Gravity interpretation : fundamentals and application of gravity inversion and geological interpretation

411 Pages·2009·7.32 MB·English

by  Peter L

Gravity Interpretation
·
Wolfgang Jacoby Peter L. Smilde
Gravity Interpretation
Fundamentals and Application of Gravity
Inversion and Geological Interpretation
WithCD-ROM
1 3
Prof.Dr.WolfgangJacoby Dr.PeterL.Smilde
Johannes FintherStr.6
Gutenberg-Universita¨tMainz 55257Budenheim
Institutfu¨rGeowissenschaften Germany
Saarstr.21 [e-mail]
55099Mainz
Germany
[e-mail]
ISBN:978-3-540-85328-2 e-ISBN:978-3-540-85329-9
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(cid:2)c Springer-VerlagBerlinHeidelberg2009
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Preface
Gravityinterpretationinvolvesinversionofdataintomodels,butitismore.Gravity
interpretation is used in a “holistic” sense going beyond “inversion”. Inversion is
like optimization within certain a priori assumptions, i.e., all anticipated models
lie in a limited domain of the a priori errors. No source should exist outside the
anticipatedmodelvolume,butthatisneverliterallytrue.Interpretationgoesbeyond
by taking “outside” possibilities into account in the widest sense. Any neglected
possibilitycarriesthedangerofseriouslyaffectingtheinterpretation.
GravityinterpretationpertainstowiderquestionssuchastheshapeoftheEarth,
thenatureofthecontinentalandoceaniccrust,isostasy,forcesandstresses,geolog-
ical structure, finding useful resources, climate change, etc. Interpretation is often
used synonymously with modelling and inversion of observations toward models.
Interpretation places the inversion results into the wider geological or economic
contextandintotheframeworkofscienceandhumanity.Modelsplayacentralrole
inscience.Theyareimagesofphenomenaofthephysicalworld,forexample,scale
images or metaphors, enabling the human mind to describe observations and rela-
tionships by abstract mathematical means. Models served orientation and survival
inacomplex,partlyinvisiblephysicalandsocialenvironment.
Inversionofgravityanomaliesisthemathematicalderivationofdensitydistribu-
tions and their confidence limits. This is a notoriously non-unique problem, while
thesocalledforwardproblemoffindingthegravityeffectsofgivenmassdistribu-
tions is perfectly unique. The ambiguity of inversion simply results from the fact
that knowledge of a sum does not imply knowledge of the addends. If you know
c=a+b, but nothing about a and b, c reveals neither a nor b. There is always
aninfinitemodelspace;theinfinityofanswerscanbereducedonlybyinvokinga
priori information. It can be of any nature and depends on the problem at hand. If
forexampleb=2a,yougeta=c/3andb=2c/3.
Thistreatiseattemptstogiveaperspectiveoftheproblemandtopreparereaders
forfindingtheirwaytosolutions.Aprioriinformationiscentraltogravityinversion.
It ranges from “hard” geological and geophysical data, such as seismic results, to
generalideasbasedonexperienceandtomodelsofprocesseswhichwouldproduce
gravitysignals.Generallytheadditionalknowledgewillbelimited,butoftenvery
importantaspectswillberevealed.Iftheaprioriinformationwerecomplete,there
wouldbenoproblemlefttobesolved.
v
vi Preface
This touches the question: what does gravity tell and what not? To endeavour
alongtheselinesistheexcitingbusinessofapproachingthetruth,butonecannever
beabsolutelysure.Naturehasbuiltintoomanyobstacles.Ifgravityismeasuredon
orabovetheEarth’ssurface,onecannottrulylookinside.Thesignalscomelargely
fromwithin,though.Thereisaphilosophicalextensionoftheseideasaboutgravity
interpretation:inourgeneralintellectualhumanconditionweareinaverysimilar
situation concerning our world views including that of ourselves. We observe and
receivesignalsfromwithinandwithout,andwecommunicate.Webuildourvirtual
worldsthatshouldbeconsistent.Thisaimfeedsbackintoourapproachtogravity
interpretation.
TextsofAppliedGeophysicsgenerallyhavean“explorationoutlook”;thepresent
book has also a strong geodynamic “inclination”. Gravity is active and passive: a
force doing work toward equilibrium from a disturbed state and generating a field
withobservablesignalstobeinterpreted.Inmanygeologicalsituationsgravityhas
doneworkandwetrytofindoutwhathappened.Forexample,avalleyhasbeenex-
cavatedandrefilledbylower-densitysediments,givinganegativegravityanomaly.
Or hot, low-density mass has risen or is rising, and cold, high-density mass has
sunk or is sinking and working against viscous forces and deflecting density sur-
facesfromtheirequilibriumlevel.Thedensitydistributionsgeneratedgivegravity
signals which can be interpreted only in view of such model ideas. Without them,
modelsofatotallydifferentnaturecan“explain”theanomalies.
This situation causes confusion. Is it worth at all to interpret gravity? Some
seem to think: not. This view is definitively wrong. Gravity plays two fundamen-
tallyusefulrolesintheearthsciences:ithelpstoinexpensivelydetect“anomalies”
worthstudying,anditfalsifiesandeliminatesmodelsbyforwardcomputations.The
methodologicalsideisthetheoryandpracticeofdatagathering,forwardmodelling
andofBayesianinversion,includingthevariouspreliminarystepsofmeasurement
anddatapreparation.Thepracticalsideisthepresentationofapplicationsandcase
histories.Thephilosophicalsideisthatitwantstoteachgeneralaspectsofapplying
observationstoscienceandtolife.
Presentationofobservationaltechniquesiskepttoaminimum,butsomediscus-
sion is unavoidable. Gravity or geoid observations are affected by errors or confi-
dence limits. Errors have an important effect on what can be learnt from gravity,
so their discussion is carried through all chapters. With the development of new
methods of terrestrial, marine, airborne and satellite-based observational methods,
andwithincreasingaccuracyoftheobservationsthescopeofinterpretationwidens.
Many methods of forward calculation of gravity effects are well known and refer-
enceisgiventoothertexts;butsomeaspectsofthebasicapproachinthistreatise
arenovel.
Muchofatextbook isconcerned withthereaderlearningtoworkingeophysi-
cal“practice”.Manytoday,especiallyscienceadministratorsonalllevels,suggest
thatteachingisthemainfunctionofuniversities,andeffortsin“purescience”and
conveyingin-depthunderstandingisnotsoimportant.Thisattitudeisshort-sighted.
Only deep understanding will produce reliable results, also in limited exploration
projects.Goodself-criticaljudgement,forexampleoftheprobabilityandpossibility
Preface vii
oferrorsinaninterpretation,requiresknowledgebeyondtechnicalskills.Thisour
own experience we wish to share. Indeed, we endeavour to make readers wonder
aboutproblems.
Probablythebestwayoflearningisfrommistakesandfromindependentefforts
in problem solving. We therefore include, as a CD, a collection of tasks or prob-
lems with some instructions for how to approach solutions. This will give readers
the chance to make their own mistakes and to correct them. Answers are listed at
theend,includingdiscussionoftheproblemsandsolutions.Someofthetasksare
applicationsofinversion(Chap.7)togeologicalortheoreticalmodellingwhichadd
principalaspectsdiscussedatsomelength.
Oneofourownexamplesservesasanillustration:whenworkingoutthesolid-
anglesolutionforacubeatoneofitscorners(seeSect.2.9.6),theassumptionthat
the gravitational vector effect points to the centre of mass led to the surprisingly
beautifulresultthattheverticalgravityeffectwouldbeprecisely1/6ofthatofthe
infinite Bouguer slab of the same thickness and density. But beauty is no proof,
and the result did not stand the test. The mistake was that, contrary to widespread
belief, the gravitational vector does not generally point to the centre of mass, ex-
ceptincertaincasesofspecialsymmetry(seeSect.2.9.1.2)whichshouldhavebeen
immediatelyevident,forexample,fromtheEarth’sellipsoidorthegeoid.Themis-
conception arose from a mix-up with mechanics where the action of a force on a
bodyisdescribedbyactiononitsbarycentre,i.e.centreofmassorbalancepoint.
Theauthorshaveconsultedothertextscoveringthesubject,especiallytheclas-
sical book (in German) by Karl Jung (1961), Schwerkraftverfahren in der Ange-
wandtenGeophysik(GravitymethodsinAppliedGeophysics;itwillbereferredto
asKJ61),andthebookInterpretationTheoryinAppliedGeophysicsbyF.S.Grant
andG.F.West(1965)(referredtoasGW65).Manyusefulideashavebeentakenup
andpartlyexpanded.Inthoseearlydaysofcomputingmachinestheirpossibilities
hadbeenclearlyseenandthefoundationshadbeenlaiddownfortheirapplication.
One of the authors (WJ) studied physics, geophysics and more and more geol-
ogy and considers himself a geophysically guided geologist, interested in how the
Earth works and concerned about how mankind treats its home planet. The other
author(PS)studiedgeodesyandbecamemoreandmoreinvolvedingeophysicsand
geology when working with WJ on his PhD thesis on gravity inversion, develop-
ingtheprogrampackageINVERT,ofwhichanexecutablecopyisattachedtothis
book on a CD. The thesis is also the basis of the most important last chapter of
the book on optimization and inversion. The cooperation led to a synthesis of the
geological-geophysicalapproachtotheproblemsofinterpretationandthegeodetic,
moremathematicallyinclinedapproach.Itisthecombinationofgeologicalimagi-
nationandexperience,ontheonehand,andabstractgeophysical-mathematicalrea-
soning, on the other, that is the basis of Earth science. Experience-based intuition
mustbecheckedbymathematicalvalidation.Indeed,scienceissuspendedbetween
the two extremes of freedom of thinking and rigorous checking. Scientists surely
endeavourtoapproachthetruthinsuchsuspense.
Many colleagues and friends in various institutions, not only from our own
study field, have participated in teaching us this lesson, from our parents, families
viii Preface
and some school teachers to our academic teachers, Karl Jung†, Kiel, and Reiner
Rummel, Delft, and to our later colleagues and students. Every one of them has
chosen her/his own way and none is responsible for ours, but the – hopefully –
mutual benefit has been immense. The intellectual challenges by colleagues and
studentsaregratefullyacknowledged.GeologicalteachingbyEugenSeybold,Kiel,
andexchangewithRichardWalcott,RichardGibb,AlanGoodacreandImreNagy
inCanadaandwithGerhardMu¨ller†,Frankfurt(Main),wereimportant.InMainz,
Georg Bu¨chel, Evariste Sebazungu, Tanya Fedorova, Ina Mu¨ller, Chris Moos,
Michaela Bock, Herbert Wallner, Hasan C¸avs¸ak, Tanya Smaglichenko and many
otherswereinfluentialonbothofus.
HerbertWallnerhelpedintellectuallybymanydiscussions,withcalculationsand
quite a number of figures. Tanya Fedorova provided some of the gravity inversion
models. Evariste Sebazungu, in his own PhD thesis on potential field inversion,
developed original ideas which entered into this treatise. Hasan C¸avs¸ak provided
gravitycalculationsforvariouspolyhedralbodiesandhelpeddiscoveringerrorsin
some theoretical derivations. Pierre Keating provided information on some of the
freemodellingsoftware.DiscussionswithMarkusKrieger(Terrasys,Hamburg)led
toseveralideasandinsightsintothepracticalsolutionofinterpretationproblems.
All of them and many more contributed thought-provoking ideas and thus in-
fluencedthepresenttreatise.Mostimportantly,themutualdiscussionsbetweenthe
authorsthroughthewholetimeoftheircooperationwerebeneficialtoboth.Finally,
lecturingongravity(andmagnetics)taughtusmorethananythingelsetoendeavour
topresenttheideasclearly.
Petra Sigl was always helpful and did an excellent job in preparing most of
the figures in this book. The book could hardly have been completed without the
manyformsofsupportbytheInstitutfu¨rGeowissenschaften,JohannesGutenberg-
Universita¨t Mainz, various grants by Deutsche Forschungsgemeinschaft, Bonn,
and by Stiftung Rheinland-Pfalz fu¨r Innovation, Mainz, the Terrasys company,
Hamburg.
Bettie Higgs, Stefan Bu¨rger, Herbert Wallner, Mark Pilkington, Pierre Keating
critically proofread parts of the draft and partly checked the mathematics. The re-
sponsibilityforanyerrorsremains,however,exclusivelywiththeauthors.Allhelp
bypersonsandinstitutionsisgratefullyacknowledged,includingthemanythatare
notnamed.
Mainz,Germany WolfgangJacoby
PeterL.Smilde
Contents
1 Introduction................................................... 1
1.1 TheSubjectandScope ...................................... 1
1.1.1 Gravity............................................. 1
1.1.2 Motivation.......................................... 2
1.1.3 Aims .............................................. 3
1.1.4 SpecialAspects...................................... 4
1.1.5 TheBookandtheReader ............................. 4
1.2 HistoricalReview .......................................... 5
1.2.1 Astronomy,Geodesy,Geophysics,18thand19thCenturies . 5
1.2.2 20thCentury ........................................ 6
1.2.3 GeodesyandGeophysics.............................. 7
1.3 PurposesofGravityMeasurements............................ 8
1.4 GravityandGravityAnomalies............................... 9
1.5 Some Important Aspects of the Terrestrial Gravity Field and
InternalMassDistribution ................................... 10
1.5.1 GeneralConsiderations ............................... 10
1.5.2 TheEarth’sFigureandConstitution..................... 11
1.5.3 ContinentsandOceans................................ 12
1.5.4 PlateTectonicsandMantleFlow ....................... 13
1.5.5 AssociatedGravityAnomalies ......................... 16
1.5.6 OtherLarge-ScaleGravityFeatures..................... 18
1.5.7 Smaller-ScaleGravityAnomaliesRelevanttoExploration
forEconomicMinerals ............................... 19
1.5.8 HarmonicSpectrumoftheGravityField................. 19
References..................................................... 21
2 FundamentalsofGravity,ElementsofPotentialTheory ............ 23
2.1 Introduction ............................................... 23
2.2 Units ..................................................... 23
2.3 Elementsofg.............................................. 24
2.4 CoordinateSystems ........................................ 25
ix
x Contents
2.4.1 SphericalCoordinates ................................ 25
2.4.2 VerticalCylinderCoordinates.......................... 26
2.4.3 CartesianCoordinates ................................ 27
2.5 Newton’s Laws: Gravitation and Inertia Plus Centrifugal
Acceleration=Gravity...................................... 29
2.6 GravityPotentialandEquipotentialSurfaces.................... 31
2.7 Laplace Equation, Field Quantities, Equivalent Stratum;
DerivationofSomeFieldQuantities,SurfaceIntegrals,Poisson
Equations,GravitationalFluxΓ............................... 34
2.7.1 Source-FreeSpace:LaplaceEquation ................... 35
2.7.2 TheFieldQuantities.................................. 36
2.7.3 TheEquivalentStratum ............................... 39
2.7.4 Applications:EstimationofFieldQuantitiesasδW ,δW ,
x y
δW .............................................. 40
zzz
2.7.5 SourceSpace:PoissonEquationandGravitationalFluxΓ .. 43
2.7.6 SurfaceIntegrals:TotalMass,CentreofMass ............ 44
2.8 TheGravityTensor(Eo¨tvo¨sTensor) ........................... 45
2.9 GravityEffectsandAnomalies–SummationandIntegration ...... 46
2.9.1 GeneralConsiderations ............................... 46
2.9.2 CoordinateSystemsandIntegration..................... 50
2.9.3 Special Mass Elements: Integration in One and Two
Dimensions,MassLinesandMassPlanes................ 53
2.9.4 Disks .............................................. 61
2.9.5 Shells.............................................. 67
2.9.6 UniformMassiveVolumes ............................ 70
2.9.7 Two-DimensionalBodies ............................. 74
2.9.8 Two-and-a-halfDimensionalModels(21D).............. 84
2
2.10 SomeTheoreticalAspectsofAnomalyAnalysis................. 86
2.10.1 GoalsofPost-reductionDataAnalysis .................. 86
2.10.2 SmoothingofSpatialSeries ........................... 87
2.10.3 Polynomials ........................................ 88
2.10.4 TheFieldQuantities:DifferentiationandIntegration....... 89
2.10.5 HarmonicFunctions.................................. 90
2.10.6 SpecialFunctions .................................... 94
2.10.7 SphericalHarmonics ................................. 99
2.10.8 Wavelets ...........................................103
2.10.9 StochasticRepresentationofAnomalies .................104
2.11 AspectsofMagnetostatics ...................................105
References.....................................................110
3 ObservationsandFieldActivities ................................113
3.1 Introduction ...............................................113
3.2 PrinciplesofGravityMeasurementandInstrumentTypes.........114
3.2.1 GeneralConsiderations ...............................114
3.2.2 Pendulums..........................................115