f ma - tel aviv universityzivalon/geodynamics/front/gravity.pdf · newton’s law of universal...

Newton’slawofuniversalgravitation:where:• Fistheforceofgravitation.• m1 andm2 arethemasses.• risthedistancebetweenthemasses.• g isthegravitationalconstantthatisequalto6.67x10-11 Nm2kg-2.

UnitsofFareN=kgms-2 .

Thebasics

€

F = γm1m2

r2 ,

Newton’ssecondlawofmotion:where:• misthemass.• aisacceleration.

BycombiningtheuniversallawofgravitationwithNewton’ssecondlawofmotion,onefindsthattheaccelerationofm2 duetoitsattractionbym1 is:

€

F = ma ,

€

a =γm1

r2 .

Gravitationalaccelerationisthus:where:• ME isthemassoftheEarth.• RE istheEarth’sradius.

Unitsofaccelerationarems-2,orgal=0.01ms-2.

Thebasics

€

g =γME

RE2 ,

• TheEarthisanoblatespheroidthatisfatterattheequatorandisthinneratthepoles.

• Thereisanexcessmassundertheequator.

• Centrifugalaccelerationreducesgravitationalattraction.Thus,thefurtheryouarefromtherotationaxis,thegreaterthecentrifugalaccelerationis.

Thebasics

From:http://principles.ou.edu/earth_figure_gravity/geoid/

Thebasics

gisavectorfield:

whererisaunitvectorpointingtowardstheearth’scenter.Thegravitationalpotential,U,isascalarfield:

• NotethatEarth’sgravitationalpotentialisnegative.• Potentialsareadditive,andthispropertymakesthemeasier(thanvectors)toworkwith.• ToverifythatUisthepotentialfieldofgtakeitsderivativewithrespecttoR.

Thegradientofascalarfieldisavectorfield.

€

g = γ ME

RE2

ˆ r ,

€

U = −γME

RE

.

Surfacegravityanomaliesduetosomeburiedbodies

Thegeneralequationis:

where:g isthegravitationalconstantDr isthedensitycontrastristhedistancetotheobservationpointa istheanglefromverticalVisthevolumeQuestion:Whyacosineterm?

€

ΔgZ = γΔρ1r2 cosαdV ,

V∫

€

ΔgZ =4πγa3Δρ

31

x 2 + z2( )z

x 2 + z2( ) .Solutionforasphere:

z

a

x/z


Infinitelylonghorizontalcylinder:


Buriedinfiniteslab:

Aninfinitelylonghorizontalcylinder

cylinder sphere

Theexpressionforahorizontalcylinderofaradiusaanddensityr:

Itisinterestingtocomparethesolutionforcylinderwiththatofasphere.

Thishighlightstheimportanceofa2-Dgravitysurvey.

€

ΔgZ = 2γπa2ρ Zx 2 + Z 2

.


Whatshouldbethespatialextentofthesurveyedregion?

Toanswerthisquestionitisusefultocomputetheanomalyhalf-distance,X1/2,i.e.thedistancefromtheanomalymaximumtoit’smedium.Forasphere,weget:

€

X1/ 2 = Z 22/ 3 −1.


• ThesignalduetoasphereburiedatadepthZcanonlybewellresolvedatdistancesoutto2-3Z.

• Thus,toresolvedetailsofdensitystructuresofthelowercrust(say20-40km),gravitymeasurementsmustbemadeoveranextensivearea.

Theambiguityofsurfacegravityanomalies

Intheprecedingslidewehavelookedattheresultofaforwardmodelingalsoreferredtoas thedirectproblem:

Inpractice,however,theinversemodeling isofgreaterimportance:

Question:Canthedatabeinvertedtoobtainthedensity,sizeandshapeofaburiedbody?

• Inspectionofthesolutionforaburiedsphererevealsanon-uniquenessofthatproblem.ThetermDra3 introducesanambiguitytotheproblem,anddifferentcombinationsofdensitiesandradii canproduceidenticalanomalies.

1. Nearsurfaceveryelongatedbody2. Shallowelongatedbody3. Deepsphere

• Thesamegravityanomalymaybeexplainedbydifferentanomalousbodies,havingdifferentshapesandlocatedatdifferentdepths: measuredgravityanomaly

Insummary,wewanttoknow:

Butactually,gravityanomalyalonecannotprovidethisinformation.

€

ρ = ρ(x,y,z).


Theobservedanomalymaybeexplainedequallywellwithdeepmodelswithsmalldensitycontrastorshallowmodelswithgreaterdensitycontrast.

Question:istheMARinisostaticequilibrium?

Here’sanexamplefromaMid-AtlanticRidge(MAR).


Geoid

• theobservedequipotentialsurfacethatdefinesthesealevel.• theshapeafluidEarthwouldhaveifithadexactlythegravityfield

oftheEarth• roughlythesea-levelsurface- dynamiceffectssuchaswaves,and

tides,mustbeexcluded• geoidoncontinentsliesbelowcontinents- correspondstolevelof

nearlymasslessfluidifnarrowchannelswerecutthroughcontinents

• geoidhighsaregravityhighs

Thegeoid

Thevectorgravity(g)isperpendiculartothegeoid.

Referencegeoid isamathematicalformuladescribingatheoreticalequipotentialsurfaceofarotating(i.e.,centrifugaleffectisaccountedfor)symmetricspheroidalearthmodelhavingrealisticradialdensitydistribution.

Thegeoid

Theinternationalgravityformula givesthetheoreticalgravitationalaccelerationonareferencegeoid: g(λ) ≈ gE 1+α sin2 λ +β sin4 λ( )

where:gE is the g at the equatorλ is the latitudeα = 5.278895×10−3

β = 2.3462×10−5

observedgeoidreferencegeoid

Thegeoidheightanomalyisthedifferenceinelevationbetweenthemeasuredgeoidandthereferencegeoid.

Notethatthegeoidheightanomalyismeasuredinmeters.

Thegeoid

Geoidanomaly

Mapofgeoidheightanomaly:

Notethat:• Thedifferencesbetweenobservedgeoidandreferencegeoidareaslargeas100meters• Incontinentalregions,theydonotcorrelatewithtopographybecauseofisostatic

compensation

Figurefrom:www.colorado.edu/geography

Question:whatgivesrisetogeoidanomaly?

Geoidanomaly

Differencesbetweengeoidandreferencegeoidaredueto:

• Topography

• Densityanomaliesatdepth

FigurefromFowler

Geoidanomaly

FigurefromMcKenzieetal.,1980

Twocompetingeffects:

1. Upwellingbringshotterandlessdensematerial,theeffectofwhichistoreducegravity.

2. Upwellingcausestopographicbulge,theeffectofwhichistoincreasegravity.

Whatistheeffectofmantleconvectiononthegeoidanomaly?

Flow

Temp.

upwellingdownwelling

Geoidanomaly

SEASATprovideswatertopography

Notethatthemostprominentfeaturesonmostgeoidmaps(dependingonfilteringused)aresubduction zones.

Geoidanomaly

Free-airgravityanomalyfromsatellitealtimetryfortheTonga-Kermadec region

Comparisonoftopographyalongeastwestprofilesacrossthesubduction zoneat20,25and30°S(thick/blue)toobservedtopography(thin/black)

Cross-sectionsacrosssubduction-zonegeoidanomaliesshowanasymmetricanomalylow(trench)andananomalyhigh(presenceofcold,denseslabinlighterasthenospere):

(From:Billen andGurnis,EPSL,2001)

Geoidanomalyandtheglacialisostaticadjustment(GIA)


MilankovitchCycles


Observationsofglacialisostaticadjustment:• present-daydeformationfromGPS• present-daysealevelchangefrom

tidegauges• pastrelativesealevelfrom

geologicalrecord• present-daygravityfieldfrom

GRACEsatellite

Amodelformasschangeduetopost-glacialreboundandthereloadingoftheoceanbasinswithseawater.

Blueandpurpleareasindicaterisingduetotheremovaloftheicesheets.Yellowandredareasindicatefallingasmantlematerialmovedawayfromtheseareasinordertosupplytherisingareas,andbecauseofthecollapseoftheforebulges aroundtheicesheets.


Geoidanomalyandcorrections

Geoidanomalycontainsinformationregardingthe3-Dmassdistribution.Butfirst,afewcorrectionsshouldbeapplied:• Free-air(required)• Bouguer (required)• Terrain(optional)


Free-aircorrection,dgFA:

ThiscorrectionaccountsforthefactthatthepointofmeasurementisatelevationH,ratherthanatthesealevelonthereferencespheroid.


with:• l isthelatitude• histhetopographicheight• g(l)isgravityatsealevel• R(l)istheradiusofthereferencespheroidatl

Thefree-aircorrection is:

Thiscorrectionamountsto3.1x10-6 ms-2 permeterelevation.

€

δgFA = g(λ,0) − g(λ,h) = g(λ,0) 2hR(λ)

.

Question:shouldthiscorrectionbeaddedorsubtracted?Thefree-airanomaly isthegeoidanomaly,withthefree-aircorrectionapplied:

gFA =measured gravity - reference gravity +δgFA .

Bouguer correction,dgB:

Thiscorrectionaccountsforthegravitationalattractionoftherocksbetweenthepointofmeasurementandthesealevel.


Aninfinitehorizontalslaboffinitethickness:

€

dgZ = 2πγρch .

Notethatthegravityanomalycausedbyaninfinitehorizontalslabofthicknesshanddensityrcisindependentofitsdistancebfromtheobserver.


dgZ = γρ(y)(r dφ dr dy) 1r2 + (y+ b)2!" #$

(y+ b)r2 + (y+ b)2!" #$

1/2∫∫∫ .

Settingr(y)=rc andintegrationwithrespecttorfromzerotoinfinityandwithrespecttoybetween0andhleadsto:


TheBouguer correctionis:

where:g istheuniversalgravitationalconstantr istherockdensityhisthetopographicheight

Forrockdensityof2.7x103kgm-3,thiscorrectionamountsto1.1x10-6 ms-2 permeterelevation.

Question:shouldthiscorrectionbeaddedorsubtracted?

TheBouguer anomaly isthegeoidanomaly,withthefree-airandBouguer correctionsapplied:

€

δgB = 2πγρh ,

gB = measured gravity - reference gravity+δgFA −δgB .


Terraincorrection,dgT:

Thiscorrectionaccountsforthedeviationofthesurfacefromaninfinitehorizontalplane.Theterraincorrectionissmall,andexceptforareaofmountainousterrain,canoftenbeignored.


TheBougueranomaly includingterraincorrectionis:

gB =measured gravity - reference gravity+δgFA −δgB +δgT .

Bougueranomalyforoffshoregravitysurvey:• Replacewaterwithrock• Applyterraincorrectionforseabedtopography

Aftercorrectingfortheseeffects,the''corrected''signalcontainsinformationregardingthe3-Ddistributionofmassintheearthinterior.

Isostasy

Thedeflectionofplumb-bobnearmountainchainsislessthanexpected.Calculationsshowthattheactualdeflectionmaybeexplainediftheexcessmassiscanceledbyanequalmassdeficiencyatgreaterdepth.

Aplumb-bobPicturefromwikipedia

Isostasy:theAiryhypothesis(applicationofArchimedes’ principal)

• Twodensities,thatoftherigidupperlayer,ru,andthatofthesubstratum,rs.• Mountainsthereforehavedeeproots.Amountainheighth1 isunderlainbyarootofthickness:

• Oceanbasindepth,h2,isunderlainbyananti-rootofthickness:

€

r1 =h1ρuρs − ρu

.

€

r3 =d(ρu − ρw )ρs − ρu

.

r1

ru

rs

h1

r3

d

• Oceanbasinwhosedepthish2 isunderlainbyahighdensitymaterial,r2,thatisgivenby:

Isostasy:thePratt’shypothesis

• Thedepthtothebaseoftheupperlayerisconstant.• Thedensityofrocksbeneathmountainsislessthanthatbeneathvalleys.• Amountainwhoseheightish1 isunderlainbyarootwhosedensityr1is:

€

ρ1 = ρuD

h1 + D .

€

ρd =ρuD− ρwdD− d

.

Isostasy:

Isostasy

Questions:

• Whichisthecorrecthypothesis?

• Doesisostatic equilibriumapplyeverywhere?

Isthepersonrestingontopofaspring-mattressinastateofisostatic equilibrium?

Isostasy:elasticflexure

Likethespringsinsidethemattress,theelasticlithospherecanalsosupportexcessmass.

Thickplatescansupportmoreexcessmassthanthinplates.


Theresponseofthelithospheretoaverticalloaddependsonthelithosphereelasticpropertiesasfollows:

€

D d4wdx 4 =V (x) ,

whereDistheflexuralrigidity,thatisgivenby:

with:EbeingYoungModulushbeingtheplatethicknessn beingPoisson’sratio

€

D =Eh3

12(1−ν) ,


Thefigurebelowshowsthesolutionforthecaseofalineload:

€

V (x) > 0 for x = 0and

V (x) = 0 for x ≠ 0 .

Notetheflexuralbulge oneithersideofthedepression.

Ofcourseinrealitytheboundaryconditionsaremorecomplex…

FigurefromFowler

Isostasy:examplefromtheHawaiichain

bathymetry

free-air

Twoeffects:• Elasticflexureduetoislandload.• Aswellduetomantleupwelling.

FigurefromFowler

Isostasy:examplefromtheMarianasubductionzone

defle

ction[km]

distance[km]

• Theaccretionary wedgeloadstheplateedgecausingittobend.• Aflexuralbulgeisoftenobservedadjacenttothetrench.• TopographyofMarianabulgeimpliesa28kmthickplate.

Fluxuralbulge

FigurefromFowler

IsostasyexamplefromtheTongasubductionzone

defle

ction[km]

distance[km]

• TheTongaslabbendsmoresteeplythancanbeexplainedbyanelasticmodel.• Itturnedoutthatanelastic-plasticmodelforthelithospherecanexplainthebathymetrydata.

FigurefromFowler

Isostasy:localversusregionalisostaticequilibrium

AccordingtoPrattandAiryhypotheses,excessmassisperfectlycompensatedeverywhere.Thissituationisreferredtoaslocalisostasy.

Thesituationwheresomeoftheloadissupportedbythestrengthofthelithosphereisreferredtoasregionalisostasy.Inthiscase,isostatic equilibriumoccursonalargerscale,butnotatanypoint.

Isostasy

Questions:

1. Isostatic equilibriummeansnoexcessmass.Doesthismeannogravitationalanomaly.

2. Canwedistinguishcompensatedfromuncompensatedtopographies?

Isostasy:gravity

100%compensated

Aruleofthumb:AregionisinisostaticequilibriumiftheBougueranomalyisamirrorimageofthetopography.

FigurefromFowler

Isostasy:gravity

Uncompensated

Aruleofthumb:AregionisNOT inisostaticequilibriumiftheBougueranomalyremainsflatundertopographichighsandlows.

FigurefromFowler

Isostasy:isostaticrebound

FigurefromFowler

Therateofisostaticrebounddependsontheelasticpropertiesofthelithosphere(includingitsthickness)aswellasthemantleviscosity.

Isostaticreboundcanbeobservedifalargeenoughloadhasbeenaddedorremovedfastenough.

Smallloads,afewkmindiameter,cantellusabouttheelasticpropertiesofthecrust.

Isostasy:isostaticreboundFigure 8. LOS velocity map (positive towards satellite). The dotted gray rectangle indicates the 565

area excluded from the calculation of the residual orbit correction (see section 3.1). Dashed 566

rectangle indicates the region of interest. Blue circle indicates the location of DRAG GPS site. X 567

marks the center of removed mass. Areas a, b and c are small isolated patches that stand out with 568

respect to their surroundings (see text and Figure 9). Gray contour marks water-level at 415 m 569

below MSL (corresponding to the year 2001). 570

571

Figure 9. West-east profile of LOS change rate within the region of interest (see Figure 8 for 572

location). Light gray dots show the velocity of all valid pixels. Dark gray dots indicate the sub-573

calculation of the residual orbit correction (see section 3.1). (b) The bi-linear ramp accounting 559

for residual orbit phase (bracketed term in equation. (1)). (c) Unwrapped interferogram after 560

corrections due to orbit uncertainties. (d) Phase versus elevation after the application of the 561

residual orbital correction. The slope of the red straight line corresponds to the phase-elevation 562

slope (O� in Equation (1). 563

564

TheDeadSea,Israel:• Duringthepastfewdecades,theDeadSeawater-levelisdroppingatarateof1m/yr.


580

Data

Model

Residual

Ground displacement due to water-level changes can be reproduced using a homogeneous elastic half-space model.

Mediumsizeloads,say~100kmdiameter,cantellusabouttheviscosityoftheasthenosphere.


LakeBonneville,Utha:• Alake300mdeepdriedup10,000yearsago.• Lakecenterhasrisenby65m.

shoreline

Imagesfrom:academic.emporia.edu/aberjame/histgeol/gilbert/gilbert.htm

Fennoscandia:• Removalof2.5kmthickiceattheendofthelasticeage10,000yearsago.• Currentpeakupliftrateis9mm/yr.

Largeloads,say~1000kmdiametertellusabouttheupperandlowermantleviscosity.


GreatBritain:• Glaciation

affectedScotland,butnotSouthernEngland.

• Upliftrateofupto10cmpercentury.

Dipolemomentofdensityanomaly:dipolemomentofdensitydistribution

Wehaveseenthatthegravityanomalyduetoahorizontallayerofthicknessyis:

thusthegravitypotentialofthislayeris:

Thedipolemomentofdensitydistribution isjust:

Weshallseethatitisthedipolemomentofdensitydistribution,whichcontainsinformationregardingthemassdistribution,andmayhelptodiscriminatebetweenthetwoisostatic models.

€

ΔgZ = 2πγΔρy ,

€

ΔU = 2πγ Δρydyy1

y2

∫ .

€

Δρydyy1

y2

∫ .

Dipolemomentofdensityanomaly:gravitypotential

€

Uobs =Uref +dUdr

"

# $

%

& ' r= rref

ΔN

Uobs =Uref + g0ΔN⇒

ΔU =Uref −Uobs = −g0ΔN .

CombiningthiswiththeexpressionforthegravitypotentialofaBouguerslab(seepreviousslide)leadsto:

€

ΔN =−2πγg0

Δρydy.y1

y2

∫

U=Uobs

U=Uref

Dipolemomentofdensityanomaly:AiryversusPratt

Airy(positivetopography):

Pratt(positivetopography):€

b =ρch

ρm − ρc .

€

ρw+h = ρww

w + h .

Inareasofisostaticequilibrium,wewouldwishtoknowwhethermassisdistributedaccordingtoAiryorPrattmodels.


Pratt(positivetopography):

€

ρw+h = ρww

w + h .

€

ΔN =−2πγg

ρw+h ydy−h

0

∫ + (ρw+h − ρw )ydy0

w

∫( ) *

+ , -

.

Replacingwithleadsto:

NotethelinearrelationbetweenDNandh.

€

ΔN =πγgρwwh .

€

ρw+h

€

ρww /(w + h)

+y


Airy(positivetopography):

€

b =ρch

ρm − ρc .

+y

€

ΔN =−2πγg

(ρc − ρm )ydyH

H +b

∫ + ρc ydy−h

0

∫( ) *

+ , -

.

Replacing withleadsto:

NotetheNON-LINEARrelationbetweenDNandh.

€

ΔN =πγρcg

2Hh + h2 ρmρm − ρc

'

( )

*

+ , .

€

b

€

ρch /(ρm − ρc )


Pratt

Airy

• DNversushforAiryandPrattmodels.

• Airymodelimpliesanearlyfactorof3differencebetweenDN/hon-landandoff-shore.

FigurefromTurcotteandSchubert

AtwhatdirectiondoesthePacificplatemoves?



FigurefromTurcotteandSchubert• DependenceoftheobservedgeoidanomalyonbathymetryacrosstheHawaiianswellandacrosstheBermudaswellcomparedwiththepredictedanomalyaccordingtoAiryandPrattmodels.

• Fair(orgood?)agreementisobtainedforPrattmodelwithacompensationdepthof100km.

• IfweacceptthePrattmodeltobeapplicable,theconclusionisthatthemantlerocksbeneaththeseswellshaveanomalouslylowdensitydowntoadepthof100km.


• AcomparisonbetweenAiry-predictedandmeasuredgeoidanomalyacrosstheAtlanticcontinentalmarginofN.America.

• ItfollowsfromthiscomparisonthatthecontinentalmarginisinastateofisostaticequilibriumaccordingtoAirymodel.

FigurefromTurcotteandSchubert

Furtherreading:

*Turcotte,D.L.andG.Schubert,Geodynamics,CambridgeUniversityPress.

*Fowler,C.M.R.,ThesolidEarth,CambridgeUniversityPress.

Practicalissues

(Thislectureisbasedlargelyon:http://www.earthsci.unimelb.edu.au/ES304/)

Theshapeofthegravityanomalydependsnotontheabsolutedensity,butonthedensitycontrast,i.e.thedifferencebetweentheanomalousdensityandthe“backgrounddensity”.

Practicalissues

Here’salistofdensitiesassociatedwithvariousearth’smaterials:

material 1000kg/m3

sediments 1.7-2.3sandstone 2.0-2.6shale 2.0-2.7limestone 2.5-2.8granite 2.5-2.8basalt 2.7-3.1metamorphic 2.6-3.0

Notethat:• Densitydifferencesarequitesmall.• There'sconsiderableoverlapinthemeasureddensities.

Practicalissues

Considerthevariationingravitationalaccelerationduetoasphericalorebodywitharadiusof10meters,buriedatadepthof25metersbelowthesurface,andwithadensitycontrastof500kgpermetercubed.

Themaximumanomalyforthisexampleis0.025mGal.

(keepinmindthat9.8m/s2 isequalto980,000mGal!!!)

Practicalissues

• Owingtothesmallvariationinrockdensity,thespatialvariationsintheobservedgravitationalaccelerationcausedbygeologicstructuresarequitesmall

• Agravitationalanomalyof0.025mGalisverysmallcomparedtothe980,000mGalsgravitationalaccelerationproducedbytheearthasawhole.Actually,itrepresentsachangeinthegravitationalfieldofonly1partin40million.

• Clearly,avariationingravitythissmallisgoingtobedifficulttomeasure.

Practicalissues

Howisgravitymeasures:

• Fallingobjects

• Pendulum

• Massonaspring

Practicalissues

Fallingobjects:

Thedistanceabodyfallsisproportionaltothetimeithasfallensquared.Theproportionalityconstantisthegravitationalacceleration,g:

g=distance/time2 .

Tomeasurechangesinthegravitationalaccelerationdownto1partin40millionusinganinstrumentofreasonablesize,weneedtobeabletomeasurechangesindistancedownto1partin10millionandchangesintimedownto1partin10thousands!!Asyoucanimagine,itisdifficulttomakemeasurementswiththislevelofaccuracy.

Practicalissues

Pendulummeasurements:

Theperiodofoscillationofthependulum,T,isproportionaltooneoverthesquarerootofthegravitationalacceleration,g.Theconstantofproportionality,l,isthependulumlength:

€

T = 2π lg

.

Heretoo,inordertomeasuretheaccelerationto1partin50millionrequiresaveryaccurateestimateoftheinstrumentconstantl,butlcannotbedeterminedaccuratelyenoughtodothis.

Practicalissues

Butallisnotlost:

• Wecouldmeasuretheperiodofoscillationofagivenpendulumbydividingthetimeofmanyoscillationsbythetotalnumberofoscillations.

• Byrepeatingthismeasurementattwodifferentlocations,wecanestimatethevariationingravitationalaccelerationwithouthavingtomeasurel.

Practicalissues

Massonaspringmeasurements:

Themostcommontypeofgravimeterusedinexplorationsurveysisbasedonasimplemass-springsystem.

AccordingtoHook’slaw:

X=mg/k,

withkbeingthespringstiffness.

Practicalissues

• Likependulumthemeasurements,wecannotdeterminekaccuratelyenoughtoestimatetheabsolutevalueofthegravitationalaccelerationto1partin40million.

• Wecan,however,estimatevariationsinthegravitationalaccelerationfromplacetoplacetowithinthisprecision.

Underoptimalconditions,moderngravimetersarecapableofmeasuringchangesintheEarth'sgravitationalaccelerationdownto1partin1000million.

Practicalissues

Variousundesiredfactorsaffectthemeasurements:

• Temporal(time-dependent)variations:

1. Instrumentaldrift2. Tidaleffects

• Spatialvariations:

1. Latitudevariations2. Altitudevariations3. Slabeffects4. Topographyeffect

Practicalissues

Instrumentaldrift:

Thepropertiesofthematerialsusedtoconstructthespringchangewithtime.Consequently,gravimeterscandriftasmuchas0.1mgalperday.

Whatcausestheoscillatorychangessuperimposedontheinstrumentaldrift?

Practicalissues

Tidaleffect:

Inthisexample,theamplitudeofthetidalvariationisabout0.15mGals,andtheamplitudeofthedriftappearstobeabout0.12mGalsovertwodays.Theseeffectsaremuchlargerthantheexamplegravityanomalydescribedpreviously.

Practicalissues

• Sincechangescausedbyinstrumentaldriftandtidaleffectsdonotreflectthemassdistributionatdepth,theyaretreatedasnoise.

• Strategiestocorrectforinstrumentaldriftandtidaleffectsarediscussedin:www.earthsci.unimelb.edu.au/ES304/MODULES/GRAV/NOTES/tcorrect.html

Practicalissues

Regionalandlocal(orresidual)gravityanomalies:

Considerasphericalorebodyembeddedinasedimentaryunitontopofa(denser)Graniticbasementthatisdippingtotheright.

Practicalissues

Thestrongestcontributiontothegravityiscausedbylarge-scalegeologicstructurethatisnotofinterest.Thegravitationalaccelerationproducedbytheselarge-scalefeaturesisreferredtoastheregionalgravityanomaly.

Practicalissues

Thesecondcontributioniscausedbysmaller-scalestructureforwhichthesurveywasdesignedtodetect.Thatportionoftheobservedgravitationalaccelerationassociatedwiththesestructuresisreferredtoasthelocal ortheresidualgravityanomaly.

Practicalissues

Thereareseveralmethodsofremovingunwantedregionalgravityanomalies.Here'sanexampleforagraphicalapproach:

Smoothingin1dimension Smoothingin2dimensions

Practicalissues

Variationsingravityaroundtheglobeareinferredfromsatelliteorbit.

Thebalancebetweenthegravitationalattractionandthecentrifugalforceiswrittenas:

Thisleadsto:

whereTisthesatellite’speriod,2pr/V.

€

γMEmr2 =

mV 2

r .

€

ME =r3

γ2πT

$

% &

'

( )

2

,

Practicalissues

Yet,thehighestresolutionwholeearthgravitymapsarederivedfromradarmeasurementoftheheightoftheseasurface.

f ma - tel aviv universityzivalon/geodynamics/front/gravity.pdf · newton’s law of universal...

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