Introduction to Measurement Science

1
Introduction to Measurement Science

• ERE 371Surveying for Engineers

2
Measurement Science Introduction

• Principles apply to any activity relying on
  measurements
• Considerations
• Surveying measurements

• Observations

3
Overview

• Basic principles of measurements
• Subsequent use of measurements
• Planning for measurements
• Units of measurements

4
Measurements

• Single measurement
• Issues

5
Measurements

• Methods for increasing confidence
• Expressions of quality

6
Precision

• Understanding precision
• Degree of precision depends on
• Express precision using statistics

7
Variance and Standard Deviation

• Purpose
• Indicate measurement spread
• Formulae

standard deviation
variance

• Example
• Set A 2.3 m, 2.4 m, 2.5 m s2 0.01 m2, s
  0.1 m
• Set B 2.1 m, 2.3 m, 2.5 m s2 0.04 m2, s
  0.2 m

8
Accuracy

• Understanding accuracy
• Example
• Accepted value is 2.31 m
• Which measurement set is more accurate, A or B?
• Set A average 2.4 m
• Set B average 2.3 m

9
ACCURACY
High Low
PRECISION
High Low
10
Expressing Accuracy

• Absolute accuracy
• Relative accuracy

11
Considering Measurement Variations

• Issue
• Two approaches
• Sources of error

• Types of error

12
Theory of Errors

• Theory
• Error
• Issues

13
Basic Principles

• Four underlying principles in Theory of Errors
• Confidence is provided by comparing to

14
Theory of Observations

• Theory
• Issues

15
Errors and Mistakes

• Mistakes are NOT errors
• Must do all possible to

16
Types of Errors

• Systematic errors (biases)
• Random errors

17
Redundancy in Measurements

• Generally make redundant measurements
• Most probable value (MPV)
• Residuals

18
Understanding Random Errors

• Frequency of occurrence
• Dispersion
• Frequency
• Histogram
• Normal distribution

19
Laws of Probability

• Laws of probability

20
Expressing Precision

• Expressing precision using standard deviation
• Interpreting standard deviation
• Percent errors
• Use percent errors to specify required precision
• E90 1.6449? E95 1.9599?
• E.g. require E95 to be less than specific value

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Readings

• Chapter 3 sections 3.1 3.16

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Applying Measurement Science

• Subsequent Use of Measured Values
• Preanalysis of Measurements
• Importance of Units

23
Subsequent Use of Measured Values

• Types of observations
• Often use measurements to compute values
• Three specific ideas

24
Significant Figures

• Two types of numeric values
• In calculations
• Issue

25
Use of Weights

• Applications
• Incorporate multiplicative factors into
  computation
• Very powerful tool

26
Weighted Measurements

• Situation have greater confidence in some values
• How to determine weight?
• Commonly use weight derived by inverse proportion

27
Weighted Measurements Example
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Weighted Measurements Example
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Propagation of Variance

• Variance of observations
• Issue
• Variance of result depends on

30
Propagation of Variance (Contd)

• Assumptions
• Theoretical formulation

Note Standard deviation ?U found by taking
square root
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Propagation of Variance Example

• Situation
• Measure slope distance and vertical angle
• Calculate horizontal distance
• Measurements made
• Slope distance (L) 124.067 m with S.D. (sL)
  0.017 m
• Angle (?) 31 05' 25" with S.D. (s?) 00 00'
  10" 0.0000485 radians
• Calculating horizontal distance
• H L cos ? 124.067 m x cos (31 05' 25")
  106.245 m

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Propagation of Variance Example

• How good is calculated value of H?
• Use propagation of variance to estimate variance
• Calculating variance

33
Propagation of Variance (Contd)

• Formulas in text
• Generalizing propagation of variance

34
Preanalysis of Measurements

• Issue
• Basically inverse of propagation of variance
• Steps

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Preanalysis of Measurements

• Situation
• Reducing slope distance to horizontal
• How well does vertical angle need to be measured?
• Steps

• Differentiate H w.r.t. ?

• Divide by original equation

• Ignore sign, rearrange to get d? (radians)

36
Preanalysis of Measurements

• Example
• Measure slope distance along 15 incline
• Want relative accuracy in H of 1 in 10,000
• How well do we need to measure ??

• Implication
• Need a 1' instrument

37
Units of Measurement

• Linear distance units
• Historic Gunters chain 1 chain 66 feet 4
  rods 100 links
• English international foot, US survey foot,
  yard, mile
• Metric meter
• Angular distance
• Sexagesimal Degrees/minutes/seconds 360
  circle
• Decimal degrees
• Grads centesimal minutes and seconds 400g
  circle
• Mils 6400 mils circles
• Radians

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Units of Measurement

• Area units
• English Acre 1 acre 1 chain x 10 chains
  43560 ft2
• Metric Hectare 1 ha 100m x 100m 10,000 m2
• Volume units
• English Cubic yards, Acre-feet
• Metric Cubic meters
• The U.S. and the metric system

39
Readings

• Chapter 2 sections 2.1 2.5
• Chapter 3 sections 3.17 3.21