Regional gravity field modeling using airborne gravimetry data

197 Pages·2009·7.24 MB·English

by  Bas Alberts

Regional gravity field modeling using airborne gravimetry data
Bas Alberts
Publications on Geodesy 70

NCG Nederlandse Commissie voor Geodesie Netherlands Geodetic Commission

Delft, September 2009
Regional gravity field modeling using airborne gravimetry data
Bas Alberts
Publications on Geodesy 70
ISBN: 978 90 6132 312 9
ISSN 0165 1706
Published by: NCG, Nederlandse Commissie voor Geodesie, Netherlands Geodetic Commission,
Delft, the Netherlands
Printed by: Optima Grafische Communicatie, Optima Graphic Communication, Rotterdam,
the Netherlands
Cover illustration: Bas Alberts
NCG, Nederlandse Commissie voor Geodesie, Netherlands Geodetic Commission
P.O. Box 5030, 2600 GA Delft, the Netherlands
T: [phone]
F: [phone]
E: [email protected]
W: www.ncg.knaw.nl
The NCG, Nederlandse Commissie voor Geodesie, Netherlands Geodetic Commission is part of
the Royal Netherlands Academy of Arts and Sciences (KNAW).
Acknowledgments
Many years ago someonementionedto me, ’Isn’t itgreat thatgravity can be measured
in such a dynamic environment as aboard an aircraft?’. This caught my attention, and
sincethenIhavebeenworkingonvariousprojectsrelatedtoairbornegravimetry,finally
resultinginthisPhDthesis. Icannowconfirmthatmeasuringgravityaboardanaircraft
isagreatfeatindeed. Abookonthistopicandtheresearchitrepresentscannotbecom-
pletedsuccessfullywithoutthesupportfromalargenumberofpeople.
Foremost,IwouldliketothankRolandKleesforhissupervisionandguidanceduringmy
PhD.Itisimpossiblenottogetaffectedbyhisenthusiasmandbothhisencouragement
and expertise have helped me very much in completing this work. Many thanks go to
PavelDitmar,whosefeedback, resourcefulnessandcreativitycontributedgreatlytomy
work. Iconsidermyselfveryluckyhavinghadtwosupervisorswhowerealwayswilling
tolistenandgivevaluableadvice.Iespeciallyappreciatedtheweeklymeetingswithboth
ofthem,whichwerethesourceofmanygoodideasandfruitfuldiscussions.
Furthermore,Ithankallmy(former)colleaguesatDEOSandespeciallythePSGgroup
for their support and for providing an enjoyable working environment. Special thanks
go to Jasper van Loon, with whom it has been a pleasure sharing an office throughout
thepastsevenyears. Ireallyenjoyedthediscussionswhichrangedfromworktosoccer,
butalsothetripstoconferencesinEuropeandAustralia. Ialsothankeveryoneinvolved
in the GAIN project, especially Brian Gunter for his support and discussions on strap-
down airborne gravimetry. I hope the GAIN project will be very successful and many
testflightswillbeperformedinthenearfuture. IthankRellyvanWingaardenandFiona
Tuynmanfortheadministrativesupport.
IamgratefultoGFZPotsdamforprovidingtheairbornegravitydataoftheAGMASCO
andCHICAGO campaigns. IthankJu¨rgenNeumayerofGFZ andUwe MeyerofBGR
for explainingthe airborne gravity pre-processing sofware (AGS) to me and Guochang
XuofGFZforhishelpwithGPSdataprocessing.
IappreciatethewillingnessofmytwoparanymphsFreekvanLeijenandLucAlbertsto
readthemanuscript,whichhelpedreducethenumberofremainingerrorssignificantly.
Thankstoallmyfriendsfor themanynicetimesoutsideworkinghourssuchasduring
i
Acknowledgments
holidays,barevenings,boardgames,cycling,soccer,snowboarding(ok,skiingaswell),
barbecuesandalltheotherrelaxingmoments. Focusingisonlypossibleiftherearemo-
mentsthatyoudonothavetofocus.
I would like to say special thanks to my mother and John, to my brother Luc, and to
my father and Anjenet, for their overall support, encouragement and showing so much
interestinmywork.
Finally, I would like to thank Hanne for helping me with the graphics, correcting my
English,butforemostforhersupportthroughoutlastyear. ThefinalyearofmyPhDwas
theoneIenjoyedmost.
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Contents
Summary vii
Samenvatting ix
Nomenclature xiii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Airbornegravimetry 7
2.1 Historicaloverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Theprincipleofairbornegravimetry . . . . . . . . . . . . . . . . . . . . 10
2.3 Mathematicalmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Measurementmodel . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Errormodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Applicationsandopportunities . . . . . . . . . . . . . . . . . . . . . . . 16
3 Processingofairbornegravitydata 21
3.1 Pre-processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Low-passfiltering . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Cross-overadjustment . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Inversionofairbornegravitydata . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Remove-restoretechnique . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Integralmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.3 Least-squarescollocation. . . . . . . . . . . . . . . . . . . . . . 31
3.2.4 Sequentialmultipoleanalysis . . . . . . . . . . . . . . . . . . . 35
3.3 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Combineddataprocessingandinversion 41
4.1 Gravityfieldrepresentation . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Inversionmethodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Least-squaresestimation . . . . . . . . . . . . . . . . . . . . . . 44
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4.2.2 Solutionstrategies . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Regularizationandparameterchoicerule . . . . . . . . . . . . . . . . . 47
4.3.1 Regularizationmethod . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.2 Regularizationerror . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.3 Relationtoleast-squarescollocation . . . . . . . . . . . . . . . . 50
4.3.4 Parameterchoicerule. . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Frequency-dependentdataweighting . . . . . . . . . . . . . . . . . . . . 54
4.4.1 ARMAfiltersandToeplitzsystems . . . . . . . . . . . . . . . . 55
4.4.2 ARMAfilteringinthepresenceofdatagaps. . . . . . . . . . . . 58
4.4.3 Descriptionofthenoisemodel . . . . . . . . . . . . . . . . . . . 60
4.5 Estimationofnon-gravitationalparameters. . . . . . . . . . . . . . . . . 64
4.5.1 Biasanddrifthandling . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.2 Estimationofscalefactors . . . . . . . . . . . . . . . . . . . . . 66
4.5.3 Testingofnon-gravitationalparameters . . . . . . . . . . . . . . 67
4.6 Edgeeffectreduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6.1 Extensionofthecomputationarea . . . . . . . . . . . . . . . . . 69
4.6.2 Modificationofthebasefunctions . . . . . . . . . . . . . . . . . 71
4.7 Combinationwithpriorinformation . . . . . . . . . . . . . . . . . . . . 73
4.7.1 Additionofpseudo-observations . . . . . . . . . . . . . . . . . . 74
4.7.2 Additionoffixedconstraints . . . . . . . . . . . . . . . . . . . . 74
5 Applicationtosimulateddata 77
5.1 Descriptionofthedata . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Computationswithnoise-freedata . . . . . . . . . . . . . . . . . . . . . 79
5.2.1 Gravityfieldrepresentationanddownwardcontinuation . . . . . 79
5.2.2 Reductionofedgeeffects. . . . . . . . . . . . . . . . . . . . . . 82
5.3 Computationswithdatacorruptedbywhitenoise . . . . . . . . . . . . . 86
5.3.1 Choiceofregularizationmatrix . . . . . . . . . . . . . . . . . . 87
5.3.2 Parameterchoicerules . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.3 ComparisonwithLSC . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Computationswithdatacorruptedbycolorednoise . . . . . . . . . . . . 93
5.4.1 Simulationofcolorednoise . . . . . . . . . . . . . . . . . . . . 93
5.4.2 Dataweightingusingtheexactnoisemodel . . . . . . . . . . . . 94
5.4.3 Comparisonwithlow-passfiltering . . . . . . . . . . . . . . . . 95
5.4.4 Noisemodelestimationfromresiduals. . . . . . . . . . . . . . . 97
5.5 Biasanddrifthandling . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5.1 Estimationandfilteringofbiasparameters . . . . . . . . . . . . 99
5.5.2 Cross-overadjustment . . . . . . . . . . . . . . . . . . . . . . . 101
5.5.3 Simultaneousestimationofbiasanddriftparameters . . . . . . . 101
5.6 Summaryoftheoptimalsolutionstrategy . . . . . . . . . . . . . . . . . 103
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Contents
6 Applicationtoairbornegravimetricsurveydata 105
6.1 Skagerrakdataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.1 Descriptionofthedata . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.2 Frequency-dependentdataweighting . . . . . . . . . . . . . . . 109
6.1.3 Outlierdetection . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.4 Estimationofbiasparameters . . . . . . . . . . . . . . . . . . . 117
6.1.5 Geoiddetermination . . . . . . . . . . . . . . . . . . . . . . . . 118
6.1.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Chiledataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.1 Descriptionofthedata . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.2 Gravityfielddetermination . . . . . . . . . . . . . . . . . . . . . 122
6.2.3 Estimationofnon-gravitationalparameters . . . . . . . . . . . . 126
6.2.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Timmins,Ontariodataset. . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3.1 Descriptionofthedata . . . . . . . . . . . . . . . . . . . . . . . 130
6.3.2 Estimationofthenoisemodel . . . . . . . . . . . . . . . . . . . 131
6.3.3 Biasestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3.4 Downwardcontinuation . . . . . . . . . . . . . . . . . . . . . . 136
6.3.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Summaryanddiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7 Conclusionsandrecommendations 141
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.2 Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A Pre-processingofairbornegravitydata 147
A.1 GPSprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.2 Gravityprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B Coordinatetransformation 153
C Least-squarescollocationandHilbertspaces 155
C.1 DefinitionofaHilbertspaceandsomeproperties . . . . . . . . . . . . . 155
C.2 ReproducingkernelHilbertspaces . . . . . . . . . . . . . . . . . . . . . 157
C.3 Least-squarescollocation . . . . . . . . . . . . . . . . . . . . . . . . . . 157
D DerivationoftheZOTregularizationmatrix 161
E Modificationofthebasefunctions 163
Bibliography 165
CurriculumVitae 179
v

Summary
Regionalgravityfieldmodelingusingairbornegravimetrydata
Airborne gravimetry is the most efficient technique to provideaccurate high-resolution
gravitydata in regionsthatlack gooddata coverageand thatare difficulttoaccess oth-
erwise. With current airborne gravimetry systems gravity can be obtained at a spatial
resolution of 2 km with an accuracy of 1-2 mGal. It is therefore an ideal technique to
complementongoingsatellitegravitymissionsandestablishthebasisformanyapplica-
tionsofregionalgravityfieldmodeling.
Gravityfield determinationusingairborne gravitydata can be dividedintwo major
steps. The first step comprises the pre-processing of raw in-flight gravity sensor mea-
surements to obtain gravity disturbances at flight level and the second step consists of
the inversion of these observations into gravity functionals at ground level. The pre-
processingofairbornegravitydataconsistsofseveralindependentstepssuchaslow-pass
filtering,across-overadjustmenttominimizemisfitsatcross-oversofintersectinglines,
andgridding. Eachofthesestepsmayintroduceerrorsthataccumulateinthecourseof
processing,whichcanlimittheaccuracyandtheresolutionoftheresultinggravityfield.
Fortheinversionoftheairbornegravitydataatflightlevelintogravityfunctionalsat
theEarth’ssurface,severalapproachescanbeused. Methodsthathavebeensuccessfully
applied to airborne gravity data are integral methods and least-squares collocation, but
bothmethodshavesomedisadvantages.Integralmethodsrequirethatthedataareavaila-
ble in a much larger area than for which the gravity functionalsare computed. A large
capsizeisrequiredtoreduceedgeeffectsthatresultfrommissingdataoutsidethetarget
area. Least-squarescollocationsuffersmuchlessfromtheseerrorsandcanyieldaccurate
results, provided that the auto-covariance function gives a good representation of data
in- and outside the area. However, the number of base functions equals the number of
observations,whichmakesleast-squarescollocationnumericallylessefficient.
Inthisthesisanewmethodologyforprocessingairbornegravitydataisproposed. It
combinesseparatepre-processingstepswiththeestimationofgravityfieldparametersin
onealgorithm. Importantly,theconceptoflow-passfilteringisreplacedbyafrequency-
dependent data weighting to handle the strong colored noise in the data. Frequencies
at which the noise level is high get a lower weight than frequencies at which the noise
levelislow.Furthermore,biasparametersareestimatedjointlywithgravityfieldparam-
etersinsteadofapplyingacross-overadjustment.Toparameterizethegravitypotentiala
spectralrepresentationisused,whichmeansthattheestimationresultsinasetofcoeffi-
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Summary
cients. Thesecoefficientsareusedtocomputegravityfunctionalsatanylocationonthe
Earth’ssurfacewithinthesurveyarea. Theadvantageofthedevelopedapproachisthat
itrequiresaminimumofpre-processingandthatalldatacanbeusedasobtainedatthe
locationswheretheyareobserved.
The performance of the developed methodology is tested using simulated data and
data acquired in airborne gravimetry surveys. The goal of the simulationsis to test the
approach in a controlled environment and to make optimal choices for the processing
of real data. For the numerical studies with simulated data, the new methodology out-
performs the more traditional approaches for airborne gravity data processing. For the
applicationofthedevelopedmethodologytorealdata,threedatasetsareused. Thefirst
data set comprises airborne gravity measurements over the Skagerrak area, obtained as
part of a joint project between several European institutionsin 1996. This survey pro-
videdaccurateairbornegravitydata,andbecausegoodsurfacegravitydataareavailable
withinthearea, thedatasetisveryusefultotesttheperformance oftheapproach. The
seconddatasetwasobtainedbytheGeoForschungsZentrumPotsdamduringasurveyoff
thecoastofChilein2002. Thisdataset,whichhasaloweraccuracythanthefirstdata
set, is used to investigatethe estimation of non-gravitational parameters such as biases
and scaling factors. The final data set that is used consists of airborne gravity data ac-
quiredbySanderGeophysicsLimitedin2003. ThesurveyareaislocatednearTimmins,
Ontario and is much smaller than the area of the other data sets. The small size of the
areaandthehighaccuracyofthedatamakeitachallengingdatasetforregionalgravity
modeling.
Thecomputationalexperimentswithreal data showthatthe performance of thede-
velopedmethodologyisatthesamelevelastraditionalmethodsintermsofgravityfield
errors. However, itprovidesa more flexible and powerful approach to airborne gravity
data processing. It requiresa minimumof pre-processingandall observationsare used
inthedeterminationofaregionalgravityfield. Thefrequency-dependentdataweighting
issuccessfullyappliedtoeachdataset. Theapproachprovidesastatisticallyoptimalso-
lutionandisaformalizedwaytohandlecolorednoise. Anoisemodelcanbeestimated
fromaposteriorileast-squaresresidualsinaniterativeway.Theprocedureispurelydata-
drivenand,unlikelow-passfiltering,doesnotdependonpreviousexperienceoftheuser.
Thedevelopedmethodologyallowsforthesimultaneousestimationofnon-gravitational
parameters with the gravity field parameters. A testing procedure should be applied,
however,toavoidinsignificantestimationsandhighcorrelations. FortheChiledataset
asignificantimprovementoftheestimatedgravityfieldisobtainedwhenbiasandscale
factorsareestimatedfromtheobservations.Theresultsofthecomputationswiththereal
datasetsshowthehighpotentialofusingairbornegravimetrytoobtainaccurategravity
forgeodeticandgeophysicalapplications.
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