# Software Development Agreement with Residuals by pjt13416

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```									MEASUREMENT ANALYSIS
Capital Project Skill Development Class
(CPSD)
G100398

By Jeremy Evans,
P.L.S.   Psomas
Supplemented by
Caltrans Staff
Introduction
“The dark side of surveying is the
measurements, precisions and
adjustments. It is not and never
will be.”
Dennis Mouland
P.O.B. Magazine
July, 2002
Introduction
Much has been written lately about least
brings to the land surveyor. To take full
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.
Introduction
“War Stories”
Class Outline
Survey Measurement Basics - A Review
Measurement Analysis
Error Propagation
Introduction to Weighted and Least
Measure First,
   Instruments are calibrated
Networks are stronger if:
   They include Redundancy
   They have Strength of Figure
Adjust only after you have followed
proper procedures!
Survey Measurement Basics
A Review of Plumb Bob 101
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.
Measurement vs. Enumeration
A lot of statistical theory
deals with enumeration,
or counting. It’s a way
to take a test sample
the total population.
The surveyor is
concerned with
Measurement. The true
dimensions can never
be known.
Instrument Testing
Pointing error of typical total station
Instrument Specifications
Instrument Specifications
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’
Calibration or “Don’t shoot
yourself in the foot.”
Leica instruments should be serviced
every 18 months.
EDM’s should be calibrated every six
months
Tribrachs should be adjusted every six
months, or more often as needed.
Levels pegged every 90 days
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.
Blunder

a gross error or mistake resulting usually from
stupidity, ignorance, or carelessness.
Blunder
• Setup over wrong point
• Incorrect settings in equipment
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
Systematic

an error that is not determined by chance
but is introduced by an inaccuracy (as of
observation or measurement) inherent in
the system
Systematic
• Glass with wrong offset
• Poorly repaired chain
• Imbalance between level
sightings

with the tape is 0.1' shorter
than recorded.
Random

an error that has a random distribution
and can be attributed to chance.
without definite aim, direction, or
method
Random
• Inexperienced Instrument
operator
• Inaccuracy in equipment
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)
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
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
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
Histograms, Sigma, & Outliers

MEAN
2s              1s                  1s               2s

Histogram: Plot of the Residuals                         Bell shaped curve
\                        /

Outlier
\

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5          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        Residuals
2 s 95 % of residuals
must fall inside area
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.
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.
Measurement Analysis
Determining Measurement Uncertainties
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).
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.
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.
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
Averages
“Measures of Central Tendency”
   The value within a data set that tends to exist at
the center.
Arithmetic Mean
Median
Mode
Averages
Most commonly used is Arithmetic Mean
Considered the “most probable value”
 meas.
mean 
n
n = number of observations
Mean = 1000 / 10
Mean = 100.00’
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
Residuals
Calculating Residuals (mean = 100.00’):
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
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.
Standard Deviation Formula
n 2
Standard deviation (σ )  
n 1

0.0048
s         0.023'
9
Standard Deviation
Standard Deviation is a comparison of the
Standard Deviation is a measure of….
Standard Deviation
Standard Deviation is a comparison of the
Standard Deviation is a measure of….

PRECISION!
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 ….
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
ACCURACY!
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 ….
Standard Deviation of the
Mean
s
Standard Error of the Mean (sm)  
n
0.023
sm              0.007'
10
Distance = 100.00’±0.007’
(1s Confidence level)
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
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

E90  (1.6449 s)         E 95  (1.96s )

E90                       E 95
E90 m                    E95 m  
n                        n
95% Probable Error
E 95  (2s )  (2  0.023)  0.046'

E 95    0.046
E 95 m                0.015 '
n       10

Distance = 100.00’±0.015’
(2s Confidence Level)
Meaning of E95

“If a measurement falls outside
of two standard deviations, it
isn’t a random error, it’s a
mistake!”

Francis H. Moffitt
How Errors Propagate

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

Eseries   E n
Error in a Sum

Esum   E1  E2  E3  ...  En
2        2        2               2

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
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?
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
Error in Redundant
Measurements
If a measurement is repeated multiple
times, the accuracy increases, even if
the measurements have the same value

E
Ered.meas.  
n
Sample of Redundancy
Expected accuracy of a given number of unique observations at a given baseline
length, at the 95% confidence interval and stated in mm.
Base                       Horizontal                     Base                        Vertical
Line   Shots                                              Line   Shots
(KM)   1 2        3   4    5    6    7    8    9    10    (KM)   1 2       3    4    5    6    7    8    9    10
0    20   14   12   10    9    8    8    7    7    6       0   40   28   23   20   18   16   15   14   13   13
1    22   16   13   11   10    9    8    8    7    7       1   42   30   24   21   19   17   16   15   14   13
2    24   17   14   12   11   10    9    8    8    8       2   44   31   25   22   20   18   17   16   15   14
3    26   18   15   13   12   11   10    9    9    8       3   46   33   27   23   21   19   17   16   15   15
4    28   20   16   14   13   11   11   10    9    9       4   48   34   28   24   21   20   18   17   16   15
5    30   21   17   15   13   12   11   11   10    9       5   50   35   29   25   22   20   19   18   17   16
6    32   23   18   16   14   13   12   11   11   10       6   52   37   30   26   23   21   20   18   17   16
7    34   24   20   17   15   14   13   12   11   11       7   54   38   31   27   24   22   20   19   18   17
8    36   25   21   18   16   15   14   13   12   11       8   56   40   32   28   25   23   21   20   19   18
9    38   27   22   19   17   16   14   13   13   12       9   58   41   33   29   26   24   22   21   19   18
10    40   28   23   20   18   16   15   14   13   13      10   60   42   35   30   27   24   23   21   20   19
Eternal Battle of Good Vs. Evil
With Errors of a Sum (or Series), each
error of the network
With Errors of Redundant
Measurement, each redundant
measurement decreases the error of
the network.
Sum vs. Redundancy
Therefore, as the network becomes
more complicated, accuracy can be
maintained by increasing the number of
redundant measurements
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
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
Positional Accuracy vs.
Precision Ratio
Or, “How good is one error ellipse
compared to all those others?
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
 Compass Rule

 Transit Rule

 Crandall's Rule

 Rotation and Scale (Grant Line Adjustment)

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
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.
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)
A = 4324’36”, 2x
B = 4712’34”, 4x
A         C = 8922’20”, 8x
Perform a weighted
above data

C         B
ANGLE     No. Meas    Mean ValueRel. Corr.      Corrections                  Adjusted Value

A        2           43 24’ 36”             4/
4 or
4/
7
4/
7X   30” = 17”   43 24’ 53”
B        4           47 12’ 34”             2/
4 or
2/
7
2/
7X   30” = 09”   47 12’ 43”
C        8           89 22’ 20”             1/
4 or
1/
7
1/
7X   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.
BM “B”
Elev. = 102.0       +7.8’, 2 mi.

BM “NEW”

+6.2’, 10 mi.

+10.0’, 4 mi.

BM “A”                                            BM “C”
Elev. = 100.0’                                    Elev. = 104.0’
Introduction to Least Squares
Simple Examples
What Least Squares Is ...
A rigorous statistical adjustment of survey
data based on the laws of probability and
statistics
measurements
Measurements can be individually weighted
to account for different error sources and
values
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.
Least Squares Example
Arithmetic Mean
Using Least squares to prove a simple
arithmetic mean solution
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
Student exercise
GROUP #1               GROUP #2
Determine the sum      Determine sum of
of the squares from    the squares from

X=2.5, Y=2.5            Mean X, Mean Y
(1.667, 1.667)
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
What Least Squares Isn’t ...
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
Least Squares
Least Squares Should Be Used for
Conventional Traverse   Theodolite & Chain
Control Networks        Total Stations
Level Networks          Levels
Resections              EDMs
Least Squares
A                       B
Observed                    E
1st Iteration

2nd Iteration           G
F

What happens?
C
Iterative Process
D
observations, working for best solution
successive iteration
Least Squares
The Iterative Process
1 Creates a calculated observation for each field
observation by inversing between approximate
coordinates.
2   Calculates a "best fit" solution of observations and
compares them to field observations to compute
residuals.
4   Calculates the amount of movement between the
coordinate positions prior to iteration and after
iteration.
5   Repeats steps 1 - 4 until coordinate movement is no
greater than selected threshold.
Least Squares
Four component that need to be addressed
prior to performing least squares adjustment
1   Errors
2   Coordinates
3   Observations
4   Weights
Errors
Blunder - Must be removed
Systematic - Must be Corrected
Random - No action needed
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
Weights
•   Each Observation Requires an Associated Weight
 Weight = Influence of the Observation on Final
Solution
• Larger Weight - Larger Influence

 Weight = 1/σ2
• σ = Standard Deviation of the Observation

 The Smaller the Standard Deviation the Greater the
Weight

σ = 0.8  Weight = 1/0.82 = 1.56            More
Influence
σ = 2.2  Weight = 1/2.22 = 0.21            Less
Influence
Methods of Establishing Weights
•   Observational Group
•   Least Desirable Method
Good for combining
Observations from •        Example: All Angles Weighted at the Accuracy of
different classes of      the Total Station
instruments.
•   Each Observation Individually Weighted
•   Best Method
Good for projects
where standard         •    Standard Deviation of Field Observations Used as
deviation is calculated    the Weight of the Mean Observation
for each observation.
•   Combination of Types
•   Assigns the Least weight possible for each observation
Least Squares
If you remember nothing else about least squares today,
remember this!
Least Squares Adjustment Is a Two Part Process
 Analyze the Observations, Observations
Weights, and the Network
 Place Coordinate Values on All Points in the
Network
•   Also Called
•   Used to Evaluate
 Observations
 Observation Weights
 Relationship of All Observations
•   Only fix the minimum required points
Flow Chart
Start
Field Observations
Edit Field Data               Perform
Setup                      Field Data
• Remove Blunders        Unconstrained Least
Observation                 Needs            Yes
Standard Deviation         Editing?
Errors

No

Print out
Unconstrained                        Statistics                Analyze
Problems                  Statistics

Yes
Perform
Constrain Fixed                                               Modify Input
Constrained Least
Control Points                                                   Data

Print out Final                                                   Least Square
Coordinate Values            Performed by
for All Points in               User
Software

Finish
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
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 ?
 > 1.0 - Observations Are Not As Good As
Weighted
 < 1.0 - Observations Are Better Than Weighted
Observation Residuals
•   Amount of adjustment applied to observation to
obtain best fit
This is the residual that is being minimized
•   Used to analyze each observation
•   Usually flags excessive adjustments (Outliers)
than 3 times the observations weight)
•   Large residuals may indicate blunders
Observation Residuals
Site      Observation          Residual        S Dev.     Flag
10-11-12    214  33’ 17.2”         1.7”           1.2”
11-12-13    174  16’ 43.8”         7.2”           1.9”       *
12-13-14    337  26’ 08.6          2.1”           1.3”

Outlier

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5    0     0.5 1.0 1.5    2.0 2.5   3.0 3.5 4.0
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
Relative Errors

Predicted amount of error
that can be expected to
occur between points when
the network.
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)
Least Squares Examples
Straight Line Best Fit
Straight Line Best Fit
Straight Line Best Fit
Straight Line Best Fit
Straight Line Best Fit
Least Squares “Rules”
Redundancy of survey data strengthens
Error Sources must be determined correctly
Each adjustment consists of two “parts”:
A Tour of the Software Package

Star*Net

Star*Net
A 3D “Grid” Adjustment using GPS and
Conventional Data

Star*Net
Beyond Control Surveys
Other Uses for Least Squares
Questions & Discussion
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.
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
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)

Esum   E1 2  E2 2  E3 2  ...  En 2
Eseries   E n
E
Ered.meas.  
n
Instrument Specifications
Angle Measurement:
 Stated Accuracy vs. Display
 What is DIN 18723?
 What is the True Accuracy of a
Measured Angle?

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