© 2014 J.V.R. Paiva & GeoLearn, LLC 1 · 2020-01-06 · Least Squares in Surveying Practice NYSAPLS Conference January 2020 Joseph V.R. Paiva CEO - 0 ... GIS in surveying and mapping

1/6/20

© 2014 J.V.R. Paiva & GeoLearn, LLC 1

Least Squares in Surveying PracticeNYSAPLS Conference

January 2020

Joseph V.R. Paiva

CEO -

0

Outline� Why consider least squares?

� Where/when does it apply?

� What is least squares?

� What you need to do in your surveying practice to use least squares effectively

© 2020 J.V.R. Paiva 1

1

Why Consider Using Least Squares?

© 2020 J.V.R. Paiva 2

2

Why?� GNSS

� More and more “mixed” technology surveying

� Black box surveying makes the analysis of results ever more complex

� Accuracy standards based on results derived from least squares analysis

© 2020 J.V.R. Paiva 3

3

1/6/20


Practice today� Can involve total stations (manual and robotic), GNSS (static and RTK),

digital cameras, LiDAR, mobile mapping, satellite imagery, laser scanning, conventional and digital leveling systems, etc.

� GIS in surveying and mapping usually involves extensive quantities of data from multiple sources, which may include data from legacy sources (taping, theodolites, EDM, etc.)

� Fundamental to all of this and least squares is the understanding (to be discussed more in this course) that there is no such thing as a true measurement. Thus all data contains errors. Being human, we also know that data will contain blunders.

© 2020 J.V.R. Paiva 4

4

Least squares is helpful� To perform analysis using statistics to determine magnitudes of the

errors so that a decision can be made if the measurements meet the standards and/or are within acceptable tolerances

� To adjust the data so that they are in compliance with geometric constraints

© 2020 J.V.R. Paiva 5

5

Least squares is helpful� Blunder detection

� Pre-survey analysis

© 2020 J.V.R. Paiva 6

6

Where & When is Least Squares Helpful?

© 2020 J.V.R. Paiva 7

7

1/6/20


Where� Any time a measurement is made, if it is done just once, there is no

analysis

� If it is repeated (redundancy), then every additional measurement is an additional degree of freedom

� Analyzing redundant data is a fundamental premise for statistical data analysis

� Most adjustments prior to least squares understood that they were all limited because of presuppositions about the spread or the scatter in repeated observations

© 2020 J.V.R. Paiva 8

8

When� Do you wish to be rigorous in your measurements (control surveys)?

� Do wish to create points with refined state plane coordinates?

� Do you wish to leave behind the best “footsteps” for another surveyor to follow?

� Are you required (implicitly or explicitly) to provide a comprehensively analyzed and adjusted set of of survey data?

© 2020 J.V.R. Paiva 9

9

It depends on you� Do you see yourself as a responsible member of the profession?

� Do you see the responsibility to demand the best standards of practice relative to the needs of the public?

© 2020 J.V.R. Paiva 10

10

Really depends on you!� Yes, least squares has been ignored by practitioners for a long time

� But with the tools we have, pretending that the option to use it is only an option may be choosing to deliver less than is good for your client

� However, using it means, as you will see later, informing yourself and your team about what it is, how to properly apply it, and how to quality check the results

© 2020 J.V.R. Paiva 11

11

1/6/20


Application of adjustments� Analysis is always a precursor to adjustments

� We all understand GIGO

� So there is no point in adjusting bad data

� But once we decide we want to adjust something, we (used to?) use things like the compass rule, transit rule for traverses, linear adjustment of level networks, etc.

© 2020 J.V.R. Paiva 12

12

Weights� Critical to use of least squares is to understand that true precision of

the measurements and to use appropriate weights

� Common to use manufacturer’s accuracy values, but rarely is this the most appropriate

� Compass rule assumes that distances and angles were measured with the same precision

� Transit rule assumes angles were measured with high precisions

� Crandall’s rule assumes near perfection in angles and performs a least squares adjustment of the traverse distances

© 2020 J.V.R. Paiva 13

13

Creating equivalents� If you are required to perform a survey with a relative accuracy of

1:50,000 what should the accuracy of the angles be? [±4”]� If you can measure angles ±3”, what should the relative accuracy of the

distances be to be comparable in accuracy? [1/70,000]

© 2020 J.V.R. Paiva 14

14

Angle-Distance relationships

dangle

ddist

A

B

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

© 2020 J.V.R. Paiva 15

15

1/6/20


Angle-Distance relationships

a

dist

ddist

© 2020 J.V.R. Paiva 16

16

Creating equivalents� If you are required to perform a survey with a relative accuracy of

1:50,000 what should the accuracy of the angles be? [±4”]� If you can measure angles ±3”, what should the relative accuracy of the

distances be to be comparable in accuracy? [1/70,000]

© 2020 J.V.R. Paiva 17

𝑡𝑎𝑛$%1

50,000= 4.13” tan3” = 0.000015 =

168,754.9

17

Angle-Distance relationships� When measuring angles and distances, what is the limiting factor?� Remember a chain is only as strong as its weakest link

© 2020 J.V.R. Paiva 18

18

More importantly, what weight?� Angles

� Distances

� Positions with total stations

� Positions with RTK

� Positions with static GNSS

� Positions used in topo with total sta

� Positions for topo with GNSS

� Etc.

� Trig-leveled elevations

� RTK-leveled elevations

� Static GNSS elevations

� Single LiDAR shots (ground, airborne, mobile)

� Planes, lines, etc. derived from LiDAR

� Features mapped with UAS

� Features on satellite imagery

� Etc.© 2020 J.V.R. Paiva 19

19

1/6/20


Most common weight (to be correct)� Use

%67

� 𝜎9 is often referred to as the variance

� The common tendency is to use the manufacturer’s values

� Seldom is the manufacturer's values adequate

© 2020 J.V.R. Paiva 20

20

What do the manufacturers say?� Total station accuracy

� RTK accuracy

� Static GNSS accuracy

© 2020 J.V.R. Paiva 21

21

What do the manufacturers say?� Total station accuracy

� RTK accuracy

� Static GNSS accuracy

© 2020 J.V.R. Paiva 22

Angles: ±0.5” to ±5” commonlyDistances: ±(1 to 3 mm + 1 to 3 ppm)

±1-2 cm + 1-2 ppm

±0.5 – 10 mm + 1-2 ppm

22

Don’t ignore the conditions� You may be using the technology differently from the methods

specified

� And…you need to be certain that the accessories and peripherals are working perfectly

© 2020 J.V.R. Paiva 23

23

1/6/20


What is Least Squares?

© 2020 J.V.R. Paiva 24

24

It is not software!� But software makes it accessible to us today

� Using it without understanding the underlying principles and assumptions is dangerous

� Imagine getting behind the controls of a plane and all you know about flying is what you saw in Top Gun or Airplane or Sully

� Many software products are available today

� But they are not all necessarily equal

� Not a bad idea to compare results from a couple of vendors

© 2020 J.V.R. Paiva 25

25

To be formal� Least squares assumes understanding of the normal distribution

� Equation for the latter:

� 𝑦 = %6 9;

𝑒$=>77?7

� v: residual of the observations (x axis), (observation minus mean)

� y: probability of the residual occurring

� 𝜎2: variance of the observations

� e: exponential number 2.718281828…

© 2020 J.V.R. Paiva 26

26

Least squares� Involves maximizing probability of the least squares function by

minimizing the sum of the weighted squared residuals

� 𝑦 = %6 9;

𝑒$=>77?7

� Process: write equations for every observation in terms of unknown parameters

� For horizontal survey this would be a system of equations representing angles, azimuths, distances written in terms of their unknown station coordinates

© 2020 J.V.R. Paiva 27

27

1/6/20


Some howevers� The system equations are non-linear

� Linearize using Taylor series expansion

� For example, distance equation with residual is…

� 𝑛@ − 𝑛B9 + 𝑒@ − 𝑒B

9 = 𝑙B@ + 𝑣F [this is non-linear]

� So, now we can go to the linearized equation

© 2020 J.V.R. Paiva 28

28

Linearizing�

GH$GIJK L

𝑑𝑛@ +NH$NIJK L

𝑑𝑒@ +GI$GHJK L

𝑑𝑛B +NI$NHJK L

𝑑𝑒B = 𝐾 + 𝑣B

� dn, de are corrections to approximate values for unknowns n and e

� zero subscripts indicate coordinates are approximate

� K is the difference between the observed and computed length, where computed length IJ is determined from the approximate coordinate values, and

� v is the residual error

© 2020 J.V.R. Paiva 29

29

Then…� To solve, v is dropped, and the solution is iterated applying corrections

dn and de to the approximate n and e until corrections become negligibly small (so, for a distance, this may be 0.005 ft or 0.0005 mm)

� This example uses the distance equation

� Azimuth and angle observations equations must be similarly written, linearized and iterated

� What makes this “easier” is to write the equations in matrix form

� The hand solution is credited to Karl Gauss in the 18th century

© 2020 J.V.R. Paiva 30

30

Matrixes: Matrices� 𝑋 = 𝐽R𝑊𝐽 $%𝐽R𝑊𝐾

� J is matrix of coefficients determined from the linearized equations (the parenthetical values in the linearized equations)

� W is matrix of weights for each observations

� K is vector of the differences between the observed and computed values for each measurement

� X is matrix of corrections to be applied to the approximate coordinates

© 2020 J.V.R. Paiva 31

Ref: Ghilani’s Adjustment Computations: Spatial Data Analysis, 6th ed., Wiley 2018

31

1/6/20


Error theory review� Error = measurement – true value

� How do you know what the true value is?

� Most of the time we don’t know true value

� So we estimate it

� How?

© 2020 J.V.R. Paiva 32

32

Estimating true value� Measure something more than once

� Understand all the factors including details about instrumentations and how the environment affects the measurement process

� What factors contribute systematic errors, derive mathematical models or introduce measurement processes to remove them (the systematic errors)

� Then analyze for blunders

� If any are found remove them, remeasure if necessary

© 2020 J.V.R. Paiva 33

33

A good data set� Data set is created

� Systematic errors are removed

� Blunders are removed

� Re-measurements made if necessary

� Then compute the mean (average)

� This mean, often referred to as 𝑥 is used as the approximator of the true value

© 2020 J.V.R. Paiva 34

34

So, instead of errors� Residual = measurement - average

� Compare with: Error = measurement – true value

� Because we make redundant measurements to create the average, we have an over-determined system, if we continue that pattern of redundancy with every measurement we make in our surveys

� If we don’t have that redundancy, and actually record the variations, then least squares may not work for you (there is a way, but you have to be extremely careful)

© 2020 J.V.R. Paiva 35

35

1/6/20


The normal distribution� Turns out that any type of measurement results in the normal

distribution if you make measurements that are redundant (although it helps to have a high number, n, of measurements, and they are precise)

© 2020 J.V.R. Paiva 36

36

The shooting analogy

© 2020 J.V.R. Paiva 37

Precise or accurate?

37

What is the [Un]certainty� In your measurements?

� Why should you care?

© 2020 J.V.R. Paiva 38

38

The Average� Can be a better value than any single measurement

� But only if you’ve removed systematic errors and the blunders have been prevented or eliminated

© 2020 J.V.R. Paiva 39

39

1/6/20


Errors• Understand them (how they are caused, when they are caused, the

magnitude?)

• Eliminate as many as possible through modeling, procedures, instrumentation, software, and in post-processing

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

© 2020 J.V.R. Paiva 40

40

Continuing about the mean� When systematic errors in the measurements have been “handled” as

best as possible through procedures and calculations, the mean is the best estimate of the true value

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

© 2020 J.V.R. Paiva 41

41

Standard deviation (S or 𝜎)� A standard way of dealing with residuals to describe the shape of the

standard normal curve

� Quantifies precision (usually)

� Though accuracy of surveying instrumentation is often specified in terms of standard deviation

© 2020 J.V.R. Paiva 42

42

Standard normal distribution• Can be understood through process of plotting a histogram

http://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/350px-Normal_Distribution_PDF.svg.png

43

43

1/6/20


Random error example• Angle measured 50 times

• Calculate mean, residuals and standard deviation

• Let’s say we calculate that s = ±7”

• Standard deviation theory: examine individual measurements, 68% will be within 7 seconds of the mean

• Also, if you make one more measurement, there is a 68% probability that it will be within 7” of the mean

44© 2020 J.V.R. Paiva

44

Histogram plot

– 0 +

Magnitude of Class IntervalsNum

ber o

f mea

sure

men

ts in

clas

s int

erva

l

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

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

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

© 2020 J.V.R. Paiva 45

45

Simple s calculationNo.

Measurement

Resi-dual

Resi-dual2

1 27°43’ 55” 2 42 27°43’ 55” 2 43 27°43’ 50” -3 94 27°43’ 52” -1 15 27°44’ 00” 7 496 27°43’ 49” -4 167 27°43’ 54” 1 18 27°43’ 56” 3 99 27°43’ 46” -7 4910 27°43’ 51” -2 4

Mean = 27° 43’ 53”

Sum of n2 = 146

n-1 = 9

146/9 = 16.2

Sq. rt. of 16.2 = ±4”

𝜎 = ±Σ𝜐9𝑛 − 1

46

46

Confidence levels (area under curve)

� 68.2% for s

� 95.4% for 2s

� 99.7% for 3s

� 50% for 0.645s

Areaundercurve

s coefficient

0.80 1.281550.90 1.644850.95 1.959960.98 2.326350.99 2.575830.995 2.807030.998 3.090230.999 3.29052

47

© 2020 J.V.R. Paiva

47

1/6/20


Understanding probability

48http://www.usmle-forums.com/images/added/attachments/inpost/bellcurve.gif© 2020 J.V.R. Paiva

48

Thus…• Measurement has three parts

� The quantity measured

� The uncertainty� The confidence level

� E.g. 118°42’12” ±2.1” std dev (68% confidence)

� E.g. 23,478.65 m ±0.324 m @ 95% confidence

© 2020 J.V.R. Paiva 49

49

Combining random errors� Four end-to-end distances in feet are measured by four teams as

follows:

� A: 432.65 ±0.08; B: 836.75 ±0.12; C: 673.91 ±0.31; D: 560.87 ±0.10

� What is the MPV? What is the total uncertainty?

© 2020 J.V.R. Paiva 50

50

MPV� Most probable value

� In the case of the problem in the previous slide, it would be the average

� 2504.18 ft

© 2020 J.V.R. Paiva 51

51

1/6/20


Uncertainty� As long as the ± are not different probabilities,

� Combine this way

� 0.356 >> ±0.036 ft

© 2020 J.V.R. Paiva 52

𝜎XNYBZNX = 𝜎%9 + 𝜎99 + 𝜎[9 +⋯+𝜎G9

52

Combining random errors (Pt. II)� Same distances as before, but each measurement’s uncertainty is

±0.11 ft

� Use this when the uncertainties are the same

� Answer: ±0.22 ft

© 2020 J.V.R. Paiva 53

𝜎XNYBZNX = 𝜎]BG F̂N 𝑛

53

Combining random errors

𝜎XNYBZNX = 𝜎%9 + 𝜎99 + 𝜎[9 +⋯+𝜎G9

𝜎XNYBZNX = 𝜎]BG F̂N 𝑛

© 2020 J.V.R. Paiva 54

54

Some Data: which is more precise?� A � B

© 2020 J.V.R. Paiva 55

Resid Resid2

21.62 0.002 0.00000421.65 0.032 0.00102421.60 -0.018 0.00032421.63 0.012 0.00014421.59 -0.028 0.000784

𝑋 = 21.612

𝜎 = 0.023874673

Resid Resid2

21.64 0.052 0.00270421.71 0.122 0.01488421.42 -0.168 0.02822421.50 -0.088 0.00774421.67 0.082 0.006724

𝑋 = 21.588

𝜎 = 0.12275993

55

1/6/20


Calculating 𝜎

© 2020 J.V.R. Paiva 56

�𝜎 = ± _`7G$%

56

Weighted mean• “Conventional” mean is found by adding values and dividing by the

number of values

• In such a calculation the weight given to each value is “1”

• When you divide by “n,” you are actually dividing by the sum of the weights

© 2020 J.V.R. Paiva 57

57

Mean with equal weights• Example: a distance is measured three times to get the

following

• 100.00; 100.05; 100.10

• What is the mean of those obs?

• Answer: 100.05 [calculated by adding the three values and dividing by three—each has a weight of 1 and the sum of the weights is 3]

© 2020 J.V.R. Paiva 58

58

Unequal weights• The same three observations are obtained, but the first with stadia, the

second with a steel tape and the third with EDM

• Let’s say that there is no reason to discard any of the measurements; they are all good

• However relative weights may be 1:10:100

© 2020 J.V.R. Paiva 59

59

1/6/20


Unequal weights� To calculate a mean this time, giving proper recognition to

the weights:

© 2020 J.V.R. Paiva 60

60

Weighted mean• Closing the horizon, angles

add up to 359°59’44”

• How to adjust if AOB was measured twice, BOC was measured three times and COA five times?

61

A

O

C

B


61

But with unequal weight• With no other information, give weight as follows

62

Angle # Reps

Weight Proportioning LCM denominator

Standardizing

AOB 2 2 1/2 15/30 15/31

BOC 3 3 1/3 10/30 10/31

COA 5 5 1/5 6/30 6/31

Sum 31/30 1/1


62

Computing adjustments• (15/31) x 16 = 7.7” ~8”

• (10/31) x 16 = 5.2” ~5”

• (6/31) x 16 = 3.1” ~3”

16”

63

Angle # Reps

Standardizing

AOB 2 15/31

BOC 3 10/31

COA 5 6/31

Sum 1/1


63

1/6/20


Even better than repetitions• Use random error statistics such as standard deviation

• To weight, use %67

64© 2020 J.V.R. Paiva

64

A leveling example• Three lines of leveling to reach same point, different routes, equipment

crews, etc.

65

A𝜎 = ±0.023 m

Elev = 738.453 m

C𝜎 = ±0.041 m

Elev = 738.607 mB

𝜎 = ±0.060 mElev = 738.742 m


65

Computing Route Elev 𝜎 1

𝜎9

A 738.453 0.023 1890.3

B 738.742 0.060 277.8

C 738.607 0.041 594.9

66

𝑊𝑡𝑑.𝑀𝑒𝑎𝑛 =0.453×1890.3 + 0.742×277.8 + 0.607×594.9

1890.3 + 277.8 + 594.9

𝑊𝑡𝑑.𝑀𝑒𝑎𝑛 =1,423.52763.0 = 0.5152

𝑆𝑜 𝑢𝑠𝑒 738.515𝑚


66

Angle example• Closing the horizon, angles

add up to 359°59’44”

• How to adjust if 𝜎 of AOB, BOC and COA are ±6.2”, ±4.2” and ±2.8”

67

A

O

C

B

Angle 𝜎 𝜎2

AOB 6.2 38.44

BOC 4.2 17.64

COA 2.8 7.84

Sum 63.92


67

1/6/20


Computing

68

Angle 𝜎 𝜎2 Weights Weights as %

Adjustment

AOB 6.2 38.44 38.84/63.92 0.6076 10”

BOC 4.2 17.64 17.64/63.92 0.2760 4”

COA 2.8 7.84 7.84/63.92 0.1227 2”

Sum 63.92 1.006 16”


68

Enough of the mumbo jumbo� Least squares method is where equations are written for each

observation in terms of the unknown parameters

� They are then weighted according to the precisions of the observations

� The weighed equations are then solved using the previous matrix equation to minimize the sum of the weighted, squared residuals

� Yields the most probable solution for any set of data

� A method for solving an over-determined set of equations that yields most probable values for the unknowns while geometric constraints are satisfied

© 2020 J.V.R. Paiva 69

69

Reiterating� The least squares method will yield the most probable solution for any

set of data

© 2020 J.V.R. Paiva 70

70

Solution by hand is difficult1. For each station we have two unknowns, x and y

2. If we have a survey of 10 stations that means 20 unknowns

3. If not for matrices and computers, chances are we wouldn’t be considering it seriously for most land surveying work

4. Does not guarantee a good solution

5. Must not forget GIGO

6. Only works when care is taken in collecting the data, and all systematic errors are removed

© 2020 J.V.R. Paiva 71

71

1/6/20


Other benefits of least squares…� Post-adjustment statistics such as standard deviations (𝜎) and error

ellipses for the stations gives us the ability to analyze the survey to see if it fits within project tolerances

� Also provides advanced statistical methods to analyze observations for blunders

© 2020 J.V.R. Paiva 72

72

Error ellipse?

© 2020 J.V.R. Paiva 73

73

OK…so that’s least squares� But there are lot of things we need to understand

� Not everyone surveys the same way

� Not everyone evaluates their work the same way

� Not everyone thinks adjustments of surveying data is worth doing

© 2020 J.V.R. Paiva 74

74

By the way� What is a matrix?

� Set of numbers and/or symbols arranges in a square or rectangular array of numbers

� Consist of m rows and n columns

� Matrices enable certain mathematical operations (such as simultaneous equations) to be solved in a systematic and efficient manner

© 2020 J.V.R. Paiva 75

75

1/6/20


Example system� A11x1 + a21x2 + a13x3 = c1

� a21x1 + a22x2 + a23x3 = c2

� a31x1 + a32x2 + a33x3 = c3

� In summation terms

� ∑Bj%[ 𝑎%B𝑥B = 𝑐%

� ∑Bj%[ 𝑎9B𝑥B = 𝑐9

� ∑Bj%[ 𝑎[B𝑥B = 𝑐[

� In matrix form

�𝑎%% 𝑎%9 𝑎%[𝑎9% 𝑎99 𝑎9[𝑎[% 𝑎[9 𝑎[[

𝑥%𝑥9𝑥[

=𝑐%𝑐9𝑐[

� In matrix notation and compact form, 𝐴𝑋 = 𝐶

© 2020 J.V.R. Paiva 76

76

Problem analysis� Vertical angle, 𝛼, to point B observed at A is 3°00’. Standard error of

the angle being ±1’.

� A to B is observed to be 1000.00 ft, standard error of ±0.05 ft

� What is the horizontal distance and the uncertainty?

� 𝐻 = 1000𝑐𝑜𝑠3° = 998.63 𝑓𝑡

� 𝑆r =srst𝑆t

9+ sr

su𝑆u

9

© 2020 J.V.R. Paiva 77

77

Analysis continued

� 𝑆r =srst𝑆t

9+ sr

su𝑆u

9

� 𝑆r = (𝑐𝑜𝑠𝛼×0.05)9+ $]BGu×t×xL”9Lx9xy.z”/Y|X

9

� 𝑆r = (0.9986×0.05)9+ $L.L}9[×%LLL×xL”9Lx9xy.z”

9

� 𝑆r = (0.04993)9+ 0.0152 9 = ±0.052 𝑓𝑡

© 2020 J.V.R. Paiva 78

78

The other way to look at it� 1000.00 cos 2°59’ = 998.645 v = 0.0155

� 1000.05 cos 2°59’ = 998.695 v = 0.0655

� 1000.00 cos 3°01’ = 998.614 v = -0.0155

� 999.95 cos 3°01’ = 998.564 v = -0.0655

� Average 998.6295

© 2020 J.V.R. Paiva 79

79

1/6/20


How To Use It Properly In Your Practice

© 2020 J.V.R. Paiva 80

80

First things first� Competent software

� Competent people

� This means training

� This means a company policy on committing to it

© 2020 J.V.R. Paiva 81

81

Instrumentation accuracy� When manufacturer says this instrument can position with an

accuracy of ±0.2 m, they actually mean standard deviation, as in 68% probability that the error will be within the value

� But they don’t include any systematic error that you, the surveyor, may introduce

� The other elephant in the room: will the random error in the measurement be the same as the random error the manufacturer had during their testing?

© 2020 J.V.R. Paiva 82

82

Instrumentation accuracy� When manufacturer says this instrument can position measure an

angle within ±5”, they mean that there is a 68% probability that the error will be within the value


© 2020 J.V.R. Paiva 83

83

1/6/20


Instrumentation accuracy� When manufacturer says this digital level can give you rod readings

within ±0.001 ft, they mean that there is a 68% probability that the error will be within the value


© 2020 J.V.R. Paiva 84

84

Automatic/Digital levels• Level it up with circular vial

• Compensator takes over and ensures line of sight is horizontal

• Benefits: fast, when it works phenomenally accurate

• Drawbacks: compensators can hang up (usually result of shock or vibration extremes or wear and tear or lack of lubrication); cross hairs can still go out of adjustment.

• Extreme case: circular vial out of adjustment

© 2020 J.V.R. Paiva 85

85

How to use effectively� Conduct two-peg test regularly (this means knowing how to do it)

� Depending on technology, adjust the crosshair or follow other instructions provided by manufacturer. May require technical shop work.

© 2020 J.V.R. Paiva 86

86

What to do?• Teach your crews to use kid gloves in transportation, handling and set up

• Learn the two peg test, yes applies even with auto levels and digital levels

• Keep a record of each time you check

• Check circular bubble

• Level up instrument, sight at rod with eyepiece over one level screw, note reading

• Now turn level screw a bit, see if LOS returns to rod reading

© 2020 J.V.R. Paiva 87

87

1/6/20


Two peg test & adjustment

A BSetup ISetup II

•Setup I in middle gives true D elev

–If BS on A is 5.00 and FS on B is 6.00, D elev = 1.00

© 2020 J.V.R. Paiva 88

•Setup II is very close to A

–If BS on A is 5.45 and FS on B is 6.35, is the line of sight high or low?

–What should it be adjusted to?

88

EDM� Do you know the most common systematic errors with EDM use?

� How do the blunders occur, and what can you do to prevent them

� What is your EDM’s specified “accuracy”?

© 2020 J.V.R. Paiva 89

89

What Field Procedures?� Do you measure forward and back?

� How many repetitions to make a “measurement”

� How do you do those repetitions?

� Measure temperature and pressure?

� How is the effect determined?

� With prism, how is the prism constant determined?

� Without a prism, what errors are possible?

© 2020 J.V.R. Paiva 90

90

Temperature and Pressure� Each 1°C change in temperature is 1 PPM

� Each 1 inch (approx) of mercury change in atmospheric pressure is about 10 PPM [therefore 0.1 inch ≈ 1PPM [0.1” Hg ≈ 100 ft elevation change also]

� If your survey is being done at 45°F and an atmospheric pressure of 28.7 inch Hg, what is the PPM value? In 1000 ft, what is the error if uncorrrected? [Error: -17 & +12 PPM or -0.017 & + 0.012 = -0.005 or 1:200,000]

� Often standard for EDM is 29.92” Hg & 25°C or 77°F (for PPM correction of zero)

© 2020 J.V.R. Paiva 91

91

1/6/20


EDM Procedures� Repeat measurements but make them as independent as possible

� Double check HI, HT, prism constant

� Check your instrument periodically or have it checked

� Then adjust or have it adjusted

� Calibration is critical part of this checking process

© 2020 J.V.R. Paiva 92

92

Angle Measurement� What errors?

� Types?

� Sources?

� All using electronic angle measurement systems?

� [additional issues with robotic instrumentation]

© 2020 J.V.R. Paiva 93

93

Total Stations� Transit on steroids

� Plus electronics

� Plus EDM

� May also be plus

� Laser pointer

� Servo motors

� Robotics

� Imagery/videos

� Scanning

© 2020 J.V.R. Paiva 94

94

DIN/ISO Test� Actually a measure of precision, but once biases and blunders are

removed, can be a measure of accuracy

� What is your angle measuring accuracy per above test?

� How often is this checked?

© 2020 J.V.R. Paiva 95

95

1/6/20


Calibration � Horizontal reticle (for vertical angles)

� Vertical reticle (for horizontal angles)

� Height of standards or trunnion axis adjustment

� Leveling and centering errors

� Target leveling and centering errors

� Horizontal and vertical refraction

� Compensator in adjustment? How to check?

© 2020 J.V.R. Paiva 96

96

Basic alignment of theodolite

Plate vial axis

Trunnion axis(horizontal)

Telescope

Verti

cal a

xis

Circles

© 2020 J.V.R. Paiva 97

97

H and V reticles� Series of observations in F1 and F2 to clearly defined target (~100-200 ft

away)

� Discrepancy between the two (averaged) is twice the error

© 2020 J.V.R. Paiva 98

98

Trunnion axis� Also called horizontal axis

� Sight stable, high point in F1 (such as sharp target on top of building)

� Depress telescope and mark point on ground 100 ft away

� Now sight high point again in F2, then depress and mark point on ground again

© 2020 J.V.R. Paiva 99

99

1/6/20


Compensator� Two-axis compensates for mislevel in sighting direction and normal to

sighting direction

� Single axis only compensates in direction of sighting

� Thus single axis only compensates zenith angles

� Dual axis compensates both angles

© 2020 J.V.R. Paiva 100

100

Compensator check� Same as plate level

� Set up over two screws

� Center bubble

� Then turn 90°and repeat

� When centered, turn 180°

� Apparent bubble movement is twice actual error

© 2020 J.V.R. Paiva 101

101

Be careful� Some think that they can be careless with leveling when they have 2-

axis compensator

� If you are out of level by 10’

� Even if the compensator corrects the angles, your “plumb” line is 0.013 ft out of plumb

� How does that affect angle accuracy?

© 2020 J.V.R. Paiva 102

102

Analyze it!

© 2020 J.V.R. Paiva 103

0.013 ft

D =

D (ft) Angle error (“)

1000 3

500 5

100 27

50

25 107

10

103

1/6/20


Total stations� Combining EDM axis with optical axis of angle measuring instrument

requires good collimation (alignment) of former to the latter, assuming latter is in adjustment

� If robotic, requires additional collimation of target tracker, target point identifier and pointing laser to optical axis of theodolite

� Otherwise errors in angles, distances, etc. can result

© 2020 J.V.R. Paiva 104

104

GNSS error sources� Systematic errors called “biases”

� Can originate at satellites

� Can originate at receiver

� Can be from signal propagation

© 2020 J.V.R. Paiva 105

105

Some GNSS error sources� Base station location (ha!)

� Antenna height

� Accessories, accessories, accessories

optical plummetantenna pole bubbleantenna cables

antenna pole straightnesstripod stabilitytribrach bubbleantenna pole height

© 2020 J.V.R. Paiva 106

106

More GNSS error eources� Effect of geoid

� Phase and range measurement errors

� Atmospheric attenuation of signal

� Phase center errors

© 2020 J.V.R. Paiva 107

107

1/6/20


GNSS continued� Satellite (space segment) errors – ephemeris, clock, etc.)

� Receiver (user segment) errors – clock, multipath, phase center, rx measurement noise

� Ionosphere, troposphere

� Geometry (DOP)

© 2020 J.V.R. Paiva 108

108

GNSS continued� Network design

� Meaningless measurements because they are NOT independent

� Most flagrant errors caused by not understanding that GNSS does NOT directly measure rover’s position—it resolves VECTOR between base and rover

© 2020 J.V.R. Paiva 109

109

GNSS� Usual optical culprits must be considered including

� Not setting up in correct place

� Not setting up correctly (lack of leveling, centering, target height measuring—both tribrach and antenna pole) on correct point

� Tropospheric errors

� Ionospheric errors

© 2020 J.V.R. Paiva 110

110

Multipath

© 2020 J.V .R . Pa iva 111

• reflections off buildings, trees Direct Signal

Reflected Signal

111

1/6/20


Static GPS� Biggest blunder is not having independent observations (after blunder of

not setting up on correct point)

3 receivers; 1 sessionOnly 2 independent baselines

One more session with 2 receivers

© 2020 J.V.R. Paiva 112

112

Kinematic� Biggest blunder (!) is false initializations

� RTK is essentially radial surveying

� Can be improved with multiple base stations or properly implemented RTN

� Still not fool proof

� So if there is a mistake, who’s the fool?

© 2020 J.V.R. Paiva 113

113

Ellipsoid, Geoid, Topography

Local Topography

GeoidEllipsoid

Mass Deficiency

Mass Excess

© 2020 J.V.R. Paiva 114

114

Geometric relationships

Topographic Surface

Geoid

Ellipsoid

H h

N

H = h + Ne = Deflection of the vertical

e

H = Ellipsoid Heighth = Orthometric HeightN = Geoid Undulation

© 2020 J.V.R. Paiva 115

115

1/6/20


Coordinate systems� GPS measures in WGS-84 Cartesian

� Surveyor could be using SPCs, UTMs, other systems—almost neverWGS-84

� Converting from “native” GPS system to surveyor’s system can be fraught with errors (and mistakes)

� “Localization,” “calibration,” “transformation” add problems of their own

© 2020 J.V.R. Paiva 116

116

Peripherals (Accessories)� Just as important as the “big two” components of the system

� Chain is only as strong as its weakest link

� Don’t forget anything!

� Be constantly evaluating

� Most important accessory: humans

© 2020 J.V.R. Paiva 117

117

Checking tribrachs, Etc.� Fit

� Cleanliness

� Check threads (grit, smoothness, lubrication)

� Moisture issues

� Vibration and shock issues

© 2020 J.V.R. Paiva 118

118

Optical plummets� If not rotatable, watch out!

� Know how to check

� Know how to adjust

� Errors in excess of 0.1 ft per setup not uncommon

© 2020 J.V.R. Paiva 119

119

1/6/20


Better option instead of O.P. tribrachs

© 2020 J.V .R . Pa iva 120

120

Tripods� Easily forgotten

� Easily fixed

� Match the tripod to the job

� Be aware of the weak points: hinges, clamps, shoes, head

© 2020 J.V.R. Paiva 121

121

Range/Prism poles� Straightness

� Sensitivity of circular bubble vial

� Degree of adjustment of circular vial

© 2020 J.V.R. Paiva 122

122

How much is the error?� Typical level vial sensitivity can vary on prism poles from 20 minutes to 60

minutes

� If a particular 6 ft pole’s vial has 30 minute sensitivity, and the bubble is 2 mm out-of-center…

� …the prism on top out of plumb 0.05 ft

� How to calculate?

© 2020 J.V.R. Paiva 123

123

1/6/20


Calculating prism/antenna pole error

e

ht

a

© 2020 J.V.R. Paiva 124

124

Level vial centering/adjustment

© 2020 J.V .R . Pa iva 125

125

Centering error� When checking optical plummets, don’t forget to check circular level vial

� Positioning error is a function of line of sight error, level vial error and height above point

© 2020 J.V.R. Paiva 126

126

Putting it all together� Evaluate your system beginning with the smallest level components of your

system

� E.g. Optical plummet on tripod, reading errors, pointing errors, EDM errors, GNSS measurement errors

© 2020 J.V.R. Paiva 127

127

1/6/20


Unmanned Airborne Systems (UAS)� Most common blunder—not enough ground control and no/not

enough check points

� Automatic exposure

� Auto focus

� Use of “easy” button too much

� Assumption that results are always good

© 2020 J.V.R. Paiva 128

128

Other issues� Targets to large or too small—must be sized based on ground

sampling distance (GSD)

� Non-prime lenses

� Not focused to infinity

� Clouds

� Shadows where it is critical to have good matching

© 2020 J.V.R. Paiva 129

129

Other drone issues� Not understanding the photogrammetric process, i.e. shortcomings

� Insufficient ground control quality

� Insufficient ground truthing

� Digital scaling up of small scale map/model

© 2020 J.V.R. Paiva 130

130

The aircraft� Test flight control components

� Pay attention to winds (vibration and excess speed)

� Insufficient accuracy of GNSS or autopilot creates gaps

� Target: all points must be imaged 12-15 times

© 2020 J.V.R. Paiva 131

131

1/6/20


RTK, PPK or Ground Control� RTK is fine

� But still have check points and some limited ground control

� PPK is fine, but know what you are doing and that data is being fully logged

© 2020 J.V.R. Paiva 132

132

Conditions Causing 0.01 ft Error in 100 ft. (calibrated steel tape)

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

© 2020 J.V.R. Paiva 133

133

Possible Errors Using Common ProceduresStandard 100 ft measurement with calibrated tape

Source Error (ft.) Error2

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

SUM 0.023 0.000145

Sq Rt of [Sum of Errors2] = 0.012 ft

1: 8,000 OR 120 PPM

© 2020 J.V.R. Paiva 134

134

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


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


1: 3,000 OR 306 PPM

© 2020 J.V.R. Paiva 135

135

1/6/20


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


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


1: 127,000 OR 8 PPM

© 2020 J.V.R. Paiva 136

136

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


Length Known 0.000000Tropo delays 0.0025 m = 0.008 ft 0.000067Centering w/O.P. 0.005 0.000025Centering w/O.P. 0.005 0.000025Mfr’s error const. 0.01 m = 0.03281 0.001076Mfr’s error scale 2 PPM = 0.005 0.000025SUM 0.05581 0.001218Sq Rt of [Sum of Errors2] = 0.0349 ft

1:72,000 OR 14 PPM

© 2020 J.V.R. Paiva 137

137

PPM� 1 PPM = 1:1,000,000

� 10 PPM = 1:100,000

� 100 PPM = 1:10,000

� 5 PPM = ?

� 20 PPM = ?

� 80 PPM = ?

� 125 PPM = ?

© 2020 J.V.R. Paiva 138

138

Many Issues: Lack of (adequate) Control

� Not necessarily inadequate number of monuments, but lack of accuracy of the control

� If you don’t know quality, i.e. accuracy, not precision, of your control, then you cannot say anything about the quality of your work that is supposed to be controlled

© 2020 J.V.R. Paiva 139

139

1/6/20


If You Don’t Want To Get “Killed”…� Then…calibrate instrumentation

� Make redundant measurements

� Measure environment and carefully log procedures used

� Apply corrections for the meaningful errors

� Develop procedures to eliminate blunders, develop procedures for identifying blunders (field and office)

� Be consistent in the practice of these rules

� But know what’s important to consider and when, & what’s not

© 2020 J.V.R. Paiva 140

140

Some thing to watch for� If your software doesn’t allow for weighting or minimizes that feature…

� If the prompts for precision ask you to input the manufacturer’s spec…

� If you use the software to adjust data that has no redundancy…

© 2020 J.V.R. Paiva 141

141

Final Slide� Know your equipment

� Keep it adjusted

� Be honest in your estimations

� Train your people well

� Strive for weighting accuracy

� Don’t just rely on the manufacturers for precision estimates

� Encourage your fellow professionals to follow your practices

© 2020 J.V.R. Paiva 142

142

Q & A

143© 2020 J.V.R. Paiva

143

Documents

© 2014 J.V.R. Paiva & GeoLearn, LLC 1 · 2020-01-06 · Least Squares in Surveying Practice NYSAPLS Conference January 2020 Joseph V.R. Paiva CEO - 0 ... GIS in surveying and mapping