Geodetic Deformation Analysis

7/24/2019 Geodetic Deformation Analysis

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GEODETIC DEFORMATION

ANALYSISShort Lecture Notes for Graduate Students

CneytAydn(YTU-GeodesyDivision)


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CONTENTS

PREFACE..................................................................................................................................... iii

1. INTRODUCTION........................................................................................................1

2. GLOBALTEST............................................................................................................ 4

2.1 TestingWholeNetworkMethodI...........................................................................42.2 TestingWholeNetworkMethodII..........................................................................82.3 TestingWholeNetworkforNonIdenticalCase.................................................... 102.4 TestingaPartofaNetwork...................................................................................11

3. TESTINGOBJECTPOINTS ....................................................................................... 14

3.1 TestingObjectPointsinAbsoluteDeformationNetworks ................................... 143.2 Localization ............................................................................................................ 173.2.1 LocalizationwithGausseliminationmethod........................................................ 173.2.2 Localizationwithimplicithypothesismethod....................................................... 203.2.3 Localizationinabsolutedeformationnetworks....................................................22

4. SENSITIVITYANALYSIS ........................................................................................... 23

4.1 GlobalSensitivityAnalysis ..................................................................................... 234.1.1 OptimizationCriteria ............................................................................................. 234.1.2 Minimumdetectabledisplacement ...................................................................... 25

4.2 SensitivityAnalysisinAbsoluteDeformationNetworks .......................................26

5. INTERPRETATION MODELS .................................................................................... 30

5.1 KinematicModel....................................................................................................305.1.1 Singlepointmodelforanobjectpointsmovement.............................................315.1.1.1 ModelI ...................................................................................................................315.1.1.2 ModelII ..................................................................................................................325.1.1.3 ModelIII................................................................................................................. 325.1.2 Singlemodelforwholeobjectpointsmovement ................................................ 345.1.3 Testingmodelparameters.....................................................................................355.1.4 Modeltest.............................................................................................................. 365.2 StrainAnalysis........................................................................................................ 375.2.1 Definition ............................................................................................................... 375.2.2 Strainmodellingingeodeticdeformationanalysis............................................... 41

APPENDIXA:GLOSSARY ........................................................................................................... 45

APPENDIXB:THRESHOLDVALUES(Fandtdistributions) .......................................................47


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PREFACE

This short lecture notes aims to give some fundamental subjects in geodetic

conventionaldeformationanalysis.IthasbeenwrittenforgraduatestudentswhotakeError

Theory and Parameter Estimation andAdjustment Computation courses in their Geodesydepartments. For reading the notes, it is highly recommended to have knowledge on

geodetic network adjustment solutions, especially tracemin, partial tracemin and Stransformation.

Theupdatedversionwithnumericalexamplesandliteraturereviewwillappearsoon.Yourcommentsonthelecturenoteswillbeappreciated.

C.Aydn

stanbul,July2014

([email protected])


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1. INTRODUCTION

Becauseofactingforces,aphysicalbodymaydisplacefromx1initialpositiontox2

presentpositionintimein3Dspace.Thedisplacementvector

d=x2x1 (1.1)

consistsoftwoparts

relativepart,

nonrelativepart.

The relative part, the socalled rigid body displacement, represents translation and

rotation.Theyarerelativebecausetheychangedependingonwhereweareobserving

thebodyfrom.For instance,abodymaynotmoveorrotateforanobservermoving

androtatingalongthesamedirectionsimultaneouslywiththebody.Thenonrelative

part, on the other hand, does not depend on the observer position. This part

representsshapechange,i.e.deformation.

Engineeringbuildings,suchasdams,bridges,tunnelsetc.,ortheEarthscrustare

suchbodiesaffectedbysomephysicalforcesallthetime.Monitoringtheirresponses

totheforces isanessentialtasknotonlyforunderstandingthebodymechanismbut

also for taking someprecautionsbeforeanypossibledamage.The responses,which

are monitored as aforementioned displacements, are very small compared to the

bodysize.Geodeticmethodsandinstrumentsmayovercomethisproblemsufficiently

havingprovidedmilimeteraccuracy inpositioningof thestationsdistributed ineven

largeareas,andtherefore,todaytheyare indispensable incrustalandconstructional


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deformationstudies.Althoughgeodeticdeformationanalysis isnowaround30years

old and there are plenty of approaches, techniques, methods etc., the surveying

principle is the same. For monitoring the bodies (the objects as commonly

pronouncedingeodesy),weestablishadeformationnetwork.Therearetwotypesof

deformationnetwork,

absolutedeformationnetworks,

relativedeformationnetworks.

An absolute deformation network consists of two parts; 1) reference points and 2)

object points. The reference points are established in a stable region, the socalled

referenceblock,whereastheobjectpointsare locatedatsomespecificplacesofthe

objectsuchthattheyareabletocharacterizethe investigateddynamicalpropertyof

the object itself. Both point groups are predefined in absolute networks. Or, if the

reference points and object points are defined after deformation analysis or

depending on a prior information beforehand, we call such deformation networks

absoluteones.On theotherhand, ifadeformationnetworkmaynotbepartitioned

intotwopartsbeforehand,thistypeofnetworkiscalledrelativedeformationnetwork.

Before realization of a deformation network, a practitioner knows naturally

(intuitivelyordependingonpreanalysisof thestudiedarea)whichpart isreference

blockandwhichpartisobject.However,afterrealizationweshouldtestwhetherthe

referenceblockhasundergoneanydeformationornot.Inotherwords,thereference

pointsarenotexact inadeformationnetwork.Therefore, in theory,alldeformation

networks are (should be!) described as relative ones unless verifying the referencepoints by some statistical tests based on the corresponding deformation

measurements.


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Fig.1.1.Configurationofanabsolutedeformationnetwork

Inthatcoursewemainlyconcentrateonconventional(geometrical)deformation

analysis for absolute and reference deformation networks, i.e., global test and

localizationprocedures;sensitivityanalysistoderivethecapacityofourdeformation

networks;kinematicmodelstoderivevelocityand/oraccelerationofthebodieswhich

moveintimeaswellasstrainanalysistointerpretethedeformationofanobject.

Object

ReferencepointsReferencepoints


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2. GLOBAL TEST

Globaltestisrealizedfortwoaims;tolearn(orverify)

whetherwholenetworkhasundergoneanydeformationornot,

whether a part of the network (for instance, reference points) has any

deformationornot.

By global test it is not possible to answer if the corresponding points have

translatedor rotatedasablock (remember thesekindofdisplacementsare relative

changes),thereforetheaimissometimesexpressedasfollows;

To learn whether the correspondingpart has somepoints whose coordinates

havesignificantchanges.

2.1 Testing Whole Network-Method I

Weheredesiretotestifawholenetworkhasanydeformationornot.Letourc

dimensional deformation network with u=cp coordinate unknowns of p points be

measured in two periods. Applying traceminimum solution to each period

observations,supposethatweget

iiTixxi ii yPAQx = , iixxQ = T1TiiTiiiTi )()( GGGGAPAAPA += + (i=1,2) (2.1a)

aswellas

iiii yxAv = , iiiTi

2i,0 /f)(s vPv= (i=1,2) (2.1b)


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where

Ai : niudesignmatrixwithrankAi=ur,

iixxQ : uucofactormatrixoftheunknowns,

Pi : niniweightmatrixoftheobservations

iy : ni1(diminished)observationvector,

()+ : denotespseudoinverse,

iv : ni1residualvector,

2i,0s : aposteriorivarianceofunitweight,

ni : numberofobservations,

r : numberofdatumparameters,

fi : degreesoffreedom(fi=niu+r)and

GT : rucoefficientmatrixofconstraintequations(datum

matrix)todefinethedatumofthenetwork.

Using the solutions given in Eq. (2.1a), the displacement vector d and its

cofactormatrix ddQ areobtainedas

d=x2x1 ,2211 xxxxdd

QQQ += (2.2)

The test procedure depends on discriminating the following null hypothesis

(H0)againstitsalternative(H1);

H0:E(d)=0 vs. H1:E(d)0 (2.3)

Forthis,wehavetwopossibleways;

F(Fisher)test

2(Chisquare)test.


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The test statistics (T) and the threshold values () of these two tests are given as

follows:

=

=

==

+

+

21h,

2

2d

ddT

2

1f,h,2d

ddT

,(h)~T

F,f)F(h,~hs

TF

dQd

dQd

(2.4)

where

h : rank ddQ (seeNote2.1),

f : totaldegreesoffreedom,i.e.,f=f1+f2,

2ds : pooledvariancefactor(seeNote2.2),

: totalsignificancelevel(TypeIerror)and

2d : apriorivariancefactor(seeNote2.3).

WesimplycomparethecorrespondingteststatisticTwithitsthresholdvalueinEq.

(2.4).Thereistwopossibleoutcomesandresults;

i) IfT


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Note2.2:Pooledvariancefactorisobtainedasfollows

2ds =

21

22,02

21,01

ff

sfsf

+

+= 22

T211

T1 vPvvPv + =

f

However, to consider the pooled variance factor in Eq. (2.4), the ratio 2 1,0s / 2

2,0s

shouldbesmallerthan 1,f,f 21F thresholdvalue(Variancetest)*.Otherwise,wemay

notputthepooledvarianceintoEq.(2.4);inotherwords,itmeansthattheperiods

arenotproperforanycomparison.

*)Thisisvalidif 2 1,0s isnumericallybiggerthan2

2,0s .Otherwise,theratio 2

2,0s / 2

1,0s

iscomparedwith 1,f,f 12F .

Note2.3:Instatisticalpointofview,2test ismorepowerfulthanFtest. Inother

words,theprobabilityofcorrectlyacceptingthealternativehypothesis(thepower

ofthetest) in2test isbigger.However,itrequiresapreciseknowledgeonthea

priorivariance 2d .Sinceitcanbederivedfromlongtimeexperienceonthedataof

the surveying methods applied in the studied area, this requirement may not be

ensuredalwaysorinashorttime.Therefore,commonlyFtestischosenbecauseit

needsonlythevariancesfromthecurrentmeasurementsoftheperiods.Herafter,

inthetestprocedureswewillconsideronlyFdistributedteststatistics.

Note2.1:Ifbothperiodsareidentical,thenh=rank ddQ =urholds.


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2.2 Testing Whole Network-Method II

Theprevious test statisticvaluesarededuced from the theoryof generalized

linear hypothesis. In deformation analysis, there is a second method which

substitutesthehypotheses implicitly intoacorrespondingGaussMarkoffmodel. It is

calledthereforeimplicithypothesismethod.

We may gather the separate adjustment models of the periods in a unique

GaussMarkoffmodelas

=

2

1

2

1

2

1E

x

x

A0

0A

l

l ,

=

2

1

P0

0PP (2.5)

NowwewillconsiderthenullhypothesisH0:E(d)=0orH0:E(x1)=E(x2) inmodel(2.5).Forthiswewritethefollowingmodel,which impliesthatthere isnoanydifference

betweentwoperiods;

H

2

1

2

1E x

A

A

=

l

l ,

=

2

1

P0

0PP (2.6)

Note2.4:Foreachperiod,adjustmentprocedureshouldberealizedwithcommon

approximate coordinates in the network. However this may not be ensuredeverytime because in some cases (for example in monitoring of landslides which

maycausebigdisplacements)iterativeadjustmentisrequired;so,theapproximate

coordinates inevitably changes in each iteration. For such cases, instead of using

diminished coordinate unknown vectors (xi), adjusted coordinates of the periods

shouldbeconsideredtoestimatethedisplacementvectorinEq.(2.2).


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Solvingmodel(25)wegetthefollowingquadraticform,whichisequivalentto

thesumoftheweightedsumofthesquaredresidualsoftheperiods,

22,02

21,0122

T211

T1 sfsf +=+= vPvvPv (2.7)

On the other hand, the solution of model (2.6), which has fH=f+ur degrees of

freedom,resultsin

HTHH Pvv= (2.8)

If there is no any difference between two periods, the difference between two

quadratic forms of the models, i.e., R=H, should go to zero. For this the test

statisticfromtestingoflinearhypothesesissetasfollows

T=2d

HH

hs

R

/f

f)/(f)( =

F(h,f) (2.9)

becauseoffHf=ur=hand(/f)=2

ds (Note2.2).Theteststatistic is identicalwiththe

oneofFtestinEq.(2.4).So,thetestprocedureissimilar.

Note 2.5: In each free adjustment method (tracemin, partial tracemin and

minimumconstrained), we get a unique residual vector. Therefore, solution of

model (2.6)mayberealizedbyany freeadjustmentmethod.Since theminimum

constrainedsolutionisanormaladjustmentprocedure,itiseasytousethissolution

intheimplicithypothesismethod.Forthis,arbitraryrcolumnsofthedesignmatrix

inmodel(2.6)aredeletedbeforeadjustment.


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2.3 Testing Whole Network for Non-Identical Case

A deformation network may be augmented or renovated with newer points in

differentperiods. In that casewe should handlewithdifferent configurations in theperiodstobecompared.Tomakethemidenticaltherearetwopossibleways;

i) Thecorrespondingperiodsareseparatelyadjustedusing theobservations

connectingtheidenticalpointsinbothperiods.

ii) The periods are readjusted such thatonly identical points in theperiods

definethedatumofthenetwork.

Thelatterismoreadvantageousbecausewedonotmodifythenetworkdesign.Letus

considerthatourcdimensionaldeformationnetworkwithppointsisaugmentedwith

k newer points in the second period and we have already their traceminimum

solutions;

Coordinates Cofactors

1stPeriod 2ndPeriod 1stPeriod 2ndPeriod

Identicalpoints1x (cp1) 2x (cp1) 11xxQ 22xx Q n2xx Q

Newpoints nx (ck1) 2nxx Q nnxx Q

Inthesecondperiod,newpointsshouldbeextractedfromthedatumdefinition.We

may use Stransformation for doing this. Let c(k+p)r coefficient matrix of the

constrainedequationsforthesecondperiodbehadthefollowingform

2~x

22x~x~Q


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)( TnTT

2 GGG = (2.10)

where TnG is the ckr datum submatrix for the new points. TakingTnG =0 above (in

other words, new points are extracted from the datum definition) we set a new

coefficientmatrix

)( TT2 0GB = (2.11)

WithEqs.(2.10)and(2.11),theStransformationmatrixiscomputedas

T2

12

T222 )( BGBGIS = (2.12)

UsingthematrixS2wedefineanewdatumforthesecondperiod

S2 2~x =

n

2

x

x and TSQS 2x~x~2 22 =

nn2n

n222

xxxx

xxxx

QQ

QQ (2.13)

The subvector x2 and the submatrix 22xxQ in Eq. (2.13) are now compatible

with 1x and 11xxQ . Hence, both periods are identical and ready for comparison as

usual.

2.4 Testing a Part of a Network

Our reference points should be stable because we define the cloud of these

pointsasourobserver tomonitor theobject.We should thereforeverifywhetherthe reference points have undergone any deformation or whether they include any

pointwhosecoordinateshavechangedsignificantly.Thisprocedureisoneofthemost

importantstageinmonitoringoftheobject.

Supposethatandrepresentpreferencepointsandp=ppobjectpoints,


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respectively.ThedisplacementvectordanditscofactormatrixQddinEq.(2.2)maybe

writtenexplicitlyforthesepointgroupsasfollows

=

d

dd ~

~ , Qdd=

QQ

QQ

~~

~~ (2.14)

To test the reference points first we should define the network datumaccording to

them.ThismayberealizedbythefollowingStransformation

==

d

ddSd , ddQ = S Qdd

TS =

QQ

QQ (2.15)

wherethetransformationmatrix S issetasfollows;

T1T )( BGBGIS = with )( TTT

= GGG and )( TT

0GB = (2.16)

Inthetest,thesubvector d andthesubmatrix Q inEq.(2.15)areused:Thetest

statistichavingFdistribution,similartotheoneinEq.(2.4),isgivenasfollows

f),F(hsh

T2d

T

+

= dQd

(2.17)

where h =hc(pp)=rank Q . The test statistic is compared with the threshold

value 1f,,hF ;

i) If T < 1f,,hF ,thenourreferencepointsareacceptedasstablewitherror.

ii) If T 1f,,hF , there is at least one point whose coordinates has changed

significantly. In that case, we should find the responsible point(s) and

extract it(them)fromthereferencedefinition.Onepossiblewayforpoint


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detectionisaddingeachreferencepointonebyonetothepointcloudin

Eq. (2.14) in each time and repeat the above test procedure for the

remainingreferencepointsuntiltheteststatisticbecomessmallerthanthe

corresponding thresholdvalue.Otherpossibleway isapplying localization

procedure,whichisgiveninSection3.2.3.

Note2.6:Thedegreesoffreedom h mustbebiggerthan0,i.e., h 1.Sincethere

may be a significantly changed point among them, we should take into account

h 2. This natural limitation gives information about the minimum number of

referencepoints(p)foranetwork:Fromtheinequality,weget

h =hc(pp)=cprcp+cp=cpr2.

Thenweseethatpshouldbeequalto(2+r)/cinaworstcase.Thismeansthatour

absolutedeformationnetworks(1,2or3D)shouldbedesignedsothatitconsistsof

approximatelyatleast3referencepoints.


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3. TESTING OBJECT POINTS

3.1 Testing Object Points in Absolute Deformation Networks

As we mentioned previously, an absolute deformation network has twoparts;

reference points and object points. The reference points should be verified by the

globaltestgiveninSection2.2suchthattheycanbedefinedasobservertomonitor

theobject.

Letourreferencepointsbealreadyverifiedasstable.Then,eachobjectpoint

may be tested to learnwhether its observed displacement relative to the reference

pointsissignificantornotbyusingthefollowingteststatistic

iT= 2

d

1T

cs

iiii

dQd F(c,f) i=1,,p (3.1)

where

id : c1ithobjectpointsdisplacementvector,whichisthe

correspondingsubvectoroftheobjectdisplacementvector d inEq.(2.15),

ii Q : cccofactormatrixoftheithobjectpointdisplacements,

whichisthecorrespondingsubmatrixoftheobjectcofactormatrix Q inEq.(2.15)and

c : dimensionofthecorrespondingnetwork.

EachteststatisticinEq.(3.1)iscomparedwiththethresholdvalue 1f,c,F ;


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i) Ifi

T< 1f,c,F ,thecorrespondingdisplacementisnotsignificant.

ii) Ifi

T 1f,c,F , it is accepted that the corresponding displacement is

significantwith1confidencelevel.

Note 3.1: The above given test procedure is equivalent to relative confidence

interval/ellipse/ellipsoid method applied in deformation analysis studies. For

example, let us consider 2D cases: First, the displacement vector (i

d ) of a

correspondingpointisplottedonamapand,then,theconfidenceellipseobtained

fromii

Q (thisellipse iscalledrelative indeformationanalysis) iscenteredon

theendpointofthedisplacementvector(seeFig3.1).Ifthedisplacementvectoron

theplotremainsoutsideoftheellipse,thanthisdisplacementofthecorresponding

pointissaidtobesignificantwiththepreassumedprobabilityofconfidencelevel.

Fig3.1.Displacementvectorandrelativeconfidenceellipse

Displacement

vector

Relative

confidenceellipse

Objectpoint


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For1Dnetworks,i.e.,levellingandgravitymonitoringnetworks,squarerootof

the test statistici

T in Eq. (3.1) becomes t (Student)distributed, because of the

probabilitydistributionfunctionproperties,

ii

iiii

i Qs

|d|

s

QdT

d

2d

12

== t(f) (3.2)

where

id and iiQ : theithpointsdisplacementandits

cofactor,respectively

Note3.2: Ifourreferencepointsareverifiedasundeformed(stable)bytheglobal

testinSection2.4,thisdoesnotmeanthatwealsoverifiedthattheyhasnotmoved

in timeasablock. Itmeans that theobserveddisplacementsof theobjectpointswiththeabovetestingprocedurecarrytherelativeeffectofthereferencepointsif

theyhasundergonearigidbodydisplacement.This factmaynotbeso important

forsomestudies,for instance intectonic/velocitystudies,whicharemostlybased

on some relative functions. However, for some cases, for example in damage

monitoring of engineering buildings, it may be a problematic issue because an

analysist may interprete mistakenly that the object is under a damaging force

system. To be clear and not to cause a wrong or over interpretation, two

different/independent relative blocks may be chosen in the studied area, if it is

possible. Depending on these two relativetested blocks two different results are

obtained for theobjectpoints.Thenwemaycheckourobjectsdisplacementsby

comparing two results. Of course, there will be some statistical unbalancies

between two results, but it may be ignorableeffect compared to early or wrong

emergencyalarmforapossibledamage.


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The threshold value is then taken as 2/1f,1f,,1c tF = = which denotes percentage

pointofthetdistribution.

3.2 Localization

Localization is a procedure to identify the points having significant coordinate

changes.Mostly it isadapted inrelativedeformationnetworkstoseparatereference

points and object points, however it may be used also in absolute deformation

networkstodetectthedisturbingpoint(s)inthereferencepointset.

Therearethreelocalizationmethodscommonlyappliedindeformationanalysis;

i) Gausseliminationmethod

ii) Implicithypothesismethod

iii) Stransformationmethod

Wediscussonlyfirsttwoofthemforrelativedeformationnetworks.Afterwards,the

localizationprocedureinabsolutedeformationnetworksbyGausseliminationmethod

isexplained.

3.2.1 Localization with Gauss-elimination method

Iftheglobaltestshowssignificantlychangedpointsinourrelativedeformation

network,thenextstepisidentifyingorlocalizingthesepoints.Foreachpointinourc

dimensionalnetworkwecomputethefollowingeffectvalues

Ri= i1

iiTi~

P with =i AiA

1iii

~~~~ dPPd + , (i=1,2,,p) (3.3)

where

i : c1reduceddisplacementvectoroftheithpoint,


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i~d : c1displacementvectoroftheithpoint

ii~P : ccweightmatrixbelongingtotheithpoint(ithblock

diagonalof += dd

QP ),

A~d : (cpc)1vectorofdisplacementsoftheremainingpoints

denotedAnotincludingthepointiand,

iA~P : c(cpc)weightmatrixbetweenpointiandthe

remainingpoints(seeNote3.3).

The point resulting in maximum effect value by Eq. (3.3) is accepted as

significantly changed point. Let thispoint be thejthpoint. Then first object point isbeingdefined;

={j} (3.5)

Nowweshoulddefinethedatumofournetworkdependingontheremaining

p1 points. Let us denote these remaining points with B. Using the transformation

matrixSBdefiningthedatumaccordingtothepointsB,weobtain

=

d

ddS ~

~B

B , BS QddTBS =

QQ

QQ

~~

~~

B

BBB , (3.6)

OurnextaimistoinvestigatetheremainingpointsB.Iftheteststatistic

Note3.3:Displacementvectorsandtheirweightmatricesused inEq.(3.3)maybe

representedas

=

i~

~

d

dd A , Pdd

+= ddQ

=

iiiA

iAAA

~~

~~

PP

PP (3.4)


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f),F(hsh

~~~T B2

d

BBBTB

B =+

B

dQd , ( Bh =rank BB

~Q =rank ddQ c) (3.7)

is smaller than the threshold value 1f,,hBF , the localization procedure is ended. The

objectpointsandthereferencepointsarebeingseparatedalreadyas={j}and=B,

respectively. Otherwise, it is decided that the group of points B has significantly

changed point(s) and a new localization procedure is required. In that case, we

consider B~d and BB

~Q asdand ddQ inEq.(3.4),

B

~d

d ,BB

~Q

Qdd

,

(3.8)

and we search the point giving the maximum effect using Eq.(3.3) among the

remainingp1points.Theprocedure isrepeateduntiltheglobaltestshowsnomore

significantlychangedpoints.IneachlocalizationsteptheobjectpointsetinEq.(3.5)

isaugmentedwiththeneweridentifiedpoints.

For1Dnetworks, theeffectRi inEq. (3.3)maybecomputeddirectly:Let the

displacementvectoranditsweightmatrixgivenbyEq.(3.4)bewrittenasfollows

d=

n

1

d~

d~

M , Pdd+= ddQ =

nn1n

n111

P~P~

P~P~

L

MOM

L

(3.9)

ThemultiplicationdPddresultsin

=dPdd=

n

1

M , (3.10)

Dividing the elements of the vector in (3.10) by the weights in (3.9) gives the


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correspondingeffect

Ri=i/ iiP~ , (i=1,2,.,p) (3.11)

Hence, for 1D networks, the computation burden of Eq. (3.6) is drastically being

reduced.

3.2.2 Localization with implicit hypothesis method

Two periods GaussMarkov models are gained into a single GaussMarkov

modelbyEq.(2.5)as

=

2

1

2

1

2

1E

x

x

A0

0A

l

l ,

=

2

1

P0

0PP

OurhypothesisnowassumesthatthegroupofpointsA,whichdoesnotcontainthe

suspected point i, has not deformed. This hypothesis may be implicity incorporated

intotheaboveGaussMarkovmodelasfollows

=

2i

1i

A

2i2A

1i1A

2

1E

x

x

x

A0A

0AA

l

l ,

=

2

1

P0

0PP (3.12)

Suppose that the solution of model (3.12) yields the weighted sum of the squared

residuals iH)( iHTH )( Pvv= .Eachpointinthenetworkisattainedas i inmodel(3.12)in

turn,andwegetpeffectsforthepointsinthenetworkasfollows

Ri=( H )i , (i=1,2,,p) (3.13)

IfRjbelongingtothejthpointistheminimumeffectvalueamongtheothers,thispoint

isacceptedasthepointwithsignificantcoordinatechange.Itisourfirstobjectpoint:


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Thenwemaydefinetheobjectpointgroupas

={j} (3.14)

Nowweshouldlearnwhethertheremainingp1points,denotedB,stillconsist

ofanysignificantlychangedpoints.ForthisweusetheminimumdeficiencyRjtoset

thefollowingteststatistic

f),F(hsh

R

/f)(

)/h(RT B2

dB

jBj

B == (3.15)

wherehB=fHfc=hc.IftheteststatisticinEq.(3.15)isbiggerthanthecorresponding

thresholdvalue,i.e.

TB 1f,,hBF , (3.16)

weshould identify theresponsiblepoint(s)among thegroupofpointsB.For the ith

pointofB,wesetourGaussMarkovmodelasfollows;

=

2

1

2i

1i

D

222D

11i1D

2

1E

x

x

x

x

x

A0A0A

0A0AA

il

l ,

=

2

1

P0

0PP (3.17)

where

D : denotesthepointsexcepttheithpointinB(D{i}=B)

: showstheobjectpointidentifiedinthepreviouslocalizationstep(wedonotchangeoftheplaceofinEq.(3.17)duringthealllocalizationstepsanymore).

Nowinthesecondlocalizationstep,eachpointinBisattainedasiinEq.(3.17)

B


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by turnand theeffectsofallp1pointsarecomputedasrealizedbyEq. (3.13).The

pointgivingminimumeffectistakenasthenewobjectpointandweupdateourobject

pointdefinitioninEq.(3.14).Iftheglobaltest,whichissetaccordingtotheeffectof

thepoint identified in that second localization step (hBbecomesh2c), showsmore

suspectedpointsamong the remainingpoints, similarlywe continue localizing these

points.Theprocedure isrepeateduntilthecorrespondingglobaltestshowsnomore

significantlychangedpointinthenetwork.

3.2.3 Localization in absolute deformation networks

If the global test in Section 2.4 results in that our reference points is notstable,weshouldidentifythechangedpoint(s)amongthem.Onewayfordoingthisis

applying localizationprocedureto thereferencepoints:Forthisweconsider d and

Q belongingtothepreferencepointsinEq.(2.15)asdandQddin(3.4),

d d , Q Qdd (3.18)

andwestart localizationprocedurewithEq. (3.3) to identify theresponsiblepoint(s)

among p reference points. The localization procedure is similarly realized until the

correspondingglobal testshows thatremaining referencepointshavenoanypoints

disturbing thestabilityofour reference.The identifiedpointsare removed from the

referencepointsetandaddedtotheobjectpoints.


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4. SENSITIVITY ANALYSIS

Sensitivity analysis is used to optimize a deformation network such that it

becomes sensitive to the expected displacement, movement or deformation or to

derive theminimum detectable displacement,movementordeformationparameter

formeasuringthequalityofourdesign.

In theory itcanbeadapted toallkindofchanges tobemonitored inastudiedarea,howeverweexpressitjustfordisplacementshere.

4.1 Global Sensitiv ity Analysis

4.1.1 Optimization Criteria

ThehypothesesgivenbyEq.(2.3)isoriginallysetas

H0:E(d)=0 vs. H1:E(d)0= (4.1)

where

: u1vectorofexpecteddisplacements.

Because we are now at the design stage, we know that the alternative

hypothesisH1inEq.(4.1)istrue.Inthatcase,secondteststatisticinEq.(2.4)followsanoncentral2distribution;

)(h,'~T 22d

ddT

=+

dQd (4.2)


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where is the noncentrality parameter computed from the vector of expected

displacements asfollows;

2d

ddT

=

+Q

(4.3)

Once we obtain the noncentrality parameter in Eq. (4.3), the power of the

global test, i.e., theprobabilityofcorrectyaccepting the truealternativehypothesis,

maybecomputablefromthedistributionfunctionofthenoncentral2distribution,

F(2= 2 1h, ;h,),as

=1F(2= 2 1h, ;h,) (4.4)

Thepowerofthetestmathematicallyincreaseswithincreasingnoncentrality

parameterandwithdecreasingdegreesof freedomh. Itmeans that thepowerof

ourtestwillbebetterformoreprecisenetworkand lesspoints (rememberthath is

relatedwithnumberofpointsinanidenticalnetwork,i.e.h=ur=cpr).

Instead of computing the power of the test by Eq. (4.4), the noncentrality

parameteriscomparedwithanoncentralityparametergivingadesiredpowerofthe

test 0 and significance level 0. This parameter is called lower bound of the non

centralityparameter0(seeTable4.1).Ifthefollowinginequalityisfulfilled,

0,h (4.5)

thenetwork isdefinedassensitive to theexpecteddisplacements.Otherwise, the

networkisredesignedsuchthatitbecomessensitive.


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Table4.1Somelowerboundofthenoncentralityparameters(0,h)for0=80and

90%,0=5%and1h100

h 0=80% 0=90%1 7.85 10.51

2 9.64 12.65

3 10.90 14.17

4 11.94 15.415 12.83 16.47

10 16.24 20.53

20 20.96 26.13

30 24.55 30.38

40 27.56 33.9450 30.20 37.07

100 40.56 49.29

4.1.2 Minimum detectable displacement

Forecastingthedirectionsofthedisplacementsiseasierthansettingthevector

ofexpecteddisplacementsitself.Letthevector betheproductofascalefactor(b)

andagivendirectionvector(g),

=bg (4.6)

SubstitutingEq.(4.6)intoEq.(4.3)andusingEq.(4.5)weobtain

bmin=gQg

+

ddT

h,0

d (4.7)

Thenwedefinetheminimumdetectabledisplacementvectorasfollows;

min=bming (4.8)

In some cases, even directions of the displacements are not be available. To


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obtaintheminimumdisplacementvectorinsuchacase,weusetheeigenvectormax

belongingtothemaximumeigenvaluemaxofQdd.Thisresultsintheminimumvalue

ofthescalefactor

bmin=maxdd

Tmax

h,0

d

+

Q= maxh,0d (4.9)

Then, ifweconsiderEq.(4.9)inEq.(4.8),weobtainthedisplacementvectorwhichis

justdetectableonthedirectionsoftheeigenvectorwithaspecifiedpowerofthetest

0andsignificancelevel0.

4.2 Sensitiv ity Analysis in Absolute Deformation Networks

Let our object points be defined related to the reference points . The

hypothesestotesteachobjectpointaresetfollows;

H0:E( id )=0 , H1:E( id )0= i (4.10)

whereiistheexpecteddisplacementvectoroftheithobjectpoint.

LetusassumethatourteststatisticsetfordiscriminatingthehypothesesinEq.

(4.10)is2distributed.Sinceweknowthatthealternativehypothesisistruenow,the

Note4.1:Thedirectionvectorgforlevelingmonitoringnetworksconsistsof1for

the uplifted points, 1 for the subsided points and 0 for stable points. In 2D

networks (or in a horizontal plane) it includes cos and sin where is the

forecastedazimuthofthedisplacementvectorofthecorrespondingpoint.


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teststatistichasanoncentral2distribution;

iT= 2d

1T

iiii

dQd

2

(c,i) (4.11)

whereiisthenoncentralityparameter,

i= 2d

1T

iiii

Q

. (4.12)

Tolearnwhetherourexpecteddisplacementsfortheithpointisdetectableornot,thenoncentralityparameteriiscomparedwithitsboundaryvalue0,c;If

i0,c (4.13)

holds,thecorrespondingdisplacementissaidtobedetectablewiththecorresponding

powerofthetest.

To derive the minimum detectable displacement, we may follow the same

methodologygivenintheprevioussection.Insteadofgivingtheformulaforaspecific

direction, we consider the eigenvector of the maximum eigenvalue imax, of ii Q .

SimilartoEq.(4.9),weobtain

(bmin)i= imax,c,0d =( d imax, ) c,0 (4.14)

With this scale factor, theminimumdetectabledisplacementvectorof the ithpoint

becomes

min)( i =(bmin)i imax, (4.15)


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where imax, istheeigenvectorbelongingtothemaximumeigenvalue imax, .

Withtheabovementionedsimplestatistics intwopreviousnotes,one isable

to speak about the capacity of the designed network. The cofactor matrices of

displacements with respect to the reference points may easily be derived and the

pooledvariancemaybeguesseddependingontheexperiencesbeforeanyrealization.

Then theminimumdetectabledisplacementofanobjectpoint isabout3 timesof

the semimajor axis of its relative confidence ellipse and displacements standard

deviationin2Dand1Dnetworks,respectively.Ifourexpectationdoesnotmatchwith

Note4.2:For2Dnetworks,firsttermoftherighthandofEq.(4.14),i.e.,d

imax,

,

is in fact the semimajor axis of the relative error ellipse of the ith point. Then,

(bmin)i maybeconsideredasthesemimajoraxisoftherelativeerrorellipseforthe

power of the test (we may call this ellipse relative power ellipse!). Furthermore,

(4.15)showsthatminimumdetectabledisplacementisonthedirectionoverlapped

withthedirectionoftherelativeerrorellipseoftheithpoint.Thenin2Dexamples,

(bmin)i denotes the minimum detectable displacement magnitude of the

correspondingpoint.Theboundaryvalue isobtainedas0,c=2=9.64for80%power

of the test fromTable4.1. It means thatadisplacementwhose magnitude is3.1

times of the semimajor axis of the relative confidence ellipse may be just

detectable with 80% power of the test in an object point of an absolute

deformationnetwork.

For1Dnetworks, imax, becomesequaltothedisplacementscofactorvalueofthe

correspondingpoint.FromTable4.1weread0,c=1=7.85for80%powerofthetest.Then it is clear that a displacement (upliftor subsidence)may bejust detectable

with 80% power, if its magnitude is 2.8 times of the displacements standard

deviation.


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thismagnitude, thenwemay redesignournetworksuch that itsobjectpointshave

smaller ellipses or smaller standard deviations depending on the network type. For

example, letusassumethatweexpect5mmverticalmovement inaregion,andour

levelingnetworksobjectpoints relative toa referencepoint sethavearound2mm

standarddeviationinaperiod.Thenthedisplacementsstandarddeviationisexpected

as 22 =2,8mm.With80%power(ormore),theminimumdetectabledisplacementis

around32,8=8.4mm.Itmeansthatwemaynotabletodetect5mmmovementwith

80%powerofthetestusingthecorrespondingdesign.Forthatreason,weshouldplan

additional observations or should measure the network with more precise levels to

improve the precision of our points. Such an optimization is called trial and errormethod; but we may also set some analytical target functions to obtain a global

solution to this optimization problem. This is called analytical optimization of

deformation networks; however, nowadays, because of our improved computing

capabilitiesbynormalPCs,trialanderrormethodsmaybemorepreferable.Theyare

morerealisticbecausewemayproducetheproblemdependingonsomeexperiments

whichwemayfacewithinreality.Inanalyticaltools,sometimes,ifwedonotconsider

the constraints realistically (it is a little bit hard to consider all conditions

mathematically!),thesolutionmaygofarawayfromtheglobalsolutionandstopata

local one, which may mislead the practitioner, or, which may cause an another

problemwaitingforadifferentsolution.


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5. INTERPRETATION MODELS

Afterdefiningthereferencepointsinournetwork,wemayinvestigatetheobject

pointsmovementsandstrainelementsofsomeobjectblockstointerpretethemotion

andthedeformationoftheobject,respectively.Forthisweusekinematicmodeland

strain model. In this section we briefly explain these models used in deformation

analysis.

5.1 Kinematic Model

An object may change its position in time continuously related to a reference

frame.Thismotionisexpressedbythefollowingwellknownequation

x)t(t

2

1x)t(t)(t(t) 2000 &&& ++= (5.1)

where

t : currenttime,

t0 : initialtime,

(t) : current(present)position,

)(t0 : initialposition,

x& : velocityand

x&& : acceleration.

If theparameters )(t0 , x& and x&& areavailable,onemaypredict theobjects

positioninanyperiodfromEq.(5.1).Reversely,ifwehavecoordinatesofthepointin


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differentperiods,wemayestimatetheparameterstocreateamodelfortheobjects

movement. This is called kinematic model in deformation analysis. To satisfy

redundancy inkinematicmodelling, thereshouldbeat least4periods.Hereafterwe

assumethereforethatwehavem4periods.

5.1.1 Single point model for an object points movement

5.1.1.1 Model I

Nowsupposethatweworkwitha1Dnetwork,orwewouldliketomodelonly

onecomponentofapointamongitsothercomponents(asrealizedmostlyforNorth,

East and Up components of a GNSS station). For this we set the following Gauss

MarkovmodelfromEq.(5.1)havingtakenthe initialperiodast1andpartitioningthe

initial(unknown)position )(t0 intotwopartsas )(t0 = + )(t1 ,

=

x

x

)t(t5.0)t(t1

)t(t5.0)t(t1

)t(t5.0)t(t1

y

y

y

E

21m1m

21212

21111

m

2

1

&&

&MMMM

,

=

1mm

122

111

20

Q00

0Q0

00Q

L

MOMM

L

L

P (5.2)

where

iy : c1(diminished)coordinatecomponent(observation),( =iy )(t)(t 0i ),

)(ti : coordinatecomponentintheithperiod,

: unknownshiftparameter,

x& : unknownvelocity,

x&& : unknownacceleration,

20 : apriorivarianceofunitweightand

Qii : cofactorvalueofthecorrespondingcoordinate

component.


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5.1.1.2 Model II

Forcdimensionalnetworks,consideringmodel(5.2)maycauseoveroptimistic

results because in that case we neglect the correlations between the coordinate

components of a point. They are in fact highly correlated therefore they may be

consideredinasingleGaussMarkovmodel

=

x

x

III

III

III

y

y

y

&&

&MMMM

21m1m

21212

21111

m

2

1

)t(t5.0)t(t

)t(t5.0)t(t

)t(t5.0)t(t

E ,

=

1mm

122

111

20

Q00

0Q0

00Q

P

L

MOMM

L

L

(5.3)

where

iy : c1(diminished)coordinate(observation)vector( =iy )(t)(t 1i ),

)(ti : c1coordinatevector,

I : ccidentitymatrix,

: c1unknownshiftparametervector,

x& : c1vectorofunknownvelocities,

x&& : c1vectorofunknownaccelerationsalongthecorrespondingaxes,


iiQ : cccofactormatrixbelongingtothecorrespondingpointintheithperiod.

5.1.1.3 Model III

In some cases previous two models may not yield satisfactory estimates

because of some unmodelled physical effects on the coordinates. They are mostly


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observed as periodic changes in time series of a points coordinate component as

showninFig5.1.

Ifwehaveapriorinformationabouttheperiodicalpartofthechanges,insteadofEq.(5.19)wemayconsider

++= x)t(t)(t(t) 00 & periodicpart (5.4)

wheretheperiodicpartisalinearfunctionoftheeffectofthecorrespondingphysical

sources. (Thepointsmovementmay includealsoanaccelerationpart;however, for

simplicitywedropitinEq.(5.4))

Fig5.1Coordinatechangesintimedomain

Theperiodicitymayhappenhourly,daily,annuallyorsemiannuallydepending

on the sources. They are commonly modelled using a1cos(2t/Tp)+a2sin(2t/Tp)

function,whereTp istheknownperiod,a1anda2aretheunknownamplitutes.Letus

consider thatourperiodicpartmaybemodelledwith this function:Then insteadof

model(5.2)wemaysetthefollowingGaussMarkovmodel

Periodicpart

Observation

Velocity

Time

Coordinate


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34

=

2

1

pmpmm

p2p22

p1p11

m

2

1

a

a

x

)/Tt2sin()/Tt2cos(t1



y

y

y

E &

MMMMM ,

=

1mm

122

111

20

Q00

0Q0

00Q

L

MOMM

L

L

P (5.5)

where =iy )(t)(t 0i and it =tit1.

5.1.2 Single model for whole object points movement

Kinematicmodel (5.1)maybeestablished forallobjectpointsunderasingle

model.Forp=ppobjectpoints inthenetwork,which ismeasuredmperiods,the

GaussMarkovmodelforsuchamodellingiswrittenasfollows

=

x

x

III

III

III

y

y

y

&&

&MMMM

21m1m

21212

21111

m

2

1

)t(t5.0)t(t

)t(t5.0)t(t

)t(t5.0)t(t

E ,

=

1xx

1xx

1xx

20

mm

22

11

Q00

0Q0

00Q

P

L

MOMM

L

L

(5.6)

where

Note5.1:Inmostproblems,theperiodTpisnotavailableornotexact.Inthatcase,

w=2/TpangularfrequencymaybeobtainedbyapplyingFastFouriertransformto

theobservationsbeforehand.


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iy : cp1(diminished)coordinate(observation)vectorfortheithperiod( iy = 1i ),

i : cp1coordinatevectoroftheithperiod;

I : cpcpidentitymatrix,

: cp1unknownshiftparametervector,

x& : cp1vectorofunknownvelocities,

x&& : cp1vectorofunknownaccelerations,


iixxQ : cpcpcofactormatrixbelongingtotheobjectpointsin

theithperiod.

5.1.3 Testing model parameters

Before publishing the kinematic model of an object point, we should test its

parameters(mostlyvelocityandacceleration)tolearnwhethertheyaresignificantor

not.Testingprocedureisthereforecalledsignificancytest.

Let us consider that the velocity estimate x& with standard deviation xs & is

desiredtobetested;forthiswesetthefollowinghypotheses

H0:E( x&)=0 , H1:E( x&)0 (5.7)

Thentheteststatisticfollowstdistribution

xx s

|x|

T &&

&

= t(f) (5.8)

where f is the degrees of freedom of the corresponding model. If xT& < 2/1f,t , the

estimatedvalue x& isnotsignificant.Otherwise, itisacceptedthat ithasasignificant

physicalmeaning.


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5.1.4 Model test

There may exist different kinematic models for the timedependent

observations. To verify which model fits better to the observations, we may apply

modeltest:Forexample,letustakethemodel(5.2)andcallitmodel1.Itsalternative

onemaybe themodelwithoutanaccelerationparameter, i.e.velocitymodel; letus

callitmodel2.Fromeachmodelweestimatetheunknownsandobtaintheweighted

squaresoftheresiduals;i.e.weobtain1and2quadraticformsindependently.The

followingteststatisticfollowsFdistribution,withf2f1andf2degreesoffreedom,

11

1212M

/f

)f/(f)(T

= F(f2f1,f2) (5.9)

Note5.2:Foreachestimatedparameterthesametestingproceduregivenaboveis

realized.Insignificantparametersmaybeextractedfromthecorrespondingmodel,

and the estimation procedure is repeated having established the corresponding

modelwiththeremainingsignificantparameters.Thiswillincreasetheredundancy,

i.e. degrees of freedom, and will result in more precise estimation. In some

applications,forexample inGNSSstudieswith longtimeseries,theredundancy is

alreadybig, therefore, the testingproceduremay be unnecessary: Theestimated

values and their standard deviations are declared, for example, as velocityits

standarddeviation.This iscalledsometimesvelocitywith1sigmaerror: Ifwe

have big redundancy, the accepted onedimensional tdistribution gets close to

normaldistribution and this interval shows a confidence interval with about 40%

probability. If we declare velocity with 2sigma error, from the normal

distributionfunction,weunderstandthatthe intervalshowsaconfidence inverval

withaprobabilitymorethan95%.


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where

f1 : degreesoffreedomofmodel1and

f2 : degreesoffreedomofmodel2.

Wecompare MT withthethresholdvalue 1,,fff 112F :

i) If MT < 1,,fff 112F , model 1 is not necessary. In other words, instead of

accelerationandvelocityparameters, it isbetter toconsideronlyvelocity

parameter.

ii) MT 1,,fff 112F , model 1 fits better to the observations: Model 2 does not

ensure the essential information to model the timedependent

observations.

Testingmodel1againstmodel2withtheabovementionedmodeltestpractically

maybedonewiththeprevioussignificancytest: Iftheaccelerationssignificancytest

fails, itmeans that, model 2 (themodel withonlyvelocity) should beconsidered to

model the observations. But, at this point, we should remind that, statistically and

theoretically, model test is more correct because the previous testing procedure

neglectsthecorrelationsbetweentheestimatedparameters.

Thegivenmodel testproceduremaybeapplied forcomparingdifferent typesof

models,notonly forcomparing thevelocitymodelandvelocity+accelerationmodel:

Weshouldjustcareaboutthatmodel1istobeattainedasanaugmentedmodelwith

additionalparameterswhicharenotincludedinmodel2.

5.2 Strain Analysis

5.2.1 Definition

Strain is defined as the ratio of increase or decrease in length to its original


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38

length.Itisanormalizedmeasurefordeformation.Forinstance,letusconsiderawire

withL1=100m lengthhasextended toL2=100.02m; then theengineeringstrain, the

socallednominalstrain,iscomputedasfollows

= 5

1

12 102100

10002.100

L

LL =

=

strain=20strain=20ppm.

Thisstrainmaybedenotedas

1L

dL

LengthOriginal

ntDisplaceme==

(5.10)

On the other hand, scale factor , the socalled stretch ratio, is related with the

engineeringstrainby

=1+ (5.11)

which is the one commonly used in geodesy to explain the deformations of the

coordinateaxes,forexampleinsimilarityandAffinetransformations.

In two dimensional, instead of a single strain measure, there exists a strain

tensor,

=

=

ydy/ydy/

ydx/xdx/

yyyx

xyxxE (5.11)

where

dxanddy : displacementsofaparticleintheobjectinxandydirections.

Since we assume the object is continuum, i.e. the object is full of homogeneous

particles, the tensor elements in Eq. (5.11) represent the deformation of the whole


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39

object. There are some other quantities obtained from the elements of this strain

tensor,suchas,

Dilation(meanstrain): mean=2

1(xx+yy) (5.12a)

Pureshear: pure=2

1(xxyy) (5.12b)

Simpleshear: simple=2

1(xy+yx) (5.12c)

Totalshear: shear= 2simple2pure + = ( )2yxxy2yyxx )()(

2

1++ (5.12d)

Differentialrotation: =2

1(yxxy) (5.12e)

Inearthsciences, insteadofthestrain tensorE,symmetricalstrain tensor sE ,

whichisderivedfromEq.(5.11),isused;

=

+

+=

yy

xx

yyyxxy

yxxyxx

s 2/)(

2/)(

simple

simpleE (5.13)

To show the object deformation in 2D, principal strain components, i.e., the

eigenvaluesof sE arederivedfrom(5.13)

)4)((2

1 2simple

2yyxxyyxxmax +++= =mean+shear (5.14a)


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40

)4)((2

1 2simple

2yyxxyyxxmin ++= =meanshear (5.14b)

withthedirectionofthemaximumprincipalaxis,clockwisefromxaxis(seeNote5.3),

= =

yyxx

simple2atan2

1

pure

simpleatan2

1 (5.14c)

max shows the greatest change while min is the smallest change of length per unit

length.TheyareplottedonthecentroidoftheobjectasshowninFig5.1.Thenegative

sign of any component shows contraction whereas positive sign denotes extensionthroughthecorrespondingdirection.

Fig.5.1.Principalstraincomponents

x

|max|

|min|

max0 max>0,min0 max,min


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5.2.2 Strain modell ing in geodetic deformation analysis

Fortheithobjectpointhavingdxianddyidisplacements,wemaywrite

dxi=tx+xxxi+xyyi and dyi=ty+yxxi+yyyi (5.15)

whicharethefundamentalequationsformodellingstrainofthecorrespondingobject.

Inadditionto4strainparameters(xx,xy,yx,yy)wehavetwotranslationparameters

tx and ty in Eq. (5.15), therefore, to obtain these 6 parameters, mathematically we

needatleast3points.

For modelling strain of the studied object, the structural properties of the

objectshouldbeknownpriorily.Fromsuchaprior information theobject isdivided

into the different blocks as demonstrated in Fig 5.2. For each block we consider

differentstrainmodel(seeNote5.4).

Note5.3:Theprincipleofcomputationoftheprincipalstrainangle issimilarto

theoneofcomputationofazimuth:First,2=atan(simple/pure)=aisobtained;i)if

simple>0andpure>0,=a/2,ii)ifsimple>0andpure


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42

Fig5.2.Objectblocksandtheirprincipalstraincomponents

Nowsuppose thatourobjectconsistsofoneblock; thenallobjectpointsare

includedinasinglestrainmodel.Forthis,ourGaussMarkovmodelissetasfollows

=

yy

yx

xy

xx

y

x

pp

pp

11

11

p

p

1

1t

t

yx0010

00yx01

yx0010

00yx01

dy

dx

dy

dx

E MMMMMMM E{d}=Mx (5.16a)

withthe2p2pmatrixofweightsofdisplacements,

Note5.4:Objectblocksshouldbeconsideredatthedesignstagesothateachblock

has its own object points characterizing the deformation to be monitored. An

attempt for deciding object blocks considering only the observed displacements

mayyieldwronginterpretations.

Afterstrainmodelling

Object

Beforerealization

BlockI

BlockII

Object


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43

P=

1

dydy

dydxdxdx

dydydxdydydy

dydxdxdxdydxdxdx

20

pp

pppp

m1p111

p1p11111

QQQ

QQQ

QQQQ

MMO

L

L

P= 20 Q (5.16b)

where

dandQ : vectorofdisplacementsanditscofactormatrix

belongingtotheobjectpoints,respectively,fromEq.(2.15),

M : 2p6coefficientmatrixand

x : 61unknownparametervector.

Solving model (5.16) by leastsquares method, the parameter vector x is

estimatedandsowegetstrainparametersxx,xy,yxandyyfortheobject.Byusing

themtheprincipalstrainparametersinEq.(5.14)areobtainedandtheyareplottedon

thecentroidoftheobjectunderconsiderationasshowninFig5.3.

Fig5.3Principalstrainparametersforanobject

Formoreblocks,theirindependentstrainmodelsmaybeconsideredinasingle

GaussMarkov model: For instance, for two object blocks (Block I and Block II) we

Object


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44

establishthefollowingGaussMarkovmodel;

=

II

I

II

I

II

I

E x

x

M

M

d

d

,

1

2

0II

II

= IIIII

III

QQ

QQ

P (5.17)

Fromthesolutionofmodel(5.17),weobtainxIandxIIparametervectorsincludingthe

blocksstrainparameters.


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APPENDIXA:GLOSSARY

Absolutedeformationnetwork:Mutlakdeformasyona

Acceleration:vme

Block:Blok

Confidencelevel:Gvendzeyi

Contraction:Klme

Constraintequations:Kouldenklemleri

Currentperiod:Mevcutperiyot

Currenttime:Mevcutzaman

Deformation:Deformasyon

Degreesoffreedom:Serbestlikderecesi

Diminishedobservation:Kltlm l

Directionvector:Ynvektr

Displacement:Yerdeiim

Extension:Genileme

Gausseliminationmethod:Gausseliminasyonyntemi

Globaltest:Globaltest

Identical:Edeer

Identitymatrix:Birimmatris

Implicithypothesis

method:Kapalhipotezyntemi

Initialperiod:Balangperiyodu

Initialtime:Balangzaman

Kinematicmodel:Kinematikmodel

Localization:Yerelletirme

Lowerboundofthenoncentralityparameter:D merkezlikparametresininsnrdeeri

Minimumdetectabledisplacement:Belirlenebilirenkkyerdeiim

Minimumconstrained:Zorlamasz

Monitoring:zleme

Modeltest:Modeltesti

Noncentral:Merkezselolmayan

Noncentralityparameter:D merkezlikparametresi

Nonidentical:Edeerolmayan

Object:Nesne


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Objectblock:Objeblou

Objectpoint:Objenoktas

Partialtraceminimum:Ksmiizminimum

Period:PeriyotPooledvariancefactor:Birletirilmi varyansarpan

Powerofthetest:Testgc

Principalstrainparameters:Asalgerinimparametreleri

Referenceblock:Referansblou

Referencepoint:Dayanaknoktas

Relativeconfidenceellipse:Balgvenelipsi

Relativedeformationnetwork:Baldeformasyona

Rotation:Dnklk

Quadraticform:Kareselbiim

Sensitivity:Duyarllk

Shiftparameter:Sfreki

Significancelevel:Yanlmaolasl

Significancytest:Anlamllktesti

Significant:Anlaml

Stable:Duraan

Strain:Gerinim

Straintensor:Gerinimtensr

Subsidence:kme

Teststatistic:Testbykl

Thresholdvalue:Karlatrmadeeri

Traceminimum:Tmizminimum

Translation:teleme

Undeformed:Deformeolmam

Uplift:Ykselme

Velocity:Hz


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APPENDIXB:THRESHOLDVALUES(Fandtdistributions)

TableB1.ThresholdvaluesforFdistribution(*)for=5%(Fa,b,1)

*)SquarerootofFvaluefora=1andbinthefirstrowyieldstb,1/2

1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100

1

2

3

4

5

6

7

8

9

10

20

30

40

50

60

70

80

90

100

161.45 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.35 4.17 4.08 4.03 4.00 3.98 3.96 3.95 3.94

199.50 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.49 3.32 3.23 3.18 3.15 3.13 3.11 3.10 3.09

215.71 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.10 2.92 2.84 2.79 2.76 2.74 2.72 2.71 2.70

224.58 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 2.87 2.69 2.61 2.56 2.53 2.50 2.49 2.47 2.46

230.16 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 2.71 2.53 2.45 2.40 2.37 2.35 2.33 2.32 2.31

233.99 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 2.60 2.42 2.34 2.29 2.25 2.23 2.21 2.20 2.19

236.77 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 2.51 2.33 2.25 2.20 2.17 2.14 2.13 2.11 2.10

238.88 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.45 2.27 2.18 2.13 2.10 2.07 2.06 2.04 2.03

240.54 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.39 2.21 2.12 2.07 2.04 2.02 2.00 1.99 1.97

241.88 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.35 2.16 2.08 2.03 1.99 1.97 1.95 1.94 1.93

248.01 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.12 1.93 1.84 1.78 1.75 1.72 1.70 1.69 1.68

250.10 19.46 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.04 1.84 1.74 1.69 1.65 1.62 1.60 1.59 1.57

251.14 19.47 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 1.99 1.79 1.69 1.63 1.59 1.57 1.54 1.53 1.52

251.77 19.48 8.58 5.70 4.44 3.75 3.32 3.02 2.80 2.64 1.97 1.76 1.66 1.60 1.56 1.53 1.51 1.49 1.48

252.20 19.48 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 1.95 1.74 1.64 1.58 1.53 1.50 1.48 1.46 1.45

252.50 19.48 8.57 5.68 4.42 3.73 3.29 2.99 2.78 2.61 1.93 1.72 1.62 1.56 1.52 1.49 1.46 1.44 1.43

252.72 19.48 8.56 5.67 4.41 3.72 3.29 2.99 2.77 2.60 1.92 1.71 1.61 1.54 1.50 1.47 1.45 1.43 1.41

252.90 19.48 8.56 5.67 4.41 3.72 3.28 2.98 2.76 2.59 1.91 1.70 1.60 1.53 1.49 1.46 1.44 1.42 1.40

253.04 19.49 8.55 5.66 4.41 3.71 3.27 2.97 2.76 2.59 1.91 1.70 1.59 1.52 1.48 1.45 1.43 1.41 1.39

b

a

Documents

Geodetic Deformation Analysis