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Analyzing Positional Accuracy

Charles “Chuck” Ghilani, Ph.D.

Professor Emeritus in Surveying Engineering

Penn State University

1

Class Etiquette

• Turn off all cell phones

– Or set them to vibrate

– Go outside the room to answer any calls

• Ask questions at any point during the class

– Simply speak up so that all can hear your question

2

Course Description

This workshop will present

1. Explore the relative positional accuracy requirements in 2016

ALTA/NSPS surveys

2. How to properly weight a least squares adjustment having optical

observations

3. How error ellipses are used to determine relative positional

accuracies

4. Field procedures that help ensure meeting the 2011 ALTA/NSPS

relative positional accuracy standards

5. Time permitting: Instrument calibration

3

Measurement Details for 2016 ALTA/NSPS

Land Title Surveys

4

i. “Relative Positional Precision” means the length of the semi-major

axis, expressed in feet or meters, of the error ellipse representing

the uncertainty due to random errors in measurements in the

location of the monument, or witness, marking any corner of the

surveyed property relative to the monument, or witness, marking

any other corner of the surveyed property at the 95 percent

confidence level. Relative Positional Precision is estimated by the

results of a correctly weighted least squares adjustment of the

survey.


Land Title Surveys

ii. Any boundary lines and corners established or retraced may have uncertainties

in location resulting from (1) the availability, condition, history and integrity of

reference or controlling monuments, (2) ambiguities in the record descriptions or

plats of the surveyed property or its adjoiners, (3) occupation or possession

lines as they may differ from the written title lines, or (4) Relative Positional

Precision. Of these four sources of uncertainty, only Relative Positional

Precision is controllable, although, due to the inherent errors in any

measurement, it cannot be eliminated. The magnitude of the first three

uncertainties can be projected based on evidence; Relative Positional Precision

is estimated using statistical means (see Section 3.E.i. above and Section 3.E.v.

below).

5


Land Title Surveys

6

iii. The first three of these sources of uncertainty must be weighed as part of the

evidence in the determination of where, in the surveyor’s opinion, the

boundary lines and corners of the surveyed property should be located (see

Section 3.D. above). Relative Positional Precision is a measure of how

precisely the surveyor is able to monument and report those positions; it is not

a substitute for the application of proper boundary law principles. A boundary

corner or line may have a small Relative Positional Precision because the

survey measurements were precise, yet still be in the wrong position (i.e.,

inaccurate) if it was established or retraced using faulty or improper

application of boundary law principles.

Meaning: Don’t put

precise coordinates

on an incorrect

corner.


Land Title Surveys

7

iv. For any measurement technology or procedure used on an

ALTA/NSPS Land Title Survey, the surveyor shall (1) use

appropriately trained personnel, (2) compensate for systematic

errors, including those associated with instrument calibration, and

(3) use appropriate error propagation and measurement design

theory (selecting the proper instruments, geometric layouts, and

field and computational procedures) to control random errors such

that the maximum allowable Relative Positional Precision outlined

in Section 3.E.v. below is not exceeded.


Land Title Surveys

8

v. The maximum allowable Relative Positional Precision for an

ALTA/NSPS Land Title Survey is 2 cm (0.07 feet) plus 50 parts per

million (based on the direct distance between the two corners being

tested). It is recognized that in certain circumstances, the size or

configuration of the surveyed property, or the relief, vegetation, or

improvements on the surveyed property, will result in survey

measurements for which the maximum allowable Relative Positional

Precision may be exceeded. If the maximum allowable Relative

Positional Precision is exceeded, the surveyor shall note the reason

as explained in Section 6.B.x. below.

Review

• Relative Positional Accuracy is one of four criteria that should be

met

• Does not substitute for applying proper boundary law principles

• Based on a 95% semi-major axis from a correctly weighted least

squares adjustment

• Based on a distance between two points being test?????

9

Conflicting information

• An error ellipse is the positional accuracy of a point

– All least squares software can compute this

• Distance accuracy between two points determines the

standard deviation for the distance

– Determined by error propagation principles

– It can be determined for any distance, observed or not, in

project

– Technically determined by error propagation

– Not aware of any software that does this between any/all points

10

Least Squares Adjustment

• Least squares adjustments are

– Simultaneous solution of observation equations to solve for the

unknown station coordinates

• Based on the principles of normally distributed data

– Except for how the equations are formed, no different than

what you did in your high-school algebra class

11

Fundamental Principle of Least

Squares

The most probable value for a measured

quantity is the value that renders the sum of the

squared residuals a minimum

12


• Advantages are

– All observations are used in the adjustment

– Each observation can be individually weighted to match

the estimated error in the observation

13


• Error ellipses are statistically determined errors of the

unknown station coordinates

– The computation of a 95% error ellipse is based on the F

distribution and the number of redundant observations in the

adjustment

• But the software does all of this!

14

Skipping all the equations for now!

15

So What is a Least Squares Adjustment?

• So we have a set of equations in either 2D or 3D that

represent our observations

• Least squares is a method of simultaneously solving for

the unknowns parameters in the equations such that we

get the most probable values for the unknown

parameters

– An option in your software

16

What Does This All Mean?

• You must perform a least squares adjustment on

observations

• The adjustment must be correctly weighted

• The size of the semi-major axis of a 95% error

ellipse for any point must be less than 0.07 ft + 50

ppm

– Since typical distance observations are so short, the 50

ppm is generally irrelevant17

What this means

• You are only allowed this much error

between any two stations

• Example: Assume property shown to right

– Between stations 1 and 4

– Between 2 and 6, and so on

• Uncertainty in coordinates increases as you

move farther from the initial computational

station

1

2

34

56

7

8

19

Error Propagation

Errors propagate

• Latitudes and departures from angle and distance

observations

– Computation of area/distance/directions from coordinates

20

Traverse Example

N

A

B

C

D

And then we adjust the traverse, which

drives errors away from the control

These errors then propagate to

coordinates for stations,

adjusted distances and azimuths,

and areas

Thus the largest error is always between the control station

and station farthest (in stations) from the control station21

First course

• Uncertainty in distance and

azimuth

Second and following courses

• Uncertainty in distance, angle, and

coordinates

Maximum Allowable Error

• Based on the sum of two random errors

– 0.07 ft

– 50 ppm

• Computed using equation

𝑆 = 0.072 +50

106𝐷

2

– where D is the distance in feet between “any” two stations

22

Maximum Allowable Errors

Distance Between Stations (ft) Maximum Size of Semimajor Axis

100 ±0.070 ft

300 ±0.072 ft

500 ±0.074 ft

700 ±0.078 ft

900 ±0.083 ft

1100 ±0.089 ft

1300 ±0.096 ft

1500 ±0.103 ft

1700 ±0.110 ft

1900 ±0.118 ft

23

Course Description



ALTA-NSPS surveys

2. How to properly weight a least squares adjustment


accuracies

4. Field procedures that help ensure meeting the 2011 ALTA-NSPS


24

Last Things First!

• The final size of the semi-major axis of a 95%

error ellipse is dependent on

1. The geometry of the survey

2. Precision of the instruments used in the survey

3. Abilities and knowledge of the field personnel

4. Weights of the observations are as individual as the

observations themselves

25

Last Things First!

• Guidelines to achieve the required relative precision in

guidelines

1. Use a theodolite with a ISO 17123-3 standard of 5″ or better and an

EDM that is 3 mm + 3 ppm or better

2. Calibrate all equipment before the survey

1. This includes levels and plummets

3. Be sure that field personnel know how to set an instrument or target

correctly over a point

4. Targets must be placed in tribrachs on tripods or on rods supported

with bipod or tripods

26

Last Things First!

4. Angel-face targets must be used when observing angles

5. Properly align the prism with the line of sight

6. Distances should be observed at both ends of the line and not

averaged

7. Close angular horizons at all stations

8. Preplan the survey to

1. Maximize sight distances

2. Minimize number of stations

27

Relative Positional Precision

“Relative Positional Precision” means the length of the semimajor

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.

28

Course Description



ALTA-NSPS surveys



accuracies

4. Field procedures that help ensure meeting the 2011 ALTA-

NSPS relative positional accuracy standards

29

Correct Weights

• Standards require a “correctly weighted least squares

adjustment”

– Weights for independent observations (optical obervations not

GNSS) computed as

𝑤𝑖 =𝜎02

𝜎𝑖2 =

1

𝜎𝑖2

where

• wi is the weight for the observation

• 𝜎𝑖2 is the observation’s standard error

30

Correct Weights

• Standard error is the uncertainty in a population of data

• But we only collect a small sample; for example 2 or 4

repeated angles or 1 or 2 distance observations

– Pushing the measure button 3x is not really an independent

observation

• Same setup errors for instrument and target

– Measure distances from both sides of line

31

Population

• Consists of all possible observations that can be made on

a particular item

– It usually can not be observed

– Often a population has an infinite number of observations

• The mean and variance of a population are the true

values

32

Sample

• Subset of data collected from a population

– It can be collected in an economical fashion

– May or may not be representative of a population

– Size of subset influences its ability to predict true values of a population

• Larger samples generally have a higher probability of predicting true values

33

Correct Weights

• Typical least squares software uses the standard

deviation computed from the repeated observations

collected in the field

• Standard deviation, S, computed from repeated

observations in the field

• Sample standard deviations are estimates for standard errors

– But how good of an estimate?

34

Sample Versus Population

• Assume we have a population of 100 elements18.2 26.4 20.1 29.9 29.8 26.6 26.2 25.7 25.2 26.3 26.7 30.6 22.6 22.330.0 26.5 28.1 25.6 20.3 35.5 22.9 30.7 32.2 22.2 29.2 26.1 26.8 25.324.3 24.4 29.0 25.0 29.9 25.2 20.8 29.0 21.9 25.4 27.3 23.4 38.2 22.628.0 24.0 19.4 27.0 32.0 27.3 15.3 26.5 31.5 28.0 22.4 23.4 21.2 27.727.1 27.0 25.2 24.0 24.5 23.8 28.2 26.8 27.7 39.8 19.8 29.3 28.5 24.722.0 18.4 26.4 24.2 29.9 21.8 36.0 21.3 28.8 22.8 28.5 30.9 19.1 28.130.3 26.5 26.9 26.6 28.2 24.2 25.5 30.2 18.9 28.9 27.6 19.6 27.9 24.921.3 26.7

• where

– Mean, μ = 26.1

– Population variance, σ2 = 17.5

– Standard error, σ = ±4.2

35

Random Samples of 10 Elements

Sample Mean Variance

Population 26.1 17.5

2 26.2 36.3

1 25.9 18.4

3 26.0 19.4

4 23.8 22.0

5 24.5 10.3

6 26.6 27.3

7 26.8 8.6

8 24.0 19.5

Remember: These are only estimates for the population values.

36

Population versus Sample

• Note that

– No sample of data has the population mean (26.1)

– No sample of data has the population variance (17.5)

– The most precise set of data does not have a mean that is

closest to that of the population

37


• To continue the experiment, the sample sizes were increased by 10 and sample means and variances computed. (Recall pop. Mean = 26.1 and variance = 17.5)

# of Elements Mean Variance

10 26.9 28.1

20 25.9 21.9

30 25.9 20.0

40 26.5 18.6

50 26.6 20.0

60 26.4 17.6

70 26.3 17.1

80 26.3 18.4

90 26.3 17.8 38


• Note how sample mean and variance approach

population values as sample size is increased

• This leads to a basic principle

– More confidence can be placed in sampling statistics as size

samples increases

• We can create ranges (confidence intervals) for sample

sizes that vary with the size of the sample

39

χ2 Distribution

• Distribution used to create a confidence interval for the population variance based on the sample variance S2

and number of redundant measurements, v– Note how distribution starts at 0 and goes to ∞.

– It is not symmetric

– Typically upper-tail areas (α) are tabulated in tables

– Used in the “Goodness of Fit Test” after a least squares adjustment

22

2

vS

40

χ2 Distribution

• In Excel

– Use “=chisq.inv(% pts,v)” for the left critical value• Example: =chisq.inv(0.05,10) = 3.94 where α = 0.05

– Use “=chisq.inv.rt(% pts,v)” for the right critical value• Example: =chisq.inv.rt(0.05,10) = 18.31

• Note that 90% is between the critical values of 3.94 and 18.31

Critical

values

41

Correct Weights

• Statistically stated our sample size is too small to reliabily

provide a good estimate for σ2

• χ2 provides a range for σ2 based on number of repetitions

and S2

– Where S is the sample standard deviation

42

Correct Weights

• From χ2 distribution the range for the standard error, σ2 is

computed as

𝑟𝑆2

χ𝛼2,𝑟

2 < 𝜎2 <𝑟𝑆2

χ1−

𝛼2,𝑟

2

– where r is the number of redundant observations

– α is the level of significance of the range, or (1 − α)% is the

probability

43

Redundancies

• Also called degrees of freedom and redundant

observations

• The number of observations in excess of the number

necessary to solve for the unknown(s)

• How many redundancies are there if a distance is

observed three times?

– 2

– But did you simply push the button three times?

44

How Many Degrees of Freedom?

A

B

C

DE

(N, E)A and AzAB are given

What observations required to compute

B?

C?

D?

E?

Redundant observations? 3

45

Correct Weights

• Example

– Assume an angle observed with 2 repeated

measurements has S of ±3.8″

Question: How good of an estimate of S = ±3.8″ for σ

when we have only 2 repetitions?

46

Correct Weights

• What is the range for the σ at 68% (α = 0.32)?

• To center our the range in the χ2 distribution, we need x-axis values

at half of α or α/2 = 0.16

• χ0.16,12 = 1.974; χ1−0.16,1

2 = χ0.84,12 = 0.041

68%

=16%

Critical values

47

Correct Weights

• What is the range for the σ at 68% (α = 0.32)?

– χ0.16,12 = 1.974; χ1−0.16,1

2 = χ0.84,12 = 0.041

• Variance range

1 3.82

1.974= 𝟕. 𝟑 < σ2 < 𝟑𝟓𝟒. 𝟑 =

1 3.82

0.041

• Standard error range

– So ±2.7″ < σ < ±18.8″

48

Correct Weights

• This means the correct weight is somewhere between

0.137 =1

2.72> 𝑤𝑖 >

1

18.82= 0.0028

• Computations usually done in radians so

450 > 𝑤𝑖 > 9.3

• Applying weights varying between 9.3 and 450 will result in varied

solutions!

49

Correct Weights

• Statistically stated our sample size is too small to provide

a good estimate for σ2

• What is a valid number of reps?

• Statisticians say more than 30!

• Too many to be profitable!

– But this may be why NGS standards required 16 DR for angles

in a 1st order survey

50

Correct Weights

• Must base weights from theory of random error

propagation and estimate σ

• This means that weights of observations are based on

– Instrumentation used in survey

– Field procedures

– Capability of field personnel

51

Uncertainty in Distance Observations

• What contributes to the uncertainty in a distance

observation?

– Manufacturer specifications for instrument

• a + b ppm

– where

» a is constant error

» b is the scalar error based on the length of the line

• Reflector setup error, σr

• Instrument setup error, σi

52

Uncertainty in Distance

• Errors are summed by squaring them, adding, and then taking a

square root of the sum

• Estimated error in distance computed as

𝜎 = 𝜎𝑖2 + 𝜎𝑟

2 + 𝑎2 +𝑏

106𝐷

2

– where D is the length of the line

• Note that handheld reflectors will greatly increase the uncertainty in an

observed distance due to σr

53

Set Up Error Analysis

• Setup error

– Well-defined monument has punch mark that

is about 2 mm in width

– Centering error (ep) ±1 mm or ~0.003 ft

• Leveling error

– Typical circular bubble is 40′

– Horizontal positioning error

σ = ℎ𝑟 × 𝜃; where

• θ in radians is leveling error

• hr is height of receiver

e

54

Analysis of Setup Error

• Centering in point

• Well-defined point has 2-mm

• Assume 1/2 of width of point for

setup error

• Handheld rods not included in

analysis

• Circular vial calibration can greatly

increase/decrease these values

Size of point

(mm)

Centering error

(ft)

1 0.002

2 0.003

3 0.005

4 0.007

5 0.008

6 0.010

For well defined points!

55

So How Well Did You Level?

• Bubble level app on tablet

• You think you are centered?

• At 2 m that is 0.046 ft in x and 0.023 ft

in y

• Or 0.05 ft off center! xx=0.4° y=0.2°

56

Analysis of Setup Errors for Reflectors

• Circular level bubbles are available as

40′, 20′, and 8′ (SECO)

• At 5.5 ft error caused by misleveling

of rod/tripod is shown in table to right

– S = Rθ; θ in radians

– σ = 5.5Div Bub″

206,264.8

Div 40′ 20′ 8′

0.1 0.006 0.003 0.001

0.2 0.019 0.010 0.004

0.3 0.032 0.016 0.006

0.4 0.045 0.022 0.009

0.5 0.058 0.029 0.012

All units in feet

57

Number of seconds in 1 radian

Analysis of Setup Errors for Reflectors

• Again assuming a 5.5 ft rod height

and a centering error of 1 mm or

0.003 ft

• Combining errors: 𝑒𝑠 = 𝑒𝑝2 + 𝑒𝑏

2

• Combined setup error for rod, σr,

shown in Table

Circular Bubble Accuracy

Div 40′ 20′ 8′

0.1 0.007 0.005 0.004

0.2 0.019 0.010 0.005

0.3 0.032 0.016 0.007

0.4 0.045 0.023 0.010

0.5 0.058 0.029 0.012

All units in feet

58

Analysis of GNSS Setup Errors

• Example:

– Rod height of 2 m or ~6.562 ft

– 40′ circular vial

• 𝑒𝑏 = 6.562div(40′)(60″)

206264.8ft

– Where div estimated division in leveling

error

• Best horizontal positions obtained with lower

setups

– Must avoid obstructions and

– be 1.5 ft above ground to avoid microclimate, which

can cause bigger errors

Misleveling (div)

for 40′ vial

Centering Error

(eb) at 2.0 m

0.1 0.008 ft (2.3 mm)

0.2 0.015 ft (4.7 mm)

0.3 0.023 ft (7.0 mm)

0.4 0.031 ft (9.3 mm)

0.5 0.038 ft (11.6 mm)

Note: This assumes a calibrated plummet

Note: Fixed height tripods

have 8′ vials typically

59

Analysis of GNSS Setup Errors

• Combining errors

– 𝑒𝑠 = 𝑒𝑝2 + 𝑒𝑏

2

• Example:

– Assume

• 1 mm centering error (ep)

• 0.3 div leveling error (eb) with 2.0 m setup

– Error in setup is

– 𝑒𝑠 = 12 + 7.02 = ±7.1 mm = ±0.023 ft

hr →

div↓

2.0 m 1.8 m 1.5 m

0.1 2.5 mm

(0.008 ft)

2.3 mm

(0.008 ft)

2.0 mm

(0.007 ft)

0.2 4.8 mm

(0.016 ft)

4.3 mm

(0.014 ft)

3.6 mm

(0.012 ft)

0.3 7.1 mm

(0.023 ft)

6.4 mm

(0.021 ft)

5.3 mm

(0.017 ft)

0.4 9.4 mm

(0.031 ft)

8.4 mm

(0.028 ft)

7.1 mm

(0.023 ft)

0.5 11.7 mm

(0.038 ft)

10.5 mm

(0.034 ft)

8.8 mm

(0.029 ft)

Centering error at varied heights, hr

60

Analysis of Total Station Setup Errors

• Instrument

– Assume

• 30″ bubble

• 5.5 ft height

– Error computed with S = Rθ

𝑆 = 5.5div ∗ 30"

206264.8

Misleveling (div)

for 30″ vial

Centering Error

at 5.5 ft

0.1 0.0001

0.3 0.0002

0.5 0.0004

0.7 0.0006

0.9 0.0007

61

Analysis of Total Station Setup Errors

Misleveling→

Centering ↓

0.1 0.2 0.3 0.4 0.5

0.002 0.002 0.002 0.002 0.002 0.002

0.003 0.003 0.003 0.003 0.003 0.003

0.005 0.005 0.005 0.005 0.005 0.005

0.007 0.007 0.007 0.007 0.007 0.007

0.008 0.008 0.008 0.008 0.008 0.008

0.010 0.010 0.010 0.010 0.010 0.010

All units are in feet

Misleveling error in units of fractional divisions with 30″ bubble.

Assumes bubble and plummet are calibrated!

Instrument at 5.5 ft

62

Analysis of Distance Errors

• Example

– Manufacturer specifications of 3 mm (~0.01 ft) + 3 ppm

assumed

– Using σi = ±0.003 ft; σr = ±0.023 ft

– 100-ft length computed as

𝜎 = 0.0032 + 0.0232 + 0.012 +3 100

1,000,000

2

= ±0.025 ft

63

Errors in Horizontal Angles

• Major random error sources

– Reading the circles

– Pointing at the target

– Miscentering of the instrument

– Miscentering of the target

– Misleveling of the instrument

Estimated with ISO 17123-3

standard

64

Errors in Horizontal Angles

• Total station instruments

– Report pointing and reading error using the ISO 17123-3

standard

• Standard based on the average of a single observation with both faces

of the instrument

• Provides an estimated single pointing and reading error under ideal

conditions

• Error in reading an angle n times is

𝜎𝛼𝑝𝑟 =2𝜎𝐼𝑆𝑂

𝑛65

Example

• An angle is observed 2 times (1DR) by an operator with a

total station instrument having a published ISO value of

±3″. What is the estimated error in the angle due to the

pointing and reading error?

𝜎𝛼𝑝𝑟 =2𝜎𝐼𝑆𝑂

𝑛=2 3″

2= ±4.2″

66

How Setup Errors Affect Coordinates

• Instrument setup

• Foresight direction and distance

• Backsight direction and distance

• End result is multiple locations

for station– Bottom line is that although each error

may be systematic, they appear to be

random in an adjustment

67

Sighting Errors

• Never sight your retro-reflector for an angle

– Causes line of sight to shift if reflector not perpendicular to

line of sight

– Sight the angle face or a chaining pin, supported plumb bob

string, etc.

– Never have a line of sight within 1.5 ft of surface due to

microclimate caused by surface

68

Sighting Errors

• Always sight the instrument from the reflector and do not

use the 0-offset except for mapping

69

Target Centering Errors

• Miscentering of target causes a different angle to be

observed

– If target is handheld, this is a random error in each pointing

– If targets are fixed, then it is a systematic error for the

particular angle, but will appear as a random error in a

resurvey and adjustment

eb

ef70


• There is no error in the angle if the target is on line with

the station shown in (a)

σd

(a)

D

71


• Error increases, (b) and (c), until it hits a maximum (d)

when the miscentering error is perpendicular to the line of

sight

σd

(b)

e

D

σd

(c)

e

D

σd

(d)

e

D

72

Target Centering Error

• For a single pointing the error in the direction is

𝑒 = ±𝜎𝑑𝐷

rad

– where

• σd is the estimated miscentering error

• D is the distance to the target

• e is the error in the angle computed in radians

e σd

D

73


• For an angle

– Must consider both backsight and foresight distances

eBSDBS

DFS

eFS

σd

σd

74


• Applying error in a sum for an angle yields

𝜎𝛼𝑡 =𝜎𝑑𝐵𝑆𝐷𝐵𝑆

2

+𝜎𝑑𝐹𝑆𝐷𝐹𝑆

2

• where

– DBS and DFS are the backsight and foresight distances, respectively

• Squaring both sides and rearranging

𝜎𝛼𝑡2 =

𝜎𝑑𝐵𝑆2

𝐷𝐵𝑆2 +

𝜎𝑑𝐹𝑆2

𝐷𝐹𝑆2

𝐷𝐹𝑆2 𝜎𝑑𝐵𝑆

2 + 𝐷𝐵𝑆2 𝜎𝑑𝐹𝑆

2

𝐷𝐵𝑆2 𝐷𝐹𝑆

2

75


• Assuming that the target miscentering error, σt, is the same for

both sights, then

𝜎𝛼𝑡″ = ±

𝐷𝐵𝑆2 + 𝐷𝐹𝑆

2

𝐷𝐵𝑆𝐷𝐹𝑆𝜎𝑡𝜌

– where

• DBS is the length of the backsight

• DFS is the length of the foresight

• σt is the estimated error in the target setup

• ρ is a units conversion from radians to seconds, 206,264.8″/rad

76

Example

• An observer’s estimated ability at centering targets over a station

is ±0.016 ft. For a particular angle observation, the backsight and

foresight distances from the instrument station to the targets are

approximately 250 ft and 450 ft, respectively.

• What is the angular error due to the error in target miscentering?

𝜎𝛼𝑡″ = ±

𝐷𝐵𝑆2 + 𝐷𝐹𝑆

2


𝜎𝛼𝑡″ = ±

2502 + 4502

250 4500.016(206,264.8"/rad) = ±15.1″

77

Example (cont.)

• What if targets are hand-held?

– Assume a centering error of σt = ±0.032 ft

• 40′ vial within 0.3 divisions

• Solution?

• 𝜎𝛼𝑡″ = ±

𝐷𝐵𝑆2 +𝐷𝐹𝑆

2


• 𝜎𝛼𝑡″ = ±

2502+4502

250 4500.032(206,264.8"/rad) = ±30.2″

78

Using Sight Lengths

• Assuming

– Equal sight lengths, BS = FS (ft)

– Varying target setup errors (ft)

– 8′ circular bubble

↓Dist/σt → ±0.005 ±0.010 ±0.016 ±0.023 ±0.029

50 ±29.2″ ±58.3″ ±93.3″ ±134.2″ ±169.2″

100 ±14.6″ ±29.2″ ±46.7″ ±67.1″ ±84.6″

200 ±7.3″ ±14.6″ ±23.3″ ±33.5″ ±42.3″

300 ±4.9″ ±9.7″ ±15.6″ ±22.4″ ±28.2″

400 ±3.6″ ±7.3″ ±11.7″ ±16.8″ ±21.1″

500 ±2.9″ ±5.8″ ±9.3″ ±13.4″ ±16.9″

1000 ±1.5″ ±2.9″ ±4.7″ ±6.7″ ±8.5″79

±0.0 "

±10.0 "

±20.0 "

±30.0 "

±40.0 "

±50.0 "

±60.0 "

±70.0 "

±80.0 "

50 100 200 300 400 500 1000

Angular Errors Due to Various Target Setup Errors and Sight Distances

0.009 ft 0.015 ft 0.023 ft 0.031 ft 0.038 ft

(feet)80Assumes 8′ vial centered to within 0.3 divisions

Using Sight Lengths

• Assuming

– Equal sight lengths, BS = FS (ft)

– Varying target setup errors (ft)

– 8′ circular bubble

↓Dist/σt → ±0.004 ±0.005 ±0.007 ±0.010 ±0.012

50 ±23.3″ ±29.2″ ±40.8″ ±58.3″ ±70.0″

100 ±11.7″ ±14.6″ ±20.4″ ±29.2″ ±35.0″

200 ±5.8″ ±7.3″ ±10.2″ ±14.6″ ±17.5″

300 ±3.9″ ±4.9″ ±6.8″ ±9.7″ ±11.7″

400 ±2.9″ ±3.6″ ±5.1″ ±7.3″ ±8.8″

500 ±2.3″ ±2.9″ ±4.1″ ±5.8″ ±7.0″

1000 ±1.2″ ±1.5″ ±2.0″ ±2.9″ ±3.5″81

Ce

nte

red

to

with

in 1

.5 d

iv

±0.0 "

±10.0 "

±20.0 "

±30.0 "

±40.0 "

±50.0 "

±60.0 "

±70.0 "

±80.0 "

50 100 200 300 400 500 1000

Angular Errors Due to Various Target Setup Errors and Sight Distances

0.009 ft 0.015 ft 0.023 ft 0.031 ft 0.038 ft

(feet)

82

Best Practices

• Keep sight distances as long as possible

• Except for mapping and stakeout always use a tripod

or bipod setup

– Use a 20′ or 8′ circular bubble

• Check and maintain calibration on level bubbles often

• Replace worn tips on rods

• Note that improvements in accuracy start to level off at

sight distance of 300 ft

83

Instrument Miscentering Error• This error is caused by not placing the instrument exactly over

the station.

– Systematic error that will appear as a random error in a resurvey

– Also appears as a random error in computed quantities

– Errors in each direction may cancel or create some combined error

2

1

α

σi

(a) P2

P1

1

σi

(b)

α

2

(c)

1

α

2

σ i

P1

P2

P1

P2

84

Estimated Error in Angle

Due to Instrument Miscentering

𝜎𝛼𝑖″ = ±

𝐷3𝐷𝐵𝑆𝐷𝐹𝑆

𝜎𝑖

2𝜌

where

– D3 is the distance between the backsight and foresight stations

– DBS is the backsight distance

– DFS is the foresight distance

– σi is the estimated miscentering error

– σα is the error in the angle in units of seconds

– ρ is the units conversion from radians to seconds, 206,264.8″/rad85

Example• An observer centers the instrument to within ±0.005 ft of a station

for an angle with backsight and foresight distances of 250 ft and

450 ft, respectively. The angle observed is 50°.

• What is the error in the angle due to the instrument centering

error?

– Distance between backsight and foresight stations

𝐷3 = 2502 + 4502 − 2 250 450 cos 50° = 346.95 ft

86

Example

• The equation is

𝜎𝛼𝑖″ = ±

𝐷3𝐷𝐵𝑆𝐷𝐹𝑆

σ𝑖

2𝜌

• Substituting in the appropriate values

𝜎𝛼𝑖″ = ±

346.95

250 450

0.005

2206,264.8 ൗ″ 𝑟𝑎𝑑 = ±2.2″

87

Angular Errors Due to Various

Instrument Setup Errors and Sight Distances

• Assuming that

– BS distance = FS distance

– Angle of 90°

– All length units in ft

↓Dist/Si → ±0.002 ±0.004 ±0.006 ±0.008 ±0.010

50 ±4.5″ ±8.9″ ±13.4″ ±17.9″ ±22.3″

100 ±2.2″ ±4.5″ ±6.7″ ±8.9″ ±11.2″

200 ±1.1″ ±2.2″ ±3.3″ ±4.5″ ±5.6″

300 ±0.7″ ±1.5″ ±2.2″ ±3.0″ ±3.7″

400 ±0.6″ ±1.1″ ±1.7″ ±2.2″ ±2.8″

500 ±0.4″ ±0.9″ ±1.3″ ±1.8″ ±2.2″

600 ±0.4″ ±0.7″ ±1.1″ ±1.5″ ±1.9″88

±0.0"

±5.0"

±10.0"

±15.0"

±20.0"

±25.0"

50 100 200 300 400 500 600

Angular Errors Due to Various Instrument Setup Errors and Sight Distances

±0.002 ±0.004 ±0.006 ±0.008 ±0.010 89

Best Practices

• Always maintain calibration on levels and optical

plummets

• Make sure that your setup is solid

• Accuracy gains begin to level off at sight distance of 300 ft

90

Estimated Error in an Angle• What is the estimated overall error in the angle observation?

• Using

– Error due to pointing and reading = ±2.4″

– Error due to target miscentering = ±4.7″

– Error due to instrument miscentering = ±2.2″

• Overall error is determined by taking the square root of the sum

of the squared errors or

σ𝑎 = 2.42 + 4.72 + 2.22 = ±5.7″

• And its estimated weight would be 𝑤 =1

5.72= 0.0306

91

Side Note

• Since we seldom set up twice on a station, we seldom

see the setup errors in angular misclosure

• Thus we would see an error of ±2.4″ at the station in the

angular misclosure

• However the larger error of ±5.7″ would be seen on a

resurvey or in an adjustment

– Explains why computed bearings often differ from one survey

to the next

92

Error in Horizontal Angles Due to Instrument

Misleveling

• When instrument is not level, scope does

not plunge along vertical axis.

• Error is only significant when sighting with

large vertical angles

– Explains why traverses that go up and down

steep hillsides do not close as well as ones on

reasonably level ground and why instrument

leveling is critical in astronomic observations

fd μ

I

D

S

PP

v

93

Instrument Misleveling

• Since the backsight and foresight elevations are typically different

𝜎𝛼𝑙 =𝑓𝑑𝜇 tan 𝑣𝐵𝑆

2 + 𝑓𝑑𝜇 tan 𝑣𝐹𝑆2

𝑛

= 𝑓𝑑𝜇tan2 𝑣𝐵𝑆+tan

2 𝑣_𝐹𝑆

𝑛

• where – vBS and vFS are the backsight and foresight altitude angles,

respectively

– n is the number of angle repetitions.

– Units of 𝜎𝛼𝑙 in the same units as sensitivity of bubble, μ

fd μ

I

D

S

PP

v

97

Instrument Misleveling

• For zenith angles

𝜎𝛼𝑙 = ±𝑓𝑑𝜇cot2 𝑧𝐵𝑆 + cot2 𝑧𝐹𝑆

𝑛

• where

– zBS and zFS are the backsight and foresight altitude

angles, respectively

– n is the number of angle repetitions.

– Units of 𝜎𝛼𝑙

are in the same units as μ

fd μ

I

D

S

PP

v

98

Example

• A horizontal angle is observed on a mountainside where the

backsight is to the peak and the foresight is in the valley. The

average zenith angles to the backsight and foresight are 70°

and 100°, respectively. The instrument has a level bubble with

a sensitivity of 30”/div and is leveled to within 0.3 div.

• For the average angle obtained from six repetitions, what is

the contribution of the leveling error to the overall angular

error?

99

Example

• μ = 30”; fd = 0.3

• vb = 90° − 70° = 20°

• vf = 90° − 100° = −10°

• n = 2 (1DR)

𝜎𝛼𝑙 = 𝑓𝑑μtan2 𝑣𝐵𝑆 + tan2 𝑣𝑓𝑠

2

𝜎𝛼𝑙 = ±0.3 30″tan2 20° + tan2(−10)

2= ±2.6″

𝜎𝛼𝑙 = ±0.3 30″cot2 70° + cot2 100°

2= ±2.6″

• Note that this error will be seen in angular misclosure! 100

Overall Angular Error

• Previously we had

– 𝑆𝛼𝑝𝑟 = ±2.4″

– 𝑆𝛼𝑡 = ±4.7″

– 𝑆𝛼𝑖 = ±2.2″

– 𝑆𝛼𝑙 = ±2.6″

• Estimated error in angle is

𝑆𝛼 = 2.42 + 4.72 + 2.22 + 2.62 = ±6.3″

101

Astronomical Observation

• Assume that the zenith angle to the backsight was 80°

and to the foresight was 47°, what is the error?

(otherwise use same values as given)

• Solution?

𝜎𝛼𝑙 = ±0.3 30″cot2 80° + cot2 47°

2= ±6.0″

102

Overall Angular Error

• Previously we had

– 𝜎𝛼𝑝𝑟 = ±2.4″

– 𝜎𝛼𝑡 = ±4.7″

– 𝜎𝛼𝑖 = ±2.2″

– Using 𝜎𝛼𝑙 = ±6.0″

• Estimated error in angle is

𝜎𝛼 = 2.42 + 4.72 + 2.22 + 6.02 = ±8.3″

103

Weights

• Bottom line is that weights of observations are as

individual as the observations themselves

– It is incorrect to assume that all angles or distance

observations have the same weight!

104

Course DescriptionThis workshop will present


ALTA/NSPS surveys



accuracies

4. Field procedures that help ensure meeting the 2011 ALTA/NSPS


105

Error Ellipse

• Uncertainties in length and direction

seldom align with cardinal axes of

coordinate system.

• Thus uncertainty in station

coordinates is larger than Sx or Sy.

• An error ellipse provides the largest

and smallest errors for any station.

B

A

Y

2S

2S x

X

y

AzAB

106

Station Uncertainty

• The uncertainty of the position of a station is defined by

two jointly distributed (x,y) coordinates.

• Thus, it follows a bivariate distribution.

Z

107

Contour Plot

• Each contour depicts estimated

error in unknowns at specific

probability level.

• Can increase probability level by

going to a lower contour on the

distribution.

108

Parts of an Error Ellipse

• The angle from the y axis to Su axis is

called the t angle

– t angle is the direction to largest uncertainty

– Su is the semimajor axis

– Sv is the semiminor axis

• Sx and Sy are the standard deviations in

the coordinates that form the standard

error rectangle

Standard error rectangle

S u

x

Standard errorellipse

S y

S x

t

S v

u

y

v

109

Probability of an Error Ellipse

• Standard error ellipse is at 35%

• Error ellipse probability can be increased to 95% by using a

multiplier

– 𝑐 = 2𝐹𝛼,2,degrees of freedom

• Su and Sv at a given probability level are computed as

– 𝑆𝑢% = 𝑆𝑢𝑐 = 𝑆𝑢 2𝐹𝛼,2,degrees of freedom

– 𝑆𝑣% = 𝑆𝑣𝑐 = 𝑆𝑣 2𝐹𝛼,2,degrees of freedom

110

Goodness of Fit Test

• The χ2 test

– Used to check the assumption that the a posteriori (after the

adjustment) computed reference variance, 𝑆02, is equal to its a

priori value, 𝜎02, which was assigned a value of 1 during the

computations of the weights

• Recall 𝑤𝑖 =𝜎02

𝜎𝑖2 =

1

𝜎𝑖2

• This is why S0 is called the standard deviation of unit

weight.

111


• How is 𝑆02 computed?

– 𝑆02 =

σ𝑤𝑖𝑣𝑖2

𝑅or in matrix terms 𝑆0

2 =𝑉T𝑊𝑉

𝑅

– where

• R is the number of redundant observations in the adjustment

• V is a matrix of the residuals, 𝑣𝑖• W is a matrix of the weights, 𝑤𝑖

112


𝑆02 =

σ𝑤𝑖𝑣𝑖2

𝑅

• When can test fail?

– A single or many observations have blunders causing large v

– A single or many observations have too large or small

assigned/estimated weights

• The weights are incorrect

113


𝑟𝑆02

χ𝛼2,𝑟

2 < 𝜎02 <

𝑟𝑆02

χ1−

𝛼2,𝑟

2

• Test is nothing more than constructing a range for 𝜎02

based on

– 𝑆02 and

– Number of redundant observations, r, in adjustment

• Range must bound 1

114

99% χ2 confidence interval

Results from a GNSS Adjustment

𝑆02,Reference Variance for Horizontal

Component

𝑆02, Reference Variance for Vertical

Component

Adjustment type: Plane + Height, Minimal constraint

Confidence level: 99 %

Number of adjusted points: 9

Number of plane control points: 1

Number of used GPS vectors: 31

A posteriori plane UWE: 1.182194 , Bounds: ( 0.7369072, 1.272843 )

Number of height control points: 1

A posteriori height UWE: 1.304971 , Bounds: ( 0.6345145, 1.385954 )


• Some will say if goodness of fit test passes

– Then we can use normal distribution multiplier of 1.96 from

normal distribution for 95%

• This explains the 2 in the ALTA-NSPS standards

• Note that using the sample multiplier (F critical value) will never be

incorrect!

• NGS always states that manufacturer software over-

estimates the quality of the survey

– You decide116

Propagation of Errors

• Errors

– Computing

• Latitudes and departures from angle and distance observations

• Computation of area/distance/directions from coordinates

Errors propagate

117

Effects of Errors on Coordinates

N

A

B

C

D

And then we adjust the traverse

Drives errors away from the control

118

Basic Principles in Network Design

• Distance observations strengthen the positions of stations in

direction collinear with the lines

• Angle and direction observations strengthen positions of

stations in direction perpendicular to sight line

• Largest errors occur farthest from the control

119

Course Description


1. Explore the relative positional accuracy

requirements in 2011 ALTA-NSPS surveys

2. How to properly weight a least squares

adjustment

3. How error ellipses are used to determine relative

positional accuracies

4. Field procedures that help ensure meeting the

2011 ALTA-NSPS relative positional accuracy

standards

120

Last Things First!

• The final size of the semi-major axis of a 95% error ellipse

is dependent on

1. The geometry of the survey

2. Precision of the instruments used in the survey

3. Abilities and knowledge of the field personnel

4. Weights of the observations are as individual as the

observations themselves

121

Last Things First!

• Guidelines to achieve the required relative precision in

guidelines

1. Use a theodolite with a ISO 17123-3 standard of 5″ or better

and an EDM that is 3 mm + 3 ppm or better

2. Calibrate all equipment before the survey

1. This includes levels and plummets

3. Be sure that field personnel know how to set an instrument or

target correctly over a point

4. Targets must be placed in tribrachs on tripods or on rods

supported with bipods/tripods122

Last Things First!

4. Angel-face targets must be used when observing angles

5. Properly align the prism with the line of sight

6. Distances should be observed at both ends of the line

and not averaged

7. Close angular horizons at all stations

8. Preplan the survey to

1. Maximize sight distances

2. Minimize number of stations123

Checking the Tripod

• Tightness of clamps on wooden tripods change with changes in humidity– Never leave the leg clamps tight when storing a tripod since this can

cause the wood fibers to be crushed when humidity increases

• Check shoes and points on shoes are tight

• Check that legs drop in a controlled fashion

• Always put cap on tripod when moving or storing it– Protects mounting screw assembly

• If your tripod is not properly adjusted, it will adversely affect your observations– You always need a good base to work from

126

Leveling Process• Leveling screws

• Coarse level

– Centering the circular bubble

• Fine level

– Centering the long

tube/electronic bubble in both

directions

127

Basic concepts

• Grasp a leveling screw with your left thumb and

forefinger

– Bubble moves in direction of left thumb

• Opposite direction of right thumb

Turn left thumb this

wayRight thumb moves opposite direction

Needed bubble motion

128

Leveling in Two Directions

• Orient instrument so that its line of

sight axis is parallel to line

connecting 2 of the leveling screws

• Center the level bubble

Line of sight

129

Leveling in Two Directions

• Rotate instrument 90° from first

position

• Center the level bubble by turning the

third screw as appropriate Lin

e o

f sig

ht

130

Checking the Level Vial

• Level vials maybe out of adjustment

• To check rotate instrument 180°

– Bubble should remain centered

• If not bring level ½ of the way back in

both directions

– Turn capstan screw on level vial to center

bubble

– For electronic bubbles follow

manufacturer’s procedures

Lin

e o

f sig

ht

131

Testing Plummet on Instrument

• Instrument must be level!

• Mark optical point on paper below instrument

• Turn instrument 180° degrees

• Marked point should still be under optical

plummet

• Correction is ¼ of distance back to original

point

• Leave corrections to a instrument technician

Should not happen

Correction

132

Vertical Wire Truly Vertical

• Test to check vertical wire for verticality

– Note that instrument must be leveled!

– Sight well-defined target high on wire

– Using the vertical tangent screw move the scope until point is at other

end

– Point should still be centered on wire

• Field Correction

– Always point on targets at intersection of horizontal and vertical wires

• Instrument Correction

– Vertical wire needs to be rotated

• Leave for instrument technician to correct

Should not happen

133

Horizontal Wire Truly Vertical

• Test to check horizontal wire

– Note that instrument must be leveled!

• Sight well-defined target near edge of horizontal wire

– Using the horizontal tangent screw move the scope until point

is at other edge

– Point should still be centered on wire

• Field Correction

– Always point on targets at intersection of horizontal and vertical

wires

• Instrument Correction

– Horizontal wire needs to be rotated

• Leave for instrument technician to correct

Should not happen

134

Line of Sight Not Perpendicular to Horizontal Axis

• Caused by vertical wire of not being

centered on line of sight (LoS) axis

• Can be adjusted by moving vertical

wire with capstan screws on reticle.

• When telescope is reversed, line of

sight scribes out a cone

Horizontal

axis

Error

exaggerated

on

lin

e

LoS

135


• When telescope is reversed, line of sight

scribes out a cone

• Results in extended line of sight is off to

one side of a straight-line extension

Horizontal

axis

Error

exaggerated

on

-lin

e

136


• Compensation

– Observe point on line in one

face and set extension by

plunging scope

– Observe point on line with

scope plunged and set second

point on extension

• Midpoint of line connecting

two points is extension of line

Horizontal

axis

on

lin

e

Face I

Face IILine

Hub

Extension

137


• Correction to vertical wire

– Bring vertical wire back to ½ of distance

between extension of line and set point

– So correction is ¼ of overall distance

between set points

– Best to leave this to a instrument

technician.

Face

I

Face

II

Line

Hub

Correction

138

Horizontal Axis Not Perpendicular to the

Vertical Axis

• Causes the line of sight to scribe a

nonvertical line when targets are at

different elevations

• Results in angle being observed to be

either too small or too large

Plu

mb

Angular error

Horizontal

139


Vertical Axis

• Compensation

– Observe angle with both faces of

instrument and average results

– Errors are the same but opposite in

both faces and cancel each other

out resulting in correct angle

Plu

mb

Angular errors

Horizontal

140


Vertical Axis

• To Check

– Set instrument close to vertical structure

with high well-defined point

– Sight point – plunge to horizontal and

mark point

– Reverse scope – Sight on same point –

Go to horizontal – Should be on marked

point

– If not, send in for repair

Plu

mb

Angular errors

Horizontal

141

Checking the Instrument-Reflector Offset

• Procedure– Set 3 points on line on nearly level ground with spacings of 10 m and 50 m

while at A

• Observe AC

– Move instrument to B

• Observe BA and BC

10 m 50 m

A B C

142

Checking the Instrument-Reflector Offset

• Check instrument-reflector constant, K

– (BA + K) + (BC + K) = AC + K

– So K = AC − (BA + BC)

• Procedure should be repeated several times to verify

value of K

• Looking at manual, enter appropriate value for K

10 m 50 m

A B C 143

Review

• Know the procedures to check your instrument’s

calibration

– Check the instruments before any important work

• Known the field procedures to compensate for

instrumental errors

– Instrument should be used as if it is out of adjustment

but always kept in adjustment

144

Review

• Instruments must be in adjustment before any

precise work can be obtained from them

• Check the adjustment of your instruments often and

send them in for repair and calibration as required

– This includes rods, tripods, tribrachs, and all level vials

145

Questions?

146

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