Errors Analysis in Surveying2019 - cdn.ymaws.com · 1/10/19 1 Errors Analysis in Surveying Joseph Paiva Saratoga Springs January 2019 Objectives •Something about errors and mistakes

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Errors Analysis in SurveyingJoseph Paiva

Saratoga SpringsJanuary 2019

Objectives• Something about errors and mistakes• Something about types and sources of errors• Standard deviation and its relevance to us• Use statistics to design surveying procedures• Analyzing what we do• Applying the results of the analysis

© 2019 J.V.R. Paiva 21/10/19

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Errors• Difference between a measurement and the true

value• ! = #$%& − ()*$• Therefore ()*$ = #$%& − !• Errors can be positive or negative depending on

whether the true value is larger or smaller than the measured value

© 2019 J.V.R. Paiva 31/10/19

It Makes The Difference• Surveying is a tough business• Especially when competitors are constantly

undercutting each other decreasing the value of a survey

• This “competition” also moves status of asurvey map or plat toward being meaningless

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Instead• Avoid “me too” work• Be proud of what you deliver when you

perform a survey• Actually describe it piece by piece• TALK to your clients, business groups, service

groups, etc.© 2019 J.V.R. Paiva 51/10/19

But…• That means talking about your work as if you

are an expert• But without obfuscation or talking down

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When It Comes to Errors• Know the language• Know the math• Know the statistics• If we can’t explain the technical aspects of

what we do, why would anyone want to pay good money for it? [and think it was worth it]

© 2019 J.V.R. Paiva 71/10/19

Definitions [1]• Error: difference between a measurement and

true value• So positive error means the measurement is

larger than the true value• BUT! How do we know what the true value is?

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Definitions [2]• Mean is the average• Usually considered more reliable than a single

measurement• Redundancy is one of the hallmarks of good

surveying measurement• Independent measurements is another© 2019 J.V.R. Paiva 91/10/19

Definitions [3]• Residual is the difference between a measurement

and the mean• ! = #$%& − ()• Where ! is residual• () is the mean• If we have n measurements, then we will have n

residuals© 2019 J.V.R. Paiva 101/10/19

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Error Sources• Nature (natural errors) – from the

environment• Technology (instrumental errors)• Humans (personal errors) – usually cause of

randomness in how something is sighted, aligned, marked, etc.

Definitions [4]

© 2019 J.V.R. Paiva 111/10/19

Error Types• Systematic– Usually varies as a function of the measurement,

but not always linear as in curvature of the Earth• Random– Varies in sign and magnitude with every

measurement– Large random errors occur rarely

Definitions [5]

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Mistakes• Also known as blunders• Usually caused by lack of care or attention in

the measurement process by humans• There are cases of technology defects that

appear as blunders in measurements

Definitions [6]

© 2019 J.V.R. Paiva 131/10/19

If We Are Going To Be Precise…• Do not interchange error and mistake• Do not call typing errors errors; they are

mistakes• Do not call errors in sports errors; mistakes!• When you write something down incorrectly,

that’s not an error MISTAKE!© 2019 J.V.R. Paiva 141/10/19

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Bias in Measurements• Your car odometer reads 10.1 miles when you

have truly gone 10.0 miles• Error or mistake?• Systematic or Random?• Source?

© 2019 J.V.R. Paiva 151/10/19

Bias in Measurements• Your EDM measurements are consistently ~0.1

ft long regardless of how long the measurement is

• Error or mistake?• Systematic or Random?• Source?© 2019 J.V.R. Paiva 161/10/19

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Bias in Measurements• You initialize your RTK system, then conduct a

survey• You check into two known monuments and each is

1.2 ft N and 3.4 ft W of your record• Error or mistake?• Systematic or Random?• Source?

© 2019 J.V.R. Paiva 171/10/19

Bias in Measurements• You are following a surveyor• You set up at a monument on map, sight along

block and turn 90°16’30” to find a corner that’s at a record distance of 567.00 ft

• You find the monument, but miss it for distance by -0.06 ft and for angle by 0.04 ft to the right

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What Do You Do?• Call existing monument good?• Set your own monument?• Something else?• [The monuments on the subdivision plat were

set in …]

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How Well• Can you define a direction?• A distance?• A position?• An elevation?

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What Is The [Un]Certainty• In your measurements?• Why should you care?• Does anybody care?

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The Average• Can be a better value than any single

measurement• But only if you’ve removed systematic errors

and the blunders have been prevented or eliminated

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Accuracy• A statement of how close a measurement or

value is to the true value• A reflection of the results of measurements

and analysisOUTCOMES

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Precision• A statement of repeatability, fineness of

measuring technology, care taken in the measurements

• A reflection of the methods used in the measurement

PROCESSES

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The Shooting Analogy

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Precise or accurate?

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Errors• Understand them (how they are caused, when

they are caused, the magnitude?)• Eliminate as many as possible through

modeling, procedures, instrumentation, software, and in post-processing

• Watch out for blunders! (operations and data analysis)

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Continuing About the Mean• When systematic errors in the measurements

have been �handled� as best as possiblethrough procedures and calculations, the mean is the best estimate of the true value

• �Best as possible� varies depending on the intended use of the survey data by the �user�

27© 2019 J.V.R. Paiva 1/10/19

Standard Normal Distribution•Can be understood through process of plotting a histogram

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http://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/350px-Normal_Distribution_PDF.svg.png© 2019 J.V.R. Paiva

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Random Error Example•Angle measured 50 times•Calculate mean, residuals and standard deviation• Let’s say we calculate that s = �7�• Standard deviation theory: examine individual

measurements, 68% will be within 7 seconds of the mean•Also, if you make one more measurement, there is a 68%

probability that it will be within 7� of the mean

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Histogram Plot

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- 0 +

Magnitude of Class IntervalsNum

ber o

f mea

sure

men

ts in

cla

ss in

terv

al

• Sort by sign and interval; determine class interval; plot bar graph

• Join tops of bars; approximates standard normal (Gauss) curve

• As measurements are increased and class intervals decreased, ideal shape can be observed

© 2019 J.V.R. Paiva 1/10/19

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Understanding Probability

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http://www.usmle-forums.com/images/added/attachments/inpost/bellcurve.gif

© 2019 J.V.R. Paiva 1/10/19

Confidence Levels (area under curve)

• 68.2% for s• 95.4% for 2s• 99.7% for 3s• 50% for 0.645s

Areaundercurve

s coefficient

0.80 1.28155

0.90 1.64485

0.95 1.95996

0.98 2.32635

0.99 2.57583

0.995 2.80703

0.998 3.09023

0.999 3.29052

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Thus…• Measurement has three parts– The quantity measured– The uncertainty– The confidence level– E.g. 118°42’12”�2.1” std dev (68% conf)– E.g. 23,478.65 m ±0.324 m @ 95% confidence

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Simple s Calculation

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No. MeasurementResi-dual

Resi-dual2

1 27�43’ 55” 2 42 27�43’ 55” 2 43 27�43’ 50” -3 94 27�43’ 52” -1 15 27�44’ 00” 7 496 27�43’ 49” -4 167 27�43’ 54” 1 18 27�43’ 56” 3 99 27�43’ 46” -7 49

10 27�43’ 51” -2 4Mean = 27� 43’ 53”

Sum of n2 = 146

n-1 = 9

146/9 = 16.2

Sq. rt. of 16.2 = �4�

! = ± Σ%&' − 1

© 2019 J.V.R. Paiva

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• The standard deviation of the 50 measurements in our �thought�experiment is �7�

• But what is the standard deviation of the mean?

• Logical that mean should be more certain than an individual measurement

The Strength of the Mean

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!" = !$

!" = 750 ≅ 1”


“Designing” Procedures

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!"#$%& =!()#*+#,-

.

. = !()#*+#,-!"#$%&

/

Eq. A

Eq. B

• Tot Sta accuracy ±5”• Specs req ±4” @ 95% conf• Requirement at 68% conf is ±2”• Substitute into eq. B to solve for n

. = 52

/= 2.5/ = 6.25

• n = 6, 7, 8?

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Combining Random Errors

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!"#$%&#" = !() + !)) + !+) + ⋯+ !-)

!"#$%&#" = !.%-/0# 1


How Much is the Error (e.g. prism pole)?• Typical level vial sensitivity can vary on prism

poles from 10 to 60 minutes• The level spec refers to the angle change to

move the position of the bubble 2 mm• If 30 minutes, and the bubble is 2 mm out-of-

center…• Prism on top of 6 ft pole is out of plumb 0.05 ft

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Calculating Prism Pole Error

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e

ht

a

! = #$%&' (ℎ(*+ℎ#

( = ℎ(*+ℎ#× tan != 61#× tan 30′= 0.052 1#


Level Vial Centering/Adjustment

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Angle-Distance Relationships

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dangle

ddist

A

B

dangle = ddist (if angles and distances have same uncertainty)


Expanded View

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a

dist

ddist

! = tan&' ()*+,-./0

tan ! = 1234./.56



• When measuring angles and distances, what is the limiting factor?

• Remember a chain is only as strong as its weakest link

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Angle–Distance Relationships / 2

• If you are required to perform a survey with a relative accuracy of 1:50,000 what should the accuracy of the angles be? [�4�]

• If you can measure angles �3�, what should the relative accuracy of the distances be to be comparable in accuracy? [1/70,000]

1/10/19 45© 2019 J.V.R. Paiva

Conditions Causing 0.01 ft error in 100 ft(calibrated steel tape/band)

Tape Length 0.01 Temperature 15o F Tension (pull) 5.4 lbsSag 7.5” at centerAlignment 1.4 ft at one end or 7.5” at centerTape Not Level 1.4 ft diff in elevationPlumbing 0.01Marking 0.01Interpolation 0.01

461/10/19© 2019 J.V.R. Paiva

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Possible Errors Using Common ProceduresStandard 100 ft measurement with calibrated tape

Source Error (ft.) Error2 \

Tape Length Known 0.000000Temp (10o F error) 0.006 0.000036Tension (5 lb error) 0.009 0.000081Alignment (0.05 ft) 0.000 0.000000Tape Not Level (0.5 ft) 0.001 0.000001Plumbing 0.005 0.000025Marking 0.001 0.000001Interpolation 0.001 0.000001SUM 0.023 0.000145

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Sq Rt of Sum of Errors2 = 0.012 ft

1:8,000 OR 120 PPM

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Possible Errors Using Common ProceduresCalibrated EDM (100 ft; accuracy 3 mm + 3 PPM)

Source Error (ft.) Error2

Length Known 0.000000Temp (10o F error) 5 PPM = 0.0005 0.00000025Pressure (1/2” Hg) 5 PPM = 0.0005 0.00000025Centering with pole 0.03 0.0009Centering w/O.P. 0.005 0.000025Mfr’s error constant 0.003 0.000009Mfr’s error scale 3 PPM = 0.0003 0.00000009SUM 0.0393 0.000093459

0.0306 ft

1: 3,000 OR 306 PPM

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Possible Errors Using Common ProceduresCalibrated EDM (5,000 ft; accuracy 3 mm + 3 PPM)


Length Known 0.000000Temp (10o F error) 5 PPM = 0.025 0.000625Pressure (1” Hg) 5 PPM = 0.025 0.000625Centering w/O.P. 0.005 0.000025Centering w/O.P. 0.005 0.000025Mfr’s error constant 0.003 0.000009Mfr’s error scale 3 PPM = 0.015 0.000225SUM 0.078 0.001534

0.03917 ft

1: 127,000 OR 8 PPM

Possible Errors Using Common ProceduresRTK GPS (pole w/bipod) 2,500 ft baseline (acc. 1 cm + 2 PPM)

501/10/19© 2019 J.V.R. Paiva


Length Known 0.000000Tropospheric delays 0.0025 m = 0.008 0.000067Centering w/O.P. 0.005 0.000025Centering w/bubble 0.005 0.000025Mfr’s error constant 0.01 m = 0.03281 0.001076Mfr’s error scale 2 PPM = 0.005 0.000025SUM 0.05581 0.001218

0.0349 ft

1:72,000 OR 14 PPM

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Typical Total Station Error

• ±3 arc seconds AND

• ±(2 mm + 2 PPM)

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Actually A Statement Of Precision • Becomes a statement of accuracy only if• Blunders have been eliminated and,• Systematic errors have been removed

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Example• Measure a traverse four times or ten times• Average the “precision” computed by dividing the

closure error by the perimeter• Let’s say the total error in the traverse averaged

out to 0.20 m and the total traverse length was 10,000 m

• Then precision is 1:50,000

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BUT• Let’s say PPM correction for temperature and

pressure (combined) as used is set to +45• But it really should have been set to -5 PPM• So, considering distances alone, they are all in

error by 1,000,000/50 = 1:20,000• This is 2.5 cm per 500 m traverse side (0.082 ft per

1,600 ft)

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So…• No matter how well the precision comes out

(in this case 1:50,000)• The accuracy can’t be better that 1:20,000• In fact it will be worse, as we have not

evaluated angle errors yet [plus, what other error sources have not yet been accounted for?]

551/10/19© 2019 J.V.R. Paiva

Typical RTK Error• ±(1 to 2 cm + 1 to 2 PPM)• Again determined from repeated

measurements• Value is standard deviation with assumption

that blunders and systematic errors have been eliminated

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Bad News For RTK-Determinations• When you use RTK-derived positions to inverse

the distance, what’s the uncertainty & confidence level?

• Answer: 1.4 cm (0.043 ft) one sigma, double if two sigma

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If line is 400 ft, precision is 1:9,000 at 68% confidence and 1:4,500 at 95% confidence

Worst Case• If we use 0.086 ft (95% conf.) for radius of

circle and L = 200 ft• Potential error in bearing is ≈ 89 arc seconds• Angle uncertainty of ±89” is equivalent to

1:2,300 or 430 PPM

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When You Determine Survey “Accuracy”

• Most of the time you are only determining precision

• UNLESS…. • What?

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Least Squares Adjustments• What do you put in for angle/distance uncertainty?• What do you put in for position and/or line

uncertainty?• How do you make allowances for number of sets

(rounds) measured?• How do you account for number of epochs,

satellites, observation time, DOP, etc.?

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What Do Manufacturers Account For?• Not your leveling error• Not your centering error• If using prism/antenna pole, not its straightness• Not centering and leveling error of target if TS• Not temp and pressure PPM error• Not iono and tropo error• Etc.

611/10/19© 2019 J.V.R. Paiva

What Will You Account For?Good Luck!

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www.geo-learn.com

AboutseminarpresenterJosephV.R.Paiva

Dr. Joseph V.R. Paiva, is principal and CEO of GeoLearn, LLC (www.geo-learn.com), an online provider of professional and technician education since February 2014. He also works as a consultant to lawyers, surveyors and engineers, and international developers, manufacturers and distributors of instrumentation and other geomatics tools, as well being a writer and speaker. One of his previous roles was COO at Gatewing NV, a Belgian manufacturer of unmanned aerial systems (UAS) for surveying and mapping during 2010-2012. Trimble acquired Gatewing in 2012. Because of this interest in drones, Joe is an FAA-licensed Remote Pilot.

Selected previous positions Joe has held includes: managing director of Spatial Data Research, Inc., a GIS data collection, compilation and software development company; senior scientist and technical advisor for Land Survey research & development, VP of the Land Survey group, and director of business development for the Engineering and Construction Division of Trimble; vice president and a founder of Sokkia Technology, Inc., guiding development of GPS- and software-based products for surveying, mapping, measurement and positioning. Other positions include senior technical management positions in The Lietz Co. and Sokkia Co. Ltd., assistant professor of civil engineering at the University of Missouri-Columbia, and partner in a surveying/civil engineering consulting firm.

Joe has continued his interest in teaching by serving as an adjunct instructor of online credit and non-credit courses at the State Technical College of Missouri, Texas A&M University-Corpus Christi and the Missouri University of Science and Technology. His key contributions in the development field are: design of software flow for the SDR2 and SDR20 series of Electronic Field Books, project manager and software design of the SDR33, and software interface design for the Trimble TTS500 total station.

He is a Registered Professional Engineer and Professional Land Surveyor, was an NSPS representative to ABET serving as a program evaluator, where he previously served as team chair, and commissioner, and has more than 30 years experience working in civil engineering, surveying and mapping. Joe writes for POB, The Empire State Surveyor and many other publications and has been a past contributor of columns to Civil Engineering News. He has published dozens of articles and papers and has presented over 150 seminars, workshops, papers, and talks in panel discussions, including authoring the positioning component of the Surveying Body of Knowledge published in Surveying and Land Information Science. Joe has B.S., M.S. and PhD degrees in Civil Engineering from the University of Missouri-Columbia. Joe’s current volunteer professional responsibilities include president of the Surveying and Geomatics Educators Society (SaGES) and various ad hoc and organized committees of NSPS, the Missouri Society of Professional Surveyors and other groups.

GeoLearn is the online learning portal provider for the Missouri Society of Professional Surveyors, the Kansas Society of Land Surveyors, the New York State Association of Professional Land Surveyors, The Texas Society of Professional Surveyors, The Pennsylvania Society of Land Surveyors, the Wisconsin Society of Land Surveyors, Arizona Professional Land Surveyors, the Oklahoma Society of Land Surveyors and the Geographic and Land Information Society. More organizations are set to partner with GeoLearn soon. Dr. Paiva can be reached at [email protected] or on Skype at joseph_paiva.

Jan 2019

Errors Analysis in Surveying2019 - cdn.ymaws.com · 1/10/19 1 Errors Analysis in Surveying Joseph Paiva Saratoga Springs January 2019 Objectives •Something about errors and mistakes

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