Error Analysis and Least Squares

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Error Analysis and Least SquaresCPSD # G100398

Presented by

The Office of Land Surveys

Division of Right of Way and Land Surveys

Developed By:

Jeremy Evans, PLS

Psomas, Inc.

And the Office of Land Surveys


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MEASUREMENT ANALYSISAND ADJUSTMENT

Capital Project Skill Development Class(CPSD)

GRW117

By Jeremy Evans,By Jeremy Evans,P.L.S.P.L.S. PsomasPsomas

Supplemented bySupplemented byCaltrans StaffCaltrans Staff

Introduce Instructor

The purpose of this class is to give Caltrans surveyors a clearunderstanding of error analysis. This is so the typical surveyor in thefield or office knows how to determine the precision and accuracyneeded to perform a task. This applies not just to control work, butthroughout the lifetime of a project.

Understand that there are three values for any measurement: themeasured value, the adjusted value, and the true value. The truevalue can never be known, but a surveyor should know how tocombine proper techniques, strength of figure, and adjustments sothat they are confident that the measured and adjusted values areclose to the true value

Areas of interest - boundary, control and adjustments, designmapping.


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Introduction

The dark side of surveying is thebelief that surveying is aboutmeasurements, precisions and

adjustments. It is not and neverwill be.

Dennis Mouland

P.O.B. Magazine

July, 2002

All measurements, no matter how accurate, are still subject to Boundary Laws and

common sense.

I once saw a Record of Survey that called 13 monuments out of position, and never

held a single one as good.

Surveying is the ART and Science of measurement! Over-reliance on numbers leadsto the Dark Side, it does.


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Introduction

Much has been written lately about leastsquares adjustment and the advantages itbrings to the land surveyor. To take fulladvantage of a least squares adjustmentpackage, the surveyor must have a basicunderstanding of the nature ofmeasurements, the equipment he uses, themethods he employs, and the environment

he works in.

Measurement analysis is the first part of this presentation.

Surveyors should have the ability to evaluate the amount of error in theirmeasurements and / or control the errors in their measurements

An understanding of measurements gives the surveyor this control

This course will present the Least Squares adjustment LAST,because a surveyor must understand a lot more about errors beforethey accept a least squares adjustment.


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Introduction

Measurements and Adjustments:War Stories

Discuss projects/situations where misunderstanding of measurements hascaused problems

A level run between benchmarks has an error of 0.25 feet in threemiles. Is that caused by random error that can be adjusted? Or by asingle bad reading that leaves a 0.30 jump in elevation betweentwo TBMs?

The 100 Hubble Space telescope was ground and polished to thesmoothest finish of any large mirror ever built. Was it any good?

Answer: The mirror was ground to the wrong prescription! Twoshuttle space flights were needed to add corrective lenses.

The surveyors understanding of measurements and datums iscritical to the success of any project.

Especially as projects get bigger in geographical terms. In the firstexample above, a bust can be hidden by the relative size of the

project. How do you know when to accept a weak adjustment, andwhen to re-measure?


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Class Outline

Survey Measurement Basics - A ReviewMeasurement Analysis

Error Propagation

Introduction to Weighted and LeastSquares Adjustments

Least Squares Adjustment Software

Sample Network Adjustments

Star*Net is one of many least squares adjustment software on the market

Terramodel

Intergraph Survey Select Cad

Trimnet or TGO

All of these softwares deliver correct results, however star*net was thefirst I used, speaks surveyor, has the greatest flexibility, etc.

The Department is moving to Trimnet from Star*net

Caltrans owns 120 Star*net licenses, so easiest to use as anexample


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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!

Leica 1103 should be turned in for servicing every 18 months. HQ has received

broken units that havent be serviced ever! Thats FOUR years!

Its not only good practice to regularly service the equipment, but cheaper in the long

run.

Are your tribrachs adjusted? Did you check the plummet and level before youpicked up a sight?

Have you cross-tied any control monuments?

Did you turn more sets when you had a weak control scheme, such as straight along

a RR?

Did you avoid as many 180 degree turns as possible?

Every Caltrans Surveyor should know Figure 5.1 !


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Survey Measurement Basics

A Review of Plumb Bob 101

Introduce books

Adjustment Computation by Wolf and Ghilani- more readable than most

star*net manual, good basics of adjustment theory and star*net

Random Errors chapter of Moffitt. Also the adjustment chapter has a goodsection on weighted means


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Surveying (Geospatial Services?)

Surveying That discipline whichencompasses all methods for measuring,processing, and disseminating informationabout the physical earth and ourenvironment. Brinker & Wolf

Surveyor - An expert in measuring,processing, and disseminating informationabout the physical earth and our

environment.

If surveyors want to be considered professionals, we need to know the theory

behind our procedures.


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This is a typical brochure of a modern total station.

Before you accept any of the statements as gospel, you must understand exactly

what the manufacturer is telling you.

DIN 18723 is the international testing standards for survey instrumentaccuracy. Other DINs may cover such things as food safety or strength offishing line.

DIN 18723 sets exact parameters for testing; such as temperature rangeduring testing, rigidity of setups, and other parameters that can only beeasily performed at the factory. None of theses tests are done while sightingover AC pavement in 100 weather.


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Instrument Specifications

The 5602, 5603, and 5605 are all essentially the same instrument! Aftermanufacture, all instruments are the tested for accuracy (DIN 18723).

Those instruments that have a standard deviation of less than 2 are labeled5602.

Those that have a standard deviation of more than 2 but less than 3 are5603s.

The 5605 is built to the same manufacturing tolerances as the 5602, butjust didnt test as well when checked at the factory.

Note: The instrument companies always try to build their equipment to thehigher standard. If you tried to order a 5605 from Trimble, they might tellyou that they dont have any available right now, and arent making any dueto the high demand for the 5602s.

What they really mean is that the factory is doing a great job, and allinstruments are passing the 2 standard.

Use (3mm + 3ppm) value (far right)

Is the ppm value here the same as the ppm value that is dialed into theinstrument dealing with temperature and pressure?


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Instrument Specifications

Distance Measurement

z m = (0.01 + 3ppm x D)

z What is the error in a 3500 footmeasurement?

z m= (0.01+(3/1,000,000 x 3500)) = 0.021

Discuss setting PPM . If you are on the beach in So Cal, you might get awaywith setting the PPM to zero. If you arent at sea level and 72, startcalibrating.

Sigma (lower case) denotes standard deviation

m is Standard deviation of the mean, a measurement of accuracy. More on thatlater.

Apply the standards for a 5605.

3 mm x 10,000 = 30m

So a single measurement less that 30m (100ft) will have an precision ration less

than 1/10,000.

Thats why we tie monuments twice!


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Calibration or Dont shootyourself in the foot.

Leica instruments should be servicedevery 18 months.

EDMs should be calibrated every sixmonths

Tribrachs should be adjusted every sixmonths, or more often as needed.

Levels pegged every 90 days

The service contract with the Leica suppliers call for 18 mos. service intervals.

Servicing doesnt cost anything, but blunders do!

more often as needed means before a control survey, after being dropped,

or any rainy equipment day.


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Using SECO Tribrach Adjusters Tech Tip Number: 12Created: January 1, 2001

Opt ical Plummet Adjustment Using: Tribrach Adjust ing Cylinder #2001 orTribrach Adjuster #2002 (see il lust rat ion 1)

EQUIPMENT NEEDED: Tribrach Adj ust ing Cyli nder #2001 or Tribrach Adj uster #2002, Tripod orinstrument stand, 2 tribrachs, target

1. Place the tr ibrach on the tripod. Put the adjuster in t he tribrach. Place the tr ibrach to beadjusted on top of t he adj uster so that i t is upsidedown looking at t he target on the ceili ng.

The distance between the t ribrach and target should be between 4 and 5 feet.

2. Using the leveling screws of the bottom tribrach, point the crosshair of the tribrach being

tested to coincide with the target .

3. Rotate t he t ribrach being tested 180 degrees on t he adjuster. Crosshairs wil l stay on thetarget of an adj usted tr ibrach.

4. Aft er 180 degree turn if the crosshair does not stay on the target , half the error should becorrected with the adjust ing screws provided by the manufacturer of t he tri brach. The

remainder should be corrected with the leveling screws of the bottom tribrach.

5. Repeat steps 2, 3 & 4 unti l the crosshair stays on target at all posit ions.

Tribrach Circular Vial Adjustment Using: Tribrach Adjuster #2002 (seeil lust ration 2)

EQUIPMENT NEEDED: Tripod or instrument stand, Tribrach Adjuster #2002 and adjusting pins.

1. Place the tribrach on the tripod and fasten to the tripod.

2. Place the #2002 in the t ribrach and level t he tr ibrach using the vial on the #2002. Ignore the

circular vial on the t ribrach.

3. To level the #2002: Point one end of the #2002 vial to any leveling screw and using thatscrew bring the vial to center.

4. Now turn t he #2002 90 degrees so that each end of t he vial is as close as possibl e to theother two leveli ng screws. Using these two leveli ng screws, center the vial.

5. Turn the vial 90 degrees back to the original leveling screw and level again if necessary.

6. Repeat 1,2 & 3 unti l the vial remains centered at both posit ions.

7. To test t he adjustment of the #2002 vial at any centered posit ion, rotate t he #2002 180degrees. The vial should stay centered within one graduat ion. If not , take half the error back

to the center with the vial mounting screw that is on the high side. 8. If the circular vial on thet ribrach is not centered, use the adjust ing screws and bring to center.


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Is It a Mistake or an Error?

Mistake - Blunder in reading, recording orcalculating a value.

Error - The difference between a measuredor calculated value and the true value.

Discuss true value. For a traverse, there is the measured value, adjustedvalue, and true value.

True Value does exist but cannot be measured or known

The best that anyone can do is a mean value or most probable value


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Blunder

a gross error or mistake resulting usually from

stupidity, ignorance, or carelessness.

Most blunders are caused by human error. If you are lucky, its someone elses error,

not yours. This is why we have specific field techniques, such as double tying

monuments, measuring all HIs, and closing traverses. If all of the procedures are

done properly, then blunders can be isolated and dealt with.


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Setup over wrong point Bad H.I.

Incorrect settings in equipment

Blunder


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Types of Errors

Systematic

Random

Error is the difference between the measured value and the true value.

Its the job of a surveyor to reduce errors to a minimum. But always accept that

there will be minor errors, and not try to fix data that is within tolerance.


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Random

Poorly adjusted tribrach

Inexperienced Instrument

operator

Inaccuracy in equipment

Many tribrachs have a centering error of +/- 2mm. They dont have to be poorly

adjusted to introduce error. A poorly adjusted tribrach creates systematic error, a

properly adjusted one will still be a source of random error.

All equipment has inherent inaccuracy. Therefore, all measurements will containrandom error.


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Nature of Random Errors

A plus or minus error will occur with thesame frequency

Minor errors will occur more often thanlarge ones

Very large errors will rarely occur (seemistake)

A Normal Distribution Curve has all of these attributes:

1. It is symmetrical about the mean

2. More data is close to the mean that farther away

3. Very little data is found at the fringe


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Normal Distribution Curve #1

A plus or minus error will occur with the samefrequency, so

Area within curve is equal on either side of the mean

The Normal Distribution curve is also known as the Bell Curve due to its shape.

Its was developed by an 18th century German mathematician and astronomer named

Karl Gauss.

If this was a chart of coin tosses, the chance of a coin land on heads is equal to the

number of coins landing tails. And the number of coins landing heads 6 out of 10

times is equal to the number of coins landing tails 6 out of 10 times, etc.

This is a Normal Curve! In real life, the data is often skewed.


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Normal Distribution Curve #3

Very large errors will rarely occur, so

The total area within 2 of the mean is 95%of the sample population

In the previous slide we mentioned that a small number of data wont result in a

smooth curve.

In a random sampling of a general population, it usually takes a minimum of sample

of about 30 to see a true curve start to form.

With numbers less than thirty, its possible that there wont be any measurementsoutside of two standard deviations.

Since surveyors only measure a sample of thirty or more with GPS equipment, ALL

conventional field measurements should fall within that limit.

If we go back to the example of the curve representing 10 coin tosses, the chance of

any person tossing 4 heads in a row is 24 (16:1) or 6.25%

The odds of 5 in a row is 32:1, or 3.125%.

So while it is possible to toss heads 10 times in a row (1024:1, or 0.1%),

measurements outside of 2 sigma of the mean arent usually relevant. For

measurement data, that means flawed.


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Histograms, Sigma, & Outliers

4.00.5 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

Residuals

1 1

Outlier

\

MEA

N

2 2

Histogram: Plot of the Residuals

\

1 : 68% of residualsmust fall inside area

2 95 % of residualsmust fall inside area

Bell shaped curve

/

This data shows the precision of a set of turned angles.

A residual is the difference between the measured value and the most likely value

(usually the mean). Thats different from the definition of error, which is the

difference between measured and true. Since the true value isnt known, you cant

calculate error. But a residual is a value that can be calculated and used for

mathematical adjustments.

Notice the Outlier. A bell curve should help identify data that should be excluded

(Blunders)

All data within 2 sigma is significant data, even if it isnt precise.

It still has statistical value, and isnt weak data

In this example, measurements within 1.75 seconds of the mean will happen 68% of

the time. So even very good measurements have a measure of uncertainty.


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Measurement Components

All measurements consist of twocomponents: the measurement andthe uncertainty statement.

1,320.55 0.05

The uncertainty statement is not aguess, but is based on testing ofequipment and methods.

Uncertainty statement is usually a statement of accuracy

In the last slide, the measurement of one standard deviation was +/- 1.75. That was

the uncertainty statement

The second bullet originally read The uncertainty statement is not a guess, but is

based on testing of equipment, personnel, methods and the surveyors judgment.Whats the difference between the two?

Answer: All properly adjusted equipment, used correctly, has random errors. The

human factors (personnel and judgment) introduce blunders and systematic errors.

The published instrument uncertainty statements (2.0 mm +/- 2 ppm) are the

expected instrument error. The actual field measurements include the systematic

errors beyond the manufacturers control.


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Accuracy Vs. Precision

Precision - agreement among readings of thesame value (measurement). A measure ofmethods.

Accuracy - agreement of observed valueswith the true value. A measure of results.

Bullseye example

Bullseye is not a very good example in surveying. A tightly grouped set ofmeasurements (precision) that misses the Bullseye (accuracy) doesnt helpthe surveyor.

Q. If several tightly grouped measurements miss the bullseye, how wouldyou know?

A. See Standard Deviation of the Mean


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Measurement Analysis

Determining Measurement Uncertainties

Now well take a data sample and show how the Bell curve applies to

measurements.


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Determining Uncertainty

Uncertainty - the positive and negative rangeof values expected for a recorded orcalculated value, i.e. the value (the secondcomponent of measurements).


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Your Assignment

Measure a line that is very close to 1000 feetlong and determine the accuracy of yourmeasurement.

Equipment: 100 tape and two plumb bobs.

Terrain: Basically level with 2 high brush.

Environment: Sunny and warm.

Personnel: You and me.

If the instructor wishes to have the class perform this exercise, see the sample

instructions in the student work book.


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Chaining Test Data Exercise

Equipment: 100 steel chain, 2 nails, 2 plumb bobs.

Setup: On level ground lay the chain out flat, and place two nails approximately 100 feet

apart. The site can be on grass, dirt, or pavement, as long as it is level.

Procedure: Have the class form 2-person teams, with each team making a single

measurement of the distance. Both chainmen should use a plumb bob, with the head

chainman holding the chain no more than waist high. If time permits, the trainees can use

a spring balance and thermometer, and adjust for sag and temperature. The tape

corrections would be part of eliminating systematic errors. If corrections for sag or

temperature arent made, students should still be aware of the correction procedures. You

can still use uncorrected measurements for the classroom exercise.

Measurements: At least 10 measurements should be made. If the class has fewer than

20 students (10 teams), then teams may switch off head and rear chainmen until a total of

10 measurements are obtained. There may be more measurements, but for simplicity it

shouldnt be much more than 10, and should be an even number. Each chaining team

should not reveal their results until all measurements have been made.

Calculations: After all measurements have been collected, the student will return to the

classroom, and use the data as shown in the PowerPoint to obtain mean, standard

deviation, and standard deviation of the mean

NOTE: After completing the exercise, DONT try to measure the distance using EDM

equipment! Students should be aware that they will never know the true value. There

are measurements that are close to the mean value, but there is no right answer. Even a

distance measured by modern equipment has its own random errors, and is not the true

value.


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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

Need to measure between two points approximately 1000 apart and needthe accuracy of the measurement

Discuss how measurements were made (chain, bobing up to waist high,etc.)

Objective is to determine error per chain length by testing, then determinethe error in the 1000 distance.


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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

Measures of Central Tendency is a corollary to the Nature of Random Errors #2.

Mean is the sum of measurements divided by the number of observations.

Median is the midpoint of the observations (half are less, half are greater).

Mode is the most common value.


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Averages

Most commonly used is Arithmetic Mean

Considered the most probable value

n = number of observations

Mean = 1000 / 10

Mean = 100.00

nmean

meas.=


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Residuals

The difference between an individual readingin a set of repeated measurements and themean

Residual () = reading - mean

Sum of the residuals squared (2) is used infuture calculations

Residuals are also called variations. Thats why v is used in the formula


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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 0.0004100.00 0 0

2 = 0.0048

The determination of mean and the resulting residuals are the beginningof a least squares adjustment. The sum of the residuals squared should besmallest where the mean was properly calculated.


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Standard Deviation

The Standard Deviation is the rangewithin which 68.3% of the residuals will fallor

Each residual has a 68.3% probability offalling within the Standard Deviation rangeor

If another measurement is made, theresulting residual has a 68.3% chance offalling within the Standard Deviation range.

Standard Deviation is sometimes referred to as standard error.

Dont get the cart before the the horse, the slide makes it sound like thedefinition of deviation is 68.3%.

In reality, the formula for Standard Deviation results in a 68.3%

probability, not the other way around.


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Standard Deviation Formula

( )1n

deviationStandard2

=

'023.09

0048.0==

Sigma (lower case) denotes standard deviation

Sigma (upper case) denotes Summation

Vee (italics) denotes residual, the difference between individual measurements and

mean

n denotes number of measurements

The pure formula for standard deviation would have just n in the denominator,

not n-1

But you cant have a standard deviation from just one measurement. So n-1

represents the number ofredundant measurements,

You can make only a single measurement if you wanted to. But you wouldnt be

able to calculate a standard deviation from a single measurement.

Note that the more redundant measurements (n-1) you have, the closer n-1

approaches n.

That is, if n=2, then n-1 is of n. But if n=100, then n-1 is 99% of n.The more redundant measurements, the more accurate the standard deviation


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Standard Deviation

Standard Deviation is a comparison of theindividual readings (measurements) to themean of the readings, therefore

Standard Deviation is a measure of.


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Standard Deviation

Standard Deviation is a comparison of theindividual readings (measurements) to themean of the readings, therefore

Standard Deviation is a measure of.

PRECISION!PRECISION!

Draw a Bell Curve that is very tall and steep, and compare it to a very low and flat

curve.

Which curve represents a higher precision?

The closer the data are to the mean, the higher precision.


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Standard Deviation of theMean

This is an uncertainty statement regarding the meanand not a randomly selected individual reading as isthe case with standard deviation.

Since the individual measurements that make up themean have error, the mean also has an associatederror.

The Standard Deviation of the Mean is the rangewithin which the mean falls when compared to thetrue value, therefore the Standard Deviation of theMean is a measure of .


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This is an uncertainty statement regardingthe mean and not a randomly selectedindividual reading as is the case withstandard deviation.

Since the individual measurements thatmake up the mean have error, the meanalso 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 theStandard Deviation of the Mean is ameasure of .

ACCURACY!

Draw a Bell Curve that is skewed, with one steep side and one gentle slope.

Is this more accurate than a symmetrical data set?

Q. What happens if you turn three sets of angles instead of two or four?

SEE EXERCISE FOR STANDARD DEVIATION OF THE MEAN


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Exercise for StandardDeviation of the MeanSlide #39An instrument man measures an angle

three times.

He gets the following results:

504538

504544

504538

Calculate the Standard Deviation andStandard Deviation of the Mean for this

of three angles.

(Hint: just use the seconds as wholenumbers)

( )1n

deviationStandard2

=

n

m

=)(MeantheofDeviationStandard

Not satisfied with the spread of the

measurements, the instrument man then

turns another set of angles:

504544

504538

504542

504536

Calculate the Standard Deviation andStandard Deviation of the Mean for theset of four angles.


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Calculation of Standard Deviation

Meas.# Angle Residual V2

1 44 4 16

2 38 -2 4

3 38 -2 4

SUM 24

1 44 4 16

2 38 -2 4

3 42 2 4

4 36 -4 16

SUM 40

Calc Standard Deviation for set #1

alc Stand. Dev. of the Mean for set #1C

Calc Standard Deviation for set #2

Calc Stand. Dev. of the Mean for set #2

Note that:

1. Both data sets have the same mean 40

2. The first set has a smaller spread thanthe second 6 vs. 8, but isnt

symmetrical about the mean

3. The first asymmetrical set has a

smaller standard deviation (precision)( ) 46.3224

#1set ==

4. The second symmetrical set has a

smaller standard deviation of the mean

(accuracy)

If your observations arent symmetrical,

it is better to take more observationsthan guess which ones are better

When turning sets with a total station,

always turn an even number!

0.2)(#1 ==

( )

3m

46.3set

65.33

40#2set ==

82.14

65.3)(#2set ==m


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Distance = 100.000.007(1 Confidence level)

n)(MeantheofErrorStandard m =

'007.010

023.0m ==

Back to the chaining example.

For every measurement that you make, there are three values.

The first value is the measured value (in this example, each of the 10measurements)

The second value is the adjusted value; i.e. the mean.The third value is the True Value.

The Standard Deviation of the Mean is your confidence in the adjustedvalue.

This calculation states I am confident that the true value lies within 0.007of the adjusted value.


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90% & 95% Probable Error

A 50% level of certainty for a measure ofprecision or accuracy is usually unacceptable.

90% or 95% level of certainty is normal forsurveying applications

)6449.1(E90 = )96.1(95 =E

n

E

E

90m90

= n

EE

95m95 =

Must calculate E90 or E95 before calculating E90m or E95m


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Meaning of E95

If a measurement falls outsideof two standard deviations, it

isnt a random error, its amistake!

Francis H. Moffitt

Were Surveyors, not statisticians. Random Errors that fall outside of E95 arent

random errors.

Time to re-check you measurements and equipment.


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How Errors Propagate

Error in a Series

Errors in a Sum

Error in RedundantMeasurement


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Error in a Sum

Esum is the square root of the sum of each ofthe individual measurements squared

It is used when there are several

measurements with differing standard errors

2222n321sum E...EEEE ++++=

Error in a series and error in a sum are basically the same.

If the variable E is the same for each of the measurements, then the result is the

Error of a Series formula.

If they arent the same value, then you use the Errors of a Sum formula.


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Exercise for Errors in a Sum

Assume a typical single point occupation. Theinstrument is occupying one point, with tripodsoccupying the backsight and foresight.

How many sources of random error are there in thisscenario?

Hint: First look at errors that would affect distance, then errors that would affect the

angle.


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Exercise for Errors in a Sum

There are three tribrachs, each with its owncentering error that affects angle and distance

Each of the two distance measurements have errors

The angle turned by the instrument has severalsources of error, including poor leveling and parallax

The combination of all of the possible random errors exceeds the amount of error

we normally associate with a single measurement

For someone to say this is a half-second gun, or The EDM is accurate to 2mm

ignores all of the other possible error sources


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Error in RedundantMeasurements

If a measurement is repeated multipletimes, the accuracy increases, even ifthe measurements have the same value

n

EE .meas.red =

If sigma= one (1), and n=1,then one over the square root of 1 = 1

If sigma= one (1), and n=2, then one over the square root of 2 = 0.707

If sigma= one (1), and n=2, then one over the square root of 4 = 0.5

What is error in 1000 distance using error value determined before

=0.015 (10) =0.047 = 0.05

Error in redundant measurement is used when a value is measured morethan one time

What is error value when 1000 distance is measured 4 times.

=0.047 4 = 0.024 = 0.02


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Eternal Battle of Good Vs. Evil

With Errors of a Sum (or Series), eachadditional variable increasesthe totalerror of the network

With Errors of RedundantMeasurement, each redundantmeasurement decreasesthe error ofthe network.

This may be the single most im portant statement i n thi s entir ecourse.

As networks become more complex, there is is a greater chance of error.

Also, a blunder can hide in a complex network, by having the error spread

out to more points. At the beginning we had the example of a level networkwith 0.10 closure per mile (0.25 in three miles). A single three mile levelrun cant isolate a bust. But three one mile loops will show whether youhave poor measurements or poor control.

Always think of redundancy when planning a network.


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Sum vs. Redundancy

Therefore, as the network becomesmore complicated, accuracy can bemaintained by increasing the number ofredundant measurements

.

Redundancy can mean:

1. Turning more sets of angles with a Total Station. This is very easy with servo

instruments turning rounds in auto mode.

2. Traverses with cross-ties and double stubbing

3. Longer occupations using GPS

4. Multiple occupations of GPS points using different configurations

5. Level runs that use several loops, instead of a single long run between two known

points.


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Coordinate StandardDeviations 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

This is why the standard errors have only a 39.4% chance of falling within the error

ellipse.

The standard deviations arent oriented the same as the ellipse.


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Positional Accuracy vs.Precision Ratio

Or, How good is one error ellipsecompared to all those others?

Older surveys use closure as a measure of accuracy. Newer adjustments dont. How

do you compare the two?

Or to put it another way, How close together can two error elipses be and still have

an accurate survey?

See Attached Document


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Positional Accuracy vs. Precision Ratio

Traditional compass rule adjustments were analyzed using precision ratios.

The length of a traverse is divided by the error in closure. The result is the precision ratio.

The standard for a control traverse run to second order accuracy is 1:20,000.The standard for a landnet traverse run to third order accuracy is 1:10,000.(See Chapter 5 Surveys Manual)

0.01 ft x 10,000 = 100.00

Therefore, any single distance measured to an accuracy of less that 0.01 per 100 ft cannot

meet the 1:10,000 ratio. This is one reason why all landnet points are double-tied.

RTK.

RTK only measures baselines between the base station and the rover. Each measurementto a monument is independent of measurements to other monuments. The vector between

two unknown stations is never measured. Each is independently measured to a known

base station. This is one reason why RTK can only be used for surveys of third order or

less.

Positional Accuracy

Least square adjustments dont publish precision ratios. Instead, each point is given aposition and an error ellipse, defining the most likely position of the point. A position can

also be defined as the circle in which the true position has a 95% chance of being located.

(E95). The question then becomes How do you determine the precision ratio of ameasurement that doesnt have a traverse closure?

The simple way to check for precision ration is to divide the distance between two pointsby the sum of the standard errors of the two points.

Errors in a Sum

The Standard Error of the sum of two quantities is equal to the square root of the sum of

the squares of the standard errors of the individual quantities .The concept can be

extended to the sum of any number of quantities that are not correlated.

-Moffitt/ Bouchard


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To determine the precision ratio between to monuments:

The ratio between the length of the line and the sum of the errors of the two point.

Y = (Distance)(A + B)Where A is the positional accuracy at the first station and B is the

positional accuracy at the second station.

1:Y is the resultant precision ratio, where Y shall be greater than or

equal to 10,000 to achieve third order accuracy.

Assume that you locate two monuments using RTK that are approximately 140 m apart.

Each has a positional accuracy of 10mm. What is the precision ratio of the measureddistance between the two monuments?

Y = (140.00)(0.010) + (0.010)

Y = (140.00)0.0002

Y = 140.00 0.014Y = 10,000 and 1:Y = 1:10,000

OR

Given two RTK monuments at E95 of 10mm,The minimum distance between the two monuments that would

achieve a 1: 10,000 ratio would be 140 meters (460 ft.)

Monuments found at distances less than 140 meters (460 ft) apart must be tied using

conventional total station methods to achieve third order standards. Monuments

between 140 and 200 meters apart should be checked for positional error before

being accepted.

The 140 meter standard applies when each monument has been occupied according tostandards (occupied twice for minimum of 15 epochs) and are within a properly boxed

control net. See Surveys Manual Chapter 6

Exercise #2

An EDM with an accuracy () of 2 mm 2.0 ppm is used to measure a distance of 40meters. The instrument and foresight are on tribrachs with an accuracy of 1.5 mm.Using the Errors of a Sum formula, calculate the total measurement error. Then calculate

the shortest distance that such a setup could measure a 1: 10,000 precision ratio (land

net), and a 1: 50,000 ratio (project control)


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Introduction to Adjustments

Adjustment - A process designed to removeinconsistencies in measured or computedquantities by applying derived corrections tocompensate for random, or accidental errors,such errors not being subject to systematiccorrections.

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

Compass rule assumes that both angles and distances are measured with equal

precision. The most common way of adjusting metes and bounds descriptions.

The Compass Rule can only solve a traverse, not redundant measurements.

Transit Rule assumes angles are more accurate than distances, but the formula

results in different corrections depending on the orientation of a figure (if you havea closed traverse, and then rotate it 45 degrees, the adjustment for each leg will

change)

Crandalls rule again assumes angles superior than distances, but is more

complicated than Transit Rule

Rotation and scale holds interior angles as fixed, and adjusts distances. This is the

same as the BLM Grant line Adjustment.

Least Squares simultaneously adjusts the angular and linear measurements to make

the sum of the squares of the residuals a minimum.

If there are no redundant measurements, the results are the same as a Compass Rule.


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Weighted Adjustments

Weight - The relative reliability (or worth) ofa quantity as compared with other values ofthe same quantity.

Definitions of Surveying and

Associated Terms,

1989 Reprint


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The concept of weighting measurements toaccount for different error sources, etc. isfundamental to a least squares adjustment.

Weighting can be based on error sources, ifthe error of each measurement is different, orthe quantity of readings that make up areading, if the error sources are equal.


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Formulas:W (1 E2) (Error Sources)

C (1 W) (Correction)

W n (repeated measurements ofthe same value)

W (1 n) (a series ofmeasurements)

Symbol means proportional

Weights are inversely proportional to the residuals. The closer a measurement is

to the mean, the more heavily weighted it should be.

Therefore, corrections are inversely proportional to the weights. The farther a

measurement is from the mean, the more it will be corrected. Weights are proportional to redundancy. The more times a value is repeated, the

stronger the weight.

Weights are inversely proportional to measurements of a series. A level run of 4

turns is stronger than a run using 8 turns. (All other factors being even)


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A

BC

A = 432436, 2x

B = 471234, 4x

C = 892220, 8x

Perform a weightedadjustment based on theabove data


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ANGLE No. Meas Mean Value Rel. Corr. Corrections Adjusted Value

A 2 43 24 36 4/4 or4/7

4/7 X 30 = 17 43 24 53

B 4 47 12 34 2/4 or2/7

2/7 X 30 = 09 47 12 43

C 8 89 22 20 1/4 or1/7

1/7 X 30 = 04 89 22 24

TOTALS 17959 30 7/4 or 7/7 = 30 180 00 00

The relative correction for the three angles are 1 : 2 : 4, the inverse proportion tothe 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 thedenominator for the second set of corrections. The sum of the second set ofrelative corrections shall always equal 1. The second set is used for corrections.

The correction to angle C should be one fourth the correction to angle A, and one

half the correction of angle B. This ration is the relative correction factors between

the measurements. This is the first correction factor.

The sum of the relative factors results in the total correction factor for the figure.

The total figure correction factor is then used to correct the measured angles.


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BM A

Elev. = 100.0

BM B

Elev. = 102.0

BM C

Elev. = 104.0

+6.2, 10 mi.

+7.8, 2 mi.

+10.0, 4 mi.

BM NEW

This exercise doesnt have a published solution. Instructors may include it as an

exercise, or save time by skipping it.

SeeMoffittfor a good example of solving this type of problem.


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Introduction to Least SquaresAdjustment

Simple Examples


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What Least Squares Is ...

A rigorous statistical adjustment of surveydata based on the laws of probability andstatistics

Provides simultaneous adjustment of allmeasurements

Measurements can be individually weightedto account for different error sources andvalues

Minimal adjustment of field measurements

Compass rule adjustment is based on proportional adjustment of data

Simultaneous adjustment of all measurements is the most importantbenefit of least squares. In multiple traverses, a compass adjustment mustsolve each traverse in order, and hold the results as fixed for the nexttraverse. Least squares can solve the entire network simultaneously

Each measurement can have its own error estimate or you can globallyset the error estimate or a combination of the two

Maintains the integrity of the field measurements, least squares tries tominimize the amount of adjustment to each measurement


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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.

What is Least Squares?

Think of ways that other adjustment methods can skew data.

The Compass Rule adjusts angles based on the length of the legs. But short sights

are less accurate than long ones, so why adjust the long sight more?

A least squares adjustment can take weighted means, redundancy, and strength offigure to adjust a network.


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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 landsurveyor

>Any survey can be manipulated to pass a least squares adjustment byfreeing up data or changing error estimates

>All adjustments must be reviewed prior to moving on to next step

>A traverse that runs 3 miles along a straight highway is inherently weak

>If you occupy the wrong monument, and dont perform a check shot, leastsquares wont help you

>A survey with no redundancy will have the same results whetheradjustment is compass rule or least squares


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Least Squares

Least Squares Shou ld Be Used f or

The Adj ust ment Of: Collect ed By:

Conventional Traverse

Control Networks

GPS Networks

Level Networks

Resections

Theodolite & Chain

Total Stations

GPS Receivers

Levels

EDMs


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Least Squares

What happens?I t erati ve Process

Each it erati on applies adjustments to

observations, w orking f or best solut ion

Adjustm ents become smaller w ith each

successive it erati on

A B

CD

E

F

G

Observed

1st Iteration

2nd Iteration

.


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1 Creates a calculated observation for each fieldobservation by inversing between approximatecoordinates.

2 Calculates a "best fit" solution of observations andcompares them to field observations to computeresiduals.

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.

Least Squares

The Iterative Process


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1 Errors

2 Coordinates

3 Observations

4 Weights

Least Squares

Four component that need to be addressedprior to performing least squares adjustment


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Errors

Blunder - Must be removed

Systematic - Must be Corrected

Random - No action needed


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Coordinates

Because the Least Squares process begins bycalculating inversed observations approximatecoordinate values are needed.

1 Dimensional Network (Level Network) - Only1 Point.

2 Dimensional Network - All Points NeedNorthing and Easting.

3 Dimensional Network - All Points NeedNorthing, Easting, and Elevation. (Except foradjustments of GPS baselines.)


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Weights Each Observation Requires an Associated Weight

Weight = Influence of the Observation on FinalSolution

Larger Weight - Larger Influence

Weight = 1/2

= Standard Deviation of the Observation

The Smaller the Standard Deviation the Greater theWeight

= 0.8 Weight =

1

/0.82 = 1.56 = 2.2 Weight = 1/2.22 = 0.21

More

InfluenceLessInfluence


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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

Methods of Establishing Weights

Good for combining

Observations from

different classes of

instruments.

Good for projects

where standard

deviation is calculated

for each observation.


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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

Least Squares

If you remember nothing else about least squares today,remember this!


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Also Called

Minimally Constrained Adjustment

Free Adjustment

Used to Evaluate

Observations

Observation Weights

Relationship of All Observations

Only fix the minimum required points

Unconstrained Adjustment


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Flow Chart

Field ObservationsSetupObservationStandard Deviation

Field DataNeedsEditing?

Edit Field Data Remove Blunders Correct Systematic

Errors

PerformUnconstrained LeastSquares Adjustment

No

AnalyzeAdjustmentStatistics

StatisticsIndicate

Problems

Modify InputData

Constrain FixedControl Points

No

PerformConstrained Least

Squares Adjustment

Print outUnconstrained

Adjustment Statistics

Print out FinalCoordinate Valuesfor All Points in

Adjustment

Yes

Yes

Least SquareAdjustment

SoftwareDecision Step

Performed byUser

Start

Finish


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Analyze the Statistical Results

There are 4 main statistical areas that need to be lookedat:

1. Standard deviation of unit weight2. Observation residuals

3. Coordinate standard deviations and error ellipses

4. Relative errors

A 5th statistic that is sometimes available that should belooked at:Chi-square Test


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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

Standard Deviation ofUnit Weight


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Observation Residuals

Amount of adjustment applied to observation toobtain best fit

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

This is the residual that is being minimized


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Observation ResidualsSite 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

4.00.5 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

Outlier

0


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Least Squares Examples

Arithmetic Mean

Straight Line Best Fit


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Least Squares Examples


Explain scenario (must be straight line thru points)

This is an example of determining a best fit alignment for a prescriptiveeasement.

In a boundary problem, it might help you reject a monument, but best fitis never to be used as a boundary solution


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Perpendicular offsets:

1 = (0,0)

2 = (100,100)

3 = (200, 400)

This example - Perpendicular offset = 141.421

1: r = 0, r sq. = 0

2: r = 0, r sq. = 0

3: r = 141.421, r sq. = 20,000

Sum r sq. = 20,000


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1: r = 63.246, r2 = 4,000

2: r = 0, r2 = 0

3: r = 0, r2 = 0

Sum r2 = 4000


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1, 2 & 3: r = 22, r2 = 484

Sum r2 = 3*484 = 1452

This has the lowest Sum r2 therefore is best result so far

Actual best result is a skewed line that runs 19.9 feet SE of point 1 to 8.4

feet SE of point 3.


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Least Squares Rules

Redundancy of survey data strengthensadjustment

Error Sources must be determined correctly

Each adjustment consists of two parts:z Minimally Constrained Adjustment

z Fully Constrained Adjustment

Redundancy is a good thing!!

Explain the necessity of two adjustments

A closed traverse that is minimally constrained (one point and bearing held)

should result in a tight closure. If it doesnt, that means that yourmeasurements were poor.

If you have a good minimally constrained adjustment, then you run a fullyconstrain the adjustment (hold all found control monuments as fixed).

If the results are poor, then you know that it is the control that is weak, notyour measurements.

Then you go back to the minimally constrained adjustment, and start addingone control monument each run, until you can isolated the poor control.


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Star*Net Adjustment Software

A Tour of the Software Package

Star*Net

1

3

2

6

4

5


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Sample Network Adjustment

A Simple 2D Network Adjustment

Star*Net1

3

2

6

4

5

Printout from this adjustment in in appendix

Run adjustment and review printout (unconstrained & constrained)

add mistake to input data and run adjustment

explain how least squares will point to potential mistake (if only one

mistake!) If inputting data by hand, input one page then run adjustment and

check for errors, input second page and check for errors, etc.

If time permits, do adjustment with GPS vectors

Show the results of traverse (linear precision) in this adjustment


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Sample Network Adjustments

A 3D Grid Adjustment using GPS andConventional Data

0012224.299

North Rock

0017209.3AZDO

0013205.450BM-9331

0051201.018

0052192.051SW Bridge

0053203.046

0018204.86

0015188.195

0016186.655

Star*NetStar*Net


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Beyond Control Surveys

Other Uses for Least SquaresAdjustments / Analysis

Thinking outside of the box!


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Questions & Discussion


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Statistics Glossary

Error the difference between a measured or computed result and the true value. In

mathematics, errors can be systematic orrandom. See Residual.

Systematic Errors an error that is not determined by chance but is introduced byan inaccuracy (as of observation or measurement) inherent in the system. If they are

cumulative, such as temperature corrections for a steel tape, applying correction factors

can compensate for the effects. If they are variable, such as error caused by a poorlyadjusted tribrach, they can be controlled by proper field procedures or calibrations.

Random Errors Often called accidental errors. They are unpredictable errorsthat remain after mistakes and systematic errors have been eliminated. They are usually

compensating, and follow the laws of probability. Present in all survey measurements.

Residual (

) The difference between a measured value and the most probable value,which is usually the mean. Residuals are similar to errors except that residuals can be

calculated and errors cant, because a true value is never known. All adjustmentcalculations therefore use residuals. The symbol is used because residuals are

sometimes referred to as variations.

Variance () The variance is a measure of the range of a set of measurements. It is a

function of the sum of the residuals. Its square root is the standard deviation. The greater

the range of measurements, the larger the standard deviation.

Standard Deviation () A measurement of the precision of a set of measurements. Alsoreferred to as standard error. In a normal distribution curve, the area within one standard

deviation is 68.27% of the total.

Standard Deviation of the Mean (m) isa measure of accuracy. The mean is the

Documents

Error Analysis and Least Squares