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Error Analysis Relative to Positional Precision

Presented by: Anthony M. Gregory, PLSIndiana Society of ProfessionalLand Surveyors Annual ConferenceIndianapolis, IndianaJanuary 15, 2016 1

Seminar Outline/Description:

Land surveyors have long been expected to understand how errors occur in their measurements, how to quantify the identified errors, and, where possible, how to control them. As technology has changed measurement methods over the past several decades, and as computers have allowed for more thorough means of error analysis and adjustment, today’s land surveyor is in a position to deal with and report measurement errors in a very different and more detailed manner than in the past. This seminar will demonstrate the importance of applying fundamental principles of the evaluation of surveying measurements to modern measurement processes.

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Seminar Objectives:

The objectives of this seminar will be to review the fundamental principles associated with the

analysis of surveying measurement errors; to relate those principles to current measuring methods;

and to relate those principles to current state and national standards associated with uncertainties

and tolerances of concerning surveying measurements.

Topics presented in this seminar will include the following:

• Review of Measurement Errors

• Review of Random Errors and Random Error Propagation

• Identification of Past and Current Measurement Uncertainties and Tolerances

• Review of Adjustment Methods Used for Surveying Measurements

• Definition of What Theoretical Uncertainty Means (or Meant)

• Definition of What Relative Positional Accuracy Means

• Definition of What Relative Positional Precision Means

• Identification of Means Used to Report Surveying Measurement Uncertainties

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What this Workshop Will Not Include:

An in depth analysis of Least Squares Adjustment

A comparison of various available Least Squares software packages

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Understanding Measurements

“The undisputable fact that no measurement is exact, must be acknowledged, for exactness is impossible in the physical world. The man who regards his measurements as being ‘right on the button’ is only deceiving himself. In fact, he may be very shocked to discover how inaccurate his measurements really are! Every measurement has a certain degree of uncertainty. Indeed, the mark of a good surveyor may well be his ability to recognize and control the uncertainty of his measurements.”

Stoughton, Herbert W., “Introduction to Least Square Adjustment for the Land Surveyor”

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Understanding Measurements

• No observation is exact

• Every observation contains errors

• The true value of an observation is never known

• The exact error present is always unknown

From Elementary Surveying, An Introduction to Geomatics, by Wolf and Ghilani

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3

2 + 2 = ?????

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Errors in Measurements

Dictionary Definition of Error ‐ the state of believing what is untrue; a wrong belief; something incorrectly done; mistake; transgression. "in error" means "wrong“

“Surveying Measurement" Definition of Error ‐ the difference between an observed or computed value of a quantity and the true value of the quantity.

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True Value

True Value ‐ actual value existing in nature, which is seldom known except when fixed by mathematics or by authority.

1. By mathematics – sum of angles in triangle = 180o

2. By authority – 1 foot = 0.3048 meters (exactly)

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Errors in Measurements

Observed Value = Reading = Observation

Error = Reading ‐ True Value

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Sources of Errors in Measurements

Natural Errors ‐ caused by wind, temperature, humidity, atmospheric pressure, refraction, etc.

Instrumental Errors ‐ caused by (1) manufacture of instrumentation, or (2) wear and/or maladjustment of instrumentation.

Personal Errors ‐ caused by inability of people to perceive anything exactly because of limitations of senses of sight and touch. (Not to be confused with mistakes!!)

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Mistakes and Blunders

According to Wolf and Ghilani, a “mistake” is “usually caused by the observer’s misunderstanding of the problem, carelessness, fatigue, missed communication, or poor judgment”.


“Blunders” are typically defined as “stupid mistakes”, or “foolish mistakes”.

“Mistakes can be avoided and most errors cannot …”

From Surveying Measurements and Their Analysis, by Buckner

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Some Examples of Mistakes

Transposition of numbers Reading an angle counterclockwise, but indicating it as clockwise

in the field notes Sighting a wrong target Miscounting of full tape lengths “Cutting a foot” from a tape, and not factoring into recorded

measurement Misreading angular value on vernier scale of a transit or theodolite

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Finding Mistakes in the Field

Mistakes must be detected by careful and systematic checking of all work.

Repeating of measurements eliminates most mistakes It is difficult to detect small mistakes – when mistakes are not

detected, they will incorrectly be treated as errors.


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Systematic Errors

Conform to known mathematical and physical laws

The algebraic sign of the error is known, but the magnitude varies

Result due to a condition

Error will occur as long as the condition exists

Can be constant errors, or variable errors

Errors repeat – therefore, are accumulating

Can often be removed by computation

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Systematic Errors

Systematic errors are generally caused by:

(1) maladjustments of the instruments which can be traced to either misuse, normal wear, or manufacture of the equipment, or

(2) natural phenomena

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Some Examples of Systematic Errors

Incorrect length of a steel tape due to:‐ temperature‐ incorrect length‐ tension applied when measuring

Line of sight in telescope of level out of adjustment, causing inclined sights rather than horizontal

Error in horizontal angle measurement due to errors in centering the theodolite over the point or station

Error in setup of target for backsight or foresight for horizontal angle measurement

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Random Errors

Follow laws of probability and compensation

Are unavoidable

Can be minimized, but never eliminated

The algebraic sign of the error is never known (usually indicated as ±)

The magnitude of the error varies and is unknown

Errors tend to cancel, or compensate, but never totally cancel out

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Random Errors

From Surveying Measurements and Their Analysis, by Buckner, p. 99 19

Random Errors

• The probability is low that large errors will occur.

• It is more probable that small errors will occur than that large errors will occur.

• It is just as probable that an error will be positive than that the error will be negative.

20From Surveying Measurements and Their Analysis, by Buckner, p. 99

Some Examples of Random Errors

The inability of a person performing taping operations to accurately align the zero mark of the tape on a physical point or mark.

Errors that result from one attempting to interpolate to hundredths of a foot on a tape graduated only to tenths.

Errors in aligning the crosshair of a telescope to the center of a target, as in horizontal angle measurement.

Errors in reading the vernier scale of a transit of theodolite.

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Accuracy vs. Precision

Accuracy is the “agreement of the measurement or measurements with the true value. A value that is closer to the true value is, by definition, more accurate than one which is farther from the true.”

Precision is the “agreement among readings of the same quantity. If the readings are very close together in size, the set is more precise than if the values are widely scattered.”

From Surveying Measurements and Their Analysis, by Buckner, p. 29

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From Elementary Surveying, by Wolf and Ghilani

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Accuracy relates to how much attention has been paid to the detection and removal of systematic errors and mistakes.

Precision relates to the refinement in manufacture and the care and refinement in making measurements. Therefore, precision and random errors have a very close relationship.

From Surveying Measurements and Their Analysis, by Buckner, p. 29

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Direct vs. Indirect Measurements

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Random Error Theory

“The primary cause of random errors is the inability of the observer to make exact measurements. Although perhaps attempting to be exact, the observer will read, point, or otherwise respond somewhat to one side of the true value, sometimes to the other side, the errors varying in magnitude, and rarely if ever being zero. The algebraic sign of random errors is, by nature, plus or minus. Due to the accidental nature of random errors, small errors are more likely to occur than large ones.”

From Surveying Measurements and Their Analysis, by Buckner, p. 88‐89

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Averages

1. Mean ‐ or arithmetic mean ‐ sum of the individual observations divided by the number of repetitions. Sometimes referred to as most probable value.

2. Median ‐ the middle observation when the data are arranged in ascending or descending order, or the mean of the two middle observations if n is an even number and the two are different.

3. Mode ‐ the value which occurs most frequently.

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Example Problem 1: Averages

A distance is measured 9 times, and the results are 538.46’, 538.47’, 538.48’, 538.48’, 538.49’, 538.54’, 538.55’, 538.57’, and 538.59’. Determine the mean, the median, and the mode.

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Example Problem 1: Averages

a) The mean value would be 538.51’ –this is determined by dividing the sum of all 9 measurements (4,846.63’) by 9.

b) The median value would be 538.49’ –this is determined by listing the values in ascending or descending order, and determining the middle observation

c) The mode (occurs most frequently) would be 538.48’ (occurred twice)

(a) 538.51’ M

9 4,846.63 M

4,846.63

538.59'

538.57’

538.55’

538.54’

538.49’

538.48’

538.48’

538.47’

538.46’

} (c)

(b)

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Basic Probability and Statistics

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Probability

“A measure of how likely it is that some event will occur; a number expressing the ratio of favorable cases to the whole number of cases possible;”

wordnet.princeton.edu/perl/webwn

“The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.”

http://en.wikipedia.org/wiki/Classical_definition_of_probability

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Some Basic Probability Termsbased on Surveying Measurements and Their Analysis, by Buckner

Most Probable Value – also termed “best value”, and is typically represented by the arithmetic mean, since it is the best available representation of the true value (remember, we generally never know the true value of a measurement).

Residual – the difference between a particular reading in a set of repeated measurements of a quantity and the arithmetic mean (or most probable value).

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Most Probable Value and Residuals

True Value Most Probable Value

Error Residual (v)

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Most Probable Value and Residuals

Error = True Value ‐ Reading

v = Mean ‐ Reading

v = Most Probable Value – Reading

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Histogram – a bar graph of repeated observations of a quantity, the readings being plotted on the abscissa and the number of occurrences (frequency) of each value being plotted on the ordinate.

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Normal Distribution Curve – or “bell curve”, or “probability curve” – curve produced by connecting the top center points of each bar in the histogram – subject to “class interval” , or abscissa interval.

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Example Problem 2: Most Probable Value and Residuals

From Elementary Surveying an Introduction to Geomatics

by Wolf and Ghilani

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The following is taken from the data given and the textbook example:

The range of observations from smallest to largest (or dispersion) is 19.5” to 30.8”, which provides a difference of 11.3”

The residuals vary from +5.5” to ‐5.8” A class interval of 0.7” was chosen to prepare a

histogram


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39From Elementary Surveying, an Introduction to Geomatics by Wolf and Ghilani

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From Elementary Surveying an Introduction to Geomatics

by Wolf and Ghilani

Standard Deviation

The area under a probability curve between the plus and minus points of inflexion represents 68.3% of the total area under the curve. These two residual values are known as the standard deviation, , which is calculated from the equation:

1nΣvσ

2

Where

v = residualsn = number of observations

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Standard Deviation

42From Elementary Surveying, an Introduction to Geomatics by Wolf and Ghilani

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Levels of Certainty

“The or m, with their inherent 68.3% probability, are perhaps not the most useful precision indexes. No surveyor would risk the liability of measuring with only approximately 2/3rds confidence in uncertainty statements. If expensive structures or high land values are involved, a professional land surveyor might rest easier at 90% or even 99%. Percent probability will hereafter be called level of certainty. It represents a level or degree of confidence regarding an error or uncertainty statement.”

“Any desired level of certainty can be calculated by σ.”

(From Surveying Measurements and Their Analysis, by Buckner)

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Levels of Certainty

Name of Error Symbol Value % Certainty

Probable E50 0.6745 50

Standard Deviation 1 68.3

90% Error E90 1.6449 90

Two Sigma E95 2 95

99% Error E99 2.5 99

Three Sigma E99.7 3 99.7

44(From Surveying Measurements and Their Analysis, by Buckner)

Example Problem 3: Standard Deviation and Levels of Certainty

A line has been measured 9 times using the same equipment and procedures. The measured lengths, in ascending order, are 1114.21’, 1114.23’, 1114.23’ 1114.25’, 1114.26’, 1114.28’, 1114.28’, 1114.29’, and 1114.32’.

Determine the standard deviation, and the 90% error, 2 error, 99% error, and 3 error.

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2.19"σ

1100472.94σ

1nΣvσ

2

By examining the residuals of the observations, 70 of the 100 residuals, or 70%, are less than 2.19”.(From Elementary Surveying an Introduction to

Geomatics, by Wolf and Ghilani)

Back to Example Problem No. 2

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Random Error Propagation

A CB D E

A E

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Random Error Propagation ‐ General

The general theory of what we've gone over regarding probability in error analysis has focused on measurements of a single quantity. Even though numerous observations were included in a set being analyzed, the set still represented one quantity. In actual practice, most surveyed quantities, or measurements, are derived or computed from more than one single measured quantity. Actually, they are indirect measurements.

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Examples of Indirect Measurements

• Taping a distance is the sum of several tape lengths and a fractional length

• Halving a double angle involves a computation

• Area computations involve length and width

• Triangulation computations involve angle and distance components

(Based on Surveying Measurements and Their Analysis, by Buckner)

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Random Error Propagation is a term that describes the process of determining how random errors of several direct, single measurements accumulate, cancel, decrease, or otherwise behave in order to produce a random error of an overall (indirect) measurement that includes all of the direct, single measurements.

(Based on Surveying Measurements and Their Analysis, by Buckner)

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Error in a Sum

This formula computes the error of a sum of single quantities, each of which contain a different random error.

2n

23

22

21sum E....EEEE

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A CB D E

A E

(E1) (E2) (E3) (E4)

(Esum)

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Example Problem No. 45Error in a Sum

A length of 1200.00 feet is measured in four segments with varying random errors: 250.00’ ± 0.04’; 300.00’ ± 0.07’; 350.00’ ±0.02’; and 300.00’ ± 0.05’. Determine the total error (error of the sum) of the overall 1200‐foot length.

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Example Problem No. 4Error in a Sum

Distance = 1200.00’ ± 0.10’

Or 1199.90’ to 1200.10’

0.10'or0.097'E

0.050.020.070.04E

E....EEEE

sum

2222sum

2n

23

22

21sum

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Error in a Series

This formula computes the resulting error of a series of repeated measurements, each having about the same individual error. This formula is also called the law of compensation, and illustrates the principle that random errors tend to cancel, but never do so completely. In a series, they accumulate in proportion to the square root of the number of opportunities for their occurrence. Note how random errors tend to accumulate at a rate of n, while systematic errors tend to accumulate at a rate of n.

nEEseries

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A CB D E

A E

(E) (E) (E) (E)

(Eseries)

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Example Problem No. 5Error in a Series

A length of 1500.00 feet is measured with a 100‐foot steel tape. Each tape length contains a random error of ± 0.015’. Determine the total error (error of the series) of the overall 1500‐foot length.

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Example Problem No. 5 ‐ Error in a Series

Distance = 1500.00’ ± 0.06’or 1499.94’ to 1500.06’

0.06'or0.058'E

150.015'E

nEE

series

series

series

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Error in a Product

where Ea and Eb are the respective errors in A and B.

This best fits the error in computing the area of a rectangular parcel, with dimensions of A and B.

2a

22b

2product E BE AE

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Error in a Product

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Example Problem No. 6 ‐ Error in a Product

A parcel of land has dimensions of 1287.55 feet by 665.89 feet. The long dimensions contains an uncertainty of ±0.15’, and the short dimension contains an uncertainty of ±0.12’. Determine the expected error in the area of the parcel.

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Example Problem No. 6 ‐ Error in a Product

SF857,550.7toSF857,182.7orSF184.0SF857,366.7parcelofArea

SF184.0E

︶︵0.15︶︵665.89︶︵0.12︶︵1287.55E

E BE AE

SF857,366.7665.89'x1287.55'AreaParcel

product

2222product

2a

22b

2product

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Identifying Random Errors

As stated previously, random errors (associated with the observer) are caused by the observer’s inability to be exact – typically tied to the observer’s senses of sight and touch. Therefore, it should be understood that the value of a random error would be based on an estimate of the observers capabilities.

Random errors (associated with instrumentation) are typically based on information and specifications provided by the instrument manufacturer.

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Taping

• Reading and marking

• Temperature• Tension• Slope• Sag

EDM

• Prism Centering

• Manufacturer’s Accuracy Specifications

• Prism Constant (Systematic)

• Atmospheric Correction PPM (Systematic)

• Reading of Atmospheric Conditions

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Angles

• Focusing• Reading• Pointing• Instrument Centering

• Target Centering


Errors in GPS??

Random??? Systematic???

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(Images from lrsurveyors.com.au and searchpp.com)

Example Problem No. 7 ‐ Random Error ‐ Taping

A taping crew tapes a distance of 1475.50 feet, and it is estimated that they can read and mark each tape end to ±0.015 feet. Determine the uncertainty in this measurement.

15x2.ft015.0erL

.ft082.0erL

n2ee rLr

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Random Error – EDMManufacturer’s Accuracy Specification

This is an uncertainty factor published in the manuals and brochures for each EDM. Values vary between models. Usually, the error is stated as a constant plus or minus value plus an additional uncertainty which is relative to the distance. Generally speaking, the more expensive the instrument, the lower the error. Interpretation of this error also displays how shorter distances determined by EDM’s have lower precision than longer distances.

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Random Error – EDMManufacturer’s Accuracy Specification

Constant Error Variable Error

ppm3mm3Error

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(Images from jerrymahun.com)

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Example Problem No. 8 ‐ Random Error ‐ EDM

Two measurements are made with an EDM with a manufacturer’s accuracy specification of ± 3mm + 3ppm. The measurements are (a) 150 feet, and (b) 2500 feet. Determine the probable errors.

70(Images from jerrymahun.com)


ppm3mm3Error

1,000,000mm/ft304.8x150'x3mm3Error

ft. 0.0103mm3.1372Error

3ppm3mmError

1,000,000mm/ft304.8x2500'x3mm3Error

ft.0.0173mm5.2860Error 71

(a) 150’ measurement:

(b) 2,500’ measurement:


14,6001

ft.150ft.0.0103Precision

144,2001

ft.2500ft.0.0173Precision

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(a)

(b)

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Example Problem No. 9Random Error – Angles ‐ Pointing

An operator experiences an error due to pointing of ± 4.2”. If 4 repetitions of the angle are measured, determine the random error due to the pointing error.

(Note that this error will occur twice in a typical angle measurement –once with the backsight, and once with the foresight.)

srepetitionn"forσerrorpointing

todueangletheindeviationstandardtheisσ

n2σ

σ

p

pα

ppα

"

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Example Problem No. 9Random Error – Angles ‐ Pointing

2.97"σ pα

424.2"σ pα

n2σ

σ ppα

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Adjustments, Uncertainties and Tolerances

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Mistakes Uncertainties

Measure Random

Errors

SystematicErrors Tolerances

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The Role of the SurveyorThe problem of measurement generally reduces to a five‐step process:

1. Selection of the method or technique to use for aspecific measuring task,

2. derivation and adoption of specifications and standards,3. execution of the measurements and recording of

the data,4. employment of various checks and controls during

execution,5. analysis and reduction of data.


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Adjustments

Adjusting (or balancing) distributes closing errors throughout the survey so that lengths, directions, angles, heights, or positions have values as close to true as analysis of the random errors will permit. Adjustments relate to random error theory. The first basic assumption is that all systematic errors and mistakes have been detected and removed – only random errors should remain before adjusting.


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Adjustments

The second basic assumption, related to the first, is that the closure required by specifications must have been achieved before adjustments are made.

It is foolhardy to apply elaborate adjustment methods before closing errors are within tolerances required.


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Adjustments

A third assumption is that the field methods used were those on which the adjustment method is based. Sometimes, for example, an adjustment program is prepared based on the relative precision of distances and angles as achieved by a certain instrument combination. If the field work does not follow the planned techniques, the adjustment method may be invalid and result in less than the best possible values for the adjusted quantities.


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Simple Adjustments

In situations where measuring conditions are assumed to be equal, simple adjustments can be made. For example, angular error in a triangle can be distributed equally amongst all three angles. In an example problem, if three measured angles sum to 12 seconds short of 180o the deficiency can be divided by the number of angles (3), and each angle would be increased by 4 seconds to balance the error.

B

C

A

81(From Surveying Measurements and Their Analysis, by Buckner)

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Weighted Adjustments

Sometimes, conditions warrant that varying precisions exist in

measurements, which would cause the need for varying adjustments.

Weighting of corrections is based on the relative reliability among

measured quantities. Quantities with higher weights would receive

less adjustment. Conversely, quantities with lower weights would

receive more adjustment.


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Take the same triangle in the previous example, and apply weights to the three angles. The weight of the angle at A is 4, at B is 2, and at C is 1. This means that the angle at A is twice as reliable as the angle at B, and 4 times as reliable as the angle at C. Since adjustments are applied in inverse proportion to the weights, angle A would receive one‐fourth of the adjustment that is applied to C.

B

C

A

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In the example,

So, therefore, corrections are applied as:

1.7511

21

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W1

W1

W1

CBA

1.71"12"x1.750.25CA

3.43"12"x1.750.5CB

6.86"12"x1.75

1CC 84

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Traverse Adjustment Methods

Transit Rule

Compass Rule

Least Squares Method

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Least Squares Adjustment

“The method of least squares is based on the theory of probability, simultaneously adjusting the angular and linear measurements to make the sum of the squares of the residuals a minimum. This method is valid for any type of traverse survey, regardless of the relative precision of angle and distance measurements, since each measured quantity can be assigned a relative weight.”

(From http://www.surpac.com/refman/default/tutorials/surveying/adjustment_14.htm)

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Least Squares Adjustment

The advantage of the least squares method is that it gives the smallest possible changes to original observed values and affords a straightforward way to apply weights and hold certain conditions fixed. It can be applied to level circuits, fitting points to lines or curves, traverse and triangulation adjustments or any situation where there are redundancies and a means to express residuals in terms of the observations or geometric conditions. For more complex situations with a high number of redundancies, matrix algebra is a valuable tool in the solutions. Computers have made it possible and practical for surveyors to make least squares adjustments.

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Theoretical Uncertainty

“Theoretical uncertainty” refers to theoretical uncertainty of measurements.

“Theoretical uncertainty of measurements” means the radius of a circle which circumscribes an area which contains the probable true location of a specified point.

Per Title 865 IAC 1‐12 (Repealed May, 2006)

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“Theoretical Uncertainty” of a Line AB

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“A” “B”


(d) The following specifications shall be used for the location of property boundaries with respect to the referenced controlling corners:

Class of Survey Theoretical Uncertainty (tu)A plus or minus 0.10 feetB plus or minus 0.25 feetC plus or minus 0.50 feetD plus or minus 1.00 feet

E (all other surveys) to be negotiated with the client


90

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(d) Angular error under Equation I of theoretical uncertainty, shown as “AE”,is the dimension, measured perpendicularly to the left or right of theobserved line, which occurs from the random errors made in measuring anangle. It is dependent of the distance observed, the accuracy (least divisionof direct reading) of the instrument used, and the number of times that theangle was observed. Equation I is as follows:

AE = dist x sin (AVE)

Where: dist = distance observedAE = angular errorAVE = error (in seconds) as shown in Table I

Per Title 865 IAC 1‐12 (Repealed May, 2006)91

Theoretical UncertaintyTABLE I (AVE)

Least Division of Theodolite or Transit

# Pairs* 1 sec 6 sec 10 sec 20 sec 30 sec 60 sec

1 4.7 4.9 5.0 7.0 8.7 10.0

2 3.9 4.1 3.8 5.3 6.6 7.6

4 3.3 3.4 3.5 4.9 6.1 7.1

6 3.0 3.1 3.2 4.5 5.5 6.4

8 2.8 2.9 3.0 4.1 5.1 5.9

*Note: Number of pairs is the number of direct/reverse pairs of angles observed.


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Theoretical Uncertainty(e) Distance error (EDM) under Equation IIa of theoretical uncertainty, shown as DE(a), is the positive or negative error along the EDM observed line. It is dependent upon the accuracy of the instrument used and the distance measured. The distance error to be expected may be found in the operation manual of the instrument and shown as a standard deviation. Equation IIa is as follows:

DE(a) = A + (B x dist)

Where: DE(a) = distance error (from EDM) measurement

A and B = constants supplied by the manufacturer

dist = distance measured


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(f) Distance error (taping) under Equation IIb of theoretical uncertainty, shown as DE(b), is the positive or negative error along the taped line. An empirical form of that equation is as follows:

DE(b) = 0.01n + n/200

Where: DE(b) = distance error (from taping)n = number of full and partial tape lengths


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(g) Theoretical uncertainty (for one (1) point) underEquation III of theoretical uncertainty is as follows:

tu = AE or DE (a or b) (whichever is larger)

Where: tu = theoretical uncertainty of one (1) point

Per Title 865 IAC 1‐12 (May, 2006)

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Example Problem 10 – Theoretical Uncertainty

In completing a boundary survey, a measurement is made to a parcel corner which involves an angle measured with an instrument with a least division of 10 seconds (2 pairs of angles are turned), and an EDM with accuracy constants of ± 3mm + 3ppm. If the length of the line measured is 1550 feet, determine the theoretical uncertainty of the location of the parcel corner.

96

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AE = dist x sin (AVE)AE = 1550’ x sin 4.2”

AE = ± 0.032’

DE(a) = A + (B x dist)DE(a) = 3mm + (3 x 1550’ x 304.8)/1,000,000

DE(a) = ± 4.417 mm = ± 0.014’

therefore, tu = ± 0.032’


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(h) Theoretical uncertainty of any point of a string under Equation IV of theoretical uncertainty is as follows:

tu = tu21 + tu22 + ..... tu2n

Where: tu = theoretical uncertainty of the points in asurvey of (n) points


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In completing a boundary survey with 6 corners, the theoretical uncertainties of the individual parcel corners are 0.105’, 0.060’, 0.110’, 0.120’, 0.085’, and 0.155’ respectively. Determine the theoretical uncertainty of the string of points.

tu = tu21 + tu22 + ..... tu2n

tu =0.1052 + 0.0602 + 0.1102 + 0.1202 + 0.0852 + 0.1552

tu = ± 0.269’

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Relative Positional Accuracy

100

(Image from searchpp.com)


Title 865 IAC 1‐12 (Rule 12)

Minimum Standard Detail Requirements for ALTA/ACSM Land Title Surveys – 2005 Version

101


“Relative Positional Accuracy” means the value expressed in feet or meters that represents the uncertainty due to random errors in measurements in the location of any point on a survey relative to any other point on the same survey at the 95 percent confidence level.

(Per “Minimum Standard Detail Requirements for ALTA/ACSM Land Title Surveys” – 2005 Version)

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103


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Relative Positional Accuracy – Rule 12

Sec. 7. (a) The purpose of this section is to prescribe precision and accuracy standards to be used by a land surveyor in conducting original and retracement surveys and route surveys.(b) The land surveyor shall select the appropriate equipment and methods and use trained personnel to assure that the acceptable relative positional accuracy specified in this section is not exceeded.(c) The degree of precision and accuracy necessary for a survey shall be based upon the intended use of the real estate. If the client does not provide information regarding the intended use, the classification of the survey shall be based on the current use of the real estate.

105

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Relative Positional Accuracy – Rule 12(d) Classifications of surveys are as follows:

(1) Urban surveys. Urban surveys are performed on land lying within or contiguous with a city or town, except for single family residential lots. Urban surveys also include:(A) commercial and industrial properties;(B) condominiums;(C) townhouses;(D) apartments; and(E) other multiunit developments;regardless of geographic location.(2) Suburban surveys. Suburban surveys are performed on residential subdivisions lots. Surveys of single family residential lots shall be suburban surveys even if the lot is located in an urban or a rural area.(3) Rural surveys. Rural surveys are performed on real estate lying in rural areas that does not otherwise meet the definition of an urban or suburban survey.

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Relative Positional Accuracy – Rule 12

(e) The acceptable relative positional accuracies for each classification of survey are as follows:(1) Urban surveys: 0.07 feet (21 millimeters) plus 50 parts per million.(2) Suburban surveys: 0.13 feet (40 millimeters) plus 100 parts per million.(3) Rural surveys: 0.26 feet (79 millimeters) plus 200 parts per million.(f) Relative positional accuracy may be tested by:(1) comparing the relative location of points in a survey as measured by an independent survey of higher accuracy; or(2) the results of a minimally constrained, correctly weighted least square adjustment of the survey.

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Relative Positional Accuracy –ALTA/ACSM Standards

The lines and corners on any property survey have uncertainty in location which is the result of (1) availability and condition of reference monuments, (2) occupation or possession lines as they may differ from record lines, (3) clarity or ambiguity of the record descriptions or plats of the surveyed tracts and its adjoiners and (4) Relative Positional Accuracy.

The first three sources of uncertainty must be weighed as evidence in the determination of where, in the professional surveyor’s opinion, the boundary lines and corners should be placed. Relative Positional Accuracy is related to how accurately the surveyor is able to monument or report those positions.

(Per “Minimum Standard Detail Requirements for ALTA/ACSM Land Title Surveys” – 2005 Version)108

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Relative Positional Accuracy –ALTA Standards (2005)

Of these four sources of uncertainty, only Relative Positional Accuracy is controllable, although due to the inherent error in any measurement, it cannot be eliminated. The first three can be estimated based on evidence; Relative Positional Accuracy can be estimated using statistical means.


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Relative Positional Accuracy –ALTA Standards

The surveyor shall, to the extent necessary to achieve the standard contained herein, (1) compensate or correct for systematic errors, including those associated with instrument calibration, (2) select the appropriate equipment and methods, and use trained personnel and (3) use appropriate error propagation and other measurement design theory to select the proper instruments, field procedures, geometric layouts and computational procedures to control random errors.


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Relative Positional Accuracy –ALTA Standards

If radial survey methods, GPS or other acceptable technologies or procedures are used to locate or establish points on the survey, the surveyor shall apply appropriate procedures in order to assure that the allowable Relative Positional Accuracy of such points is not exceeded.


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Angle Measurement

Random Errors Distances (EDM)

Static (± 5mm + 1ppm)

GPS

RTK (± 10mm + 1ppm)

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Relative Positional Precision

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(Image from searchpp.com)


“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.

(Per “Minimum Standard Detail Requirements for ALTA/NSPS Land Title Surveys” – 2016 Version)

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The maximum allowable Relative Positional Precision for an ALTA/NSPS Land Title Survey is 2cm (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 of 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 not e the reason as explained in Section 6.B.x below.


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Section 6. Plat or Map – A plat or map of an ALTA/NSPS Land Title Survey shall show the following information. …..

B. Boundary, Descriptions, Dimensions, and Closures

x. A note on the face of the plat or map explaining the site conditions that resulted in a Relative Positional Precision that exceeds the maximum allowed pursuant to Section 3.E.v.


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Thank You!!!

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Professor Tony Gregory, PLSPurdue University CalumetDepartment of Construction Scienceand Organizational Leadership

[email protected]

error analysis relative to positional precision - handout ...c.ymcdn.com/sites/ · pdf...

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