Preliminaries, Planning, & General RulesPreliminaries, Planning, & General Rules

Detailed planning:Detailed planning:

Essential before performing actual survey Essential before performing actual survey work, large or smallwork, large or small..

Common mistake for inexperienced people:Common mistake for inexperienced people: AArrive on site completely unprepared, not rrive on site completely unprepared, not

knowing what to do and/or how to do itknowing what to do and/or how to do it

OftenOften ends up costing much extra time ends up costing much extra time & effort, due to idling of crew (while & effort, due to idling of crew (while trying to come to an agreement), trying to come to an agreement), inappropriate methods used, and inappropriate methods used, and unnecessary mistakes made. unnecessary mistakes made.

Time spent on careful planning = time savedTime spent on careful planning = time saved

  

Tips on preliminaries & planning:Tips on preliminaries & planning:

Study existing maps &/or aerial photos Study existing maps &/or aerial photos of area if available. Relatively few of area if available. Relatively few surveys are performed in unmapped surveys are performed in unmapped areas, & it is not cheating but common areas, & it is not cheating but common sense to utilize all available informationsense to utilize all available information..

Distances & angles to be measured: Distances & angles to be measured: estimated on a scaled map beforehandestimated on a scaled map beforehand

Rough height differences: obtained using Rough height differences: obtained using contour lines / photogrammetric techniques contour lines / photogrammetric techniques

These estimates can also be obtained (e.g. These estimates can also be obtained (e.g. by pacing or a compass) during by pacing or a compass) during reconnaissance visit, to be discussed next.reconnaissance visit, to be discussed next.

Pay a reconnaissance visit to site, if at Pay a reconnaissance visit to site, if at all possibleall possible

Look for suitable locations for survey Look for suitable locations for survey stations forming control framework & stations forming control framework & mark them if possible, noting possible mark them if possible, noting possible lines of sight & potential obstructions (e.g. lines of sight & potential obstructions (e.g. clouds or vehicles) to them. clouds or vehicles) to them.

Which detail features to include in the Which detail features to include in the survey, what instruments & techniques to survey, what instruments & techniques to use, how to divide labor, & what part of use, how to divide labor, & what part of work to carry out on which day, etc.: work to carry out on which day, etc.: should plan at this stage, in a way that should plan at this stage, in a way that optimizes manpower & equipment useoptimizes manpower & equipment use. .

Every survey should be designed Every survey should be designed in such a way that it is impossible in such a way that it is impossible for errors to go undetected for errors to go undetected

Introducing redundancies into scheme (e.g. Introducing redundancies into scheme (e.g. measure all three angles in a triangle, or measure all three angles in a triangle, or both diagonals of a quadrilateral) both diagonals of a quadrilateral)

Cannot be achieved by merely repeating Cannot be achieved by merely repeating the same measurement many times the same measurement many times

redundancies: not only guard against redundancies: not only guard against blunders, but also allow overall statistical blunders, but also allow overall statistical treatment of data to improve accuracy treatment of data to improve accuracy

Design & produce well-organized booking Design & produce well-organized booking forms or electronic spreadsheets for recording forms or electronic spreadsheets for recording data, & include entries for redundant data, & include entries for redundant measurements & geometric checks to remind measurements & geometric checks to remind the observer.the observer.

Before leaving base for the actual survey:Before leaving base for the actual survey:

Be completely familiar with all instruments to Be completely familiar with all instruments to be used be used

Make sure they are properly calibrated by Make sure they are properly calibrated by qualified personnel such as instruments’ qualified personnel such as instruments’ manufacturers manufacturers

Read instruction manuals if available, & Read instruction manuals if available, & become familiar with instruments through become familiar with instruments through practice before the surveypractice before the survey. .

During actual measurements: compare with During actual measurements: compare with previously estimated values &/or geometric previously estimated values &/or geometric constraints to detect gross errors right away. constraints to detect gross errors right away.

Although instruments well adjusted, make Although instruments well adjusted, make measurements as though the equipment is measurements as though the equipment is not well adjusted, using techniques so that not well adjusted, using techniques so that instrument maladjustment errors will be instrument maladjustment errors will be significantly reduced even if present significantly reduced even if present (techniques: later chapters) (techniques: later chapters)

Be absolutely sure that no measurement Be absolutely sure that no measurement is missed: very time-consuming & is missed: very time-consuming & expensive to go back & repeat any expensive to go back & repeat any measurement(s) after a survey is measurement(s) after a survey is completed, if possible at all.completed, if possible at all.

Above allAbove all:: SafetySafety －－ NoNo. 1 concern during fieldwork. 1 concern during fieldwork:: Hostile environments due to landscape, Hostile environments due to landscape,

traffic, animals or people at sitetraffic, animals or people at site Take extreme caution to avoid injuries or Take extreme caution to avoid injuries or

damage to instruments. damage to instruments.

Classification of errors; accuracy & precisionClassification of errors; accuracy & precision

SurveyingSurveying －－ measurement science measurement science

All measurements contain errors All measurements contain errors

Identify sources of errorIdentify sources of error

Devise methods of dealing with themDevise methods of dealing with them

Three main kinds of errors in surveying:Three main kinds of errors in surveying:

BlundersBlunders (or mistakes) (or mistakes) – “gross errors” – “gross errors”

e.g. misreading by a whole meter or a degree; e.g. misreading by a whole meter or a degree; due to carelessness or lack of attention.due to carelessness or lack of attention.

OftenOften detected by proper checks detected by proper checks TTeam members: check on each other for eam members: check on each other for

mistakes mistakes CCommon source of careless mistakes: ommon source of careless mistakes:

copying numbers from one place to copying numbers from one place to another.another.

SystematicSystematic ErrorsErrors – –

DueDue to some persistent cause, generally in the to some persistent cause, generally in the instrument, but sometimes in a habit of the instrument, but sometimes in a habit of the observer.observer.

Thermal expansion of tapes & collimation Thermal expansion of tapes & collimation errors in a level errors in a level

Can be reduced by better technique, but Can be reduced by better technique, but not by averaging readingsnot by averaging readings

Do not obey “cancel out” laws of Do not obey “cancel out” laws of probabilityprobability

All distances measured with an All distances measured with an inaccurate tape: same percentage error inaccurate tape: same percentage error with same sign, however many times with same sign, however many times they are measured they are measured

Only remedy: calibrate the tape more Only remedy: calibrate the tape more carefully carefully

Most serious sort of error; techniques of Most serious sort of error; techniques of surveying & instrument calibration are surveying & instrument calibration are mainly directed against it. mainly directed against it.

Random Errors –Random Errors –

Without mistakes & systematic errors: still Without mistakes & systematic errors: still remain small random errors remain small random errors

Due to causes beyond control of observer (e.g. Due to causes beyond control of observer (e.g. temperature & wind)temperature & wind)

A matter of chance & subject to laws of A matter of chance & subject to laws of probability probability

Magnitude: depends on precision of Magnitude: depends on precision of instrument & observer’s skills instrument & observer’s skills

Cannot be corrected Cannot be corrected + & + & -- errors equally probable errors equally probable small errors more frequent than large onessmall errors more frequent than large ones very large errors do not occur very large errors do not occur

normal distribution to describe errors as random normal distribution to describe errors as random variables variables

Taking average of many readings will help in Taking average of many readings will help in reducing this type of error reducing this type of error

Rigorous statistical methods: Rigorous statistical methods: Least squares adjustment (LSA): in each Least squares adjustment (LSA): in each

relevant chapter with reference to particular relevant chapter with reference to particular survey figuressurvey figures

Accuracy Accuracy ≡ ≡ closeness of measurement to closeness of measurement to true value true value

PrecisionPrecision ≡≡ agreement between repeated agreement between repeated measurements of same quantitymeasurements of same quantity

If systematic errors dominate results in spite of If systematic errors dominate results in spite of greatest precautions taken, close agreement among greatest precautions taken, close agreement among data set may only indicate a consistent instrument data set may only indicate a consistent instrument & steady observer, i.e. high precision, not & steady observer, i.e. high precision, not necessarily accurate result. necessarily accurate result.

Fig. 1.3Fig. 1.3 Center of dartboard: “true value” Center of dartboard: “true value” which observer is trying to aim atwhich observer is trying to aim at

Precise Accurate

Precisions of instruments: standard errors Precisions of instruments: standard errors (or standard deviation, “(or standard deviation, “”)”)

Theodolite: “Theodolite: “ = 5” in user menu: = 5” in user menu: indicates reliability of observationsindicates reliability of observations

WeightsWeights ：： attached to observations attached to observations 1 /1 /22

WWeighted observations (next section)eighted observations (next section)

Least Squares Adjustment of Random ErrorsLeast Squares Adjustment of Random Errors

To determine: three lengths xTo determine: three lengths x11, x, x22 & x & x33 by by various direct or indirect measurementsvarious direct or indirect measurements

Fig. 1.4: Generic problem Fig. 1.4: Generic problem

x2 + x3

x1 + x2

x1

x2 + x3

x1 + x2 + x3

Data: assume no blunders/ systematic Data: assume no blunders/ systematic errorserrors

More observed than minimum necessaryMore observed than minimum necessary

Contain random errorsContain random errors

Different sets of data yields different Different sets of data yields different answers answers

Observations carry different Observations carry different weightsweights (i.e. (i.e. degree of reliability): degree of reliability):

Table 1.1Table 1.1

Quantity Observed Value Weight of Observation

x1 3.0 1

x1 + x2 6.1 2

x1 + x2 + x3 11.2 3

x2 + x3 8.1 2

Assignment of weightsAssignment of weights observers’ skills and experienceobservers’ skills and experience precisions of different instruments used precisions of different instruments used

No need to worry about such details No need to worry about such details here except to work with here except to work with givengiven weights weights

How to get How to get uniqueunique set of “ set of “bestbest” values ” values for xfor x11, x, x22, x, x33? ?

Principle of least squares (LS)Principle of least squares (LS)

Best estimates: “most probable values” (MPV’sBest estimates: “most probable values” (MPV’s ））

Difference between observed value (#’s) & Difference between observed value (#’s) & corresponding MPV (unknown) corresponding MPV (unknown) ≡≡ residualresidual::

Residual = (Observed Value – its MPV)Residual = (Observed Value – its MPV)

(1.1)(1.1)

LS condition: LS condition:

The most probable values of observed quantities are such that they minimize the total weighted sum

of square residuals,

SSR = weight (residual)2 (1.2)

LS principle: starting point LS principle: starting point (could be derived based on other (could be derived based on other

principles, e.g. “maximum likelihood principles, e.g. “maximum likelihood estimators”: probability theory)estimators”: probability theory)

Practical concern: numerical answers for xPractical concern: numerical answers for x11, x, x22 & & xx33 from (1.2) from (1.2)

Minimization problem, solved by calculusMinimization problem, solved by calculus (see Ex. 1.5) (see Ex. 1.5)

Matrix approach: only involves matrix algebraMatrix approach: only involves matrix algebra

Advantage: readily performed on a spreadsheet.Advantage: readily performed on a spreadsheet.

Pretend: all weights = unity Pretend: all weights = unity

(will remove this restriction later) (will remove this restriction later)

Vector of unknowns, Vector of unknowns, [[xx11, , xx22, , xx33]]T T ≡≡ xx

(superscript (superscript TT = “transpose”) = “transpose”)

Left-most column in Table 1.1: Left-most column in Table 1.1:

re-write as matrix product: re-write as matrix product:

3

2

1

32

321

21

1

110

111

011

001

x

x

x

xx

xxx

xx

x

Table 1.1 rephrased in matrix form:Table 1.1 rephrased in matrix form:

““~” means “have the observed values of”~” means “have the observed values of”

1.8

2.11

1.6

3

~

110

111

011

001

3

2

1

x

x

x

LetLet

AA ≡≡ aforesaid 4aforesaid 43 matrix of 0’s & 1’s 3 matrix of 0’s & 1’s

(i.e. coefficients on (i.e. coefficients on xx) (“) (“design design matrixmatrix”);”);

Vector Vector kk ≡≡ observed values observed values

AAxx ~ ~ kk

LS principleLS principle: :

“ “best” best” xx: must minimize overall “discrepancy” : must minimize overall “discrepancy” between Abetween Axx & & kk, i.e. (squared) magnitude of , i.e. (squared) magnitude of residual vector, hence residual vector, hence

Minimize Minimize rr Trr where where rr = = AAxx – – kk (1.3) (1.3)

Well known problem, solved in linear algebra Well known problem, solved in linear algebra

Take partial derivatives w.r.t. Take partial derivatives w.r.t. x,x, followed by matrix followed by matrix algebra: can showalgebra: can show: :

xx must satisfy must satisfy

    ((AATTAA))xx = = AATTkk (1.4)(1.4)

(1.4): “(1.4): “normal equations”normal equations”

Solution: obviously Solution: obviously

xx = ( = (AATTAA))–1–1AATTkk (1.5) (1.5)

((AATTAA))–1–1AATT ≡≡ “left pseudo-inverse” of “left pseudo-inverse” of AA (why?) (why?)

  

Formal proof of (1.4) & (1.5): Formal proof of (1.4) & (1.5): Skeel & Keiper Skeel & Keiper (1993)(1993)

We will simply use this result to solve practical We will simply use this result to solve practical problems problems

Return to original problem: unequal Return to original problem: unequal weightsweights

Only minor modifications needed: Only minor modifications needed:

Introduce diagonal Introduce diagonal

“ “weight matrix”, weight matrix”,

nw

w

w

w

W

0...00

0...0...

0...00

0...0

3

2

1

and “square root of W”,and “square root of W”,

Simple algebra Simple algebra (1.2) is minimization of (1.2) is minimization of

= = (W(W1/21/2r)r)TT(W(W1/21/2 r) r)

i.e. minimize dot product of “weighted residual i.e. minimize dot product of “weighted residual vector” vector” rr’ = ’ = WW1/21/2rr with itself with itself

2

1i

n

ii rw

nw

w

w

w

W

0...00

0...0...

0...00

0...0

3

2

1

2/1

In terms of W, A & x:In terms of W, A & x:

  rr’’ = = ((WW1/21/2AA)) xx – – ((WW1/2 1/2 kk)) to be minimizedto be minimized

NoteNote: : Quantities in ( ): all Quantities in ( ): all constantsconstants independent of independent of xx

same problem described in (1.3) exceptsame problem described in (1.3) except::

““AA”: now ”: now WW1/21/2AA,,

“ “kk”: now ”: now WW1/2 1/2 kk..

(1.5): (1.5): still valid after making aforementioned still valid after making aforementioned

replacementsreplacements

   xx = [( = [(WW1/21/2AA))TT((WW1/21/2AA)])]–1–1((WW1/21/2AA))TTWW1/2 1/2 kk

RecallRecall: : ((BCBC))TT = = CCTTBBT T for matrices for matrices BB & & CC,,

(for diagonal matrix) (for diagonal matrix) WWTT = = WW, (, (WW1/21/2))TT = = WW1/21/2, ,

& & WW1/2 1/2 WW1/21/2 = = WW, ,

Solution simplifies to Solution simplifies to

xx = ( = (AATTWAWA))–1–1AATTWWkk (1.6) (1.6)

Sub. numerical values for Sub. numerical values for AA, , kk, and , and WW = Diag = Diag [1,2,3,2] [1,2,3,2]

= [3.055, 3.045, 5.082]= [3.055, 3.045, 5.082]TT

Most probable values: Most probable values: xx11 = 3.055, = 3.055, xx22 = 3.045, = 3.045, xx33 = 5.082. = 5.082.

1.8

2.11

1.6

3

2000

0300

0020

0001

110

111

011

001

110

111

011

001

2000

0300

0020

0001

110

111

011

0011 TT

x

Matrices arise Matrices arise frequently, not only in frequently, not only in CIVL 102 (LSA of survey figures), but CIVL 102 (LSA of survey figures), but also inalso in

CIVL 337 (matrix structural analysis)CIVL 337 (matrix structural analysis) CIVL 253 (hydrology: regression analysis)CIVL 253 (hydrology: regression analysis) Geotechnical reliability analysis Geotechnical reliability analysis etc. etc.

Become efficient in handling matrix Become efficient in handling matrix computations with some form of computations with some form of

software as earlysoftware as early as possibleas possible

Ch.2:Ch.2:

lots of LSA using matrix approach.lots of LSA using matrix approach.

spreadsheet for matrix spreadsheet for matrix transposetranspose, , multiplicationmultiplication & & inversioninversion, , as well as as well as other useful mathematical tasks for other useful mathematical tasks for surveying. surveying.