Analysis of Geodetic Time Series Using Allan Variances

72 Pages·2010·3.02 MB·English

Universität Stuttgart
Geodätisches Institut
Analysis of Geodetic Time Series
Using Allan Variances
Studienarbeit im Studiengang
Geodäsie und Geoinformatik
an der Universität Stuttgart
Thomas Friederichs
Stuttgart, November 2010
Betreuer: Prof. Dr.-Ing. NicoSneeuw
UniversitätStuttgart

Erklärung der Urheberschaft
IcherklärehiermitanEidesstatt, dassichdievorliegendeArbeitohneHilfeDritterundohne
BenutzungandereralsderangegebenenHilfsmittelangefertigthabe;dieausfremdenQuellen
direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Arbeit
wurde bisher in gleicher oder ähnlicher Form in keiner anderen Prüfungsbehörde vorgelegt
undauchnochnichtveröffentlicht.
Ort,Datum Unterschrift
III

Summary
The Allan variance is a statistical measure, developed in the 1960’s by the American physicist
David W. Allan. With its aid, data series measured by devices like oscillators or gyroscopes
can be analyzed with regard to their stability. In contrast to the Allan variance, the standard
varianceasameasureoftotalsignalpower,isnotabletocharacterizesignalstability.
ThereexistfurtherdevelopmentsoftheAllanvariance. Thisstudentresearchprojectconsiders
mainlynon-overlapping,overlappingandmodifiedAllanvariances.
The result of an Allan variance computation is the so-called σ-τ-diagram. This diagram pro-
vides information about the stability and beyond, it allows identification of various random
processesthatexistintheseriesofmeasurement.
TheAllanvariancemaybecomputeddirectlyinthetimedomainaswellasviathefrequency
domainusingthepowerspectraldensityofthetimeseriesandatransferfunction.
AdomainconversionbetweentheAllanvarianceandthepowerspectraldensityisonlyunidi-
rectional. Moreprecisely, onecancomputetheAllanvariancebymeansofthepowerspectral
density,butnotviceversa.
This student research project takes up the challenge of applying the concept of the Allan
variance to geodetic time series (pole coordinates as part of the Earth orientation parameters,
GPS measured coordinates of one position, Scintrex CG-5 gravimeter data and GOCE gravity
gradients,inadditiontooscillatorfrequencies).
TheAllanvarianceturnsouttobeareasonablestatisticalmeasureforanalysisofgeodetictime
series. The Allan variance, or better the Allan deviation, especially in an entire diagram, can
be considered as a form of spectral analysis. Having said this, it is possible to consider the
averagingintervalτ astheinvertedfrequency.
V

VII
Contents
1 Introduction 1
2 TimeDomainStabilityAnalysis 3
2.1 TimingSignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 StandardVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 AllanVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3 OverlappingAllanVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.4 ModifiedAllanVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.5 Overviewofdevelopedvariances . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 TheresultofanAllanvariancecomputation . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 TheSigma-Tau-Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 ComparisonofthreedifferentAllanvarianceplotsonthebasisofashow-
casedataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Computationtimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.4 AccuracyversusStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 FrequencyDomainStabilityAnalysis 19
3.1 NoiseSpectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 SpectralAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 DomainConversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Applicationtogeodetictimeseries 25
4.1 Oscillatorfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 EarthOrientationParameters: Polecoordinates . . . . . . . . . . . . . . . . . . . . 29
4.2.1 x-componentofpolecoordinates . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.2 y-componentofpolecoordinates . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 GPSmeasuredcoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.1 AbsolutevaluesofcomplexGPSpositiondata . . . . . . . . . . . . . . . . 35
4.3.2 ArgumentsofcomplexGPSpositiondata . . . . . . . . . . . . . . . . . . . 36
4.4 ScintrexCG-5Gravimeterdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5 GOCEgravitygradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.1 GOCEgravitygradients T , T and T . . . . . . . . . . . . . . . . . . . 40
xx yy zz
4.5.2 GOCEgravitygradients T . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
xy
5 Discussion 49
A Appendix XV
B Appendix XVII
VIII
C Appendix XXI
C.1 Calculation of the sigma-tau diagrams for non-overlapping, overlapping and
modifiedAllandeviation(Time-Domain-Based) . . . . . . . . . . . . . . . . . . . XXI
C.2 Calculation of the sigma-tau diagrams for overlapping and modified Allan de-
viation(Frequency-Domain-Based) . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI
C.3 Mainprogramsfortheanalysisofgeodetictimeseries . . . . . . . . . . . . . . . . XXI
C.4 Additionalhelperfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXII
IX
List of Figures
1.1 Example for random processes, for which the standard deviation does not con-
verge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Samplingtime,observationtimeanddeadtime. . . . . . . . . . . . . . . . . . . . 4
2.2 Simulatedtimedeviation x(t)andfractionalfrequencyploty(t). . . . . . . . . . 7
2.3 Comparisonofnon-overlappingandoverlappingsampling . . . . . . . . . . . . 8
2.4 DerivationmodelfortheoverlappingAllanvariance . . . . . . . . . . . . . . . . 9
2.5 Frequencyvaluesofanoscillatorinone-secondcycles . . . . . . . . . . . . . . . . 12
2.6 Frequencyvaluesofanoscillatorintwo-secondcycles . . . . . . . . . . . . . . . . 12
2.7 Frequencydataofanoscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 σ-τ diagramwithnon-overlappingAllandeviationonthebasisofoscillatordata 15
2.9 σ-τ diagramwithoverlappingAllandeviationonthebasisofoscillatordata . . . 15
2.10 σ-τ diagramwithmodifiedAllandeviationonthebasisofoscillatordata . . . . 16
2.11 Accuracyandstabilityarenotthesame! . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Noisetypeandtimeseriesforasetofsimulatedphasedata. . . . . . . . . . . . . 20
3.2 Slopesofcommonpowerlawnoiseprocesses . . . . . . . . . . . . . . . . . . . . . 21
3.3 TransferfunctionoftheAllan(two-sample)timedomainstability . . . . . . . . . 23
3.4 Overviewofdomainconversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Frequencydataofa10MHzreferenceofanAgilentN9020Aspectrumanalyzer . 26
4.2 σ-τ-diagramwithAllandeviationsforanalyzedoscillator . . . . . . . . . . . . . 27
4.3 σ-τ-diagramwithmodifiedAllandeviationsforanalyzedoscillator . . . . . . . . 28
4.4 Polarmotionfrom1990upto2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 x-coordinateofpolefrom1990upto2007 . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 σ-τ-diagramwithAllandeviationsforx-coordinatesofpole . . . . . . . . . . . . 30
4.7 σ-τ-diagramwithmodifiedAllandeviationsforx-coordinatesofpole . . . . . . 31
4.8 y-coordinateofpolefrom1990upto2007 . . . . . . . . . . . . . . . . . . . . . . . 32
4.9 σ-τ-diagramwithAllandeviationsfory-coordinatesofpole . . . . . . . . . . . . 32
4.10 σ-τ-diagramwithmodifiedAllandeviationsfory-coordinatesofpole . . . . . . 33
4.11 ScatterplotofGPSmeasuredpositiondata . . . . . . . . . . . . . . . . . . . . . . 34
4.12 Acomplexnumberinthecomplexplane . . . . . . . . . . . . . . . . . . . . . . . 34
4.13 TimeserieswithabsolutevaluesofcomplexGPSpositiondata . . . . . . . . . . . 35
4.14 σ-τ-diagramsforabsolutevalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.15 TimeserieswithargumentsofcomplexGPSpositiondata . . . . . . . . . . . . . 36
4.16 σ-τ-diagramsforarguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.17 Preprocessedgravimeterdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.18 σ-τ-diagramwithAllandeviationsforgravimeterdata . . . . . . . . . . . . . . . 38
4.19 σ-τ-diagramwithmodifiedAllandeviationsforgravimeterdata . . . . . . . . . 38
4.20 DifferenceinADEVcomputationbetweentrendreducedandtrendaffecteddata 39
X
4.21 GravitygradientsinalongtrackdirectionandanappropriateFourierseries . . . 41
4.22 Reducedgravitygradientsinalongtrackdirection . . . . . . . . . . . . . . . . . . 41
4.23 OriginalandreducedgravitygradientsofthethreemaindiagonaltermsT ,T
xx yy
and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
zz
4.24 σ-τ-diagramsofthethreemaindiagonalterms T , T and T . . . . . . . . . . 44
xx yy zz
4.25 OriginalgravitygradientsofthethreemaindiagonaltermsT ,T ,T andtheir
xx yy zz
correspondingtime-domainbasedsigma-tauplots . . . . . . . . . . . . . . . . . . 45
4.26 OriginalgravitygradientsofthethreemaindiagonaltermsT ,T ,T andtheir
xx yy zz
correspondingsigma-tauplotsviaPSDandtransferfunction . . . . . . . . . . . . 46
4.27 Originalandreducedgravitygradientsoftheoff-diagonalterm T . . . . . . . . 47
xy
4.28 σ-τ-diagramsoftheoff-diagonalterm T . . . . . . . . . . . . . . . . . . . . . . . 48
xy
4.29 σ-τ-diagramsoftheoff-diagonalterm T . . . . . . . . . . . . . . . . . . . . . . . 48
xy
B.1 Sinussignal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII
B.2 sigma-tauplotforthesinussignal . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII
B.3 Graphicalsolutionofthedeterminationofthemaximum . . . . . . . . . . . . . . XIX