MEASUREMENT ANALYSIS AND ADJUSTMENT

1
MEASUREMENT ANALYSIS AND ADJUSTMENT

• Capital Project Skill Development Class (CPSD)
• G100398

By Jeremy Evans, P.L.S. Psomas Supplemented
by Caltrans Staff
2
Introduction

• The dark side of surveying is the belief that
  surveying is about measurements, precisions and
  adjustments. It is not and never will be.
• Dennis Mouland
• P.O.B. Magazine
• July, 2002

3
Introduction

• Much has been written lately about least squares
  adjustment and the advantages it brings to the
  land surveyor. To take full advantage of a least
  squares adjustment package, the surveyor must
  have a basic understanding of the nature of
  measurements, the equipment he uses, the methods
  he employs, and the environment he works in.

4
Introduction

• Measurements and Adjustments
  War Stories

5
Class Outline

• Survey Measurement Basics - A Review
• Measurement Analysis
• Error Propagation
• Introduction to Weighted and Least Squares
  Adjustments
• Least Squares Adjustment Software
• Sample Network Adjustments

6
Measure First, Adjustment Last

• Adjustment programs assume that
• Instruments are calibrated
• Measurements are carefully made
• Networks are stronger if
• They include Redundancy
• They have Strength of Figure
• Adjust only after you have followed proper
  procedures!

7
Survey Measurement Basics

• A Review of Plumb Bob 101

8
Surveying (Geospatial Services?)

• Surveying That discipline which encompasses
  all methods for measuring, processing, and
  disseminating information about the physical
  earth and our environment. Brinker Wolf
• Surveyor - An expert in measuring, processing,
  and disseminating information about the physical
  earth and our environment.

9
Measurement vs. Enumeration

• A lot of statistical theory deals with
  enumeration, or counting. Its a way to take a
  test sample instead of a census of the total
  population.
• The surveyor is concerned with Measurement. The
  true dimensions can never be known.

10
Instrument Testing

• Pointing error of typical total station

11

12
Instrument Specifications
13
Instrument Specifications
14
Instrument Specifications

• Distance Measurement
• sm (0.01 3ppm x D)
• What is the error in a 3500 foot measurement?
• sm (0.01(3/1,000,000 x 3500)) 0.021

15
Calibration or Dont shoot yourself in the foot.

• Leica instruments should be serviced every 18
  months.
• EDMs should be calibrated every six months
• Tribrachs should be adjusted every six months, or
  more often as needed.
• Levels pegged every 90 days

16
Is It a Mistake or an Error?

• Mistake - Blunder in reading, recording or
  calculating a value.
• Error - The difference between a measured or
  calculated value and the true value.

17
Blunder
a gross error or mistake resulting usually from
stupidity, ignorance, or carelessness.
18
Blunder

• Setup over wrong point
• Bad H.I.
• Incorrect settings in equipment

19
Types of Errors

• Systematic
• Random
• An error is the difference between a measured
  value and the true value. Later we will compare
  this to the definition of residual

20
Systematic
an error that is not determined by chance but is
introduced by an inaccuracy (as of observation or
measurement) inherent in the system
21
Systematic

• Glass with wrong offset
• Poorly repaired chain
• Imbalance between level
• sightings

Each measurement made with the tape is 0.1'
shorter than recorded.
22
Random

• an error that has a random distribution and can
  be attributed to chance.
• without definite aim, direction, or method

23
Random

• Poorly adjusted tribrach
• Inexperienced Instrument
• operator
• Inaccuracy in equipment

24
Nature of Random Errors

• A plus or minus error will occur with the same
  frequency
• Minor errors will occur more often than large
  ones
• Very large errors will rarely occur (see mistake)

25
Normal Distribution Curve 1

• A plus or minus error will occur with the same
  frequency, so
• Area within curve is equal on either side of the
  mean

26
Normal Distribution Curve 2

• Minor errors will occur more often than large
  ones, so
• The area within one standard deviation (s) of the
  mean is 68.3 of the total

27
Normal Distribution Curve 3

• Very large errors will rarely occur, so
• The total area within 2s of the mean is 95 of
  the sample population

28
Histograms, Sigma, Outliers
MEAN
Histogram Plot of the Residuals \
Bell shaped curve /
Outlier \
4.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
1 s 68 of residuals must fall inside
area 2 s 95 of residuals must fall
inside area
Residuals
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Measurement Components

• All measurements consist of two components the
  measurement and the uncertainty statement.
• 1,320.55 0.05
• The uncertainty statement is not a guess, but is
  based on testing of equipment and methods.

30
Accuracy Vs. Precision

• Precision - agreement among readings of the same
  value (measurement). A measure of methods.
• Accuracy - agreement of observed values with the
  true value. A measure of results.

31
Measurement Analysis

• Determining Measurement Uncertainties

32
Determining Uncertainty

• Uncertainty - the positive and negative range of
  values expected for a recorded or calculated
  value, i.e. the value (the second component of
  measurements).

33
Your Assignment

• Measure a line that is very close to 1000 feet
  long and determine the accuracy of your
  measurement.
• Equipment 100 tape and two plumb bobs.
• Terrain Basically level with 2 high brush.
• Environment Sunny and warm.
• Personnel You and me.

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Planning the Project

• Test for errors in one tape length.
• Measure 1000 foot distance using same methods as
  used in testing.
• Determine accuracy of 1000 foot distance.

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Test Data Set

• Measured distances
• 99.96 100.02
• 100.04 100.00
• 100.00 99.98
• 100.02 100.00
• 99.98 100.00

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Averages

• Measures of Central Tendency
• The value within a data set that tends to exist
  at the center.
• Arithmetic Mean
• Median
• Mode

37
Averages

• Most commonly used is Arithmetic Mean
• Considered the most probable value
• n number of observations
• Mean 1000 / 10
• Mean 100.00

38
Residuals

• The difference between an individual reading in a
  set of repeated measurements and the mean
• Residual (n) reading - mean
• Sum of the residuals squared (Sn2) is used in
  future calculations

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Residuals

• Calculating Residuals (mean 100.00)
• Readings residual residual2
• 99.96 -0.04 0.0016
• 100.02 0.02 0.0004
• 100.04 0.04 0.0016
• 100.00 0 0
• 100.00 0 0
• 99.98 -0.02 0.0004
• 100.02 0.02 0.0004
• 100.00 0 0
• 99.98 -0.02 0.0004
• 100.00 0 0
• Sn2
  0.0048

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Standard Deviation

• The Standard Deviation is the range within
  which 68.3 of the residuals will fall or
• Each residual has a 68.3 probability of falling
  within the Standard Deviation range or
• If another measurement is made, the resulting
  residual has a 68.3 chance of falling within the
  Standard Deviation range.

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Standard Deviation Formula
42
Standard Deviation

• Standard Deviation is a comparison of the
  individual readings (measurements) to the mean of
  the readings, therefore
• Standard Deviation is a measure of.

43
Standard Deviation

• Standard Deviation is a comparison of the
  individual readings (measurements) to the mean of
  the readings, therefore
• Standard Deviation is a measure of.

PRECISION!
44
Standard Deviation of the Mean

• This is an uncertainty statement regarding the
  mean and not a randomly selected individual
  reading as is the case with standard deviation.
• Since the individual measurements that make up
  the mean have error, the mean also has an
  associated error.
• The Standard Deviation of the Mean is the range
  within which the mean falls when compared to the
  true value, therefore the Standard Deviation
  of the Mean is a measure of .

45
Standard Deviation of the Mean

• This is an uncertainty statement regarding the
  mean and not a randomly selected individual
  reading as is the case with standard deviation.
• Since the individual measurements that make up
  the mean have error, the mean also has an
  associated error.
• The Standard Error of the Mean is the range
  within which the mean falls when compared to the
  true value, therefore the Standard Deviation
  of the Mean is a measure of .

ACCURACY!
46
Standard Deviation of the Mean

• Distance 100.000.007 (1s
  Confidence level)

47
Probable Error

• Besides the value of s 68.3, other error values
  are used by statisticians
• An error value of 50 is called Probable Error
  and is shown as E or E50
• E50 (0.6745)s

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90 95 Probable Error

• A 50 level of certainty for a measure of
  precision or accuracy is usually unacceptable.
• 90 or 95 level of certainty is normal for
  surveying applications
•  

49
95 Probable Error

• Distance 100.000.015 (2s Confidence
  Level)

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Meaning of E95

• If a measurement falls outside of two standard
  deviations, it isnt a random error, its a
  mistake!
• Francis H. Moffitt

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How Errors Propagate

• Error in a Series
• Errors in a Sum
• Error in Redundant Measurement

52
Error in a Series

• Describes the error of multiple measurements with
  identical standard deviations, such as measuring
  a 1000 line with using a 100 chain.

53
Error in a Sum

• Esum is the square root of the sum the errors of
  each of the individual measurements squared
• It is used when there are several measurements
  with differing standard deviations

54
Exercise for Errors in a Sum

• Assume a typical single point occupation. The
  instrument is occupying one point, with tripods
  occupying the backsight and foresight.
• How many sources of random error are there in
  this scenario?

55
Exercise for Errors in a Sum

• There are three tribrachs, each with its own
  centering error that affects angle and distance
• Each of the two distance measurements have errors
  
• The angle turned by the instrument has several
  sources of error, including poor leveling and
  parallax

56
Error in Redundant Measurements

• If a measurement is repeated multiple times, the
  accuracy increases, even if the measurements have
  the same value

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Sample of Redundancy
58
Eternal Battle of Good Vs. Evil

• With Errors of a Sum (or Series), each
  additional variable increases the total error of
  the network
• With Errors of Redundant Measurement, each
  redundant measurement decreases the error of the
  network.

59
Sum vs. Redundancy

• Therefore, as the network becomes more
  complicated, accuracy can be maintained by
  increasing the number of redundant measurements

60
Error Ellipses

• Used to described the accuracy of a measured
  survey point.
• Error Ellipse is defined by the dimensions of the
  semi-major and semi-minor axis and the
  orientation of the semi-major axis
• Assuming standard errors, the measurements have a
  39.4 chance of falling within the Error Ellipse
• E95 2.447s

61
Coordinate Standard Deviations and Error Ellipses
Coordinate Standard Deviations and Error
Ellipses Point Northing Easting N SDev E
SDev 12 583,511.320 2,068,582.469 0.021 0.017
Northing Standard Deviation

Easting Standard Deviation
62
Positional Accuracy vs. Precision Ratio

• Or, How good is one error ellipse compared to
  all those others?

63
Introduction to Adjustments

• Adjustment - A process designed to remove
  inconsistencies in measured or computed
  quantities by applying derived corrections to
  compensate for random, or accidental errors, such
  errors not being subject to systematic
  corrections.
• Definitions of Surveying and
• Associated Terms,
• 1989 Reprint

64
Introduction to Adjustments

• Common Adjustment methods
• Compass Rule
• Transit Rule
• Crandall's Rule
• Rotation and Scale (Grant Line Adjustment)
• Least Squares Adjustment

65
Weighted Adjustments

• Weight - The relative reliability (or worth) of
  a quantity as compared with other values of the
  same quantity.
• Definitions of Surveying and
• Associated Terms,
• 1989 Reprint

66
Weighted Adjustments

• The concept of weighting measurements to account
  for different error sources, etc. is fundamental
  to a least squares adjustment.
• Weighting can be based on error sources, if the
  error of each measurement is different, or the
  quantity of readings that make up a reading, if
  the error sources are equal.

67
Weighted Adjustments

• Formulas
• W ? (1 ? E2) (Error Sources)
• C ? (1 ? W) (Correction)
• W ? n (repeated measurements of the same
  value)
• W ? (1 ? n) (a series of
  measurements)

68
Weighted Adjustments
A 43?2436, 2x B 47?1234, 4x C
89?2220, 8x Perform a weighted adjustment based
on the above data
A
B
C
69
ANGLE No. Meas Mean Value Rel. Corr.
Corrections Adjusted Value   A 2 43?
24 36 4/4 or 4/7 4/7 X 30 17 43? 24
53 B 4 47? 12 34 2/4 or 2/7 2/7 X 30
09 47? 12 43 C 8 89? 22 20 1/4 or
1/7 1/7 X 30 04 89? 22 24 TOTALS 179?59
30 7/4 or 7/7 30 180?
00 00
The relative correction for the three angles are
1 2 4, the inverse proportion to the number
of turned angles. This is the first set of
relative corrections. The sum of the relative
corrections is 1 2 4 7 , This is used as
the denominator for the second set of
corrections. The sum of the second set of
relative corrections shall always equal 1. The
second set is used for corrections.
70
Weighted Adjustments
BM B Elev. 102.0
7.8, 2 mi.
BM NEW
6.2, 10 mi.
10.0, 4 mi.
BM A Elev. 100.0
BM C Elev. 104.0
71
Introduction to Least Squares Adjustment

• Simple Examples

72
What Least Squares Is ...

• A rigorous statistical adjustment of survey data
  based on the laws of probability and statistics
• Provides simultaneous adjustment of all
  measurements
• Measurements can be individually weighted to
  account for different error sources and values
• Minimal adjustment of field measurements

73
What is Least Squares?

• A Least Squares adjustment distributes random
  errors according to the principle that the Most
  Probable Solution is the one that minimizes the
  sums of the squares of the residuals.

• This method works to keep the amount of
  adjustment to the observations and, ultimately
  the movement of the coordinates to a minimum.

74
Least Squares Example

• Arithmetic Mean
• Using Least squares to prove a simple arithmetic
  mean solution

75
Least Squares Example   A point is measured for
location 3 times. The measurements give the
following NE coordinates   a. 0,0 b. 0,5 c.5,
0     c 5,0 ? What is the best
solution for an average? ? How
can you prove it?   ? ? a 0,0
b 0,5
76
Student exercise

• GROUP 1
• Determine the sum of the squares from
• X2.5, Y2.5

• GROUP 2
• Determine sum of the squares from
• Mean X, Mean Y
• (1.667, 1.667)

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Solution
If ? 1.667, 1.667, then Distance a-? 2.357,
b-? 3.727, c-?3.727      c 5,0 ? N
(0 0 5) ? 3 1.667 E (0 5 0) ? 3
1.667 2.357² 3.727² 3.727²
33.333   ? ? ? a
0,0 b 0,5
78
What Least Squares Isnt ...

• A way to correct a weak strength of figure
• A cure for sloppy surveying - Garbage in /
  Garbage out
• The only adjustment available to the land surveyor

79
Least Squares

• Least Squares Should Be Used for
• The Adjustment Of Collected By

Conventional Traverse Control Networks GPS
Networks Level Networks Resections
Theodolite Chain Total Stations GPS
Receivers Levels EDMs
80
Least Squares
A
B
Observed
E
1st Iteration
G
2nd Iteration
F

• What happens?

C

• Iterative Process
• Each iteration applies adjustments to
  observations, working for best solution
• Adjustments become smaller with each successive
  iteration

D
81
Least Squares
The Iterative Process

• Creates a calculated observation for each field
  observation by inversing between approximate
  coordinates.
• Calculates a "best fit" solution of observations
  and compares them to field observations to
  compute residuals.
• Updates approximate coordinate values.
• Calculates the amount of movement between the
  coordinate positions prior to iteration and after
  iteration.
• Repeats steps 1 - 4 until coordinate movement is
  no greater than selected threshold.

82
Least Squares
Four component that need to be addressed prior to
performing least squares adjustment

• Errors
• Coordinates
• Observations
• Weights

83
Errors

• Blunder - Must be removed
• Systematic - Must be Corrected
• Random - No action needed

84
Coordinates

• Because the Least Squares process begins by
  calculating inversed observations approximate
  coordinate values are needed.
• 1 Dimensional Network (Level Network) - Only 1
  Point.
• 2 Dimensional Network - All Points Need Northing
  and Easting.
• 3 Dimensional Network - All Points Need Northing,
  Easting, and Elevation. (Except for adjustments
  of GPS baselines.)

85
Weights

• Each Observation Requires an Associated Weight
• Weight Influence of the Observation on Final
  Solution
• Larger Weight - Larger Influence
• Weight 1/s2
• s Standard Deviation of the Observation
• The Smaller the Standard Deviation the Greater
  the Weight

s 0.8 ? Weight 1/0.82 1.56 s 2.2 ?
Weight 1/2.22 0.21
More Influence Less Influence
86
Methods of Establishing Weights

• Observational Group
• Least Desirable Method
• Example All Angles Weighted at the Accuracy of
  the Total Station
• Each Observation Individually Weighted
• Best Method
• Standard Deviation of Field Observations Used as
  the Weight of the Mean Observation
• Combination of Types
• Assigns the Least weight possible for each
  observation

Good for combining Observations from different
classes of instruments.
Good for projects where standard deviation is
calculated for each observation.
87
Least Squares
If you remember nothing else about least squares
today, remember this!

• Least Squares Adjustment Is a Two Part Process
• - Unconstrained Adjustment
• Analyze the Observations, Observations Weights,
  and the Network
• - Constrained Adjustment
• Place Coordinate Values on All Points in the
  Network

88
Unconstrained Adjustment

• Also Called
• Minimally Constrained Adjustment
• Free Adjustment
• Used to Evaluate
• Observations
• Observation Weights
• Relationship of All Observations
• Only fix the minimum required points

89
Flow Chart
90
Analyze the Statistical Results

• There are 4 main statistical areas that need to
  be looked at
• 1. Standard deviation of unit weight
• 2. Observation residuals
• 3. Coordinate standard deviations and error
  ellipses
• 4. Relative errors
• A 5th statistic that is sometimes available that
  should be looked at
• Chi-square Test

91
Standard Deviation of Unit Weight

• Also Called
• Standard Error of Unit Weight
• Error Total
• Network Reference Factor
• The Closer This Value Is to 1.0 the Better
• The Acceptable Range Is ? to ?
• gt 1.0 - Observations Are Not As Good As Weighted
• lt 1.0 - Observations Are Better Than Weighted

92
Observation Residuals

• Amount of adjustment applied to observation to
  obtain best fit
• Used to analyze each observation
• Usually flags excessive adjustments (Outliers)
• (Starnet flags observations adjusted more
• than 3 times the observations weight)
• Large residuals may indicate blunders

This is the residual that is being minimized
93
Observation Residuals
Outlier
4.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
0
94
Coordinate Standard Deviations and Error Ellipses

• Coordinate standard deviations represent the
  accuracy of the coordinates
• Error ellipses are a graphical representation of
  the standard deviations
• The better the network the rounder the error
  ellipses
• High standard deviations can be found in networks
  with a good standard deviation of unit weight and
  well weighted observations due to effects of the
  network geometry

95
Relative Errors
Predicted amount of error that can be expected to
occur between points when an observation is made
in the network.
96
Chi-square Test

• noun (ki'skwâr) a statistic that is a sum of
  terms each of which is a quotient obtained by
  dividing the square of the difference between the
  observed and theoretical values of a quantity by
  the theoretical value
• In other words A statistical analysis of the
  statistics.
• 10 coins 6 to 4 (6-5) or 100 coins 60-40 (60-50)

97
Least Squares Examples

• Straight Line Best Fit

98
Straight Line Best Fit
99
Straight Line Best Fit
100
Straight Line Best Fit
101
Straight Line Best Fit
102
Least Squares Rules

• Redundancy of survey data strengthens adjustment
• Error Sources must be determined correctly
• Each adjustment consists of two parts
• Minimally Constrained Adjustment
• Fully Constrained Adjustment

103
StarNet Adjustment Software

• A Tour of the Software Package

StarNet
104
Sample Network Adjustment

• A Simple 2D Network Adjustment

StarNet
105
Sample Network Adjustments

• A 3D Grid Adjustment using GPS and Conventional
  Data

106
Sample Network Adjustments

• A 3D Grid Adjustment using GPS and Conventional
  Data

StarNet
107
Beyond Control Surveys

• Other Uses for Least Squares Adjustments /
  Analysis

108
Questions Discussion
109
Systematic vs. Random Error

• Systematic Error - An error whose magnitude and
  algebraic sign can be determined or corrected by
  procedure. Example temperature correction for
  steel tape or balanced level distances.
• Random Error - An error whose magnitude and
  algebraic sign cannot be determined. They tend
  to be small and compensating. Measurement
  Analysis is the study of random errors.

110
Weighted Adjustments
A 43?2436 5 B 47?1234 15 C
89?2220 30 Perform a weighted adjustment
based on the above data
A
B
C
111
What Least Squares Is...

• Adjustment report provides details of survey
  measurements
• A TOOL to be used by the Surveyor to complement
  his knowledge of measurements

112
Random Error Propagation

• Error in a Sum (Esum) (E12 E22 E32
  .. En2)1/2
• Error in a Series (Eseries) (E (n)1/2)
• Error in Redundant Measurement (Ered.) (E /
  (n)1/2)

113
Instrument Specifications

• Angle Measurement
• Stated Accuracy vs. Display
• What is DIN 18723?
• What is the True Accuracy of a Measured Angle?