute of an object relative to a standard unit
The application of a device or apparatus for the purpose of determining an unknown quantity
An observation made to determine an unknown quantity
Characteristics of measurements
No measurements are exact
True value is never known
All measurements contain errors

5. Measurements Types of measurements Direct Indirect
Direct measurement’ refers to measuring exactly the thing that you’re looking to measure
Taping, rope
Indirect
Deducing the measurement from measurements of other quantities.
measuring something by measuring something else
EDM, stadia tacheometry

6. Errors
Every measuring technique is subject to unavoidable error. A surveyor must know the quality of their measurements and whether they meet requirements.
An error is the difference between a measured quantity and its true value
 = y - 
True value
Error
Measured value

7. Correction
C = - =  - y
Correction = (estimated value) true value – measured value

8. Sources of error Instrumental Natural errors Personal errors
Caused by imperfections in instrument construction or adjustment
Wear and tear
Natural errors
caused by changing conditions in the surrounding environment.
Temperature, wind, moisture, etc
Personal errors
Caused by limitations in human sense
Natural errors: These are caused due to variations in nature i.e., variations in wind, temperature, humidity, refraction, gravity and magnetic field of the earth.
Instrumental: These result from imperfection in the construction or adjustment of surveying instruments, and movement of their individual parts.
Personal Errors: These arise from limitations of the human senses of sight, touch and hearing.

9. Types of errors Mistakes/blunders/Gross errors
Caused by confusion or carelessness of the observer
Faults in equipment
Use of wrong technique
Misinterpretation
Can be spotted by check measurements and then eliminated
Examples
Wrong booking
Misreading the tape
Miscounting tape lengths
The blunders or mistakes result into large errors and thus can easily be detected by comparing with other types of errors (generally small in value). The maximum permissible error in an observation is ± 3.29 s (where s is the standard deviation of sample distribution) and is used to separate mistakes or blunders from the random errors. If any error deviates from the mean by more than the maximum permissible error, it is considered as a gross error and the measurement is rejected.
After mistakes have been detected and eliminated from the measurements, the remaining errors are usually classified either as systematic or random error depending on the characteristics of errors.
Systematic errors occur according to a system. These errors follow a definite pattern. Thus, if an experiment is repeated, under the same condition, same pattern of systematic errors reoccur. These errors are dependent on the observer, the instrument used, and on the physical environment of the experiment. Any change in one or more of the elements of the system will cause a change in the character of the systematic error. Depending on the value and sign of errors in successive observation, systematic errors are divided into two types.
Cummulative Error
Compensating Error
Systematic errors are dealt with mathematically using functional relationships or models.

10. Types of errors Systematic errors Follow some physical law
Stays constant in sign and magnitude when repeated under the same condition.
They are cumulative in nature
Corrections:
applying some mathematical corrections.
Can also be removed by calibrating the observing equipment and quantifying the errors
Proper selection of measuring procedure
The blunders or mistakes result into large errors and thus can easily be detected by comparing with other types of errors (generally small in value). The maximum permissible error in an observation is ± 3.29 s (where s is the standard deviation of sample distribution) and is used to separate mistakes or blunders from the random errors. If any error deviates from the mean by more than the maximum permissible error, it is considered as a gross error and the measurement is rejected.
After mistakes have been detected and eliminated from the measurements, the remaining errors are usually classified either as systematic or random error depending on the characteristics of errors.
Systematic errors occur according to a system. These errors follow a definite pattern. Thus, if an experiment is repeated, under the same condition, same pattern of systematic errors reoccur. These errors are dependent on the observer, the instrument used, and on the physical environment of the experiment. Any change in one or more of the elements of the system will cause a change in the character of the systematic error. Depending on the value and sign of errors in successive observation, systematic errors are divided into two types.
Cummulative Error
Compensating Error
Systematic errors are dealt with mathematically using functional relationships or models.

11. Types of errors Examples Wrong length of tape Poor ranging
poor straightening
slope
sag
temperature variation
wrong tensioning
The blunders or mistakes result into large errors and thus can easily be detected by comparing with other types of errors (generally small in value). The maximum permissible error in an observation is ± 3.29 s (where s is the standard deviation of sample distribution) and is used to separate mistakes or blunders from the random errors. If any error deviates from the mean by more than the maximum permissible error, it is considered as a gross error and the measurement is rejected.
After mistakes have been detected and eliminated from the measurements, the remaining errors are usually classified either as systematic or random error depending on the characteristics of errors.
Systematic errors occur according to a system. These errors follow a definite pattern. Thus, if an experiment is repeated, under the same condition, same pattern of systematic errors reoccur. These errors are dependent on the observer, the instrument used, and on the physical environment of the experiment. Any change in one or more of the elements of the system will cause a change in the character of the systematic error. Depending on the value and sign of errors in successive observation, systematic errors are divided into two types.
Cummulative Error
Compensating Error
Systematic errors are dealt with mathematically using functional relationships or models.

12. Random errors
Errors that remain in a measurement after systematic and gross errors have been removed.
Magnitude and direction of the error beyond the control of the surveyor
Compensating
Normally distributed
After mistakes are eliminated and systematic errors are corrected, a survey measurement is associated with random error only. This error is small and is equally liable to be plus or minus thus partly compensating in nature. Random errors are unpredictable and they cannot be evaluated or quantified exactly.
Random errors are determined through statistical analysis based on following assumptions :
Small variations from the mean value occur more frequently than large ones.
Positive and negative variations of the same size are about equal in frequency, rendering their distribution symmetrical about a mean value.
Very large variations seldom occur.
Thus, to eliminate random error in a measurement, observations are repeated for number of times. The mean (average) of observations is considered to be the true (or estimated) value of the measurement. Normal or Gaussian distribution typifies the distribution of samples of any measurement.

13. 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.

14. Examples holding and marking variation in tension Error frequency 10
-10 10
Magnitude of Error
Error
frequency

15. Reliability of measurements
Mean
Standard deviation

16. x
f(x)
measurements
High precision
small 
Low precision
high  2 1

17. Quality of measurements
Accuracy
Degree of perfection obtained in a measurement
Absolute nearness of a measured quantity to its true value
Precision
The closeness of one measurement to another
Degree of consistency between measurements
Based on the size of discrepancies in a data set

18. IN SURVEYING WE WANT OUR MEASUREMENTS TO BE ACCURATE AND PRECISE
Accuracy is telling the truth
Precision is repeating the same story over and over again.
IN SURVEYING WE WANT OUR MEASUREMENTS TO BE ACCURATE AND PRECISE

20. Significant Figures
The number of significant digits in a number/value.
Measurements can only be accurate to the degree that the measuring instrument is precise.

21. Rules and conventions :
All non-zero digits in a number are significant.
Example: Numbers , , 21.6 and 216 have the same number of significant figures namely three (2, 1, 6).
All zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.
Example : In the numbers , , 20.6 and 206, the zero lying between the digits 2 and 6 is only significant.
If the number is less than 1, the zeroes on the right of decimal point but to the left of the first non-zero digit are not significant.
Example : In , the four zeros after decimal and before the digit 2 has no significance. Similarly, in , the zero after decimal and before the digit 2 has no significance. So the number of significant figures of these numbers are three (2, 0 and 6).

22. Rules and conventions :
The terminal or trailing zeros in a number without a decimal point are significant depending on accuracy of measurement.
Example : In 2360 m, the terminal zero has no significance, if the accuracy of measurement is 10 m then the number of significant figures of this number is three (2, 3 and 6). If the accuracy of measurement is 1 m, the terminal zero is significant figures of this same number will be four (i.e. 2, 3, 6 & 0).
The digit 0 conventionally put on the left of a decimal for a number less than 1 is never significant. However, the zeros at the end of such number are significant in a measurement.
Example : The number has three significant numbers. The zero before the decimal point is not significant.
The terminal or trailing zeros in a number with a decimal point are significant.
Example : In m, the terminal zero has significance, so the number of significant figures in this number is four (2, 3, 6 and 0).

23. Error propagation Propagation of Error
Error Propagation in a Sum or a Difference of Measurments
Error Propagation in a Product of Measurments
Error Propagation in a Division
Error Propagation due to the Power of a Measured Quantity

24. Addition and Subtraction
When two or more quantities are added or subtracted, the error in result (Es) is the square root of the sum of the square of the errors (e1, e2, .....) of the individual quantity i.e.,
For example : The total distance D = (120 ± 2) mm + (321 ± 5) mm

25. Product
When two or more quantities are multiplied, the error in result (Eproduct) is the square root of the sum of the square of the fractional errors of the individual quantity. Thus, where EA and EB are errors in observed values of A and B respectively.

26. Product
For example: A rectangle is measured ± cm long and ± cm wide. The error in its area (12,067 cm2) is

27. Division
When two or more quantities are divided, the error in result is the square root of the sum of the square of the fractional errors in the individual quantity.

28. Division
For example: If the area of a rectangular plot is somehow known to be 49,650 ± 10 m2 and the width dimension measured several times found to be ± 0.46 m, the calculated length dimension is

29. Power
The error in a physical quantity raised to the power is the power times the fractional errors in the individual quantity.
For example: If a sphere's radius is measured as ± 0.08 m, the calculated volume is m3 and the error will be

30. END