geodetic deformation analysis
TRANSCRIPT
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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
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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,
20 : apriorivarianceofunitweightand
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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=
2
1
pmpmm
p2p22
p1p11
m
2
1
a
a
x
)/Tt2sin()/Tt2cos(t1
)/Tt2sin()/Tt2cos(t1
)/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,
20 : apriorivarianceofunitweightand
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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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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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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)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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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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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