Download Relative Positional Precision Explained in Everyday Language

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Definitions
Relative Positional
Precision
Explained in
Everyday Language
Todd W. Horton, PE, PLS
•
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.
February 2015
Measurement vs. Enumeration
•
A lot of statistical theory deals
with enumeration, or counting.
It’s 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.
ALTA / ACSM
Standards
and
Kentucky
Standards of Practice
2011 ALTA/ACSM Land Title Survey
• The American Land Title Association (ALTA) and the American
Congress on Surveying and Mapping (ACSM) are two organizations
that represent the Title Insurance Industry and the Land Surveying
Industry respectively.
• In 1962, ALTA and ACSM came together for the first time to develop
a survey product that would meet the needs of the title insurer to
delete the standard survey exceptions from their title policy.
2011 ALTA/ACSM
Land Title Survey
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• The product that was developed is titled an ALTA/ACSM Land Title
Survey, and the land surveyor’s responsibilities are outlined in the
“Minimum Standard Detail Requirements for ALTA/ACSM Land
Title Surveys.”
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A brief history of ALTA/ACSM Standards
• 1962 - First version published
• 1986 - Minimum angle & distance
requirements
• 1988 - Optional Table A Items added
• 1992 - Miscellaneous revisions
• 1997 - Positional Tolerance requirements
• 1999 - Table A and other miscellaneous
revisions
• 2005 - Effective since January 1, 2006
• 2011 - Effective since February 23, 2011
Measurement Standards
8/17/2014
Changes in the 2011 Standards
American Land Title Association (ALTA)
and American Congress on Surveying and Mapping(ACSM)
MINIMUM STANDARD DETAIL REQUIREMENTS FOR
ALTA/ACSM LAND TITLE SURVEYS
(Effective February 23, 2011)
Measurement Standards
E. Measurement Standards - The following measurement standards address
Relative Positional Precision for the monuments or witnesses marking the
corners of the surveyed property.
E. Measurement Standards - The following measurement standards address
Relative Positional Precision for the monuments or witnesses marking the
corners of the surveyed property.
i. “Relative Positional Precision” means the length of the semi-major axis,
expressed in feet or meters, of the error ellipse representing the uncertainty
due to random errors in measurements in the location of the monument, or
witness, marking any corner of the surveyed property relative to the monument,
or witness, marking any other corner of the surveyed property at the 95 percent
confidence level (two standard deviations). Relative Positional Precision is
estimated by the results of a correctly weighted least squares adjustment of the
survey.
v. The maximum allowable Relative Positional Precision for an ALTA/ACSM
Land Title Survey is 2 cm (0.07 feet) plus 50 parts per million (based on the
direct distance between the two corners being tested). It is recognized that in
certain circumstances, the size or configuration of the surveyed property, or the
relief, vegetation or improvements on the surveyed property will result in survey
measurements for which the maximum allowable Relative Positional Precision
may be exceeded. If the maximum allowable Relative Positional Precision is
exceeded, the surveyor shall note the reason as explained in Section 6.B.ix
below.
•
Accuracy:
–
–
•
agreement of observed values with the “true value”.
A measure of results.
Precision:
–
–
agreement among readings of the same value
(measurement).
A measure of methods.
IMPROVING ACCURACY
Accuracy versus Precision
IMPROVING PRECISION
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New Accuracy Standards
for NGS Datasheets
(http://fgdc.er.usgs.gov/standards/status/swgstat.html)
• Local Accuracy:
• Network Accuracy:
•
•
•
•
adjacent points
relative to CORS
Numeric quantities, units in cm (or mm)
Both are relative accuracy measures
Will not use distance dependent expression
Order/Class codes will no longer be used
8/17/2014
Order & Class Codes
LC1766 ***********************************************************************
LC1766 CBN
- This is a Cooperative Base Network Control Station.
LC1766 DESIGNATION - ALEXANDER 2
LC1766 PID
- LC1766
LC1766 STATE/COUNTY- IL/PIATT
LC1766 USGS QUAD - SEYMOUR (1970)
LC1766
LC1766
*CURRENT SURVEY CONTROL
LC1766 ___________________________________________________________________
LC1766* NAD 83(1997)- 40 06 20.45303(N) 088 29 42.90776(W) ADJUSTED
LC1766* NAVD 88 231.3 (meters)
759. (feet) GPS OBS
LC1766 ___________________________________________________________________
LC1766 X
- 128,287.167 (meters)
COMP
LC1766 Y
- -4,883,624.381 (meters)
COMP
LC1766 Z
- 4,087,096.112 (meters)
COMP
LC1766 LAPLACE CORR-0.58 (seconds)
DEFLEC99
LC1766 ELLIP HEIGHT199.36 (meters)
GPS OBS
LC1766 GEOID HEIGHT-31.94 (meters)
GEOID99
LC1766
LC1766 HORZ ORDER - B
LC1766 ELLP ORDER - FOURTH CLASS I
Relative Accuracy
LC1766 ***********************************************************************
LC1766 CBN
- This is a Cooperative Base Network Control Station.
LC1766 DESIGNATION - ALEXANDER 2
LC1766 PID
- LC1766
LC1766 STATE/COUNTY- IL/PIATT
LC1766 COUNTRY - US
LC1766 USGS QUAD - SEYMOUR (1970)
LC1766
LC1766
*CURRENT SURVEY CONTROL
LC1766 ______________________________________________________________________
LC1766* NAD 83(2011) POSITION- 40 06 20.45302(N) 088 29 42.90701(W) ADJUSTED
LC1766* NAD 83(2011) ELLIP HT- 199.346 (meters)
(06/27/12) ADJUSTED
LC1766* NAD 83(2011) EPOCH - 2010.00
LC1766* NAVD 88 ORTHO HEIGHT - 231.3 (meters)
759. (feet) GPS OBS
LC1766 ______________________________________________________________________
LC1766
LC1766 FGDC Geospatial Positioning Accuracy Standards (95% confidence, cm)
LC1766 Type
Horiz Ellip Dist(km)
LC1766 ------------------------------------------------------------------LC1766 NETWORK
0.77 2.21
LC1766 ------------------------------------------------------------------LC1766 MEDIAN LOCAL ACCURACY AND DIST (039 points)
1.00 2.67 69.56
LC1766 -------------------------------------------------------------------
Mistake ≠ Error
•
•
Mistake - Blunder in reading, recording or
calculating a value.
Error - The difference between a measured
or calculated value and the true value.
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Error Types
Blunder
• a gross error or mistake resulting usually from
stupidity, ignorance, or carelessness.
• Setup over wrong point
• Bad H.I.
• Incorrect settings in equipment
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Surveying errors
• Systematic errors
– Can be measured / quantified
– Have a positive or negative value
– Can be determined or corrected by procedure
• Random errors
– Cannot be measured / quantified
– Tend to be small and compensating
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Systematic Error
• Error that is not determined by chance but
is introduced by an inaccuracy (as of
observation or measurement) inherent in
the system.
– Prism with wrong offset
– Poorly repaired tape
– Imbalance between level sightings
Random Error
Nature of Random Errors
• an error that has a random distribution and
can be attributed to chance.
• without definite aim, direction, or method
• Positive and negative errors will occur with
the same frequency.
– Poorly adjusted tribrach
– Inexperienced Instrument operatorInaccuracy
in equipment
• Minor errors will occur more often than
large ones.
• Very large errors will rarely occur.
Error management
Error Sources
• Some amount of error is acceptable.
– Acceptable error is determined by the
intended use of the measurement.
• Good surveying procedures are designed
to minimize systematic and random errors.
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Error sources in leveling
• Systematic
• Random
– Earth curvature &
refraction
– Maladjusted
instrument
– Temperature effect on
level rod
– Level rod not plumb
–
–
–
–
–
–
Instrument not level
Bad rod reading
Poor turning point
Parallax
Heat wave effects
Wind effects
GNSS Errors Summary
• Orbital error
– Predicted versus as-flown trajectories
– Dilution of precision
• Atmospheric error
• Clock synchronization error
• Multipath error
GNSS Errors: Clock Sync
GNSS Errors: Orbit Data
• Problem:
– Receiver clocks are
not as accurate as
satellite clocks
AS-FLOWN
– Satellite and receiver
clocks are not
synchronized
• Broadcast ephemeris (almanac file from satellite)
– 100 cm GPS orbit accuracy PREDICTED
• Ultra-rapid ephemeris (6-hour latency from IGS)
– 5 cm GPS orbit accuracy
• Rapid ephemeris (13-hour latency from IGS)
– 2.5 cm GPS orbit accuracy
• Final ephemeris (12 to 14 day latency from IGS)
– 2.5 cm GPS orbit accuracy
– 5 cm GLONASS orbit accuracy
Good PDOP
GNSS Errors:
Atmospheric
Delay
12,500 mi
Ionosphere
125 mi
31 mi
Troposphere
A
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GNSS Error: Multipath
Error
Computations
=
EXTRA DISTANCE
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.
Normal Distribution
• Positive and negative errors will occur with the same
frequency.
• Area under curve is equal on either side of the mean.
Minor Errors
Large Errors
68%
95%
• Minor errors will occur more often than large ones.
• Very large errors will rarely occur.
• The area within one standard deviation (s) of the
mean is 68.3% of the total area.
• The total area within 2s of the mean is 95% of the
sample population.
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2s
MEAN
Histograms, Sigma, & Outliers
1s
2s
1s
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
1 s : 68% of residuals
must fall inside area
2 s 95 % of residuals
must fall inside area
0.5 1.0 1.5
2.0 2.5
3.0 3.5 4.0
Figure 3.5
Residuals
Measurement Components
Determining Uncertainty
•
•
All measurements consist of two
components: the measurement and the
uncertainty statement.
Uncertainty - the positive and negative range of
values expected for a recorded or calculated
value, i.e. the ± value (the second component of
measurements).
1,320.55 ft ± 0.05 ft
•
The uncertainty statement is not a guess,
but is based on testing of equipment and
methods.
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
100.04
100.00
100.02
99.98
Averages
•
100.02
100.00
99.98
100.00
100.00
–
•
•
•
Averages
•
•
Most commonly used is Arithmetic Mean
Considered the “most probable value”
mean 
•
•
•
•
n = number of observations
Mean = 1000 / 10
Mean = 100.00’
Residuals
•
Calculating Residuals (mean = 100.00’):
Readings
99.96’
100.02’
100.04’
100.00’
100.00’
99.98’
100.02’
100.00’
99.98’
100.00’
residual
-0.04
+0.02
+0.04
0
0
-0.02
+0.02
0
-0.02
0
Sn2 =
residual2
0.0016
0.0004
0.0016
0
0
0.0004
0.0004
0
0.0004
0
0.0048
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The value within a data set that tends to exist at the
center.
Arithmetic Mean
Median
Mode
Residuals
•
 meas.
n
“Measures of Central Tendency”
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
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
Standard Deviation Formula
Standard deviation (σ )  
n 2
n 1
•
•
s
•
Standard Deviation is a measure of….
0.0048
 0.023'
9
Standard Deviation of the Mean
•
Standard Deviation is a comparison of the
individual readings (measurements) to the
mean of the readings, therefore…
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 ….
PRECISION!
Standard Deviation of the Mean
Standard Error of the Mean (sm)  
sm  
0.023
10
s
n
 0.007'
Distance = 100.00’±0.007’
(1s Confidence level)
ACCURACY!
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”
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  (1.6449s)
E50= (0.6745)s
E90m  
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E90
n
E 95  (1.96s )
E95m  
E95
n
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95% Probable Error
Meaning of E95
“If a measurement falls outside of
E 95  (2s )  (2  0.023)  0.046'
E 95m  
•
two standard deviations, it isn’t a
random error, it’s a mistake!”
E 95
0.046

 0.015'
n
10
Francis H. Moffitt
Distance = 100.00’±0.015’
(2s Confidence Level)
Random Error Propagation
How Errors Propagate
• Errors in a Series
• Errors in a Sum
• Error in Redundant Measurement
•
•
•
Error in a Sum (Esum) =
±(E12 + E22 + E32 + ….. + En2)1/2
Error in a Series (Eseries) =
(n)1/2)
Error in Redundant Measurement
(Ered.) = ±(E / (n)1/2)
±(E
Esum   E1 2  E2 2  E3 2  ...  En 2
Eseries   E n
Ered.meas.  
E
n
Error in a Series
Error in a Sum
Eseries   E n
Esum   ( E12  E 2 2  E 3 2  ...  En 2 )
• Describes the error of multiple measurements
with identical standard deviations, such as
measuring a 1000’ line with using a 100’
chain.
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• 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
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Exercise for Errors in a Sum
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.
• 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 .
• How many sources of random error are there
in this scenario?
Instrument Specifications
Impact of
Equipment
Precision
Specifications
Instrument Specifications
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•
Angle Measurement:
–
–
–
Stated Accuracy vs. Display
What is DIN 18723?
What is the True Accuracy of a Measured
Angle?
Instrument Specifications
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Instrument Specifications
•
Distance Measurement
l sm = ±(0.01’ + 3ppm x D)
l What is the error in a 3500 foot
measurement?
l sm= ±(0.01’+(3/1,000,000 x 3500)) = ±
0.021’
Calibration
• Total station 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
Trimble R8 Accuracy Specs
Constant
error
RTK with single base
Scalar
error
RTK positional error increases
with distance from the base.
http://trl.trimble.com/docushare/dsweb/Get/Document-140079/022543-079J_TrimbleR8GNSS_DS_1109_LR.pdf
RTK Vector Errors
• Vector is the line from base to rover.
– 10 mm + 1 ppm horizontal error for RTK
vectors is typical.
– 10 mm = 0.033 ft = constant error
– 1 ppm = scalar error (distance dependent)
• 1 part error to 1,000,000 parts measurement
• 1 mm error / 1 km RTK vector
• 0.005 ft error / 1 mile RTK vector
Differential Position Errors
RTK vector
(miles)
1
2
4
8
16
32
E const
E scalar
E dist
0.033
0.033
0.033
0.033
0.033
0.033
0.005
0.011
0.021
0.042
0.084
0.169
0.033
0.034
0.039
0.053
0.091
0.172
(feet)
Edist 
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(feet)
(feet)
2
2
( Econst
 Escalar
)
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RTK relative accuracy
8/17/2014
Absolute & relative accuracy
A: known
B: unknown
Relative accuracy = ±0.05’
Absolute
accuracy = ±5’
Absolute
accuracy = ±5’
A: known
B: unknown
Relative accuracy = ±0.05’
Base can occupy a known point or an unknown (assumed) point.
Metadata
Poor absolute accuracy
Good relative accuracy
Absolute
accuracy = ±0.05’
Good absolute accuracy
Good relative accuracy
Absolute
accuracy = ±0.07’
Observed positions and errors
• Data describing data creation and data quality
• Key measure of GPS data reliability
• Position quality expressed in terms of standard
deviations
Positions lose their credibility
without error estimates.
Mean = 2000.00
1999.90
1999.80
2000.10
2000.20
• 100 distance measurements
• Mean = 2000.00 feet
• Standard deviation = ±0.10 feet
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Mean = 2000.00
8/17/2014
Standard Deviation
99%
1999.96
2000.04
95%
68%
1999.92
2000.08
• 100 distance measurements
• Mean = 2000.00 feet
• Standard deviation = ±0.04 feet
2D Position Quality
68%
confidence
39%
confidence
Observed
Position
(address)
DRMS  ( E12  E22 )
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Positional Errors
Commonly Known As
• HRMS
HRMS = 0.03 ft
– Trimble
– Topcon
1000.00 ft calculated
• 2DCQ
– Leica
999.94 ft
Observed
Position
(address)
1000.06 ft
DRMS  ( E12  E22 )
Effects of Positional Errors
Weakness of GPS
HRMS = 0.03 ft
1000.00 ft calculated
HRMS = 0.03 ft
100.00 ft calculated
99.94 ft
Any of these lines
are possible.
100.06 ft
Match the tool to the task
80.00ft
80.00ft
80.00ft
80.00ft
80.00ft
80.00ft
• Can you stake this straight boundary line
accurately with GPS?
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Match the tool to the task
• GPS methods
give greater
accuracy over
long distances.
• Total station
methods give
greater accuracy
over shorter
distances.
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Confidence Levels
Ground
Truth
99% 95%
68%
Observed
Position
8/17/2014
Reported Precision
• 68% confidence = 68%
probability that the TRUTH
falls within 1 standard
deviation of the ADDRESS
(mean).
• 95% confidence = 95%
probability that the TRUTH
falls within 2 standard
deviations of the ADDRESS
(mean).
Trustworthy?
Displayed at
data collector
Specified in contracts
Reported in NGS
datasheets
68% confidence overlap
Observed
Position
Published
Position
95% confidence overlap
Strength of Figure
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Error in Redundant Measurements
Redundancy
• If a measurement is repeated multiple
times, the accuracy increases, even if the
measurements have the same value
Ered.meas.  
Sample of Redundancy
0
1
2
3
4
5
6
7
8
9
10
Horizontal
Shots
1
2
3
4
5
6
7
8
9
10
Base
Line
(KM)
20
22
24
26
28
30
32
34
36
38
40
12
13
14
15
16
17
18
20
21
22
23
10
11
12
13
14
15
16
17
18
19
20
9
10
11
12
13
13
14
15
16
17
18
8
9
10
11
11
12
13
14
15
16
16
8
8
9
10
11
11
12
13
14
14
15
7
8
8
9
10
11
11
12
13
13
14
7
7
8
9
9
10
11
11
12
13
13
6
7
8
8
9
9
10
11
11
12
13
0
1
2
3
4
5
6
7
8
9
10
14
16
17
18
20
21
23
24
25
27
28
Vertical
Shots
1
2
3
4
5
6
7
8
9
10
40
42
44
46
48
50
52
54
56
58
60
23
24
25
27
28
29
30
31
32
33
35
20
21
22
23
24
25
26
27
28
29
30
18
19
20
21
21
22
23
24
25
26
27
16
17
18
19
20
20
21
22
23
24
24
15
16
17
17
18
19
20
20
21
22
23
14
15
16
16
17
18
18
19
20
21
21
13
14
15
15
16
17
17
18
19
19
20
13
13
14
15
15
16
16
17
18
18
19
28
30
31
33
34
35
37
38
40
41
42
n
Eternal Battle of Good Vs. Evil
Expected accuracy of a given number of unique observations at a given baseline
length, at the 95% confidence interval and stated in mm.
Base
Line
(KM)
E
Sum vs. Redundancy
• Therefore, as the network becomes more
complicated, accuracy can be maintained
by increasing the number of redundant
measurements
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• 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.
Redundancy
• Check known points
before, during, and
after session.
• Use averaged
positions to improve
confidence.
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Kentucky Association of Professional Surveyors
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MOLA to RV22 10.8 Km
Day 264
dh
(m)
Day 265
dh
(m)
-10.281
Importance
of-10.279
-10.278
-10.270
-10.281
-10.278
Redundancy
-10.291
-10.274
14:00-14:30
17:00-17:30
14:30-15:00
17:30-18:00
15:00-15:30
18:00-18:30
15:30-16:00
18:30-19:00
19:00-19:30
20:30-21:00
-10.274
-10.287
-10.279
-10.270
-10.277
-10.271
-10.277
-10.271
-10.259
-10.254
14:30-15:00
-10.274
-10.276
-10.261
-10.251
-10.270
-10.276
-10.278
-10.286
-10.278
-10.295
14:00-21:00
-10.275
14:00-21:00
-10.276
16:00-16:30
16:30-17:00
17:00-17:30
17:30-18:00
18:00-18:30
18:30-19:00
19:00-19:30
19:30-20:00
20:00-20:30
19:30-20:00
20:00-20:30
20:30-21:00
15:00-15:30
15:30-16:00
16:00-16:30
16:30-17:00
14:00-14:30
Mean dh
(m)
-10.280
-10.274
-10.280
-10.283
-10.274
-10.282
-10.270
-10.261
-10.274
-10.274
-10.278
-10.279
-10.269
-10.275
Mean
-10.276
8/17/2014
Two Days / Same Time
-10.254
> -10.253
-10.251
Spread = 0.003 m
Mean = -10.276
Difference = 0.023 m
Two Days/
Different Times
-10.254
> -10.275
-10.295
Spread = 0.041 m
Mean = -10.276
Difference
= 0.001 m
Field Methods
for Compliance
with Standards
Least Squares
Adjustment
Basic Concepts
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!
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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
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Introduction to Adjustments
•
Common Adjustment methods:
– Compass Rule
– Transit Rule
– Crandall's Rule
– Rotation and Scale (Grant Line Adjustment)
– Least Squares Adjustment
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
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.
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)
Weighted Adjustments
A = 4324’36”, 2x
B = 4712’34”, 4x
A
C = 8922’20”, 8x
Perform a weighted
adjustment based on the
above data
ANGLE
No. Meas
Mean Value Rel. Corr.
Corrections
A
2
43 24’ 36”
4/
4 or
4/
B
4
47 12’ 34”
2/
4 or
2/
C
8
89 22’ 20”
1/
4 or
17959’ 30”
7/
4 or
TOTALS
Adjusted Value
7
4/
7 X 30” = 17”
43 24’ 53”
7
2/
7X
30” = 09”
47 12’ 43”
1/
7
1/
7X
30” = 04”
89 22’ 24”
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.
C
B
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What Least Squares Is ...
Weighted Adjustments
•
BM “B”
+7.8’, 2 mi.
Elev. = 102.0
•
BM “NEW”
•
+6.2’, 10 mi.
•
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
+10.0’, 4 mi.
BM “A”
BM “C”
Elev. = 100.0’
Elev. = 104.0’
Least Squares Example
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.
A point is measured for location 3 times. The measurements give the
following NE coordinates:
c 5,0
b. 0,5
c.5, 0
• GROUP #1
• Determine the sum of
the squares from
• GROUP #2
• Determine sum of the
squares from
X=2.5, Y=2.5
Mean X, Mean Y
(1.667, 1.667)

What is the best solution for
an average?
How can you prove it?
?

a 0,0
•
Arithmetic Mean
Using Least squares to prove a simple
arithmetic mean solution
Student exercise
Least Squares Example
a. 0,0
•

b 0,5
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What Least Squares Isn’t ...
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 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
?


a 0,0
b 0,5
Least Squares Examples
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
Straight Line Best Fit
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•
Straight Line Best Fit
Straight Line Best Fit
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Straight Line Best Fit
Straight Line Best Fit
Least Squares “Rules”
Least Squares
A
•
•
•
l
l
Minimally Constrained Adjustment
Fully Constrained Adjustment
B
Observed
Redundancy of survey data strengthens
adjustment
Error Sources must be determined correctly
Each adjustment consists of two “parts”:
E
1st Iteration
G
2nd Iteration
F
What happens?
Iterative Process
C
D
Each iteration applies adjustments to
observations, working for best solution
Adjustments become smaller with each
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.
3 Updates approximate coordinate values.
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.
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Least Squares
Four components that need to be addressed
prior to performing least squares adjustment
1
2
3
4
Errors
Coordinates
Observations
Weights
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Errors
Coordinates
• Blunder - Must be removed
• Systematic - Must be Corrected
• Random - No action needed
Weights
• 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.)
Methods of Establishing 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
σ = 2.2  Weight = 1/2.22 = 0.21
More
Influence
Less
Influence
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
• Observational Group
Good for combining
Observations from
different classes of
instruments.
• Least Desirable Method
• Example: All Angles Weighted at the
Accuracy of the Total Station
• Each Observation Individually Weighted
Good for projects
where standard
deviation is calculated
for each observation.
• 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
Least Squares
If you remember nothing else about least squares today,
remember this!
• Least Squares Adjustment Is a Two Part Process
1 - Unconstrained Adjustment
• Analyze the Observations, Observations
Weights, and the Network
2 - Constrained Adjustment
• Place Coordinate Values on All Points in the
Network
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Unconstrained Adjustment
Flow Chart
Start
• Also Called
• Minimally Constrained Adjustment
Field Observations
Setup
Observation
Standard Deviation
Field Data
Needs
Editing?
• Free Adjustment
Yes
Edit Field Data
• Remove Blunders
• Correct Systematic
Errors
Perform
Unconstrained Least
Squares Adjustment
No
• Used to Evaluate
Print out
Unconstrained
Adjustment Statistics
• Observations
No
• Observation Weights
• Relationship of All Observations
Statistics
Indicate
Problems
Analyze
Adjustment
Statistics
Yes
Perform
Constrained Least
Squares Adjustment
Modify Input
Data
Constrain Fixed
Control Points
• Only fix the minimum required points
Print out Final
Coordinate Values
for All Points in
Adjustment
Performed by
User
Decision Step
Least Square
Adjustment
Software
Finish
Carlson Adjustment Software
Least Squares
Adjustment
•
A Tour of the Software Package
Demonstration
Project
Sample Network Adjustment
•
A Simple 2D Network Adjustment
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Sample Network Adjustments
•
A 3D “Grid” Adjustment using GPS and
Conventional Data
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Beyond Control Surveys
•
Least Squares
Adjustment
Other Uses for Least Squares Adjustments
/ Analysis
Interpreting
Results
Analyze the Statistical Results
Standard Deviation of Unit Weight
Also Called
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 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
Observation Residuals
Site
10-11-12
11-12-13
Observation
214  33’ 17.2”
174  16’ 43.8”
Residual
1.7”
7.2”
S Dev.
1.2”
1.9”
12-13-14
337  26’ 08.6
2.1”
1.3”
This is the residual that is being minimized
• Used to analyze each observation
• Usually flags excessive adjustments (Outliers)
(Star*net flags observations adjusted more
than 3 times the observations weight)
• Large residuals may indicate blunders
*
Outlier
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5
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Flag
0
0.5 1.0 1.5
2.0 2.5
3.0 3.5 4.0
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Relative Errors
Coordinate Standard Deviations
and Error Ellipses
Predicted amount of error
that can be expected to
occur between points when
an observation is made in
the network.
• 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
Chi-square Test
Reporting
Compliance
with Standards
• 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)
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
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Coordinate Standard Deviations and Error Ellipses:
Point
12
Northing
583,511.320
Easting
2,068,582.469
N SDev
0.021
E SDev
0.017
{
Northing Standard Deviation
}
•
Coordinate Standard Deviations
and Error Ellipses
Easting Standard Deviation
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Kentucky Association of Professional Surveyors
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Truth versus Address
• Physical monument
– Truth
– Relatively stable (in
most of CONUS)
8/17/2014
Error circles: 1dRMS & 2dRMS
• Contrary to one-dimensional statistics,
there is no fixed probability level for this
error measure. The confidence level
depends on the ratio of standard
deviations.
• Point coordinate
– Merely an address
– Contains error
– Subject to change
• Owing to the low probability content of the
dRMS error circle, 95% is generally
required for position-finding errors.
1dRMS & 2dRMS
Confidence
Confidence
2*dRMS
@ 1*dRMS
@ 2*dRMS
σy/σx
1*dRMS
0.0
1.0
68.27%
2.0
95.45%
0.25
1.0308
68.15%
2.0616
95.91%
0.5
1.1180
66.29%
2.2361
96.97%
0.75
1.25
63.92%
2.5
97.87%
1.0
1.4142
63.20%
2.8284
98.16%
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