chapter2 Earth grav field

Chapter2 The Earth’ sGravitational field 2.1GlobalGravity,Potentials,FigureoftheEarth,Geoid Introduction Historically ,gravityhasplayed acentral roleinstudies ofdynamic processes intheEarth’ sinterior andisalsoimportant inexploration geophysics. Theconcept ofgravityisrelati velysimple, high- precision measurements ofthegravityfield areinexpensi veandquick, andspatial variations inthe gravitational acceleration giveimportant information about thedynamical state ofEarth. However, thestudy ofthegravity ofEarth isnoteasy since manycorrections havetobemade toisolate thesmall signal duetodynamic processes, andtheunderlying theory —although perhaps more elegant than, forinstance, inseismology —iscomple x.Withrespect todetermining thethree- dimensional structure oftheEarth’ sinterior ,anadditional disadv antage ofgravity,indeed, ofany potential field, overseismic imaging isthatthere islargerambiguity inlocating thesource of gravitational anomalies, inparticular intheradial direction. Ingeneral thegravity signal hasacomple xorigin: theacceleration duetogravity,denoted by,(
invector notation) isinfluenced bytopography ,aspherical variation ofdensity within the Earth, andtheEarth’ srotation. Ingeophysics, ourtask istomeasure, characterize, andinterpret thegravity signal, andthereduction ofgravity data isaveryimportant aspect ofthisscientific field. Gravitymeasurements aretypically givenwith respect toacertain reference, which canbut does nothavetobeanequipotential surface .Animportant example ofanequipotential surfaceis thegeoid (which itself represents deviations from areference spher oid). TheGravityField Thelawofgravitational attraction wasformulated byIsaac Newton (1642-1727) andpublished in 1687, thatis,about three generations after Galileo haddetermined themagnitude ofthegravita- tional acceleration andKepler haddisco vered hisempirical “laws”describing theorbits ofplanets. Infact,astrong argument forthevalidity ofNewton’ slawsofmotion andgravitywasthatthey could beused toderiveKepler’ slaws. Forourpurposes, gravitycanbedefined astheforce exerted onamass duetothecombination of (1)thegravitational attraction oftheEarth, with mass or  and(2)therotation oftheEarth. Thelatter hastwocomponents: thecentrifugal acceleration duetorotation with angular velocity andtheexistence ofanequatorial bulge thatresults from thebalance between self-gra vitation and 25 26 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD rotation. These contrib utions are,infact,components ofaseries thatcanbedescribed elegantly bymeans ofspherical harmonics. Thegravitational forcebetween anytwoparticles with (point) massesatposition 
andat position separated byadistance isanattraction along alinejoining theparticles (see Figure 2.1):  (2.1) or,invector form:          (2.2) Figure 2.1:Vector diagram showing thegeometry ofthegravitational attraction. where isaunitvector inthedirection of   .Theminus sign accounts forthefactthatthe force vector points inward(i.e., towards)whereas theunitvector points outw ard(away from ).Inthefollowing wewillplace attheorigin ofourcoordinate system andtake attosimplify theequations (e.g.,   andtheunitvector becomes )(seeFigure 2.2). Figure 2.2:Simplified coordinate system.istheuniversal gravitational constant :  "! !#%$'&)(+*-,/.0.m  kg ,/.s , (orNm kg , ), which hasthesame value forallpairs ofparticles. must notbeconfused with
,thegravita- tional acceleration ,orforce ofaunitmass duetogravity,forwhich anexpression canbeobtained byusing Newton’ slawofmotion. Ifisthemass ofEarth: '1 
   32
    and  
  (2.3) The acceleration isthelength ofavector
(the gravitational acceleration perunit mass) and isbydefinition alwayspositi ve: 54 *.Wedefine thevector
asthegravity field andtake,by convention,
positi vetowards thecenter oftheEarth, i.e.,inthe direction. 2.1.GLOBALGRAVITY,POTENTIALS, FIGURE OFTHEEARTH,GEOID 27 Thegravitational acceleration wasfirstdetermined byGalileo; themagnitude of
varies over thesurfaceofEarth butauseful ball-park figure is =9.8ms , (orjust10ms , )(inS.I.— Syst`eme International d’Unit ´es—units). Inhishonor ,theunit often used ingravimetry isthe Gal.1Gal=1cms , =0.01 ms , 10 , .Gravityanomalies areoften expressed inmilliGal , i.e.,10 ,
ormicroGal ,i.e.,10 ,.This precision caneasily beachie vedbymodern gravimeters. Analternati veunitisthegravity unit,1gu=0.1mGal =10 ,. When wasdetermined byCavendish in1778 (with theCavendish torsion balance) themass of theEarth could bedetermined anditwasfound thattheEarth’ smean density ,   * *kgm ,  , ismuch largerthan thedensity ofrocks attheEarth’ ssurface. This observ ations wasoneofthe firststrong indications thatdensity must increase substantially towards thecenter oftheEarth. In thedecades following Cavendish’ measurement, manymeasurements were done of atdifferent locations onEarth andthevariation of with latitude wassoon established. Inthese early days of“geodesy” onefocused onplanet wide structure; inthemidtolate1800’ sscientists started to analyze deviations ofthereference values, i.e.local andregional gravityanomalies. Gravitypotential Byvirtue ofitsposition inthegravityfield
duetomass,anymasshasgravitational po- tential energy.This energycanberegarded asthework done onamass bythegravitational force dueto inmoving from to where oneoften takes  .Thegravitational potentialisthepotential energyinthefield duetoperunitmass. Inother words, it’sthe workdone bythegravitational force
perunitmass. (One candefine aseither thepositive or negative oftheworkdone which translates inachange ofsign. Beware!). Thepotential isascalar field which istypically easier tohandle than vector fields. And, aswewillseebelow,from the scalar potential wecanreadily derivethevector field anyway. (The gravityfield isaconser vativefield sojusthow themassismovedfrom to isnot relevant: theworkdone only depends ontheinitial andfinal position.) Following thedefinition forpotential asiscommon inphysics, which considers Earth asapotential well —i.e.negative —wegetforU:  
"!$# %    !&#  '( (  #   (2.4) Note that !$# #because and # point inopposite directions. Figure 2.3:Bydefinition, thepotential iszero atinfinity anddecreases towards themass.represents thegravitational potential atadistancefrom mass.Notice thatitisassumed that    *(seeFigure 2.3). 28 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD Thepotential istheintegration overspace (either aline, asurfaceoravolume) ofthegravityfield. Viceversa, thegravityfield, thegravityforce perunitmass, isthespatial derivative(gradient) of thepotential.
   
        (2.5) Intermezzo 2.1THEGRADIENTOFTHEGRAVITATIONALPOTENTIAL Wemay easily seethisinamore general waybyexpressing (the incremental distance along theline joining twopoint masses) into some setofcoordinates, using theproperties ofthedotproduct andthe total derivativeofasfollows(byourdefinition, moving inthesame direction asaccumulates negative potential):    "!#$%&"'(*)+ (2.6) Bydefinition, thetotal derivativeof isgivenby:-,/. .  102. . $ $302. . ) *) (2.7) Therefore, thecombination ofEq.2.6andEq.2.7yields:4
. . 5 . . $65 . . ) 798 :<;=>?,7>@A (2.8) One cannowseethatthefactthatthegravitational potential isdefined tobenegativemeans that when mass approaches theEarth, itspotential (ener gy)decreases while itsacceleration dueto attraction theEarth’ scenter increases. Theslope ofthecurveisthe(positi ve)value of ,andthe minus sign makessure thatthegradient points inthedirection ofdecreasing ,i.e.towards thecenter ofmass. (The plus/minus convention isnotunique. Intheliterature oneoften sees CB and
 .) Some general properties:DThegradient ofascalar fieldisavector thatdetermines therateanddirection ofchange in .Letanequipotential surface Ebethesurfaceofconstant and .and bepositions onthatsurface(i.e., with  .  ).Then, thecomponent of along Eisgivenby  . B  .    *.Thus
 F hasnocomponents alongE:thefield is perpendicular totheequipotential surface. This isalways thecase, asderivedinIntermezzo 2.2.DSince fluids cannot sustain shear stress —theshear modulusG  *,theforces acting onthe fluid surfacehavetobeperpendicular tothissurfaceinsteady state, since anycomponent ofaforce along thesurfaceofthefluid would result inflowuntil thiscomponent vanishes. Therestoring forces aregivenby  asinFigure 2.4;afluid surfaceassumes an equipotential surface.DForaspherically symmetric Earth theequipotential would beasphere and
would point towards thecenter ofthesphere. (Eveninthepresence ofaspherical structure androtation thisisaverygood approximation of
.However,iftheequipotential isanellipsoid,
 2.1.GLOBALGRAVITY,POTENTIALS, FIGURE OFTHEEARTH,GEOID 29 Figure 2.4:    provides therestoring force thatlevelstheseasurfacealong anequipoten- tialsurface. does notpoint to   *;thisliesattheorigin ofthedefinition ofgeographic and geocentric latitudes.)DUsing gravitypotentials, onecaneasily provethatthegravitational acceleration ofaspheri- cally symmetric mass distrib ution, atapoint outside themass, isthesame astheacceleration obtained byconcentrating allmass atthecenter ofthesphere, i.e.,apoint mass. This seems trivial, butfortheuseofpotential fields tostudy Earth’ sstructure ithasseveral important implications: 1.Within aspherically symmetric body ,thepotential, andthus thegravitational accel- eration
isdetermined only bythemass between theobserv ation point atrandthe center ofmass. Inspherical coordinates:  
  '
    # (2.9) This isimportant intheunderstanding ofthevariation ofthegravityfield asafunction ofradius within theEarth; 2.The gravitational potential byitself does notcarry information about theradial dis- tribution ofthemass. Wewillencounter thislater when wediscuss more properties ofpotentials, thesolutions oftheLaplace andPoisson equations, andtheproblem of non-uniqueness ingravityinterpretations. 3.ifthere arelateral variations ingravitational acceleration onthesurfaceofthesphere, i.e.iftheequipotential isnotasphere there must beaspherical structure (departure from spherical geometry; canbeintheshape ofthebody aswell asinternal distrib ution ofdensity anomalies). 30 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD Intermezzo 2.2GEOMETRICINTERPRETATIONOFTHEGRADIENT Let beacurvewith parametric representation 
,avector function. Let beascalar function of multiple variables. Thevariation of ,confined tothecurve ,isgivenby: 

  ?@ 

   
(2.10) Therefore, if isacurveofconstant, 

 willbezero. Nowlet 
beastraight lineinspace:
0 (2.11) then, according tothechain rule(2.10), at  : 

   ?@ 
   (2.12) Itisusefule todefine thedirectionalderivativeof inthedirection of atpoint as:!"
 ?@ 
  # # (2.13) From thisrelation weinfer thatthegradient vector @ 
 at givesthedirection inwhich thechange of ismaximum. Nowlet $beanequipotential surface ,i.e.thesurfaceofconstant .Define asetof curves &%
onthissurface $.Clearly , 
&%
   ?@ 
   %
  (2.14) foreach ofthose curves.Since the '%
liecompletely onthesurface $,the  ()
 willdefine aplane tangenttothesurface$atpoint.Therefore, thegradient vector@A isperpendicular tothesurface$of constant .Or:thefield isperpendicular totheequipotential surface. Inglobal gravityoneaims todetermine andexplain deviations from theequipotential surfaces, or more precisely thedifference (height) between equipotential surfaces. This difference inheight is related tothelocal
.Inpractice onedefines anomalies relati vetoreference surfaces. Important surfaces are: Geoid theactual equipotential surfacethatcoincides with theaverage sealevel(ignoring tides andother dynamical effects inoceans) (Refer ence) spher oid:empirical, longitude independent (i.e., zonal) shape ofthesealevelwith asmooth variation inlatitude thatbest fitsthegeoid (ortheobserv edgravity data). This forms thebasis oftheinternational gravityformula thatprescribes asafunction oflatitude thatforms thereference value forthereduction ofgravitydata. Hydr ostatic Figur eofShape ofEarth :theoretical shape oftheEarth ifweknowdensityand rotation (ellipsoid ofrevolution). Wewillnowderivetheshape ofthereference spheroid; thisconcept isveryimportant forgeodesy since itunderlies thedefinition oftheInternational GravityFormula. Also, itintroduces (zonal, i.e.longitude independent) spherical harmonics inanatural way. 2.2Gravitational potentialduetonearlysphericalbody Howcanwedetermine theshape ofthereference spheroid? Theflattening oftheearth wasalready disco vered andquantified bytheendofthe18thcentury .Itwasnoticed thatthedistance between 2.2.GRAVITATIONALPOTENTIAL DUETONEARLYSPHERICAL BODY 31 adegree oflatitude asmeasured, forinstance with asextant, differsfrom thatexpected from a sphere:  
. 
  #
,with theradius oftheEarth,
.and
twodifferent latitudes (seeFigure 2.5). Figure 2.5: Ellipticity oftheEarth measured bythe distance between latitudes oftheEarth andasphere. In1743, Clairaut1showed thatthereference spheroid canalso becomputed directly from the measured gravityfield
.Thederivation isbased onthecomputation ofapotential  atpointduetoanearly spherical body ,anditisonly validforpoints outside (or,inthelimit, onthe surfaceof)thebody . Figure 2.6:Thepotential oftheaspherical body iscalcu- lated atpoint ,which isexternal tothemass # ; ,thedistance from theobserv ation point tothe center ofmass. Note that isconstant andthat , ,and
arethevariables. There isnorotation so  represents thegravitational potential. Thecontrib ution # tothegravitational potential at duetoamass element # atdistance from isgivenby#    #  (2.15) Typically ,thepotential isexpanded inaseries. This canbedone intwoways,which lead tothe same results. One canwrite   directly interms oftheknownsolutions ofLaplace’ sequation (   *),which arespherical harmonics. Alternati vely,onecanexpand theterm (B and integrate theresulting series term byterm. Here, wewill dothelatter because itgivesbetter understanding ofthephysical meaning oftheterms, butwewill showhowthese terms are,in 1Inhisbook, Th´eorie delaFiguredelaTerre. 32 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD fact,directly related to(zonal) spherical harmonics. Aformal treatment ofsolutions ofspherical harmonics assolutions ofLaplace’ sequation followslater.Thederivation discussed here leads to what isknownasMacCullagh’ sformula2andshowshowthegravity measurements themselv es areused todefine thereference spheroid. Using thelawofcosines (seeFigure 2.6)wecanwrite  
 
sothat#    ( ' 
'   #  (2.16) WecanusetheBinomial Theorem toexpand thisexpression intoapowerseries of  B . Intermezzo 2.3BINOMIALTHEOREM
0  0    0"! #$%
! %&  0! '$(
! 
  # %& *) )0 +(+(+ (2.17) for +,.-!.Here wetake   /0  # 1/0 325416*7 and !. Sowecanwrite:8(
  

  
:9 ,  (  (

 
  
$

  <;
= h.o.t. (
  
(

  <;$
 (= h.o.t. (2.18) andforthepotential:    *>#     8(
  
(

  ;$
 (=?9#    #     
#   
   ;$
 (=#  (2.19) InEquation 2.18 and2.19, wehaveignored thehigher order terms (h.o.t). 2After James MacCullagh (1809–1847). 2.2.GRAVITATIONALPOTENTIAL DUETONEARLYSPHERICAL BODY 33 Intermezzo 2.4EQUIVALENCEWITH(ZONAL)SPHERICALHARMONICS Note thatequation (2.19) is,infact,apowerseries of

 :
     ! ;  = 025416 7 ;  = 0 ; '#2 416 7 ! # = ;  =          
2 416 7  ;  =   (2.20) Inspectral analysis there arespecial names forthefactors   multiplying

   andthese areknownas Legendrepolynomials ,which define thezonal surface spherical harmonicsa. Wewilldiscuss spherical harmonics indetail later buthere itisuseful topoint outthesimilarity between the aboveexpression ofthepotential 
 asapowerseries of

 and2 416*7andthelowerorder spherical- harmonics. Legendre polynomials aredefined as 
 ! # $  
 !     (2.21) with some function. Inourcase wetake 254 6*7sothatthesuperposition oftheLegendre polynomials describes thevariation ofthepotential with latitude. Atthisstage weignore variations with longitude. Surfacespherical harmonics thatdepend onlatitude only areknownaszonal spherical harmonics. For 5 !5 #wegetfor  
254 6*7 !(2.22)
254 6*7 2 416*7(2.23)
254 6*7  '#254 6 7  ! # (2.24) which arethesame astheterms derivedbyapplication ofthebinomial theorem. Theequivalence between thepotential expression inspherical harmonics andtheonethatwearederiving by expanding!
isnocoincidence: thepotentialsatisfies Laplace’ sequation andinaspherical coordinate system spherical harmonics arethegeneral solutions ofLaplace’ sequation. aSurfacespherical harmonics areatthesurfaceofasphere what aFourier series istoatime series; itcan bethought ofasa2DFourier series which canbeused torepresent anyquantity atthesurfaceofasphere (geoid, temperature, seismic wavespeed). Letusrewrite eq.(2.19) byusing theidentity

(:     #     
#      #  $
   
#  (2.25) Wecangetinsight inthephysics ifwelook ateach term ofeq.(2.25) separately: 1. ' #   'isessentially thepotential ofapoint mass at .This term will dominate forlarge;atalargedistance thepotential duetoanaspherical density distrib ution isclose tothatofaspherical body (i.e., apoint mass inO). 2.  
# represents atorque ofmass &distance, which also underlies thedefinition of thecenter ofmass    # CB # .Inourcase, wehavechosen asthecenter of mass and   with respect to .Another waytoseethatthisintegralmust vanish isto realize thattheintegration over# isessentially anintegration over
between *and
 andthat 
 "! 
.Integration over
takes 
back andforth overtheline 34 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD between and (within thebody) with equal contrib utions from each sideof ,since  isthecenter ofmass. 3.  # represents thetorque ofadistance squared andamass, which underlies thedefini- tionofthemoment ofinertia (recall thatforahomogeneous sphere with radius themo- ment ofinertia is0.4MR ).Themoment ofinertia isdefined as   # .When talk- ingabout moments ofinertia onemust identify theaxisofrotation. Wecanunderstand the meaning ofthethird integralbyintroducing acoordinate system
  sothat 
   , 
  sothat  #   
  #   (B
    #  
 #  
 # andbyrealizing that    #  
 #  and 
 # arethemoments ofinertia around the
-, -,and -axis respecti vely. SeeIntermezzo 2.5formore onmoments ofinertia. Withthemoments ofinertia defined asinthebox wecanrewrite thethird term inthe potential equation    #   
    (2.26) 4.  
# .Here, 
projects onaplane perpendicular to andthisintegralthus represents themoment ofinertia ofthebody around .This moment isoften denoted by. 2.2.GRAVITATIONALPOTENTIAL DUETONEARLYSPHERICAL BODY 35 Intermezzo 2.5MOMENTSANDPRODUCTSOFINERTIA Amoment ofinertia ofarigid body isdefined with respect toacertain axisofrotation. Fordiscrete masses: 
   0
  0
)  )0 +(++
% % andforacontinuum:     Themoment ofinertia isatensor quantity   
     (2.27) Note: wereverttomatrix notation andmanipulation oftensors. isasecond-order tensor .
   isaprojection operator: forinstance,
   projects thevectoronthe(,$)plane, i.e., perpendicular to .This isveryuseful inthegeneral expression forthemoments ofinertia around different axis. !  !  !and    0 $ 0 )    0 $ 0 )    0 $ 0 ) (2.28) and           >(  $)  $ )    $ )$* $ $ )) )$ ) (2.29) Sothat: 
    $ 0 ) 9 $ 9 )9$*  0 ) 9$ ) )  )$  0 $ (2.30) Thediagonal elements arethefamiliarmomentsofinertia around the,$,and)axis. (The off-diagonal elements areknownastheproductsofinertia ,which vanish when wechoose,$,and)astheprincipal axes.) Moment ofInertia around -axis around$-axis around )-axis   
$ 0 )   !(! 
 0 )   '<'  
 0 $    Eq.(2.25) canthen berewritten as      
   !  $#"(2.31) which isknownasMacCullagh’ sformula .      
   !  $#"(2.32) Atfacevalue thisseems tobetheresult ofastraightforw ardandrather boring derivation, butit does revealsome interesting andimportant properties ofthepotential andtherelated field. Equa- tion(2.25) basically showsthatinabsence ofrotation thegravitational attraction ofanirregular body hastwocontrib utions; thefirstistheattraction ofapoint mass located atthecenter ofgrav- ity,thesecond term depends onthemoments ofinertia around theprincipal axes,which inturn depend completely ontheshape ofthebody ,or,more precisely ,onthedeviations oftheshape from aperfect sphere. This second term decays as (B  sothatatlargedistances thepotential approaches thatofapoint mass andbecomes lessandlesssensiti vetoaspherical variations 36 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD intheshape ofthebody .This simply implies thatifyou’reinterested insmall scale deviations from spherical symmetry youshould notbetofarawayfrom thesurface: i.e.it’sbetter touse data from satellites with arelati velyloworbit. This phenomenon isinfactanexample ofup(or down)w ardcontinuation, which wewill discuss more quantitati velyformally when introducing spherical harmonics. Wecanpursue thedevelopment further byrealizing thatthemoment ofinertia "around anygeneral axis (here )canbeexpressed asalinear combination ofthemoments ofinertia around the principal axes.Let ,  ,and
bethesquares ofthecosines oftheangle oftheline with the
-, -,and -axis, respecti vely.With 
(wecanwrite "    
(seeFigure 2.7). Figure 2.7:Definition ofdirection cosines. Sofarwehavenotbeen specific about theshape ofthebody ,butfortheEarth itisrelevantto consider rotational geometry sothat    .This leads to:"    
(2.33) Here,

with
theangle between andthe -axis, thatis
istheco-latitude .(
 *3,where isthelatitude )."    
(2.34) and        
$

 (
(2.35) Itiscustomary towrite thedifference inmoments ofinertia asafraction of ,withthe Earth’ sradius attheequator .    (2.36) sothat        
$

 (
(2.37) isameasure ofellipticity; forasphere ,  *,andthepotential   reduces tothe expression ofthegravitational potential ofabody with spherical symmetry . 2.2.GRAVITATIONALPOTENTIAL DUETONEARLYSPHERICAL BODY 37 Intermezzo 2.6ELLIPTICITYTERMS Let’sbriefly return totheequivalence with thespherical harmonic expansion. Ifwetake 25416 7(see box) wecanwrite for 
 
  
57     
2 416 7 0  ;  = 
25416:7 0  ;  = 
254 6 7  (2.38) The expressions (2.25), rewritten as(2.37), and(2.38) areidentical ifwedefine thescaling factors  as follows.Since 
254 6*7 !, must be1because  
isthefarfield term;  ifthecoordinate origin coincides with thecenter ofmass (see above);and isasdefined above.This term isofparticular interest since itdescribes theoblate shape ofthegeoid. (The higher order terms ( 
, etc.) aresmaller by afactor oforder 1000 andarenotcarried through here, buttheyareincorporated inthecalculation ofthe reference spheroid.) Thefinal steptowards calculating thereference gravityfield istoaddarotational potential. Let   betheangular velocity ofrotation around the -axis. Thechoice ofreference frame isimportant togettheplus andminus signs right. Aparticle thatmoveswith therotating earth is influenced byacentripetal force  '1,which canformally bewritten interms ofthecross products between theangular velocityandtheposition vector as & & .This showsthat thecentripetal acceleration points totherotation axis. Themagnitude oftheforce perunitmass is     .Thesource of  is,infact,thegravitational attraction
(
   
). Theeffectivegravity
  
  (seeFigure 2.8). Since wearemainly interested intheradial component (thetangential component isverysmall wecanwrite       . Figure 2.8:Thegravitational attraction deliversthecentripetal force needed tosustain therotation oftheEarth. Interms ofpotentials, therotational potential hastobeadded tothegravitational potential          !,with ! %   #  (
     (
  
(2.39) (which isinfactexactly therotational kinetic energy( "  . "  .    )perunitmass ofa rigid body  .     . # ,eventhough weused anapproximation byignoring thecomponent of
 inthedirection ofvarying latitude # .Why? Hint: usetheabovediagram andconsider the symmetry oftheproblem) Thegeopotential cannowbewritten as 
     
$

 (
  (
  
(2.40) 38 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD which describes thecontrib ution tothepotential duetothecentral mass, theoblate shape ofthe Earth (i.e.flattening duetorotation), andtherotation itself. Wecanalsowrite thegeopotential interms ofthelatitude bysubstituting (   
):        
$
   (
  (
   (2.41) Wenowwanttousethisresult tofindanexpression forthegravitypotential andacceleration atthe surfaceofthe(reference) spheroid. Theflattening isdetermined from thegeopotential bydefining theequipotential 
,thesurfaceofconstant . Since
isanequipotential,must bethesame (
)forapoint atthepole andattheequator . Wetake forthepolar radius and fortheequatorial radius andwrite:

      * 

    *(2.42)         (2.43)        
    (
  (2.44) (2.45) andafter some reordering toisolateand weget    $
    (
   B   $
 (
 (2.46) Which basically showsthatthegeometrical flattening asdefined bytherelati vedifference be- tween thepolar andequatorial radius isrelated totheellipticity coefficient andtheratio between therotational (   )tothegravitational (  , )component ofgravityattheequator . Thevalue fortheflattening canbeaccurately determined from orbital data; infactwithin ayear after thelaunch ofthefirstartificial satellite —bythesoviets —thisvalue could bedetermined with much more accurac ythan byestimates givenbymanyinvestigators inthepreceding cen- turies. Thegeometrical flattening issmall (  (B
 
# (B $%* *)(butlargerthan expected from equilibrium flattening ofarotating body). Thedifference between thepolar andequatorial radii isthus about   ! $# (km B $%* *
(km. Inorder togettheshape ofthereference geoid (orspher oid)onecanusetheassumption thatthe deviation from asphere issmall, andwecanthus assume thevector from theEarth’ scenter toa point atthereference geoid tobeoftheform   
# 
 (or,with
,     ((2.47) Itcanbeshownthat canbewritten asafunction of andlatitude asgivenby:    (   and(from binomial expansion)  ,   ,  (
 . Geoid anomalies, i.e.thegeoid “highs” and“lows” thatpeople talkabout aredeviations from thereference geoid andtheyaretypically oftheorder ofseveraltensofmeters (with amaximum (absolute) value ofabout 100mnear India), which issmall (often lessthan 0.5%) compared tothe latitude dependence oftheradius (see above).Sothereference geoid with according to (2.47) does apretty good jobinrepresenting theaverage geoid. 2.2.GRAVITATIONALPOTENTIAL DUETONEARLYSPHERICAL BODY 39 Finally ,wecandetermine thegravityfield atthereference geoid with ashape asdefined by(2.47) calculating thegradient ofeqn. (2.41) andsubstituting thepositiondefined by(2.47). Inspherical coordinates:
   
 ( (2.48) 


 
(    (2.49) because .' issmall. Sowecanapproximate themagnitude ofthegravityfield by:    $   
$
   (
    (2.50) and, with   (       (     $     (   
$
   (
    (    (2.51) or,with thebinomial expansion givenabove(2.47)      (
   $
$
   (
   (      ( $
  

 
      ( $
   (

 B
 ( $B
       ( $
   (  (2.52) Eqn. (2.52) showsthatthegravityfield atthereference spheroid canbeexpressed assome latitude- dependent factor times thegravityacceleration attheequator:     *  
( $
 (2.53) Information about theflattening canbederiveddirectly from therelati vechange ingravityfrom thepole totheequator .      (         (2.54) Eq.2.54 iscalled Clairaut’ stheor em3.Theabovequadratic equation forthegravityasafunction oflatitude (2.52) forms thebasis fortheinternational gravityformula. However,thisinternational 3After AlexisClaude Clairaut (1713–1765). 40 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD reference forthereduction ofgravitydata isbased onaderivation thatincludes some ofthehigher order terms. Atypical form is   (


(2.55) with thefactor ofproportionality and
depending on ,,,and .Thevalues ofthese parameters arebeing determined more andmore accurate bytheincreasing amounts ofsatellite data andasaresult theinternational gravityformula isupdated regularly .Theaboveexpression (2.55) isalsoatruncated series. Aclosed form expression forthegravityasfunction oflatitude is givenbytheSomigliana Equation4   
( (    (2.56) This expression hasnowbeen adopted bytheGeodetic Reference System andforms thebasis for thereduction ofgravitydata tothereference geoid (orreference spheroid).   # *$*
!# # ( ms , ;   * * * ( $ ( ( $ ! $;   * * *! ! $#   *-( $. 2.3ThePoissonandLaplaceequations Thegravitational field oftheEarth iscaused byitsdensity .Themass distrib ution oftheplanet isinherently three-dimensional, butwemortals willalwaysonly scratch atthesurface. Themost wecandoismeasure thegravitational acceleration attheEarth’ ssurface. However,thanks to afundamental relationship knownasGauss’ sTheor em5,thelinkbetween asurfaceobserv able andtheproperties ofthewhole body inquestion canbefound. Gauss’ stheorem isoneofa class oftheorems invector analysis thatrelates integrals ofdifferent types (line, surface, volume integrals). Stokes’s,Greens andGauss’ stheorem arefundamental inthestudy ofpotential fields. Theparticularly useful theorem duetoGauss relates theintegraloverthevolume ofsome property (most generally ,atensor )toasurfaceintegral. Itisalsocalled thedivergence theorem. Let beavolume bounded bythesurface E  (seeFigure 2.9). Adifferential patch ofsurface # canberepresented byanoutw ardly pointing vector with alength corresponding tothearea ofthe surfaceelement. Interms ofaunitnormal vector ,itisgivenby  # . Figure 2.9: Surfaceen- closing avolume. Unit normal vector . Gauss’ stheorem (forgeneric “stuf f” )isasfollows: 4After C.Somigliana. 5After Carl-Friedrich Gauss (1777–1855). 2.3.THEPOISSON ANDLAPLACEEQUATIONS 41> !  #   > !  # E (2.57) Let’sseewhat wecaninfer about thegravitational potential within theEarth using only informa- tionobtained atthesurface. Remember wehad   and
  (2.58) Suppose wemeasure
everywhere atthesurface,andsum theresults. What wegetisthefluxof thegravityfield >
"! (2.59) Atthispoint, wecanalready predict thatif Eisthesurfaceenclosing theEarth, thefluxofthe gravityfield should bedifferent from zero, andfurthermore, thatitshould havesomething todo with thedensity distrib ution within theplanet. Why? Because thegravitational field lines allpoint towards thecenter ofmass. Ifthefluxwaszero, thefield would besaidtobesolenoidal .Unlik e themagnetic field thegravity field isessentially amonopole. Forthemagnetic field, field lines both leaveandenter thespherical surfacebecause theEarth hasapositi veandanegativepole. The gravitational field isonly solenoidal inregions notoccupied bymass. Anyway,we’llstart working with Eq.2.59 andseewhat wecome upwith. Ontheonehand (we useEq.2.57 andEq.2.58), >
"! # E  > !
# %> !  # %> %# (2.60) Ontheother hand6(weusethedefinition ofthedotproduct andEq.2.58): >
"! # E   >
# E       > # (2.61) We’veassumed that Eisaspherical surface,butthederivation willworkforanysurface. Equating Eq.2.60 and2.61, wecanstate that    
   %Poisson’ sEquation (2.62) andinthehomogeneous case     *Laplace’ sEquation (2.63) Theinterpretation interms ofsources andsinks ofthepotential fields anditsrelation with thefield lines issummarized inFigure 2.10: Poisson’ sequation isafundamental result. Itimplies 1.thatthetotal mass ofabody (say,Earth) canbedetermined from measurements of  
atthesurface(seeEq. 2.61) 6Note thattheidentity / @@A istrueforscalar fields, butforavector fieldweshould havewritten  ?@
@   @
@  . 42 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD Figure 2.10: Poisson’ sandLaplace’ sequations. 2.noinformation isrequired about howexactly thedensity isdistrib uted within ,and 3.thatifthere isnopotential source (orsink) enclosed by ELaplace’ sequation should be applied tofindthepotential atapoint outside thesurface Ethatcontains allattracting mass, forinstance thepotential atthelocation ofasatellite. Butinthelimit, itisalsovalid attheEarth’ ssurface. Similarly ,wewillseethatwecanuseLaplace’ sequation todescribe Earth’ smagnetic field aslong asweareoutside theregion thatcontains thesource forthe magnetic potential (i.e., Earth’ score). Weoften havetofind asolution for ofLaplace’ sequation when only thevalue of ,orits derivatives  areknownatthesurfaceofasphere. Forinstance ifonewantstodetermine theinternal mass distrib ution oftheEarth from gravitydata. Laplace’ sequation iseasier tosolve than Poisson’ sequation. Inpractice onecanusually (re)define theproblem insuch awaythatone canuseLaplace’ sequation byintegrating overcontrib utions from small volumes # (containing thesource ofthepotential # ,i.e.,mass # ),seeFigure 2.11 orbyusing Newton’ sLawof Gravityalong with Laplace’ sequation inaniterati veway(notdiscussed here). Figure 2.11: Applicability ofPoisson’ sandLaplaces’ equations. SeeIntermezzo 2.7. 2.4.CARTESIAN ANDSPHERICAL COORDIN ATESYSTEMS 43 Intermezzo 2.7NEWENVIRONMENT Onecanprovethatthesolution ofLaplace’ sequation canbeuniquely determined iftheboundary conditions areknown(i.e. ifdata coverage atthesurfaceisgood); inother words, ifthere aretwosolutions and thatsatisfy theboundary conditions,and canbeshowntobeidentical. Thegood newshere is thatonce youfindasolution for of  / thatsatisfies theBC’syoudonothavetobeconcerned about thegenerality ofthesolution. Thebadnewsis(seealsopoint (2)above)thatthesolution ofLaplace’ s equation does notconstrain thevariations ofdensity within .This leads toafundamental non-uniqueness which istypical forpotentials offorce fields. Wehaveseen thisbefore: thepotential atapoint outside a spherically symmetric body with total mass isthesame asthepotential ofapoint mass located inthe center
.Inbetween
and thedensity inthespherical shells canbedistrib uted inaninfinite number of different ways,butthepotential at remains thesame. 2.4Cartesian andsphericalcoordinate systems InCartesian coordinates wewrite for  (theLaplacian) 
  (2.64) FortheEarth, itisadvantageous tousespherical coordinates. These aredefined asfollows(see Figure 2.12):
 
  
  
(2.65) Figure 2.12: Definition of ,
and inthespherical coordinate system. where
 * =co-latitude,   *
=longitude. Itisveryimportant torealize that, whereas theCartesian frame isdescribed bytheimmobile unit vectors  ,  and ,theunitvectors , and aredependent ontheposition ofthepoint. They arelocal axes.Atpoint , points inthedirection ofincreasing radius from theorigin, inthe direction ofincreasing colatitude
and inthedirection ofincreasing longitude. One cangobetween coordinate axesbythetransformation      
 
 

 
 
  *       (2.66) Furthermore, weneed toremember thatintegration overavolume element#
# # becomes, after changing ofvariables  
# #
#.This may beremembered bythefactthat  
is 44 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD thedeterminant oftheJacobian matrix, i.e.thematrix obtained byfilling a3 &3matrix with all partial derivativesofEq.2.65. After (some) algebra, wecanfinally write thespherical Laplacian:   ( 
   ( 




 ( 
    *(2.67) 2.5Spherical harmonics Wenowattempt tosolveLaplace’ sEquation  *,inspherical coordinates. Laplace’ sequa- tionisobeyedbypotential fields outside thesources ofthefield. Remember howsines andcosines (oringeneral, exponentials) areoften solutions todifferential equations, oftheform 
or 
,whereby cantakeanyintegervalue. Thegeneral solution isanycombination ofsines andcosines ofallpossible ’swith weights thatcanbedetermined bysatisfying theboundary conditions (BC’ s).Theparticular solution isconstructed byfinding alinear combination ofthese (basis) functions with weighting coefficients dictated bytheBC’s:itisaseries solution. Inthe Cartesian case theyareFourier Series. InFourier theory ,asignal, sayatime series  ,forin- stance aseismogram, canberepresented bythesuperposition of and functions andweights canbefound which approximate thesignal tobeanalyzed inaleast-squares sense. Spherical harmonics aresolutions ofthespherical Laplace’ sEquation: theyarebasically anadap- tionofFourier analysis toaspherical surface. Justlikewith Fourier series, thesuperposition of spherical harmonics canbeused torepresent andanalyze physical phenomena distrib uted onthe surfaceon(orwithin) theEarth. Still inanalogy with Fourier theory ,there exists asampling the- orem which requires thatsufficient data areprovided inorder tomakethesolution possible. In geophysics, oneoften talks about (spatial) data coverage,which must beadequate. Wecanfindasolution forof   *bythegood oldtrick ofseparation ofvariables. Welook forasolution with thefollowing structure: 
      

 (2.68) Let’stakeeach factor separately .Inthefollowing, anoutline isgivenofhowtofindthesolution ofthiselliptic equation, butworking thisoutrigorously requires some more effortthan youmight bewilling tospend. Butlet’snottrytolosethephysical meaning what wecome upwith. Radialdependence:   Itturns outthatthefunctions satisfying Laplace’ sEquation belong toaspecial class ofhomoge- neous7harmonic8functions. Afirstproperty ofhomogeneous functions thatcanbeused toour advantage isthatingeneral, ahomogeneous function canbewritten intwodifferent forms: .
     
  (2.69) 
   
(    . 
  (2.70) This ofcourse givestheform ofourradial function: 7Ahomogenous function ofdegree satisfies 
 5  $5  )   
5 $5 ) . 8Bydefinition, afunction which satisfies Laplace’ sequation iscalled homogeneous. 2.5.SPHERICAL HARMONICS 45     ;.' =  .
(2.71) Thetwoalternati ves    and    (B   .describe thebehavior offoranexternal andinternal field, respecti vely(in-andoutside themass distrib ution). Whether touse    and    (B   .depends ontheproblem you’reworking onandontheboundary conditions. Iftheproblem requires afinite value forat *than weneed touse    .Howeverif werequire   *for   then wehavetouse    (B   ..Thelatter isappropriate forrepresenting thepotential outside thesurfacethatencloses allsources ofpotential, such as thegravity potential     ,/..However,both areneeded when wedescribe themagnetic potential atpointduetoaninternal andexternal field. Longitudinal dependence:   Substitution ofEq.2.68 intoLaplace’ sequation with  givenbyEq.2.71, anddividing Eq. 2.68 outagain yields anequation inwhich
-and -derivativesoccur onseparate sides ofthe equation sign. Forarbitrary
and thismust mean:   

constant(2.72) which isbestsolvedbycalling theconstant andsolving for
as:
        (2.73) Indeed, allpossible constants and givevalidsolutions, and must beapositi veinteger. Latitudinal dependence:   Thecondition issimilar ,except itinvolvesboth and.After some rearranging, itispossible to arriveat
##


##
 
   (
   
  *(2.74) This isawell knowntranscendental equation: itistheassociated Legendr eEquation .Itturns outthatthespace ofthehomogeneous functions hasadimension
(,hence *  . Ifwesubstitute 
,Eq.2.74 becomes (   # #      
 ##   8 (  (  9      *(2.75) Eq.2.75 isinstandard form andcanbesolvedusing avariety oftechniques. Most commonly , thesolutions arefound aspolynomials  
.Theassociated Legendre Equation reduces to theLegendre Equation incase  "*.Inthelatter case, thelongitudinal dependence islostas also Eq.2.73 revertstoaconstant. Theresulting functions  
havearotational symmetry around the -axis. Theyarecalled zonal functions. 46 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD Itispossible tofindexpressions ofthe(associated) Legendre polynomials thatsummarize their behaviorasfollows:     (
# #     ( (2.76)       (  

#  #       ( (2.77) written interms ofthe     derivativeand  
.Itiseasy tomakeasmall table with these polynomials (note thatinTable 2.1,wehaveused some trigrules tosimplify theexpressions.) — thisshould getyoustarted inusing Eqs. 2.76 or2.77. l     
 01 1 1z 
2 .  $  ( .  $

( 3 .    $  .  $
$
 Table 2.1:Legendre polynomials. Some Legendre functions areplotted inFigure 2.13. 0 0.5 1 1.5 2 2.5 3−1−0.8−0.6−0.4−0.200.20.40.60.81 θP(cosθ) P0 P1 P2 P3 Figure 2.13: Legendre polynomials. Spherical harmonics The generic solution foristhus found bycombining theradial, longitudinal andlatitudinal behaviors asfollows: 
      ;.' =  .
              
(2.78) 2.5.SPHERICAL HARMONICS 47 These arecalled thesolid spherical harmonics ofdegr ee andorder . Thespherical harmonics form acomplete orthonormal basis. Weimplicitly assume thatthefull solution isgivenbyasummation overallpossible and indices, asin: 
    (

  ;.' =  .
             
(2.79) Theconstants need tobedetermined from theboundary conditions. Because thespherical har- monics form acomplete orthonormal basis, anarbitrary realfunction 
  canbeexpanded in terms ofspherical harmonics by 
    (

            
 (2.80) The process ofdetermining thecoefficients   and  isanalogous tothattodetermine the coefficients inaFourier series, i.e.multiply both sides ofEq. 2.80 by     


or    


,integrate, andusetheorthogonality relationship —outcomes   .Forun- equal data distrib utions, thecoefficients may befound inaleast-squares sense. Visualization Itisimportant tovisualize thebehaviorofspherical harmonics, asinFigure 2.14. Some terminology toremember isthatonthebasis ofthevalues of andoneidentifies three types ofharmonics.DThe zonal harmonics aredefined tobethose oftheform 

  
.The superposition ofthese Legendre polynomials describe variations with latitude; theydonot depend onlongitude. Zonal harmonics vanish at small circles ontheglobe, dividing the spheres intolatitudinal zones.DThe sectorial harmonics areoftheform     
or     
.As theyvanish at
meridians (longitudinal lines, so great circles), theydivide thesphere intosectors.DThetesseral harmonics arethose oftheform     
or      
 for  .Theamplitude ofasurfacespherical harmonic ofacertain degree andorder  vanishes at
meridians oflongitude andon   parallels oflatitude. Inother words thedegree givesthetotal number ofnodal lines andtheordercontrols how thisnumber isdistrib uted overnodal meridians andnodal parallels. The higher thedegree and order thefiner thedetail thatcanberepresented, butincreasing and only makessense ifdata coverage issufficient toconstrain thecoefficients ofthepolynomials. Adifferent rendering isgiveninFigure 2.15. Finally ,animportant property followsfrom thedepth dependence ofthesolution: From eqn. (2.79) wecanseethat(1)theamplitude ofallterms will decrease with increasing distance from theorigin (i.e., theinternal source ofthepotential) andthat(2)therateofdecay increases with increasing degree . This hasthefollowing consequences: 48 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD Figure 2.14: Some spherical harmonics. 1.with increasing distance thelowerorder harmonics become increasingly dominant since the signal from small-scale structure (large ’sand ’s)decays more rapidly .Recall thatthe perturbations ofsatellite orbits constrain thelowerorders veryaccurately .Thefinedetails atdepth aredifficult todiscern atthesurfaceoftheEarth (orfurther outinspace) owing to thisspatial attenuation. 2.Conversely ,thiscomplicates thedownw ardcontinuation of  
  from theEarth’ ssur- facetoasmaller radius, since thisprocess introduces higher degree components intheso- lutions thatarenotconstrained bydata atthesurface. This problem isimportant inthe downw ardcontinuation ofthemagnetic field tothecore-mantle-boundary . 2.6.GLOBALGRAVITYANOMALIES 49 Zonal P5,0Sectoral P5,5Tesseral P5,3 Figure 2.15: Zonal, tesseral andsectoral spherical harmonics. 2.6Globalgravityanomalies Gravityinandoutsideasphericalmasssheet Thefullsolution toLaplaces equation wasgiveninEquation 2.79. We’vetalkedabout thechoice ofradial functions. Inside themass distrib ution, weuse 
                 
(2.81) andoutside, weuse  
    (  .              
(2.82) From nowon,we’lladdanormalization factor withtheradius attheequator:  
     (


   .             
(2.83) Sointerms ofasurfacespherical harmonic potential   ontheunitcircle, wegetthefollowing equations forthefield in-andoutside themass distrib ution:   
        
9  .  (2.84) Forthegravity,thisbecomes:
      ,/.   
     . ( (  .    (2.85) What isthegravityduetoathinsheet ofmass ofspherical harmonic degree ?Let’srepresent this asasheet with vanishing thickness, andcall  themass density perunitarea. This waywecan 50 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD workatconstant andusetheresults forspherical symmetry .Weknowfrom Gauss’ slawthatthe fluxthrough anysurfaceenclosing abitofmass isequal tothetotal enclosed mass (times   ). Soconstructing aboxaround apatch ofsurface with area # E,enclosing abitofmass # ,we candeduce that       (2.86) Onthisshell —giveitaradius a,wecanuseEqs. 2.85 tofind     (B and      B ,andsolvefor   using Eq.2.86 as        B 
(.Plugging thisintoEqs. 2.85 again wegetforthegravityin-andoutside thismass distrib ution     
(    ,/. ,/.        (
(       (2.87) Lengthscales Measurements ofgravitational attraction are—aswehaveseen —useful inthedetermination oftheshape androtational properties oftheEarth. This isimportant forgeodesy .Inaddition, theyalso provide information about aspherical density variations inthelithosphere andmantle (important forunderstanding dynamical processes, interpretation ofseismic images, orforfinding mineral deposits). However,before gravitymeasurements canbeused forinterpretation several corrections willhavetobemade: thedata reductions plays animportant roleingravimetry since thesignal pertinent tothestructures weareinterested inisverysmall. Let’stakeastepback andgetafeelforthedifferent length scales andprobable sources involved. Ifweuseaspherical harmonic expansion ofthefieldwecanseethatit’stheup-ordownw ard continuation ofthefield anditsdependence on anddegree thatcontrols thebehavior ofthe solution atdifferent depths (orradius) (remeber Eq.2.83). Withincreasing from thesource theamplitude ofthesurfaceharmonics become smaller and smaller ,andthedecay inamplitude (spatial attenuation) isstronger forthehigher degrees (i.e., thesmall-scale structures). 2.6.GLOBALGRAVITYANOMALIES 51 Intermezzo 2.8CARTESIANVSSPHERICALREPRESENTATION Ifyouworkonasmall scale with local gravity anomalies (forinstance inexploration geophysics) itis notefficient touse(global) basis functions onasphere because thenumber ofcoefficients thatyou’dneed would simply betoolarge.Forexample togetresolution oflength scales of100km(about 1 )youneed toexpand uptodegree  =360 which with allthecombinations -
-  involvesseveralhundreds of thousands ofcoefficients (howmanyexactly?). Instead youwould useaFourier Series. The concept is similar tospherical harmonics. AFourier series isjustasuperposition ofharmonic functions (sine and cosine functions) with different frequencies (orwavenumbers
 #
, thewavelength): '254 6 ;  6 

    '254 6 ;  6  ; # =  (2.88) (Fora2Dfield theexpression includes ybutisotherwise beverysimilar .)Or,inmore general form"'>  254 6 ;  =0  6  ;  =  (2.89) (compare totheexpression ofthespherical harmonics). Inthisexpression  specifies thesizeofthegrid atwhich themeasurements aremade, andtheup-anddownw ardcontinuation ofthe1Dor2Dharmonic field iscontrolled byanexponential form. Theproblem with downw ardcontinuation becomes immediately clear from thefollowing example. Suppose inamarine gravityexpedition toinvestigate density variation in thesediments beneath theseafloor,say,at2kmdepth, gravitymeasurements aretakenat10minterv alson theseasurface. Upon downw ardcontinuation, thesignal associated with thesmallest wavelength allowed bysuch gridspacing would beamplified byafactor of
#   
!    
#   "!!  $#).(The waterdoes notcontain anyconcentrations ofmass thatcontrib utetothegravity anomalies andintegration overthesurfaceenclosing thewatermass would addonly aconstant value tothe gravitypotential butthatisirrele vantwhen studying anomalies, andLaplace’ sequation canstillbeused.) Soitisimportant tofilter thedata before thedownw ardcontinuation sothatinformation ismaintained only onlength scales thatarenottoomuch smaller than thedistance overwhich thedownw ardcontinuation has totakeplace. Table 2.2givesandidea about therelationship between length scales, theprobable source regions, andwhere themeasurements havetobetaken. wavelength short wavelength long wavelength ( &% (+* * *kmor 4 $ !)( 4 (+* * *kmor % $ !) Source region shallo w: probably deep crust, lithosphere (lowermantle) butshallo wer source cannot beexcluded Measurement: close tosource: surface, Largerdistance from origin how,where? sealevel,”loworbit” ofanomalies; perturbations satellites, planes ofsatellite orbits Representation values atgridpoints; spherical harmonics 2DFourier series Coordinate system cartesian spherical Table 2.2:Wavelength ranges ofgravityanomalies Thefree-airgravityanomaly Let’sassume thatthegeoid height'with respect tothespheroid isduetoananomalous mass# .If # represents excess mass, theequipotential iswarped outw ards andthere willbeageoid 52 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD high ( ' 4 *);conversely ,if # represents amass deficienc y, ' %*andthere willbeageoid low. Figure 2.16: Mass deficit leads togeoid undulation. Wecanrepresent thetwopotentials asfollows: theactual geoid, 
  isanequipotential surfacewith thesame potential 
asthereference geoid 
,only 
   

    
  (2.90) Wedefine thefree-air gravityanomaly asthegravity  measured atpoint minus thegravity attheprojection
ofthispoint onto thereference geoid at
,

.Neglecting thesmall differences indirection, wecanwrite forthemagnitudes:   

(2.91) Figure 2.17: Derivation. Note thatinthisfigure, the sign convention forthegravityisreversed; wehave used andareusing thatthegravity isthenegative gradient ofthepotential. Interms ofpotentials:
  


# 
#    '

 %
' (2.92) (Remember that 
isthemagnitude ofthenegativegradient ofandtherefore appears with a positi vesign.) Weknewfrom Eq.2.90 that 2.6.GLOBALGRAVITYANOMALIES 53   
    

 %
'   (2.93) Butalso, since thepotentials ofand
were equal,   

andwecanwrite
'    (2.94) This result isknownasBrun’ sformula .Nowforthegravityvectors
and

,theyaregivenby thefamiliar expressions
  
   
(2.95) andthegravity disturbance vector


canbedefined asthedifference between those twoquantities:
    
# #(2.96) Ontheother hand, from afirst-order expansion, welearn that
  


#
#  '  '  %


# %
#  '  ' # #(2.97) (2.98) Nowwedefine thefree-air gravity anomaly asthedifference ofthegravitational accelartion measured ontheactual geoid (ifyou’reonamountain you’llneed torefer tosealevel)minus the reference gravity:   

(2.99) This translates into 
#
#  '  ' # ###
  '  ' # # 



 ' # # 54 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD 

%
' # # 

 # #(2.100) (2.101) Soatthisarbitrary point onthegeoid, thegravityanomaly duetotheanomalous mass arises from twosources: thedirect contrib ution# duetotheextraacceleration bythemass# itself, andanadditional contrib ution # thatarises from thefactthat ismeasured onheight 'above thereference spheroid. Thelatter term isessentially afreeaircorrection ,similar totheoneone hastoapply when referring themeasurement (onamountain, say) totheactual geoid (sealevel). Note thateq.(2.94) contains theboundary conditions of   *.Thegeoid height'atany point depends onthetotal effectofmass excesses anddeficiencies overtheEarth. 'canbe determined uniquely atanypoint 
  from measurements ofgravityanomalies takenoverthe surfaceofthewhole Earth —thiswasfirst done byStokes(1849) —butitdoes notuniquely constrain thedistrib ution ofmasses. Gravityanomalies fromgeoidalheights Aconvenient waytodetermine thegeoid heights' 
  from either thepotential field anomalies 
  orthegravity anomalies 
  isbymeans ofspherical harmonic expansion of' 
  interms of  
  or 
  . It’sconvenient tojustgivethecoefficients ofEq.2.83 since thebasic expressions arethesame. Let’sseehowthatnotation would workforeq.(2.83):

  
   (2.102) Note thatthesubscripts and areused tolabel thecoefficients ofthe   and   parts, respecti vely.Note also thatwehavenowtakenthefactor  , asthescaling factor ofthe coefficients. Wecanalsoexpand thepotential
onthereference spheroid:





  

 * (2.103) (Note thatwedidnotdrop the,eventhough  *forthezonal harmonics used forthe reference spheroid. Wejustrequire thecoefficient 
 tobezero for    *.Bydoing thiswe cankeeptheequations simple.) Thecoefficients oftheanomalous potential  
  arethen givenby:

   
   
   (2.104) Wecannowexpand 
  inasimilar series using eq.(2.100). Forthe ,wecanseeby inspection thattheradial derivativeasprescribed hasthefollowing effectonthecoefficients (note thatthereference radius 
from earlier definitions): 2.6.GLOBALGRAVITYANOMALIES 55# #   (
(2.105) andtheother term ofEq.2.100 brings down

 
(2.106) Asaresult, weget

   
 ( 
   
   (2.107)
  (
   
   (2.108) Theproportionality with   (
means thatthehigher degree terms aremagnified inthegravity field relati vetothose inthepotential field. This leads totheimportant result thatgravity maps typically contain much more detail than geoid maps because thespatial attenuation ofthehigher degree components issuppressed. Using eq.(2.94) wecanexpress thecoefficients oftheexpansion of' 
  interms ofeither the coefficients oftheexpanded anomalous potential

'
'   
   
   (2.109) which, ifwereplace by andbyassuming that getsthefollowing form
'
'  
   
   (2.110) orinterms ofthecoefficients ofthegravityanomalies (eqns. 2.108 and2.110)
'
'     (


 
   
   (2.111) The geoid heights canthus besynthesized from theexpansions ofeither thegravity anomalies (2.111) ortheanomalous potential (2.110). Geoid anomalies havebeen constructed from both surfacemeasurements ofgravity(2.111) andfrom satellite observ ations (2.110). Equation (2.111) indicates thatrelati vetothegravity anomalies, thecoefficients of ' 
  aresuppressed bya factor of (B   (.Asaresult, shorter wavelength features aremuch more prominent ongravity maps. Inother words, geoid (and geoid height) maps essentially depict thelowharmonics ofthe gravitational field. Afinal note thatisrelevantforthereduction ofthegravitydata. Gravitydata aretypically reduced tosea-le vel,which coincides with thegeoid andnotwith theactual reference spheroid. Eq.8canthen beused tomaketheadditional correction tothereference spheroid, which effectivelymeans thatthelong wavelength signal isremo ved.This results inveryhigh resolution gravitymaps. 56 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD 2.7Gravityanomalies andthereductionofgravitydata Thecombination ofthereduced gravityfield andthetopography yields important information on themechanical state ofthecrust andlithosphere. Both gravity andtopography canbeobtained byremote sensing andinmanycases theyform thebasis ofourknowledge ofthedynamical state ofplanets, such asMars, andnatural satellites, such asEarth’ sMoon. Data reduction plays an important roleingravitystudies since thesignal caused bytheaspherical variation indensity that onewants tostudy areverysmall compared notonly totheobserv edfield butalso other effects, such astheinfluence oftheposition atwhich themeasurement ismade. Thefollowing sum shows thevarious components totheobserv edgravity,with thename ofthecorresponding corrections thatshould bemade showninparenthesis: Observed gravity =attraction ofthereference spher oid,PLUS:Deffects ofelevation abovesealevel(FreeAircorrection ),whichshould include theeleva- tion(geoid anomaly) ofthesealevelabovethereference spher oidDeffectof”normal” attracting mass between observation point andsealevel(Bouguer and terrain correction )Deffectofmasses thatsupport topographic loads (isostatic correction )Dtime-dependent chang esinEarth’ sfigureofshape (tidal correction)Deffectofchang esintherotation term duetomotion oftheobservation point (e.g.when measur ements aremade fromamoving ship. (E¨otv¨oscorrection)Deffects ofcrust andmantle density anomalies (”geology”or”geodynamic processes”). Only thebold corrections willbediscussed here. Thetidal correction issmall, butmust beac- counted forwhen high precision data arerequired. Theapplication ofthedifferent corrections is illustrated byasimple example ofasmall density anomaly located inatopography high thatis isostatically compensated. Seeseries ofdiagrams. FreeAirAnomaly Sofarithasbeen assumed thatmeasurements atsealevel(i.e. theactual geoid) were available. This isoften notthecase. If,forinstance, ismeasured ontheland surfaceatanaltitude one hastomakethefollowing correction :# 
 
(2.112) For atsealevelthiscorrection amounts to# 
  * $%* ! mgal or * $%* ! &3(+*,ms , ( in meter). Note thatthisassumes nomass between theobserv erandsealevel,hence thename “free- air”correction. Theeffectofellipticity isoften ignored, butonecanuse   ( 
. NBpermeter elevation thiscorrection equals $ ( & (+*,
ms ,  $ ( & (+*,:thisisonthe limit oftheprecision thatcanbeattained byfield instruments, which showsthatuncertainties in elevation arealimiting factor intheprecision thatcanbeachie ved.(Arealistic uncertainty is1 mgal). 2.7.GRAVITYANOMALIES ANDTHEREDUCTION OFGRAVITYDATA 57 Makesurethecorrection isapplied correctly,since therecanbeconfusion about thesign ofthe correction, whichdepends onthedefinition ofthepotential. Theobjective ofthecorrection isto compensate forthedecrease ingravity attraction with increasing distance fromthesource(center oftheEarth). Formally ,given theminus sign in(2.113), thecorrection hastobesubtr acted, but itisnotuncommon totakethecorrection asthepositive number inwhichcase itwillhave tobe added. (Justbear inmind thatyouhave tomakethemeasur edvalue largerby“adding” gravity soitcompar esdirectly tothereference value atthesame height; alternatively ,youcanmakethe reference value smaller ifyouareabovesealevel;ifyouareinasubmarine youwill, ofcourse, have todotheopposite). TheFreeAiranomaly isthen obtained bythecorrection forheight abovesealevelandbysub- traction ofthereference gravityfield
  
#
 
   
* $%* ! & (+* ,  
 (2.113) (Note thatthere could beacomponent duetothefactthatthesealevel(thegeoid) does not coincide with thereference spheroid; anadditional correction canthen bemade totakeoutthe extragravityanomaly .One cansimply apply (2.113) anduse   'astheelevation, which isequivalent toadding acorrection to 
 sothatitrepresents thereference value atthegeoid. This correction isnotimportant ifthevariation ingeoid issmall across thesurveyregion because then thecorrection isthesame foralldata points.) Bougueranomaly Thefreeaircorrection does notcorrect foranyattracting mass between observ ation point andsea level.However,onland, atacertain elevation there willbeattracting mass (eventhough itisoften compensated -isostasy (see below)). Instead ofestimating thetrueshape of,say,amountain on which themeasurement ismade, oneoften resorts towhat isknownasthe”slab approximation” in which onesimply assumes thattherocks areofinfinite horizontal extent. TheBouguer correction isthen givenby#  
  (2.114) where isthegravitational constant, istheassumed mean density ofcrustal rock and isthe height abovesealevel.For   ! !# &)(+*,/.0.m  kg ,  s , and 
 #* *kgm ,  weobtain a correction of ( ( & (+* ,
ms , permeter ofelevation (or * ( ( mgal, inmeter). Iftheslab approximation isnotsatisf actory ,forinstance near thetopofmountains, onhastoapply anaddi- tional terrain correction .Itisstraightforw ardtoapply theterrain correction ifonehasaccess to digital topography/bathymetry data. The Bouguer anomaly hastobesubtracted, since onewants toremo vetheeffects oftheextra attraction. TheBouguer correction istypically applied after theapplication oftheFree Aircor- rection. Ignoring theterrain correction, theBouguer gravityanomaly isthen givenby  
#
 
  #  
 # (2.115) Inprinciple, with theBouguer anomaly wehaveaccounted fortheattraction ofallrock between observ ation point andsealevel,and thusrepresents thegravitational attraction ofthematerial belowsealevel.Bouguer Anomaly maps aretypically used tostudy gravityoncontinents whereas theFree AirAnomaly ismore commonly used inoceanic regions. 58 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD Isostasyandisostaticcorrection Ifthemass between theobserv ation point andsealevelisallthatcontrib utes tothemeasured gravityonewould expect thattheFree Airanomaly islarge,andpositi veovertopography highs (since thismass isunaccounted for)andthattheBouguer anomaly decreases tozero. This rela- tionship between thetwogravityanomalies andtopography isindeed what would beobtained in case themass iscompletely supported bythestrength oftheplate (i.e.noisostatic compensation). Inearly gravity surveys,however,theyfound thattheBouguer gravity anomaly overmountain ranges was,some what surprisingly ,largeandnegative.Apparently ,amass deficienc yremained after themass abovesealevelwascompensated for.Inother words, theBouguer correction sub- tracted toomuch! This observ ation inthe19   century lead Airy andPratt todevelop theconcept ofisostasy .Inshort, isostasy means thatatdepths largerthan acertain compensation depth the observ edvariations inheight abovesealevelnolonger contrib utetolateral variations inpressure. Incase ofAiry Isostasy thisisachie vedbyacompensation root, such thatthedepth totheinter- facebetween theloading mass (with constant density) andtherestofthemantle varies. This is,in factArchimedes’ Law,andagood example ofthismechanism isthefloating iceber g,ofwhich we seeonly thetopabovethesealevel.Inthecase ofPratt Isostasy thecompensation depth does not varyandconstant pressure isachie vedbylateral variations indensity .Itisnowknownthatboth mechanisms play arole. Figure 2.18: Airy andPratt isostasy . Thebasic equation thatdescribes therelationship between thetopographic height andthedepth of thecompensating body is:  
 
(2.116) Figure 2.19: Airy isostasy . 2.8.CORRELA TIONBETWEEN GRAVITYANOMALIES ANDTOPOGRAPHY . 59 Assuming Airy Isostasy andsome constant density forcrustal rock onecancompute 
  from known(digital) topography 
  andthus correct forthemass deficienc y.This results inthe Isostatic Anomaly .Ifallisdone correctly theisostatic anomaly isolates thesmall signal dueto thedensity anomaly thatisnotcompensated (local geology ,orgeodynamic processes). 2.8Correlationbetweengravityanomalies andtopograph y. Thecorrelation between Bouguer andFree Airanomalies ontheonehand andtopography onthe other thus contains information astowhat levelthetopography isisostatically compensated. Inthecase ofAiry Isostasy itisobvious thatthecompensating rootcauses themass deficienc ythat results inanegativeBouguer anomaly .Ifthetopography iscompensated themass excess above sealeveliscanceled bythemass deficienc ybelowit,andasaconsequence theFree AirAnomaly issmall; usually ,itisnotzero since theattracting mass iscloser totheobserv ation point andis thus lessattenuated than thecompensating signal ofthemass deficienc ysothatsome correlation between theFree AirAnomaly andtopography canremain. Apart fromthiseffect(whic halso plays arolenear theedgesoftopographic featur es),theFree airanomaly isclose tozeroandtheBouguer anomaly largeandnegative when thetopography is completely compensated isostatically (also referr edtoas“inisostatic equilibrium”). Incase thetopography isNOTcompensated, theFreeairanomaly islargeandpositive ,andthe Bouguer anomaly zero. (This also depends onthelength scale oftheload andthestrength ofthesupporting plate). Whether ornotatopographic load isorcanbecompensated depends largely onthestrength (and thethickness) ofthesupporting plate andonthelength scale oftheloading structure. Intuiti velyit isobvious thatsmall objects arenotcompensated because thelithospheric plate isstrong enough tocarry theload. This explains why impact craters cansurvi veoververylong periods oftime! (Largecraters may beisostatically compensated, butthenarro wrims ofthecrater willnotdisap- pear byflow!) Incontrast, loading overlargeregions, i.e.much largerthan thedistance tothe compensation depth, results inthedevelopment ofacompensating root. Itisalso obvious why thestrength (viscosity )oftheplate enters theequation. Iftheviscosity isverysmall, isostatic equilibrium canoccur evenforverysmall bodies (consider ,forexample, thefloating iceber g!). Willdiscuss therelationship between gravityanomalies andtopography inmore (theoretical) de- taillater.Wewillalso seehowviscosity adds atime dependence tothesystem. Also thisiseasy tounderstand intuiti vely;lowstrength means thatisostatic equilibrium canoccur almost instantly (iceber g!),butforhigher viscosity therelaxation time ismuch longer .Theflowrateofthematerial beneath thesupporting plate determines howquickly thisplate canassume isostatic equilibrium, andthisflowrateisafunction ofviscosity .Forlargeviscosity ,loading orunloading results ina viscous delay; forinstance therebound after deglaciation. 2.9Flexureandgravity. Thebending ofthelithosphere combined with itslargestrength is,infact,oneofthecompensation mechanisms forisostasy .When wediscussed isostasy wehaveseen thatthedepth tothebottom oftheroot, which islessdense than surrounding rock atthesame depth, canbecalculated from Archimedes’ Principle :ifcrustal material with density
replaces denser mantle material with density amountain range with height hasacompensating rootwith thickness 60 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD  
 
(2.117) This type ofcompensation isalsoreferred toasAiry Isostasy .Itdoes notaccount foranystrength oftheplate. However,itisintuiti velyobvious thatthedepression decreases ifthestrength (or theflexural rigidity) ofthelithosphere increases. Theconsideration oflithospheric strength for calculates based onisostasy isimportant inparticular fortheloading onnottoolong atime scale. Anelegant andveryuseful waytoquantify theeffectofflexure isbyconsidering theflexure due toaperiodic load.Let’sconsider aperiodic load duetotopography with maximum amplitude
andwavelength : 
 

B .Thecorresponding load isthen givenby 
 




(2.118) sothattheflexure equation becomes# 
#
  
 




(2.119) Thesolution canbeshowntobe

 
 ; ! = 
  (   ; ! =  




(2.120) From eq.(2.120) wecanseethatforverylargeflexural rigidity (orverylargeelastic thickness oftheplate) thedenominator willpredominate theequation andthedeflection willbecome small (
 *for  );inother words, theload hasnoeffectonthedepression. The same is trueforshort wavelengths, i.e.for 
 B
 . .Incontrast, forverylong wavelengths ( 
 B
 . )orforaveryweak (orthin) plate themaximum depression becomes



 
(2.121) which isthesame asforacompletely compensated mass (seeeq.2.117). Inother words, theplate “has nostrength” forlong wavelength loads. Theimportance ofthisformulation isevident ifyourealize thatanytopography canbedescribed bya(Fourier) series ofperiodic functions with different wavelengths. One canthus useFourier Analysis toinvestigate thedepression orcompensation ofanyshape ofload. Eq.(2.120) canbeused tofindexpressions fortheinfluence offlexure ontheFree Airandthe Bouguer gravityanomaly .Thegravityanomalies depend ontheflexural rigidity inverymuch the same wayasthedeflection in(2.120). free-air gravityanomaly:   

  (  , !
( 
,   ; ! =  




(2.122) Bouguer gravityanomaly: 2.9.FLEXURE ANDGRAVITY. 61   

 , !
( 
,   ; ! = 



(2.123) where isthedepth totheMoho (i.e. thedepressed interf acebetween
and)andthe exponential inthenumerator accounts forthefactthatthisinterf aceisatacertain depth (this factor controls, infact,thedownw ardcontinuation). Theimportant thing toremember isthelinear relationship with thetopography andthepropor - tionality with ,/..One canfollowasimilar reasoning asabovetoshowthatforshort wave- lengths thefreeairanomaly islarge(and positi ve)andthattheBouguer anomaly isalmost zero. This canbeexplained bythefactthattheflexure isthen negligible sothattheBouguer correc- tionsuccessfully remo vesallanomalous structure. However,forlong wavelength loads, theload iscompletely compensated sothatafter correction tozero elevation, theBouguer correction still ’feels’ theanomalously lowdensity root (which isnotcorrected for). TheBouguer anomaly is largeandnegativeforacompletely compensated load. Complete isostasy means alsothatthere is nonetmass difference sothatthefreeairgravityanomaly isverysmall (practically zero). Gravity measurements thus contain information about thedegree ofisostatic compensation. Thecorrelation between thetopography andthemeasured Bouguer anomalies canbemodeled by means ofeq.(2.123) andthisgivesinformation about theflexural rigidity ,andthus the(effective !) thickness oftheelastic plate. Thediagram belowgivestheBouguer anomaly asafunction ofwave length (i.e. topography wassubjected toaFourier transformation). Itshowsthattopography with wavelengths lessthan about 100kmisnotcompensated (Bouguer anomaly iszero). The solid curvesarethepredictions according toeq.2.123 fordifferent values oftheflexural parameter.Theparameters used forthese theoretical curvesare   $ * *kgm ,  , 

#* *kgm ,  , $%*km,   B  
 .  5,10,20,and50km.There isconsiderable scatter but avalue for ofabout 20seems tofittheobserv ations quite well, which, with
=60GPaand = 0.25, givesaneffectiveelastic thickness  !km. Figure 2.20: Bouguer anomalies andtopography . 62 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD 2.9.1Post-glacial reboundandviscosity. Sofarwehavelookedatthebending orflexure oftheelastic lithosphere toloading, forinstance byseamounts. Todetermine thedeflection

weused theprinciple ofisostasy .Inorder for isostasy toworkthemantle beneath thelithosphere must beable toflow.Conversely ,ifweknow thehistory ofloading, orunloading, soifweknowthedeflection asafunction oftime

 ,we caninvestigate theflowbeneath thelithosphere. Therateofflowisdependent ontheviscosity of themantle material. Viscosity plays acentral roleinunderstanding mantle dynamics. Dynamic viscosity canbedefined astheratio oftheapplied (deviatoric) stress andtheresultant strain rate; here wemostly consider Newtonian viscosity ,i.e.,alinear relationship between stress andstrain rate. Theunitofviscosity isPascal Second [Pas]. Aclassical example ofasituation where thehistory of(un)loading issufficiently well knownis thatofpost-glacial rebound .Theconcept issimple: 1.thelithosphere isdepressed upon loading ofanicesheet (viscous mantle flowawayfrom depression makethispossible) 2.theicesheet melts attheendofglaciation andthelithosphere starts rebound slowlytoits original state (mantle flowtowards thedecreasing depression makesthispossible). The uplift iswell documented from elevated (and dated) shore lines. From therateofreturn flowonecanestimate thevalue fortheviscosity . Tworemarks: 1.thedimension oftheload determines tosome extend thedepth overwhich themantle is involvedinthereturn flow thecomparison ofrebound history fordifferent initial load dimensions givessome constraints onthevariation ofviscosity with depth. 2.Onlong time scales thelithosphere hasno“strength”, butinsophisticated modeling ofthe post glacial rebound theflexural rigidity isstilltakenintoaccount. Also takenintoaccount inrecent models isthehistory ofthemelting andtheretreat oftheicecapitself (including thechanges inshore linewith time!). Inolder models oneonly investigated theresponse to instantaneous remo valoftheload. 2.9.FLEXURE ANDGRAVITY. 63 Typical values forthedynamic viscosity intheEarth’ smantle are10 . Pasfortheupper mantle to(+* .Pasforthelowermantle. Thelithosphere iseven“stiffer”, with atypical viscosity ofabout 10 (forcomparison: wateratroom temperature hasaviscosity ofabout 10 ,  Pas;thisseems small butifyou’veeverdivedofa10mboard youknowit’snotnegligible!) Things toremember about these values: 1.verylargeviscosity intheentire mantle 2.lowermantle (probably) more viscous than upper mantle, 3.thedifference isnotverylargecompared tothelargevalue oftheviscosity itself. Animportant property ofviscosity isthatitistemper aturedependent ;theviscosity decreases ex- ponentially with increasing temperature as 
, 



,whereisthemelting temperature and-30isanempirical, material dependent value. 64 CHAPTER 2.THEEARTH'SGRAVITATIONALFIELD This temperature dependence ofviscosity explains why onegets convection beneath thecooling lithosphere. Aswehavediscussed before, with typical values forthegeothermal gradient (e.g., 20Kkm ,/.)asdeduced from surfaceheat flowusing Fourier’ sLawthetemperature would quickly risetonear thesolidus, thetemperature where therock starts tomelt. However,weknowfrom severalobserv ations, forinstance from thepropagation ofS-waves,thatthetemperature isbelow thesolidus inmost parts ofthemantle (with thepossible exception inthelowvelocity zone beneath oceanic andparts ofthecontinental lithosphere). Sothere must beamechanism thatkeeps the temperature down,or,inother words, thatcools themantle much more efficiently than conduction. That mechanism isconvection .Wesawabovethattheviscosity ofthelithosphere isveryhigh, and upper mantle viscosity isabout 5orders ofmagnitude lower.This islargely duetothetemperature dependence oftheviscosity (asmentioned above):when thetemperature getscloser tothesolidus ( )theviscosity drops andthematerial starts toflow.