3.02 Potential Theory and Static Gravity Field of the Earth

Transcription

1 3.02 Potential Theoy and Static Gavity Field of the Eath C. Jekeli, The Ohio State Univesity, Columbus, OH, USA ª 2007 Elsevie B.V. All ights eseved Intoduction Histoical Notes Coodinate Systems Peliminay Definitions and Concepts Newton s Law of Gavitation Bounday-Value Poblems Geen s Identities Uniqueness Theoems Solutions by Integal Equation Solutions to the Spheical BVP Spheical Hamonics and Geen s Functions Invese Stokes and Hotine Integals Vening-Meinesz Integal and Its Invese Concluding Remaks Low-Degee Hamonics: Intepetation and Refeence Low-Degee Hamonics as Density Moments Nomal Ellipsoidal Field Methods of Detemination Measuement Systems and Techniques Models The Geoid and Heights 38 Refeences 4 Glossay deflection of the vetical Angle between diection of gavity and diection of nomal gavity. density moment Integal ove the volume of a body of the poduct of its density and intege powes of Catesian coodinates. distubing potential The diffeence between Eath s gavity potential and the nomal potential. eccenticity The atio of the diffeence of squaes of semimajo and semimino axes to the squae of the semimajo axis of an ellipsoid. ellipsoid Suface fomed by otating an ellipse about its mino axis. equipotential suface Suface of constant potential. flattening The atio of the diffeence between semimajo and semimino axes to the semimajo axis of an ellipsoid. geodetic efeence system Nomal ellipsoid with defined paametes adopted fo geneal geodetic and gavimetic efeencing. geoid Suface of constant gavity potential that closely appoximates mean sea level. geoid undulation Vetical distance between the geoid and the nomal ellipsoid, positive if the geoid is above the ellipsoid. geopotential numbe Diffeence between gavity potential on the geoid and gavity potential at a point. gavitation Attactive acceleation due to mass. gavitational potential Potential due to gavitational acceleation. gavity Vecto sum of gavitation and centifugal acceleation due to Eath s otation. gavity anomaly The diffeence between Eath s gavity on the geoid and nomal gavity on the

2 2 Potential Theoy and Static Gavity Field of the Eath ellipsoid, eithe as a diffeence in vectos o a diffeence in magnitudes. gavity distubance The diffeence between Eath s gavity and nomal gavity, eithe as a diffeence in vectos o a diffeence in magnitudes. gavity potential Potential due to gavity acceleation. hamonic function Function that satisfies Laplace s field equation. linea eccenticity The distance fom the cente of an ellipsoid to eithe of its foci. mean Eath ellipsoid Nomal ellipsoid with paametes closest to actual coesponding paametes fo the Eath. mean tide geoid Geoid with all time-vaying tidal effects emoved. multipoles Stokes coefficients. Newtonian potential Hamonic function that appoaches the potential of a point mass at infinity. non-tide geoid Mean tide geoid with all (diect and indiect defomation) mean tide effects emoved. nomal ellipsoid Eath-appoximating efeence ellipsoid that geneates a gavity field in which it is a suface of constant nomal gavity potential. nomal gavity Gavity associated with the nomal ellipsoid. nomal gavity potential Gavity potential associated with the nomal ellipsoid. othometic height Distance along the plumb line fom the geoid to a point. potential Potential enegy pe unit mass due to the gavitational field; always positive and zeo at infinity. sectoial hamonics Suface spheical hamonics that do not change in sign with espect to latitude. Stokes coefficients Constants in a seies expansion of the gavitational potential in tems of spheical hamonic functions. suface spheical hamonics Basis functions defined on the unit sphee, compising poducts of nomalized associated Legende functions and sinusoids. tesseal hamonics Neithe zonal no sectoial hamonics. zeo-tide geoid Mean tide geoid with just the mean diect tidal effect emoved (indiect effect due to Eath s pemanent defomation is etained). zonal hamonics Spheical hamonics that do not depend on longitude Intoduction Gavitational potential theoy has its oots in the late Renaissance peiod when the position of the Eath in the cosmos was established on moden scientific (obsevation-based) gounds. A study of Eath s gavitational field is a study of Eath s mass, its influence on nea objects, and lately its edistibuting tanspot in time. It is also fundamentally a geodetic study of Eath s shape, descibed lagely (70%) by the suface of the oceans. This initial section povides a histoical backdop to potential theoy and intoduces some concepts in physical geodesy that set the stage fo late fomulations Histoical Notes Gavitation is a physical phenomenon so pevasive and incidental that humankind geneally has taken it fo ganted with scacely a second thought. The Geek philosophe Aistotle ( BC) allowed no moe than to asset that gavitation is a natual popety of mateial things that causes them to fall (o ise, in the case of some gases), and the moe mateial the geate the tendency to do so. It was enough of a self-evident explanation that it was not yet to eceive the scutiny of the scientific method, the beginnings of which, ionically, ae cedited to Aistotle. Almost 2000 yeas late, Galileo Galilei ( ) finally took up the challenge to undestand gavitation though obsevation and scientific investigation. His expeimentally deived law of falling bodies coected the Aistotelian view and divoced the effect of gavitation fom the mass of the falling object all bodies fall with the same acceleation. This tuly monumental contibution to physics was, howeve, only a local explanation of how bodies behaved unde gavitational influence. Johannes Keple s (57 630) obsevations of planetay obits pointed to othe types of laws, pincipally an invese-squae law accoding to which bodies ae attacted by foces that vay with the invese squae of distance. The genius of Issac Newton ( ) bought it all togethe in his Philosophiae Natualis Pincipia Mathematica of 687 with a single and simple all-embacing law that in

3 Potential Theoy and Static Gavity Field of the Eath 3 one bold stoke explained the dynamics of the entie univese (today thee is moe to undestanding the dynamics of the cosmos, but Newton s law emakably holds its own). The mass of a body was again an essential aspect, not as a self-attibute as Aistotle had implied, but as the souce of attaction fo othe bodies: each mateial body attacts evey othe mateial body accoding to a vey specific ule (Newton s law of gavitation; see Section ). Newton egetted that he could not explain exactly why mass has this popety (as one still yeans to know today within the standad models of paticle and quantum theoies). Even Albet Einstein ( ) in developing his geneal theoy of elativity (i.e., the theoy of gavitation) could only impove on Newton s theoy by incopoating and explaining action at a distance (gavitational foce acts with the speed of light as a fundamental tenet of the theoy). What actually mediates the gavitational attaction still intensely occupies moden physicists and cosmologists. Gavitation since its ealy scientific fomulation initially belonged to the domain of astonomes, at least as fa as the obsevable univese was concened. Theoy successfully pedicted the obseved petubations of planetay obits and even the location of peviously unknown new planets (Neptune s discovey in 846 based on calculations motivated by obseved petubations in Uanus obit was a majo tiumph fo Newton s law). Howeve, it was also discoveed that gavitational acceleation vaies on Eath s suface, with espect to altitude and latitude. Newton s law of gavitation again povided the backdop fo the vaiations obseved with pendulums. An ealy achievement fo his theoy came when he successfully pedicted the pola flattening in Eath s shape on the basis of hydostatic equilibium, which was confimed finally (afte some contovesy) with geodetic measuements of long tiangulated acs in by Piee de Maupetuis and Alexis Claiaut. Gavitation thus also played a dominant ole in geodesy, the science of detemining the size and shape of the Eath, pomulgated in lage pat by the fathe of moden geodesy, Fiedich R. Helmet (843 97). Teestial gavitation though the twentieth centuy was consideed a geodetic aea of eseach, although, of couse, its geophysical exploits should not be ovelooked. But the advancement in modeling accuacy and global application was pomoted mainly by geodesists who needed a well-defined efeence fo heights (a level suface) and whose astonomic obsevations of latitude and longitude needed to be coected fo the iegula diection of gavitation. Today, the moden view of a height efeence is changing to that of a geometic, mathematical suface (an ellipsoid) and thee-dimensional coodinates (latitude, longitude, and height) of points on the Eath s suface ae eadily obtained geometically by anging to the satellites of the Global Positioning System (GPS). The equiements of gavitation fo GPS obit detemination within an Eath-centeed coodinate system ae now lagely met with existing models. Impovements in gavitational models ae motivated in geodesy pimaily fo apid detemination of taditional heights with espect to a level suface. These heights, fo example, the othometic heights, in this sense then become deived attibutes of points, athe than thei cadinal components. Navigation and guidance exemplify a futhe specific niche whee gavitation continues to find impotant elevance. While GPS also dominates this field, the vehicles equiing completely autonomous, self-contained systems must ely on inetial instuments (acceleometes and gyoscopes). These do not sense gavitation (see Section ), yet gavitation contibutes to the total definition of the vehicle tajectoy, and thus the output of inetial navigation systems must be compensated fo the effect of gavitation. By fa the geatest emphasis in gavitation, howeve, has shifted to the Eath sciences, whee detailed knowledge of the configuation of masses (the solid, liquid, and atmospheic components) and thei tanspot and motion leads to impoved undestanding of the Eath systems (climate, hydologic cycle, tectonics) and thei inteactions with life. Oceanogaphy, in paticula, also equies a detailed knowledge of a level suface (the geoid) to model suface cuents using satellite altimety. Clealy, thee is an essential tempoal component in these studies, and, indeed, the tempoal gavitational field holds cente stage in many new investigations. Moeove, Eath s dynamic behavio influences point coodinates and Eath-fixed coodinate fames, and we come back to fundamental geodetic concens in which the gavitational field plays an essential ole! This section deals with the static gavitational field. The theoy of the potential fom the classical Newtonian standpoint povides the foundation fo modeling the field and thus deseves the focus of the exposition. The tempoal pat is a natual extension that is eadily achieved with the addition of the time vaiable (no new laws ae needed, if we neglect

4 4 Potential Theoy and Static Gavity Field of the Eath geneal elativistic effects) and will not be expounded hee. We ae pimaily concened with gavitation on and extenal to the solid and liquid Eath since this is the domain of most applications. The intenal field can also be modeled fo specialized puposes (such as submaine navigation), but intenal geophysical modeling, fo example, is done usually in tems of the souces (mass density distibution), athe than the esulting field. u z b E δ Confocal ellipsoid Sphee x Coodinate Systems Modeling the Eath s gavitational field depends on the choice of coodinate system. Customaily, owing to the Eath s geneal shape, a spheical pola coodinate system seves fo most applications, and vitually all global models use these coodinates. Howeve, the Eath is slightly flattened at the poles, and an ellipsoidal coodinate system has also been advocated fo some nea-eath applications. We note that the geodetic coodinates associated with a geodetic datum (based on an ellipsoid) ae neve used in a foundational sense to model the field since they do not admit to a sepaation-of-vaiables solution of Laplace diffeential equation (Section ). Spheical pola coodinates, descibed with the aid of Figue, compise the spheical colatitude,, the longitude,, and the adial distance,. Thei elation to Catesian coodinates is x ¼ sin cos y ¼ sin sin z ¼ cos Consideing Eath s pola flattening, a bette appoximation, than a sphee, of its (ocean) suface is an ellipsoid of evolution. Such a suface is ½Š Figue 2 Ellipsoidal coodinates. geneated by otating an ellipse about its mino axis (pola axis). The two focal points of the best-fitting, Eath-centeed ellipsoid (ellipse) ae located in the equato about E ¼ 522 km fom the cente of the Eath. A given ellipsoid, with specified semimino axis, b, and linea eccenticity, E, defines the set of ellipsoidal coodinates, as descibed in Figue 2. The longitude is the same as in the spheical case. The colatitude,, is the complement of the so-called educed latitude; and the distance coodinate, u, is the semimino axis of the confocal ellipsoid though the point in question. We call ð; ; uþ ellipsoidal coodinates; they ae also known as spheoidal coodinates, o Jacobi ellipsoidal coodinates. Thei elation to Catesian coodinates is given by p x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ E 2 sin cos p y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ E 2 sin sin ½2Š z ¼ u cos Points on the given ellipsoid all have u ¼ b; and, all sufaces, u ¼ constant, ae confocal ellipsoids (the analogy to the spheical case, when E ¼ 0, should be evident). z Peliminay Definitions and Concepts The gavitational potential, V, of the Eath is geneated by its total mass density distibution. Fo applications on the Eath s suface it is useful to include the potential,, associated with the centifugal acceleation due to Eath s otation. The sum, W ¼ V þ, is then known, in geodetic teminology, as the gavity potential, distinct fom gavitational potential. It is futhe advantageous to define a elatively simple efeence potential, o nomal potential, that accounts fo the bulk of the gavity potential Figue θ λ x Spheical pola coodinates. y

5 Potential Theoy and Static Gavity Field of the Eath 5 (Section ). The nomal gavity potential, U, is defined as a gavity potential associated with a bestfitting ellipsoid, the nomal ellipsoid, which otates with the Eath and is also a suface of constant potential in this field. The diffeence between the actual and the nomal gavity potentials is known as the distubing potential: T ¼ W U; it thus excludes the centifugal potential. The nomal gavity potential accounts fo appoximately % of the total potential. The gadient of the potential is an acceleation, gavity o gavitational acceleation, depending on whethe o not it includes the centifugal acceleation. Nomal gavity,, compises % of the total gavity, g, although the diffeence in magnitudes, the gavity distubance, dg, can be as lage as seveal pats in 0 4. A special kind of diffeence, called the gavity anomaly, g, is defined as the diffeence between gavity at a point, P, and nomal gavity at a coesponding point, Q, whee W P ¼ U Q, and P and Q ae on the same pependicula to the nomal ellipsoid. The suface of constant gavity potential, W 0,that closely appoximates mean sea level is known as the geoid. If the constant nomal gavity potential, U 0,on the nomal ellipsoid is equal to the constant gavity potential of the geoid, then the gavity anomaly on the geoid is the diffeence between gavity on the geoid and nomal gavity on the ellipsoid at espective points, P 0, Q 0, shaing the same pependicula to the ellipsoid. The sepaation between the geoid and the ellipsoid is known as the geoid undulation, N, o also the geoid height (Figue 3). A simple Taylo expansion of the nomal gavity potential along the ellipsoid pependicula yields the following impotant fomula: N ¼ T This is Buns equation, which is accuate to a few millimetes in N, and which can be extended to N ¼ T= ðw 0 U 0 Þ= fo the geneal case, W 0 6¼ U 0. The gavity anomaly (on the geoid) is the gavity distubance coected fo the evaluation of nomal gavity on the ellipsoid instead of the geoid. This coection is N q=qh ¼ ðq=qhþðt=þ, whee h is height along the ellipsoid pependicula. We have g ¼ qt=qh, and hence g ¼ qt qh þ q qh T The slope of the geoid with espect to the ellipsoid is also the angle between the coesponding pependiculas to these sufaces. This angle is known as the deflection of the vetical, that is, the deflection of the plumb line (pependicula to the geoid) elative to the pependicula to the nomal ellipsoid. The deflection angle has components,,, espectively, in the noth and east diections. The spheical appoximations to the gavity distubance, anomaly, and deflection of the vetical ae given by g ¼ qt q ; ¼ qt q ; qt g ¼ q 2 T ¼ qt sin q whee the signs on the deivatives ae a matte of convention. ½3Š ½4Š ½5Š Newton s Law of Gavitation P ellipsoidal Ellipsoidal height, h h H, othometic height W = W 0 Deflection of the vetical Topogaphic suface In its oiginal fom, Newton s law of gavitation applies only to idealized point masses. It descibes the foce of attaction, F, expeienced by two such solitay masses as being popotional to the poduct of the masses, m and m 2 ; invesely popotional to the distance,,, between them; and diected along the line joining them: Figue 3 P 0 N, geoid undulation U = U g 0 Q 0 P 0 Q 0 Geoid Ellipsoid Relative geomety of geoid and ellipsoid. F ¼ G m m 2, 2 n ½6Š G is a constant, known as Newton s gavitational constant, that takes cae of the units between the left- and ight-hand sides of the equation; it can be

6 6 Potential Theoy and Static Gavity Field of the Eath detemined by expeiment and the cuent best value is (Goten, 2004): G ¼ ð6: :00030Þ0 m 3 kg s 2 ½7Š The unit vecto n in eqn [6] is diected fom eithe point mass to the othe, and thus the gavitational foce is attactive and applies equally to one mass as the othe. Newton s law of gavitation is univesal as fa as we know, equiing efomulation only in Einstein s moe compehensive theoy of geneal elativity which descibes gavitation as a chaacteistic cuvatue of the space time continuum (Newton s fomulation assumes instantaneous action and diffes significantly fom the geneal elativistic concept only when vey lage velocities o masses ae involved). We can ascibe a gavitational acceleation to the gavitational foce, which epesents the acceleation that one mass undegoes due to the gavitational attaction of the othe. Specifically, fom the law of gavitation, we have (fo point masses) the gavitational acceleation of m due to the gavitational attaction of m 2 : g ¼ G m 2, 2 n ½8Š The vecto g is independent of the mass, m, of the body being acceleated (which Galileo found by expeiment). By the law of supeposition, the gavitational foce, o the gavitational acceleation, due to many point masses is the vecto sum of the foces o acceleations geneated by the individual point masses. Manipulating vectos in this way is cetainly feasible, but fotunately a moe appopiate concept of gavitation as a scala field simplifies the teatment of abitay mass distibutions. This moe moden view of gavitation (aleady adopted by Gauss ( ) and Geen (793 84)) holds that it is a field having a gavitational potential. Lagange (736 83) fully developed the concept of a field, and the potential, V, of the gavitational field is defined in tems of the gavitational acceleation, g, that a test paticle would undego in the field accoding to the equation ÑV ¼ g whee Ñ is the gadient opeato (a vecto). Futhe elucidation of gavitation as a field gew fom Einstein s attempt to incopoate gavitation into his special theoy of elativity whee no efeence fame has special significance above all othes. It was necessay to conside that gavitational foce is not a eal ½9Š foce (i.e., it is not an applied foce, like fiction o populsion) athe, it is known as a kinematic foce, that is, one whose action is popotional to the mass on which it acts (like the centifugal foce; see Matin, 988). Unde this pecept, the geomety of space is defined intinsically by the gavitational fields contained theein. We continue with the classical Newtonian potential, but intepet gavitation as an acceleation diffeent fom the acceleation induced by eal, applied foces. This becomes especially impotant when consideing the measuement of gavitation (Section ). The gavitational potential, V, is a scala function, and, as defined hee, V is deived diectly on the basis of Newton s law of gavitation. To make it completely consistent with this law and thus declae it a Newtonian potential, we must impose the following conditions: lim,v ¼ Gm and lim V ¼ 0 ½0Š,!,! Hee, m is the attacting mass, and we say that the potential is egula at infinity. It is easy to show that the gavitational potential at any point in space due to a point mass, in ode to satisfy eqns [8] [0], must be V ¼ Gm, ½Š whee, again,, is the distance between the mass and the point at which the potential is expessed. Note that V fo, ¼ 0 does not exist in this case; that is, the field of a point mass has a singulaity. We use hee the convention that the potential is always positive (in contast to physics, whee it is usually defined to be negative, conceptually close to potential enegy). Applying the law of supeposition, the gavitational potential of many point masses is the sum of the potentials of the individual points (see Figue 4): V P ¼ G X j m j, j ½2Š And, fo infinitely many points in a closed, bounded egion with infinitesimally small masses, dm, the summation in eqn [2] changes to an integation, V P Z ¼ G mass dm, ½3Š o, changing vaiables (i.e., units), dm ¼ dv, whee epesents density (mass pe volume) and dv is a volume element, we have (Figue 5)

7 Potential Theoy and Static Gavity Field of the Eath 7 m l z l 2 P l j spheically symmetic density we may choose the integation coodinate system so that the pola axis passes though P. Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, ¼ 9 2 þ 2 29 cos 9; d, ¼ 9 sin 9d9, m 2 l3 m j ½5Š y It is easy to show that with this change of vaiables (fom to,) the integal [4] becomes simply x Figue 5 n V P m 3 Figue 4 Discete set of mass points (supeposition pinciple). dv ρ Continuous density distibution. Z ¼ G volume dv ½4Š, In eqn [4],, is the distance between the evaluation point, P, and the point of integation. In spheical pola coodinates (Section ), these points ae ð; ; Þ and ð9; 9; 9Þ, espectively. The volume element in this case is given by dv ¼ 9 2 sin 9 d9 d9 d9. V and its fist deivatives ae continuous eveywhee even in the case that P is on the bounding suface o inside the mass distibution, whee thee is the appaent singulaity at, ¼ 0. In this case, by changing to a coodinate system whose oigin is at P, the volume element becomes dv ¼, 2 sin c d dc d, (fo some diffeent colatitude and longitude c and ); and, clealy, the singulaity disappeas the integal is said to be weakly singula. Suppose the density distibution ove the volume depends only on adial distance (fom the cente of mass): ¼ ð9þ, and that P is an exteio evaluation point. The suface bounding the masses necessaily is a sphee (say, of adius, R) and because of the l s P V ð; ; Þ ¼ GM ; R ½6Š whee M is the total mass bounded by the sphee. This shows that to a vey good appoximation the extenal gavitational potential of a planet such as the Eath (with concentically layeed density) is the same as that of a point mass. Besides volumetic mass (density) distibutions, it is of inteest to conside suface distibutions. Imagine an infinitesimally thin laye of mass on a suface, s, whee the units of density in this case ae those of mass pe aea. Then, analogous to eqn [4], the potential is V P ¼ G s ds ½7Š, In this case, V is a continuous function eveywhee, but its fist deivatives ae discontinuous at the suface. O, one can imagine two infinitesimally close density layes (double laye, o laye of mass dipoles), whee the units of density ae now those of mass pe aea times length. It tuns out that the potential in this case is given by (see Heiskanen and Moitz, 967, p. 8) V P ¼ G q ds ½8Š s qn, whee q=qn is the diectional deivative along the pependicula to the suface (Figue 5). Now, V itself is discontinuous at the suface, as ae all its deivatives. In all cases, V is a Newtonian potential, being deived fom the basic fomula [] fo a point mass that follows fom Newton s law of gavitation (eqn [6]). The following popeties of the gavitational potential ae useful fo subsequent expositions. Fist, conside Stokes s theoem, fo a vecto function, f, defined on a suface, s: I ðf Þ? n ds ¼ f? d ½9Š s whee p is any closed path in the suface, n is the unit vecto pependicula to the suface, and d is a p

8 8 Potential Theoy and Static Gavity Field of the Eath diffeential displacement along the path. Fom eqn [9], we find g ¼ 0 ½20Š since ¼ 0; hence, applying Stokes s theoem, we find with F ¼ mg that I w ¼ F? ds ¼ 0 ½2Š That is, the gavitational field is consevative: the wok, w, expended in moving a mass aound a closed path in this field vanishes. In contast, dissipating foces (eal foces!), like fiction, expend wok o enegy, which shows again the special natue of the gavitational foce. It can be shown (Kellogg, 953, p. 56) that the second patial deivatives of a Newtonian potential, V, satisfy the following diffeential equation, known as Poisson s equation: 2 V ¼ 4G ½22Š whee 2 ¼ Ñ? Ñ fomally is the scala poduct of two gadient opeatos and is called the Laplacian opeato. In Catesian coodinates, it is given by 2 ¼ q2 qx 2 þ q2 qy 2 þ q2 qz 2 ½23Š Note that the Laplacian is a scala opeato. Eqn [22] is a local chaacteization of the potential field, as opposed to the global chaacteization given by eqn [4]. Poisson s equation holds wheeve the mass density,, satisfies cetain conditions simila to continuity (Hölde conditions; see Kellogg, 953, pp ). A special case of eqn [22] applies fo those points whee the density vanishes (i.e., in fee space); then Poisson s equation tuns into Laplace equation, 2 V ¼ 0 ½24Š It is easily veified that the point mass potential satisfies eqn [24], that is, 2 ¼ 0 ½25Š, whee, ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x 0 Þ 2 þðy y 0 Þ 2 þðz z 0 Þ 2 and the mass point is at ðx9; y9; z9þ. The solutions to Laplace equation [24] (that is, functions that satisfy Laplace equations) ae known as hamonic functions (hee, we also impose the conditions [0] on the solution, if it is a Newtonian potential and if the mass-fee egion includes infinity). Hence, evey Newtonian potential is a hamonic function in fee space. The convese is also tue: evey hamonic function can be epesented as a Newtonian potential of a mass distibution (Section ). Whethe as a volume o a laye density distibution, the coesponding potential is the sum o integal of the souce value multiplied by the invese distance function (o its nomal deivative fo the dipole laye). This function depends on both the souce points and the computation point and is known as a Geen s function. It is also known as the kenel function of the integal epesentation of the potential. Functions of this type also play a dominant ole in epesenting the potential as solutions to bounday-value poblems (BVPs), as shown in subsequent sections Bounday-Value Poblems If the density distibution of the Eath s inteio and the bounday of the volume wee known, then the poblem of detemining the Eath s gavitational potential field is solved by the volume integal of eqn [4]. In eality, of couse, we do not have access to this infomation, at least not the density, with sufficient detail. (The Peliminay Refeence Eath Model (PREM), of Dziewonsky and Andeson (98), still in use today by geophysicists, epesents a good pofile of Eath s adial density, but does not attempt to model in detail the lateal density heteogeneities.) In this section, we see how the poblem of detemining the exteio gavitational potential can be solved in tems of suface integals, thus making exclusive use of accessible measuements on the suface Geen s Identities Fomally, eqn [24] epesents a patial diffeential equation fo V. Solving this equation is the essence of the detemination of the Eath s extenal gavitational potential though potential theoy. Like any diffeential equation, a complete solution is obtained only with the application of bounday conditions, that is, imposing values on the solution that it must assume at a bounday of the egion in which it is valid. In ou case, the bounday is the Eath s suface and the exteio space is whee eqn [24] holds (the atmosphee and othe celestial bodies ae neglected fo the moment). In ode to study the solutions to these BVPs (to show that solutions exist and ae

9 Potential Theoy and Static Gavity Field of the Eath 9 unique), we take advantage of some vey impotant theoems and identities. It is noted that only a athe elementay intoduction to BVPs is offeed hee with no attempt to addess the much lage field of solutions to patial diffeential equations. The fist, seminal esult is Gauss divegence theoem (analogous to Stokes theoem, eqn [9]), Z? f dv ¼ f n ds ½26Š v whee f is an abitay (diffeentiable) vecto function and f n ¼ n? f is the component of f along the outwad unit nomal vecto, n (see Figue 5). The suface, s, encloses the volume, v. Equation [26] says that the sum of how much f changes thoughout the volume, that is, the net effect, ultimately, is equivalent to the sum of its values pojected othogonally with espect to the suface. Conceptually, a volume integal thus can be eplaced by a suface integal, which is impotant since the gavitational potential is due to a volume density distibution that we do not know, but we do have access to gavitational quantities on a suface by way of measuements. Equation [26] applies to geneal vecto functions that have continuous fist deivatives. In paticula, let U and V be two scala functions having continuous second deivatives, and conside the vecto function f ¼ UÑV. Then, since n? Ñ ¼ q=qn, and? ðuv Þ ¼U? V þ U 2 V ½27Š we can apply Gauss divegence theoem to get Geen s fist identity, Z v U? V þ U 2 V dv ¼ s s U qv qn ds ½28Š Intechanging the oles of U and V in eqn [28], one obtains a simila fomula, which, when subtacted fom eqn [28], yields Geen s second identity, Z v U 2 V V 2 U dv ¼ s U qv qn V qu ds qn ½29Š This is valid fo any U and V with continuous deivatives up to second ode. Now let U ¼ =,, whee, is the usual distance between an integation point and an evaluation point. And, suppose that the volume, v, is the space exteio to the Eath (i.e., Gauss divegence theoem applies to any volume, not just volumes containing a mass distibution). With efeence to Figue 6, conside the evaluation point, P, to be inside the volume (fee Figue 6 identity. space) that is bounded by the suface, s; P is thus outside the Eath s suface. Let V be a solution to eqn [24], that is, it is the gavitational potential of the Eath. Fom the volume, v, exclude the volume bounded by a small sphee,, centeed at P. This sphee becomes pat of the suface that bounds the volume, v. Then, since U, by ou definition, is a point mass potential, 2 U ¼ 0 eveywhee in v (which excludes the inteio of the small sphee aound P); and, the second identity [29] gives qv s, qn V q ds qn, qv þ, qn V q d ¼ 0 qn, ½30Š The unit vecto, n, epesents the pependicula pointing away fom v. On the small sphee, n is opposite in diection to, ¼, and the second integal becomes n Eath qv q þ V q q d ¼ V d ¼ qv q 2 d qv q d 4 V ½3Š whee d ¼ 2 d, is the solid angle, 4, and V is an aveage value of V on. Now, in the limit as the adius of the small sphee shinks to zeo, the ighthand side of eqn [3] appoaches 0 4V P. Hence, eqn [30] becomes (Kellogg, 953, p. 29) V P ¼ 4 s ds Geomety fo special case of Geen s thid qv, qn V q qn l s σ n, ds l P dv ½32Š

10 20 Potential Theoy and Static Gavity Field of the Eath with n pointing down (away fom the space outside the Eath). This is a special case of Geen s thid identity. A change in sign of the ight-hand side tansfoms n to a nomal unit vecto pointing into v, away fom the masses, which confoms moe to an Eath-centeed coodinate system. The ight-hand side of eqn [32] is the sum of single- and double-laye potentials and thus shows that evey hamonic function (i.e., a function that satisfies Laplace equation) can be witten as a Newtonian potential. Equation [32] is also a solution to a BVP; in this case, the bounday values ae independent values of V and of its nomal deivative, both on s (Cauchy poblem). Below and in Section , we encounte anothe BVP in which the potential and its nomal deivative ae given in a specified linea combination on s. Using a simila pocedue and with some exta cae, it can be shown (see also Couant and Hilbet, 962, vol. II, p. 256 (footnote)) that if P is on the suface, then V P ¼ 2 s qv, qn V q qn, ds ½33Š whee n points into the masses. Compaing this to eqn [32], we see that V is discontinuous as one appoaches the suface; this is due to the doublelaye pat (see eqn [8]). Equation [32] demonstates that a solution to a paticula BVP exists. Specifically, we ae able to measue the potential (up to a constant) and its deivatives (the gavitational acceleation) on the suface and thus have a fomula to compute the potential anywhee in exteio space, povided we also know the suface, s. Othe BVPs also have solutions unde appopiate conditions; a discussion of existence theoems is beyond the pesent scope and may be found in (Kellogg, 953). Equation [33] has deep geodetic significance. One objective of geodesy is to detemine the shape of the Eath s suface. If we have measuements of gavitational quantities on the Eath s suface, then conceptually we ae able to detemine its shape fom eqn [33], whee it would be the only unknown quantity. This is the basis behind the wok of Molodensky et al. (962), to which we etun biefly at the end of this section Uniqueness Theoems Often the existence of a solution is poved simply by finding one (as illustated above). Whethe such as solution is the only one depends on a coesponding uniqueness theoem. That is, we wish to know if a cetain set of bounday values will yield just one potential in space. Befoe consideing such theoems, we classify the BVPs that ae typically encounteed when detemining the exteio potential fom measuements on a bounday. In all cases, it is an exteio BVP; that is, the gavitational potential, V, is hamonic ( 2 V ¼ 0) in the space exteio to a closed suface that contains all the masses. The exteio space thus contains infinity. Inteio BVPs can be constucted, as well, but ae not applicable to ou objectives. Diichlet poblem (o, BVP of the fist kind ). Solve fo V in the exteio space, given its values eveywhee on the bounday. Neumann poblem (o, BVP of the second kind ). Solve fo V in the exteio space, given values of its nomal deivative eveywhee on the bounday. Robin poblem (mixed BVP, o BVP of the thid kind ). Solve fo V in the exteio space, given a linea combination of it and its nomal deivative on the bounday. Using Geen s identities, we pove the following theoems fo these exteio poblems; simila esults hold fo the inteio poblems. Theoem. If V is hamonic (hence continuously diffeentiable) in a closed egion, v, and if V vanishes eveywhee on the bounday, s, then V also vanishes eveywhee in the egion, v. Poof. Since V ¼ 0 on s, Geen s fist identity (eqn [28]) with U ¼ V gives Z ðv Þ 2 dv ¼ s V qv qn ds ¼ 0 ½34Š The integal on the left side is theefoe always zeo, and the integand is always non-negative. Hence, ÑV ¼ 0 eveywhee in v, which implies that V ¼ constant in v. Since V is continuous in v and V ¼ 0ons, that constant must be zeo; and so V ¼ 0inv. This theoem solves the Diichlet poblem fo the tivial case of zeo bounday values and it enables the following uniqueness theoem fo the geneal Diichlet poblem. Theoem 2 (Stokes theoem). If V is hamonic (hence continuously diffeentiable) in a closed egion, v, then V is uniquely detemined in v by its values on the bounday, s. Poof. Suppose the detemination is not unique: that is, suppose thee ae V and V 2, both hamonic

11 Potential Theoy and Static Gavity Field of the Eath 2 in v and both having the same bounday values on s. Then the function V ¼ V 2 V is hamonic in v with all bounday values equal to zeo. Hence, by Theoem, V 2 V ¼ 0 identically in v, ov 2 ¼ V eveywhee, which implies that any detemination is unique based on the bounday values. Theoem 3. IfV is hamonic (hence continuously diffeentiable) in the exteio egion, v, with closed bounday, s, then V is uniquely detemined by the values of its nomal deivative on s. Poof. We begin with Geen s fist identity, eqn [28], as in the poof of Theoem to show that if the nomal deivative vanishes eveywhee on s, then V is a constant in v. Now, suppose thee ae two hamonic functions in v: V and V 2, with the same nomal deivative values on s. Then the nomal deivative values of thei diffeence ae zeo; and, by the above demonstation, V ¼ V 2 V ¼ constant in v. Since V is a Newtonian potential in the exteio space, that constant is zeo, since by eqn [0], lim,! V ¼ 0. Thus, V 2 ¼ V, and the bounday values detemine the potential uniquely. This is a uniqueness theoem fo the exteio Neumann BVP. Solutions to the inteio poblem ae unique only up to an abitay constant. Theoem 4. Suppose V is hamonic (hence continuously diffeentiable) in the closed egion, v, with bounday, s; and, suppose the bounday values ae given by g ¼ V j s þ qv ½35Š qn s Then V is uniquely detemined by these values if = > 0. Poof. Suppose thee ae two hamonic functions, V and V 2, with the same bounday values, g, ons. Then V ¼ V 2 V is hamonic with bounday values ðv 2 V Þj s þ qv 2 qn qv qn ¼ 0 ½36Š s With U ¼ V ¼ V 2 V, Geen s fist identity, eqn [28], gives Z ððv 2 V ÞÞ 2 dv ¼ ðv 2 V Þ ð V 2 V Þds v Then Z ððv 2 V ÞÞ 2 dv þ s s ½37Š ðv 2 V Þ 2 ds ¼ 0 ½38Š Since = > 0, eqn [38] implies that ðv 2 V Þ ¼ 0 in v; and V 2 V ¼ 0ons. Hence V 2 V ¼ constant in v; and V 2 ¼ V on s. By the continuity of V and V 2, the constant must be zeo, and the uniqueness is poved. The solution to the Robin poblem is unique only in cetain cases. The most famous poblem in physical geodesy is the detemination of the distubing potential, T, fom gavity anomalies, g, on the geoid (Section ). Suppose T is hamonic outside the geoid; the second of eqns [5] povides an appoximate fom of bounday condition, showing that this is a type of Robin poblem. We find that ¼ 2=, and, ecalling that when v is the exteio space the unit vecto n points inwad towad the masses, that is, q=qn ¼ q=q, weget ¼. Thus, the condition in Theoem 4 on = is not fulfilled and the uniqueness is not guaanteed. In fact, we will see that the solution obtained fo the spheical bounday is abitay with espect to the coodinate oigin (Section ) Solutions by Integal Equation Geen s identities show how a solution to Laplace s equation can be tansfomed fom a volume integal, that is, an integal of souce points, to a suface integal of BVPs, as demonstated by eqn [32]. The uniqueness theoems fo the BVPs suggest that the potential due to a volume density distibution can also be epesented as due to a genealized density laye on the bounding suface, as long as the esult is hamonic in exteio space, satisfies the bounday conditions, and is egula at infinity like a Newtonian potential. Molodensky et al. (962) supposed that the distubing potential is expessible as T ¼ ds ½39Š, whee is a suface density to be solved using the bounday condition. With the spheical appoximation fo the gavity anomaly, eqn [5], one aives at the following integal equation 2 cos s q q s, þ 2, ds ¼ g ½40Š The fist tem accounts fo the discontinuity at the suface of the deivative of the potential of a density laye, whee is the deflection angle between the nomal to the suface and the diection of the (adial) deivative (Heiskanen and Moitz, 967, p. 6; Günte, 967, p. 69). This Fedholm integal equation of the second kind can be simplified with futhe appoximations, and, a solution fo the density,, ultimately

12 22 Potential Theoy and Static Gavity Field of the Eath leads to the solution fo the distubing potential (Moitz, 980). Othe foms of the initial epesentation have also been investigated, whee Geen s functions othe than =, lead to simplifications of the integal equation (e.g., Petovskaya, 979). Nevetheless, most pactical solutions ely on appoximations, such as the spheical appoximation fo the bounday condition, and even the fomulated solutions ae not stictly guaanteed to convege to the tue solutions (Moitz, 980). Futhe teatments of the BVP in a geodetic/mathematical setting may be found in the volume edited by Sansò and Rummel (997). In the next section, we conside solutions fo T as suface integals of bounday values with appopiate Geen s functions. In othe wods, the bounday values (whethe of the fist, second, o thid kind) may be thought of as souces, and the consequent potential is again the sum (integal) of the poduct of a bounday value and an appopiate Geen s function (i.e., a function that depends on both the souce point and the computation point in some fom of invese distance in accodance with Newtonian potential theoy). Such solutions ae eadily obtained if the bounday is a sphee. appoximation based on an ellipsoid of evolution is biefly examined in Section fo the nomal potential. In spheical pola coodinates, ð; ; Þ,the Laplacian opeato is given by Hobson (965, p. 9) 2 ¼ q 2 2 q þ q q q 2 sin q sin q q þ q 2 2 sin 2 q 2 ½4Š A solution to 2 V ¼ 0 in the space outside a sphee of adius, R, with cente at the coodinate oigin can be found by the method of sepaation of vaiables, wheeby one postulates the fom of the solution, V,as V ð; ; Þ ¼ f ðþg ðþh ðþ ½42Š Substituting this and the Laplacian above into eqn [24], the multivaiate patial diffeential equation sepaates into thee univaiate odinay diffeential equations (Hobson, 965, p. 9; Mose and Feshbach, 953, p. 264). Thei solutions ae well-known functions, fo example, o V ð; ; Þ ¼ P nm ðcos Þ sin m nþ ½43aŠ Solutions to the Spheical BVP V ð; ; Þ ¼ P nm ðcos Þcos m nþ ½43bŠ This section develops two types of solutions to standad BVPs when the bounday is a sphee: the spheical hamonic seies and an integal with a Geen s function. All thee types of poblems ae solved, but emphasis is put on the thid BVP since gavity anomalies ae the most pevalent bounday values (on land, at least). In addition, it is shown how the Geen s function integals can be inveted to obtain, fo example, gavity anomalies fom values of the potential, now consideed as bounday values. Not all possible invese elationships ae given, but it should be clea at the end that, in pinciple, vitually any gavitational quantity can be obtained in space fom any othe quantity on the spheical bounday Spheical Hamonics and Geen s Functions Fo simple boundaies, Laplace s equation [24] is elatively easy to solve povided thee is an appopiate coodinate system. Fo the Eath, the solutions commonlyelyonappoximatingtheboundaybyasphee. This case is descibed in detail and a moe accuate whee P nm ðþis t the associated Legende function of the fist kind and n, m ae integes such that 0 m n, n 0. Othe solutions ae also possible (e.g., gðþ ¼ e a ðaprþ and hðþ ¼ n ), but only eqns [43] ae consistent with the poblem at hand: to find a eal-valued Newtonian potential fo the exteio space of the Eath (egula at infinity and 2-peiodic in longitude). The geneal solution is a linea combination of solutions of the foms given by eqns [43] fo all possible integes, n and m, and can be witten compactly as V ð; ; Þ ¼ X n¼0 X n m¼ n R nþ v nm Y nm ð; Þ ½44Š whee the Y nm ae suface spheical hamonic functions defined as ( cos m; m 0 Y nm ð; Þ ¼ P nm j j ðcos Þ sinjmj; m < 0 ½45Š and P nm is a nomalization of P nm so that the othogonality of the spheical hamonics is simply

13 Potential Theoy and Static Gavity Field of the Eath 23 4 ( ¼ Y nm ð; ÞY n9m9 ð; Þd ; n ¼ n9 and m ¼ m9 0; n 6¼ n9 o m 6¼ m9 ½46Š and whee ¼ fð; Þj0 ; 0 2g epesents the unit sphee, with d ¼ sin d d. Fo a complete mathematical teatment of spheical hamonics, one may efe to Mülle (966). The bounding spheical adius, R, is intoduced so that all the constant coefficients, v nm, also known as Stokes constants, have identical units of measue. Applying the othogonality to the geneal solution [44], these coefficients can be detemined if the function, V, is known on the bounding sphee (bounday condition): v nm ¼ V ð; ; RÞY nm ð; Þd ½47Š 4 Equation [44] is known as a spheical hamonic expansion of V and with eqn [47] it epesents a solution to the Diichlet BVP if the bounday is a sphee. The solution thus exists and is unique in the sense that these bounday values geneate no othe potential. We will, howeve, find anothe equivalent fom of the solution. In a moe fomal mathematical setting, the solution [46] is an infinite linea combination of othogonal basis functions (eigenfunctions) and the coefficients, v nm,ae the coesponding eigenvalues. One may also intepet the set of coefficients as the spectum (Legende spectum) of the potential on the sphee of adius, R (analogous to the Fouie spectum of a function on the plane o line). The integes, n, m, coespond to wave numbes, and ae called degee (n) and ode (m), espectively. The spheical hamonics ae futhe classified as zonal (m ¼ 0), meaning that the zeos of Y n0 divide the sphee into latitudinal zones; sectoial (m ¼ n), whee the zeos of Y nn divide the sphee into longitudinal sectos; and tesseal (the zeos of Y nm tessellate the sphee) (Figue 7). While the spheical hamonic seies has its advantages in global epesentations and spectal intepetations of the field, a Geen s function epesentation povides a moe local chaacteization of the field. Changing a bounday value anywhee on the globe changes all coefficients, v nm, accoding to eqn [47], which poses both a numeical challenge in applications, as well as in keeping a standad model up to date. Howeve, since the Geen s function essentially depends on the invese distance (o highe powe theeof), a emote change in bounday value geneally does not appeciably affect the local detemination of the field. When the bounday is a sphee, the solutions to the BVPs using a Geen s function ae easily deived fom the spheical hamonic seies epesentation. Moeove, it is possible to deive additional integal elationships (with appopiate Geen s functions) among all the deivatives of the potential. To fomalize and simultaneously simplify these deivations, conside hamonic functions, f and h, whee h depends only on and, and function g, defined on the sphee of adius, R. Thus let f ð; ; Þ ¼ X n¼0 X n m¼ n R nþ f nm Y nm ð; Þ ½48Š hð; Þ ¼ X ð2n þ Þ R nþ h n P n ðcos Þ ½49Š n¼0 gð; ; RÞ ¼ X n¼0 X n m¼ n g nm Y nm ð; Þ ½50Š p whee P n ðcosþ ¼ P n0 ðcosþ= ffiffiffiffiffiffiffiffiffiffiffiffiffi 2n þ is the nth degee Legende polynomial. Constants f nm and g nm ae the espective hamonic coefficients of f and g when these functions ae esticted to the sphee of adius, R. Then, using the decomposition fomula fo Legende polynomials, P 80 (cos θ) P 88 (cos θ) cos 8λ P 87 (cos θ) cos 7λ Figue 7 Examples of zonal, sectoial, and tesseal hamonics on the sphee.

14 24 Potential Theoy and Static Gavity Field of the Eath P n ðcos cþ ¼ 2n þ whee X n m¼ n Y nm ð; ÞY nm ð9; 9Þ ½5Š cos c ¼ cos cos 9 þ sin sin 9cosð 9Þ ½52Š it is easy to pove the following theoem. Theoem (convolution theoem in spectal analysis on the sphee). f ð; ; Þ ¼ gð9; 9; RÞhðc; Þd 4 ½53Š if and only if f nm ¼ g nm h n Hee, and in the following, d ¼ sin 9d9d9. The angle, c, is the distance on the unit sphee between points ð; Þ and ð9; 9Þ. Poof. The fowad statement [53] follows diectly by substituting eqns [5] and [49] into the fist equation [53], togethe with the spheical hamonic expansion [50] fo g. A compaison with the spheical hamonic expansion fo f yields the esult. All steps in this poof ae evesible, and so the evese statement also holds. Conside now f to be the potential, V, outsidethe sphee of adius, R, and its estiction to the sphee to be the function, g : gð; Þ ¼ V ð; ; RÞ. Then, clealy, h n ¼, fo all n;bythetheoemabove,wehave V ð; ; Þ ¼ 4 V ð9; 9; RÞUðc; Þd ½54Š whee Uðc; Þ ¼ X ð2n þ Þ R nþ P n ðcos cþ ½55Š n¼0 Fo the distance p, ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ R 2 2R cos c ½56Š between points ð; ; Þ and ð9; 9; RÞ, with R, the identity (the Coulomb expansion; Cushing, 975, p. 55),, ¼ X R nþ P n ðcos cþ ½57Š R n¼0 yields, afte some aithmetic (based on taking the deivative on both sides with espect to ), Uðc; Þ ¼ R2 ð R 2 Þ, 3 ½58Š Solutions [44] and [54] to the Diichlet BVP fo a spheical bounday ae identical (in view of the convolution theoem [53]). The integal in [54] is known as the Poisson integal and the function U is the coesponding Geen s function, also known as Poisson s kenel. Fo convenience, one sepaates Eath s gavitational potential into a efeence potential (Section ) and the distubing potential, T. The distubing potential is hamonic in fee space and satisfies the Poisson integal if the bounday is a sphee. In defeence to physical geodesy whee elationships between the distubing potential and its deivatives ae outinely applied, the following deivations ae developed in tems of T, but hold equally fo any exteio Newtonian potential. Let Tð; ; Þ ¼ GM R X X n n¼0 m¼ n R nþ C nm Y nm ; ð Þ ½59Š whee M is the total mass (including the atmosphee) of the Eath and the dc nm ae unitless hamonic coefficients, being also the diffeence between coefficients fo the total and efeence gavitational potentials (Section ). The coefficient, C 00,is zeo unde the assumption that the efeence field accounts completely fo the cental pat of the total field. Also note that these coefficients specifically efe to the sphee of adius, R. The gavity distubance is defined (in spheical appoximation) to be the negative adial deivative of T, the fist of eqns [5]. Fom eqn [59], we have gð; ; Þ ¼ q q ð ¼ GM X R 2 T ; ; Þ X n R nþ2 ðn þ n¼0 m¼ n ÞC nm Y nm ð; Þ ½60Š and, applying the convolution theoem [53], we obtain Tð; ; Þ ¼ R gð9; 9; RÞHðc; Þd ½6Š 4 whee with g nm ¼ ðn þ ÞdC nm =R and f nm ¼ dc nm,we have h n ¼ f nm =g nm ¼ R= ðn þ Þ, and hence (taking cae to keep the Geen s function unitless) Hðc; Þ ¼ X n¼0 2n þ n þ R nþ P n cos c ð Þ ½62Š The integal in [6] is known as the Hotine integal, the Geen s function, H, is called the Hotine kenel, and with a deivation based on equation [57], it is given by (Hotine 969, p. 3) Hðc; Þ ¼ 2R, ln þ, 2R sin 2 ½63Š c=2

15 Potential Theoy and Static Gavity Field of the Eath 25 Equation [6] solves the Neumann BVP when the bounday is a sphee. The gavity anomaly (again, in spheical appoximation) is defined by eqn [5] gð; ; Þ ¼ q q 2 Tð; ; Þ ½64Š o, also, gð; ; Þ ¼ GM X R 2 n¼0 X n m¼ n R nþ2 ðn ÞC nm Y nm ð; Þ ½65Š In this case, we have g nm ¼ ðn ÞdC nm =R and h n ¼ R= ðn Þ. The convolution theoem in this case leads to the geodetically famous Stokes integal, Tð; ; Þ ¼ R gð9; 9; RÞSðc; Þd ½66Š 4 whee we define Geen s function to be Sðc; Þ ¼ X n¼2 2n þ n R nþ P n ðcos cþ ¼ 2 R, þ R 3 R, 2 5 R2 2 cos c 3 R2, þ R cos c cos c ln 2 2 ½67Š moe commonly called the Stokes kenel. Equation [66] solves the Robin BVP if the bounday is a sphee, but it includes specific constaints that ensue the solution s uniqueness the solution by itself is not unique, in this case, as poved in Section Indeed, eqn [65] shows that the gavity anomaly has no fist-degee hamonics fo the distubing potential; theefoe, they cannot be detemined fom the bounday values. Conventionally, the Stokes kenel also excludes the zeo-degee hamonic, and thus the complete solution fo the distubing potential is given by Tð; ; Þ ¼ GM C 00 X þ GM R þ R 4 m¼ R 2 C m Y m ð; Þ gð9; 9; RÞSðc; Þd ½68Š The cental tem, C 00, is popotional to the diffeence in GM of the Eath and efeence ellipsoid and is zeo to high accuacy. The fist-degee hamonic coefficients, C m, ae popotional to the cente-ofmass coodinates and can also be set to zeo with appopiate definition of the coodinate system (see Section ). Thus, the Stokes integal [66] is the moe common expession fo the distubing potential, but it embodies hidden constaints. We note that gavity anomalies also seve as bounday values in the hamonic seies fom of the solution fo the distubing potential. Applying the othogonality of the spheical hamonics to eqn [65] yields immediately R 2 C nm ¼ gð9; 9; RÞY nm ð9; 9Þd; 4ðn ÞGM n 2 ½69Š A simila fomula holds when gavity distubances ae the bounday values (n in the denominato changes to n þ ). In eithe case, the bounday values fomally ae assumed to eside on a sphee of adius, R. An appoximation esults if they ae given on the geoid, as is usually the case Invese Stokes and Hotine Integals The convolution integals above can easily be inveted by consideing again the spectal elationships. Fo the gavity anomaly, we note that f ¼ g is hamonic with coefficients, f nm ¼ GMðn ÞdC nm =R. Letting g nm ¼ GM dc nm =R, we find that h n ¼ n ; fom the convolution theoem, we can wite gð; ; Þ ¼ 4R whee Tð9; 9; RÞẐðc; Þd ½70Š Ẑðc; Þ ¼ X ð2n þ Þðn Þ R nþ2 P n ðcos cþ ½7Š n¼0 The zeo- and fist-degee tems ae included povisionally. Note that Ẑðc; Þ ¼ R q q þ 2 Uðc; Þ ½72Š that is, we could have simply used the Diichlet solution [54] to obtain the gavity anomaly, as given by [70], fom the distubing potential. It is convenient to sepaate the kenel function as follows: Ẑðc; whee Þ ¼ Zðc; Þ X n¼0 ð2n þ Þ R nþ2 P n ðcos cþ ½73Š Zðc; Þ ¼ X ð2n þ Þn R nþ2 P n ðcos cþ ½74Š n¼

16 26 Potential Theoy and Static Gavity Field of the Eath We find that gð; ; Þ ¼ ð þ 4R T ; ; Þ Tð9; 9; R ÞZðc; Þd ½75Š Now, since Z has no zeo-degee hamonics, its integal ove the sphee vanishes, and one can wite the numeically moe convenient fomula: gð; ; Þ ¼ T ; ; ð Þþ 4R Tð; ; RÞÞZ c; ðtð9; 9; RÞ ð Þd ½76Š This is the invese Stokes fomula. Given T on the sphee of adius R (e.g., in the fom of geoid undulations, T ¼ N), this fom is useful when the gavity anomaly is also desied on this sphee. It is one way to detemine gavity anomalies on the ocean suface fom satellite altimety, whee the ocean suface is appoximated as a sphee. Analogously, fom eqns [60] and [64], it is eadily seen that the invese Hotine fomula is given by gð; ; Þ ¼ T ; ; ð Þþ ðtð9; 9; RÞ 4R Tð; ; RÞÞZðc; Þd ½77Š Note that the diffeence of eqns [76] and [77] yields the appoximate elationship between the gavity distubance and the gavity anomaly infeed fom eqns [5]. Finally, we ealize that, fo ¼ R, the seies fo Z ðc; RÞ is not unifomly convegent and special numeical pocedues (that ae outside the pesent scope) ae equied to appoximate the coesponding integals Vening-Meinesz Integal and Its Invese Othe deivatives of the distubing potential may also be detemined fom bounday values. We conside hee only gavity anomalies, being the most pevalent data type on land aeas. The solution is eithe in the fom of a seies simply the deivative of the seies [59] with coefficients given by eqn [69], o an integal with appopiate deivative of the Geen s function. The hoizontal deivatives of the distubing potential ae often intepeted as the deflections of the vetical to which they ae popotional in spheical appoximation (eqn [5]): 8 9 ( ) q ;; ð Þ ¼ >< >= q Tð; ; Þ ;; ð Þ ; ð Þ>: q >; sin q ½78Š whee ; ae the noth and east deflection components, espectively, and is the nomal gavity. Clealy, the deivatives can be taken diectly inside the Stokes integal, and we find ( ) ;; ð Þ R ¼ gð9; 9; RÞ ;; ð Þ 4 ; ð Þ ( ) q cos qc Sðc; Þ d sin whee and qc q ¼ sinc q qc >< >= q ¼ >< >= q q >: q >; qc qc >: >; sin q sin q ð sin cos 9 cos sin 9cos ð 9 Þ qc sin q ¼ sin c Þ ¼ cos ½79Š ½80Š sin 9sinð9 Þ ¼ sin ½8Š The angle,, is the azimuth of ð9; 9Þ at ð; Þ on the unit sphee. The integals [79] ae known as the Vening-Meinesz integals. Analogous integals fo the deflections aise when the bounday values ae the gavity distubances (the Geen s functions ae then deivatives of the Hotine kenel). Fo the invese Vening-Meinesz integals, we need to make use of Geen s fist identity fo suface functions, f and g: Z f ðþds g þ f? g ds ¼ f g? n db s s b ½82Š whee b is the bounday (a line) of suface, s; and ae the gadient and Laplace Beltami opeatos, which fo the spheical suface ae given by ¼ 0 q q q sin q C A ; q 2 ¼ q2 q 2 þ cot q q þ sin 2 q 2 ½83Š

17 Potential Theoy and Static Gavity Field of the Eath 27 and whee n is the unit vecto nomal to b. Fo a closed suface such as the sphee, the line integal vanishes, and we have f ðþd g ¼ f? g d ½84Š The suface spheical hamonics, Y nm ð; Þ, satisfy the following diffeential equation: Y nm ð; Þþnnþ ð ÞY nm ð; Þ ¼ 0 ½85Š Theefoe, the hamonic coefficients of T ð; ; Þ on the sphee of adius, R, ae ½ Tð; ; ÞŠ nm ¼ nnþ ð Þ GM R C nm ½86Š Hence, by the convolution theoem (again, consideing the hamonic function, f ¼ g), gð; ; Þ ¼ 4R whee W ðc; Þ ¼ X n¼2 Tð9; 9; R ÞW ðc; Þd ½87Š ð2n þ Þðn Þ R nþ2 P n ðcos cþ ½88Š nnþ ð Þ and whee the zeo-degee tem of the gavity anomaly must be teated sepaately (e.g., it is set to zeo in this case). Using Geen s identity [84] and eqns [80] and [8], we have gð; ; Þ ¼ 4R ¼ R 0 4R Tð9; 9; R ðð9; 9; RÞcos Þ? W ðc; Þd þ 9; ð 9; RÞsin Þ q qc W ð c; Þd ½89Š whee nomal gavity on the sphee of adius, R, is appoximated as a constant: ;R ð Þ. 0. Equation [89] epesents a second way to compute gavity anomalies fom satellite altimety, whee the along-tack and coss-tack altimetic diffeences ae used to appoximate the deflection components (with appopiate otation to noth and east components). Employing diffeences in altimetic measuements benefits the estimation since systematic eos, such as obit eo, cancel out. To speed up the computations, the poblem is efomulated in the spectal domain (see, e.g., Sandwell and Smith, 996). Clealy, following the same pocedue fo f ¼ T, we also have the following elationship: Tð; ; Þ ¼ R 0 ðð9; 9; RÞcos 4 þ 9; ð 9; RÞsin Þ q qc B ð c; Þd ½90Š whee Bðc; Þ ¼ X n¼2 ð2n þ Þ R nþ P n ðcos cþ ½9Š nnþ ð Þ It is inteesting to note that instead of an integal ove the sphee, the invese elationship between the distubing potential on the sphee and the deflection of the vetical on the same sphee is also moe staightfowad in tems of a line integal: Tð; ; RÞ ¼Tð 0 ; 0 ; RÞþ 0 4 whee Z ð; Þ ð 0; 0Þ ðð9; 9; RÞds 9; ð 9; RÞds Þ ½92Š ds ¼ R d; ds ¼ R sin d ½93Š Concluding Remaks The spheical hamonic seies, [59], epesents the geneal solution to the exteio potential, egadless of the way the coefficients ae detemined. We know how to compute those coefficients exactly on the basis of a BVP, if the bounday is a sphee. Moe complicated boundaies would equie coections o, if these ae omitted, would imply an appoximation. If the coefficients ae detemined accuately (e.g., fom satellite obsevations (Section ), but not accoding to eqn [69]), then the spheical hamonic seies model fo the potential is not a spheical appoximation. The spheical appoximation entes when appoximate elations such as eqns [5] ae used and when the bounday is appoximated as a sphee. Howeve detemined, the infinite seies conveges unifomly fo all > R c, whee R c is the adius of the sphee that encloses all teestial masses. It may also convege below this sphee, but would epesent the tue potential only in fee space (above the Eath s suface, whee Laplace s equation holds). In pactice, though, convegence is not an

18 28 Potential Theoy and Static Gavity Field of the Eath issue since the seies must be tuncated at some finite degee. Any tend towad divegence is then pat of the oveall model eo, due mostly to tuncation. Model eos exist in all the Geen s function integals as they depend on spheical appoximations in the bounday condition. In addition, the suface of integation in these fomulas is fomally assumed to be the geoid (whee nomal deivatives of the potential coincide with gavity magnitude), but it is appoximated as a sphee. The spheical appoximation esults, in the fist place, fom a neglect of the ellipsoid flattening, which is about 0.3% fo the Eath. When woking with the distubing potential, this level of eo was easily toleated in the past (e.g., if the geoid undulation is 30 m, the spheical appoximation accounts fo about 0 cm eo), but today it equies attention as geoid undulation accuacy of cm is pusued. The Geen s functions all have singulaities when the evaluation point is on the sphee of adius, R. Fo points above this sphee, it is easily veified that all the Legende seies fo the Geen s functions convege unifomly, since jp n ðcos cþj. When ¼ R, the coesponding singulaities of the integals ae eithe weak (Stokes integal) o stong (e.g., Poisson integal), equiing special definition of the integal as Cauchy pincipal value. Finally, it is noted that the BVP solutions also equie that no masses eside above the geoid (the bounday appoximated as a sphee). To satisfy this condition, the topogaphic masses must be edistibuted appopiately by mathematical eduction and the gavity anomalies, o distubances measued on the Eath s suface must be educed to the geoid (see Chapte 0.05). The mass edistibution must then be undone (mathematically) in ode to obtain the coect potential on o above the geoid. Details of these pocedues ae found in Heiskanen and Moitz (967, chapte 3). In addition, the atmosphee, having significant mass, affects gavity anomalies at diffeent elevations. These effects may also be emoved pio to using them as bounday values in the integal fomulas Low-Degee Hamonics: Intepetation and Refeence The low-degee spheical hamonics of the Eath s gavitational potential lend themselves to intepetation with espect to the most elemental distibution of the Eath s density, which also leads to fundamental geometic chaacteizations, paticulaly fo the second-degee hamonics. Let Cnm ðþ a ¼ a GM R nþ v nm a ½94Š be unitless coefficients that efe to a sphee of adius, a. (Recall that coefficients, v nm, eqn [44], efe to a sphee of adius, R.) Relative to the cental hamonic coefficient, C ðþ a 00 ¼, the next significant hamonic, C ðþ a 20, is moe than 3 odes of magnitude smalle; the emaining hamonic coefficients ae at least 2 3 odes of magnitude smalle than that. The attenuation afte degee 2 is much moe gadual (Table ), indicating that the bulk of the potential can be descibed by an ellipsoidal field. The nomal gavitational field is such a field, but it also adhees to a geodetic definition that equies the undelying ellipsoid to be an equipotential suface in the coesponding nomal gavity field. This section examines the low-degee hamonics fom these two pespectives of intepetation and efeence Low-Degee Hamonics as Density Moments Retuning to the geneal expession fo the gavitational potential in tems of the Newtonian density integal (eqn [4]), and substituting the spheical hamonic seies fo the ecipocal distance (eqn [57] with [5]), Table Spheical hamonic coefficients of the total gavitational potential a Degee (n) Ode, (m) (a) C nm (a) C n, m E E E E E E E E E E E E E E E E E E E E E 07 a GRACE model GGM02S (Tapley et al., 2005).

19 Potential Theoy and Static Gavity Field of the Eath 29 Z V ð; ; Þ ¼ G X 9 nþ volume 0 n¼0 ¼ X n¼0 m¼ n X n 2n þ m¼ n X n R nþ Z volume Y nm ð;! ÞY nm ð9; 9Þ dv G R nþ ð2n þ Þ! ð9þ n Y nm ð9; 9Þdv Y nm ð; Þ ½95Š yields a multipole expansion (so called fom electostatics) of the potential. The spheical hamonic (Stokes) coefficients ae multipoles of the density distibution (cf. eqn [44]), Z G v nm ¼ R nþ ð2n þ Þ v ð9þ n Y nm ð9; 9Þdv ½96Š One may also conside the nth-ode moments of density (fom the statistics of distibutions) defined by Z ðþ n ¼ ðx 0 Þ ðy 0 Þ ðz 0 Þ dv; n ¼ þ þ ½97Š v The multipoles of degee n and the moments of ode n ae elated, though not all ðn þ Þðn þ 2Þ=2 moments of ode n can be detemined fom the 2nþ multipoles of degee n, when n 2 (clealy isking confusion, we defe to the common nomenclatue of ode fo moments and degee fo spheical hamonics). This indeteminacy is diectly connected to the inability to detemine the density distibution uniquely fom extenal measuements of the potential (Chao, 2005), which is the classic geophysical invese poblem. The zeo-degee Stokes coefficient is coodinate invaiant and is popotional to the total mass of the Eath: v 00 ¼ G R Z volume dv ¼ GM R ½98Š It also epesents a mass monopole, and it is popotional to the zeoth moment of the density, M. The fist-degee hamonic coefficients (epesenting dipoles) ae popotional to the coodinates of the cente of mass, ðx cm ; y cm ; z cm Þ, which ae popotional to the fist-ode moments of the density, as veified by ecalling the definition of the fist-degee spheical hamonics: 8 9 9sin 9sin 9; m ¼ v m ¼ p G Z >< >= ffiffi 9cos 9; m ¼ 0 dv 3 R 2 volume >: >; 9sin 9cos 9; m ¼ 8 9 y cm ; m ¼ ¼ p GM >< >= ffiffi z 3 R 2 cm ; m ¼ 0 ½99Š >: >; x m ; m ¼ Nowadays, by tacking satellites, we have access to the cente of mass of the Eath (including its atmosphee), since it defines the cente of thei obits. Ignoing the small motion of the cente of mass (annual amplitude of seveal millimetes) due to the tempoal vaiations in the mass distibution, we may choose the coodinate oigin fo the geopotential model to coincide with the cente of mass, thus annihilating the fist-degee coefficients. The second-ode density moments likewise ae elated to the second-degee hamonic coefficients (quadupoles). They also define the inetia tenso of the body. The inetia tenso is the popotionality facto in the equation that elates the angula momentum vecto, H, and the angula velocity, w, of a body, like the Eath: and is given by H ¼ I! 0 I xx I xy I xz I ¼ B I yx I yy I yz A I zx I zy I zz ½00Š ½0Š It compises the moments of inetia on the diagonal Z I xx ¼ Z I yy ¼ Z I zz ¼ volume volume volume y9 2 þ z9 2 dv; z9 2 þ x9 2 dv; ½02Š x9 2 þ y9 2 dv and the poducts of inetia off the diagonal Note that Z I xy ¼ I yx ¼ Z I xz ¼ I zx ¼ Z I yz ¼ I zy ¼ volume volume volume x9y9 dv; x9z9 dv; y9z9 dv ½03Š

20 30 Potential Theoy and Static Gavity Field of the Eath I xx ¼ ðþ þ ðþ ; I xy ¼ 2 ðþ 0 ; etc ½04Š and thee ae as many (six) independent tenso components as second-ode density moments. Using the explicit expessions fo the second-degee spheical hamonics, we have fom eqn [96] with n ¼ 2: pffiffiffiffiffi pffiffiffiffiffi 5 G 5 G v 2; 2 ¼ 5R 3 I xy ; v 2; ¼ 5R 3 I yz pffiffiffiffiffi pffiffi 5 G 5 G v 2; ¼ 5R 3 I xz v 2; 0 ¼ 0R 3 I xx þ I yy 2I zz ½05Š pffiffiffiffiffi 5 G v 2; 2 ¼ 0R 3 I yy I xx These ae also known as MacCullagh s fomulas. Not all density moments (o, moments of inetia) can be detemined fom the Stokes coefficients. If the coodinate axes ae chosen so as to diagonalize the inetia tenso (poducts of inetia ae then equal to zeo), then they ae known as pincipal axes of inetia, o also figue axes. Fo the Eath, the z-figue axis is vey close to the spin axis (within seveal metes at the pole); both axes move with espect to each othe and the Eath s suface, with combinations of vaious peiods (daily, monthly, annually, etc.), as well as seculaly in a wandeing fashion. Because of these motions, the figue axis is not useful as a coodinate axis that defines a fame fixed to the (suface of the) Eath. Howeve, because of the poximity of the figue axis to the defined efeence z-axis, the second-degee, fist-ode hamonic coefficients of the geopotential ae elatively small (Table ). The abitay choice of the x-axis of ou Eathfixed efeence coodinate system cetainly did not attempt to eliminate the poduct of inetia, I xy (the x-axis is defined by the intesection of the Geenwich meidian with the equato, and the y-axis completes a ight-handed mutually othogonal tiad). Howeve, it is possible to detemine whee the x-figue axis is located by combining values of the second-degee, second-ode hamonic coefficients. Let u, v, w be the axes that define a coodinate system in which the inetia tenso is diagonal, and assume that I ww ¼ I zz. A otation by the angle, 0, about the w- (also z-) figue axis bings this ideal coodinate system back to the conventional one in which we calculate the hamonic coefficients. Tensos tansfom unde otation, defined by matix, R, accoding to I xyz ¼ R I uvw R T ½06Š With the otation about the w-axis given by the matix, 0 cos 0 sin 0 0 R ¼ B sin 0 cos 0 0 A 0 0 ½07Š and with eqns [05], it is staightfowad to show that pffiffiffiffiffi 5 G v 2; 2 ¼ 0R 3 ði uu I vv Þsin 2 0 pffiffiffiffiffi ½08Š 5 G v 2; 2 ¼ 0R 3 ði uu I vv Þcos 2 0 Hence, we have 0 ¼ 2 tan v 2; 2 v 2; 2 ½09Š whee the quadant is detemined by the signs of the hamonic coefficients. Fom Table, we find that 0 ¼ 4:929 ; that is, the u-figue axis is in the mid-atlantic between South Ameica and Afica. The second-degee, second-ode hamonic coefficient, v 2; 2, indicates the asymmety of the Eath s mass distibution with espect to the equato. Since v 2; 2 > 0 (fo the Eath), equations [08] show that I vv > I uu and thus the equato bulges moe in the diection of the u-figue axis; convesely, the equato is flattened in the diection of the v-figue axis. This flattening is elatively small: Finally, conside the most impotant seconddegee hamonic coefficient, the second zonal hamonic, v 2; 0. Iespective of the x-axis definition, it is popotional to the diffeence between the moment of inetia, I zz, and the aveage of the equatoial moments, I xx þ I yy =2. Again, since v2; 0 < 0, the Eath bulges moe aound the equato and is flattened at the poles. The second zonal hamonic coefficient is oughly 000 times lage than the othe second-degee coefficients and thus indicates a substantial pola flattening (owing to the Eath s ealy moe-fluid state). This flattening is appoximately Nomal Ellipsoidal Field Because of Eath s dominant pola flattening and the nea symmety of the equato, any meidional section of the Eath is close to an ellipse than a cicle. Fo this eason, ellipsoidal coodinates have often been advocated in place of the usual spheical coodinates. In fact, fo geodetic positioning and geogaphic mapping, because of this flattening, the conventional (geodetic) latitude and longitude ae coodinates that define the diection of the pependicula to an

21 Potential Theoy and Static Gavity Field of the Eath 3 ellipsoid, not the adial diection fom the oigin. These geodetic coodinates ae not the ellipsoidal coodinates defined in Section (eqn [2]) and would be athe useless in potential modeling because they do not sepaate Laplace diffeential equation. Hamonic seies in tems of the ellipsoidal coodinates, ð; ; uþ, howeve, can be developed easily. They have not been adopted in most applications, pehaps in pat because of the nonintuitive natue of the coodinates. Nevetheless, it is advantageous to model the nomal (o efeence) gavity field in tems of these ellipsoidal coodinates since it is based on an ellipsoid. Laplace equation in these ellipsoidal coodinates can be sepaated, analogous to spheical coodinates, and the solution is obtained by successively solving thee odinay diffeential equations. Applied to the exteio gavitational potential, V, the solution is simila to the spheical hamonic seies (eqn [44]) and is given by V ð; ; uþ ¼ X n¼0 X n Q nm Q m¼ n nm j j ðiu=eþ j j ðib=eþ ve nmy nm ð; Þ ½0Š whee E is the linea eccenticity associated with the coodinate system and Q nm is the associated Legende function of the second kind. The coefficients of the seies, vnm e, efe to an ellipsoid of semimino axis, b; and, with the seies witten in this way, they ae all eal numbes with the same units as V. An exact elationship between these and the spheical hamonic coefficients, v nm, was given by Hotine (969) and Jekeli (988). With this fomulation of the potential, Diichlet s BVP is solved fo an ellipsoidal bounday using the othogonality of the spheical hamonics: vnm e ¼ 4 V ð; ; bþy nm ð; Þd ½Š whee d ¼ sin dd and ¼ fðd; Þj0 d ; 0 2g. Note that while the limits of integation and the diffeential element, d, ae the same as fo the unit sphee, the bounday values ae on the ellipsoid. Unfotunately, integal solutions with analytic foms of a Geen s function do not exist in this case, because the invese distance now depends on two suface coodinates and thee is no coesponding convolution theoem. Howeve, appoximations have been fomulated fo all thee types of BVPs (see Yu et al., 2002, and efeences theein). Foms of ellipsoidal coections to the classic spheical integals have also been developed and applied in pactice (e.g., Fei and Sideis, 2000). The simplicity of the bounday values of the nomal gavitational potential allows its extension into exteio space to be expessed in closed analytic fom. Analogous to the geoid in the actual gavity field, the nomal ellipsoid is defined to be a level suface in the nomal gavity field. In othe wods, the sum of the nomal gavitational potential and the centifugal potential due to Eath s otation is a constant on the ellipsoid: V e ð; ; bþþ;b ð Þ ¼ U 0 ½2Š Hence, the nomal gavitational potential on the ellipsoid, V e ð; ; bþ, depends only on latitude and is symmetic with espect to the equato. Consequently, it consists of only even zonal hamonics, and because the centifugal potential has only zeo- and second-degee zonals, the coesponding ellipsoidal seies is finite (up to degee 2). The solution to this Diichlet poblem is given in ellipsoidal coodinates by V e ð; ; uþ ¼ GM E tan E u þ q 2!2 e a2 cos 2 q 0 3 ½3Š whee a is the semimajo axis of the ellipsoid,! e is Eath s ate of otation, and q ¼ þ 3 u2 2 E 2 tan E u 3 u ; q 0 ¼ qj E u¼b ½4Š Heiskanen and Moitz (967) and Hofmann- Wellenhof and Moitz (2005) povide details of the staightfowad deivation of these and the following expessions. The equivalent fom of V e in spheical hamonics is given by V e ð; ; Þ ¼ GM X J 2n n¼! a 2nP2n ðcos Þ ½5Š whee J 2n ¼ ð Þ nþ 3e 2n n þ 5n ð2n þ Þð2n þ 3Þ e 2 J 2 ; n ½6Š and e ¼ E=a is the fist eccenticity of the ellipsoid. The second zonal coefficient is given by J 2 ¼ e2 3 2! 2 e a2 E 5q 0 GM ¼ I e zz I e xx Ma 2 ½7Š

22 32 Potential Theoy and Static Gavity Field of the Eath whee the second equality comes diectly fom the last of eqns [05] and the ellipsoid s otational symmety (Ixx e ¼ I yy e ). This equation also povides a diect elationship between the geomety (the eccenticity o flattening) and the mass distibution (diffeence of second-ode moments) of the ellipsoid. Theefoe, J 2 is also known as the dynamic fom facto the flattening of the ellipsoid can be descibed eithe geometically o dynamically in tems of a diffeence in density moments. The nomal gavitational potential depends solely on fou adopted paametes: Eath s otation ate,! e ; the size and shape of the nomal ellipsoid, fo example, a, J 2 ; and a potential scale, fo example, GM. The mean Eath ellipsoid is the nomal ellipsoid with paametes closest to actual coesponding paametes fo the Eath. GM and J 2 ae detemined by obseving satellite obits, a can be calculated by fitting the ellipsoid to mean sea level using satellite altimety, and Eath s otation ate comes fom astonomical obsevations. Table 2 gives cuent best values (Goten, 2004) and adopted constants fo the Geodetic Refeence Systems of 967 and 980 (GRS67, GRS80) and the Wold Geodetic System 984 (WGS84). In modeling the distubing potential in tems of spheical hamonics, one natually uses the fom of the nomal gavitational potential given by eqn [5]. Hee, we have assumed that all hamonic coefficients efe to a sphee of adius, a. Coesponding coefficients fo the seies of T ¼ V V e ae, theefoe, 8 0; n ¼ 0 >< Cnm ðþ a ¼ C ðþ a n0 J p ffiffiffiffiffiffiffiffiffiffiffiffiffi n ; n ¼ 2; 4; 6;... ½8Š 2n þ >: Cnm ðþ a ; othewise Fo coefficients efeing to the sphee of adius, R,as in eqn [59], we have C nm ¼ ða=rþ n Cnm ðþ a. The hamonic coefficients, J 2n, attenuate apidly due to the facto, e 2n, and only hamonics up to degee 0 ae significant. Nomal gavity, being the gadient of the nomal gavity potential, is used only in applications tied to an Eath-fixed coodinate system Methods of Detemination In this section, we biefly exploe the basic technologies that yield measuements of the gavitational field. Even though we have educed the poblem of detemining the exteio potential fom a volume integal to a suface integal (e.g., eithe eqns [66] o [69]), it is clea that in theoy we can neve detemine the entie field fom a finite numbe of measuements. The integals will always need to be appoximated numeically, and/o the infinite seies of spheical hamonics needs to be tuncated. Howeve, with enough effot and within the limits of computational capabilities, one can appoach the ideal continuum of bounday values as closely as desied, o make the numbe of coefficients in the seies epesentation as lage as possible. The expended computational and measuement effot has to be balanced with the ability to account fo inheent model eos (such as the spheical appoximation) and the noise of the measuing device. To be useful fo geodetic and geodynamic puposes, the instuments must possess a sensitivity of at least a few pats pe million, and, in fact, many have a sensitivity of pats pe billion. These sensitivities often come at the expense of polonging the measuements (integation time) in ode to aveage out andom noise, thus educing the achievable tempoal esolution. This is paticulaly citical fo moving-base instumentation such as on an aicaft o satellite whee tempoal esolution tanslates into spatial esolution though the velocity of the vehicle. Table 2 Defining paametes fo nomal ellipsoids of geodetic efeence systems Refeence system a (m) J 2 GM (m 3 s 2 )! e (ad s 2 ) GRS E E E 05 GRS E E E 05 WGS E E E 05 Best cuent values a (mean-tide system) ( )E 03 (zeo-tide system) ( )E4 (includes atmosphee) E 5 (mean value) a Goten E (2004) Fundamental paametes and cuent (2004) best estimates of the paametes of common elevance to astonomy, geodesy, and geodynamics. Jounal of Geodesy 77: (0 ) (The Geodesist s Handbook pp ).

23 Potential Theoy and Static Gavity Field of the Eath Measuement Systems and Techniques Detemining the gavitational field though classical measuements elies on thee fundamental laws. The fist two ae Newton s second law of motion and his law of gavitation. Newton s law of motion states that the ate of change of linea momentum of a paticle is equal to the totality of foces, F, acting on it. Given moe familialy as m i d 2 x=dt 2 ¼ F, it involves the inetial mass, m i ; and, conceptually, the foces, F, should be intepeted as action foces (like populsion o fiction). The gavitational field, which is pat of the space we occupy, is due to the pesence of masses like the Eath, Sun, Moon, and planets, and induces a diffeent kind of foce, the gavitational foce. It is elated to gavitational acceleation, g, though the gavitational mass, m g, accoding to the law of gavitation, abbeviated hee as m g g¼f g. Newton s law of motion must be modified to include F g sepaately. Though the thid fundamental law, Einstein s equivalence pinciple, which states that inetial and gavitational masses ae indistinguishable, we finally get d 2 x dt 2 ¼ a þ g ½9Š whee a is the specific foce (F=m i ), o also the inetial acceleation, due to action foces. This equation holds in a nonotating, feely falling fame (that is, an inetial fame), and vaiants of it can be deived in moe complicated fames that otate o have thei own dynamic motion. Howeve, one can always assume the existence of an inetial fame and poceed on that basis. Thee exists a vaiety of devices that measue the motion of an inetial mass with espect to the fame of the device, and thus technically they sense a; such devices ae called acceleometes. Conside the special case that an acceleomete is esting on the Eath s suface with its sensitive axis aligned along the vetical. In an Eath-centeed fame (inetial, if we ignoe Eath s otation), the fee-fall motion of the acceleomete is impeded by the eaction foce of the Eath s suface acting on the acceleomete. In this case, the left-hand side of eqn [9] applied to the motion of the acceleomete is zeo, and the acceleomete, sensing the eaction foce, indiectly measues (the negative of) gavitational acceleation. This acceleomete is given the special name, gavimete. Gavimetes, especially static instuments, ae designed to measue acceleation at vey low fequencies (i.e., aveaged ove longe peiods of time), wheeas acceleometes typically ae used in navigation o othe motion-sensing applications, whee acceleations change apidly. As such, gavimetes geneally ae moe accuate. Eath-fixed gavimetes actually measue gavity (Section ), the diffeence between gavitation and centifugal acceleation due to Eath s spin (the fame is not inetial in this case). The simplest, though not the fist invented, gavimete utilizes a vetically, feely falling mass, measuing the time it takes to fall a given distance. Applying eqn [9] to the falling mass in the fame of the device, one can solve fo g (assuming it is constant): xt ðþ¼0:5gt 2. This fee-fall gavimete is a special case of a moe geneal gavimete that constains the fall using an attached sping o the am of a pendulum, whee othe (action) foces (the tension in the am o the sping) thus ente into the equation. The fist gavimete, in fact, was the pendulum, systematically used fo gavimety as ealy as the 730s and 740s by P. Bougue on a geodetic expedition to measue the size and shape of the Eath (meidian ac measuement in Peu). The pendulum seved well into the twentieth centuy (until the ealy 970s) both as an absolute device, measuing the total gavity at a point, o as a elative device indicating the diffeence in gavity between two points (Toge, 989). Today, absolute gavimetes exclusively ely on a feely falling mass, whee exquisitely accuate measuements of distance and time ae achieved with lase intefeometes and atomic clocks (Zumbege et al., 982; Niebaue et al., 995). Accuate elative gavimetes ae much less expensive, equiing a measuement of distance change only, and because many eos that cancel between measuements need not be addessed. They ely almost exclusively on a sping-suspended test mass (Nettleton, 976; Toge, 989). Developed ealy in the twentieth centuy in esponse to oilexploation equiements, the elative gavimete has changed little since then. Moden instuments include electonic ecoding capability, as well as specialized stabilization and damping fo deployment on moving vehicles such as ships and aicaft. The accuacy of absolute gavimetes is typically of the ode of pats pe billion, and elative devices in field deployments may be as good but moe typically ae at least ode of magnitude less pecise. Laboatoy elative (in time) gavimetes, based on cyogenic

24 34 Potential Theoy and Static Gavity Field of the Eath instuments that monito the vitual motion of a test mass that is electomagnetically suspended using supeconducting pesistent cuents (Goodkind, 999), ae as accuate as potable absolute devices (o moe), owing to the stability of the cuents and the contolled laboatoy envionment (see Chaptes 3.03 and 3.04). On a moving vehicle, paticulaly an aicaft, the elative gavitational acceleation can be detemined only fom a combination of gavimete and kinematic positioning system. The latte is needed to deive the (vetical) kinematic acceleation (the left-hand side of eqn [9]). Today, the GPS best seves that function, yielding centimete-level pecision in elative thee-dimensional position. Such combined GPS/ gavimete systems have been used successfully to detemine the vetical component of gavitation ove lage, othewise-inaccessible aeas such as the Actic Ocean (Kenyon and Fosbeg, 2000) and ove othe aeas that ae moe economically suveyed fom the ai fo oil-exploation puposes (Hamme, 982; Gumet, 998). The aibone gavimete is specially designed to damp out high-fequency noise and is usually stabilized on a level platfom. Thee-dimensional moving-base gavimety has also been demonstated using the tiad of acceleometes of a high-accuacy inetial navigation system (the type that ae fixed to the aicaft without special stabilizing platfoms). The oientation of all acceleometes on the vehicle must be known with espect to inetial space, which is accomplished with pecision gyoscopes. Again, the total acceleation vecto of the vehicle, d 2 x=dt 2, can be ascetained by time diffeentiation of the kinematic positions (fom GPS). One of the most citical eos is due to the coss-coupling of the hoizontal oientation eo, c, with the lage vetical acceleation (the lift of the aicaft, essentially equal to g). This is a fist-ode effect (g sin dc) in the estimation of the hoizontal gavitation components, but only a second-ode effect (gð cos dcþ) on the vetical component. Details of moving-base vecto gavimety may be found in Jekeli (2000a, chapte 0) and Kwon and Jekeli (200). The ultimate global gavimete is the satellite in fee fall (i.e., in obit due to sufficient fowad velocity) the satellite is the inetial mass and the device is a set of efeence points with known coodinates (e.g., on the Eath s suface, o anothe satellite whose obit is known; see Chapte 3.05). The measuing technology is an accuate anging system (ada o lase) that tacks the satellite as it obits (falls to) the Eath. Eve since Sputnik, the fist atificial satellite, launched into the Eath obit in 957, Eath s gavitational field could be detemined by tacking satellites fom pecisely known gound stations. Equation [9], with gavitational acceleation expessed as a tuncated seies of spheical hamonics (gadient of eqn [44]), becomes d 2 x dt 2 ¼ Xnmax n¼0 X n v nm R! nþ Y nm ð; þ! e tþ þ R m¼ n ½20Š whee! e is Eath s ate of otation and dr epesents esidual acceleations due to action foces (sola adiation pessue, atmospheic dag, Eath s albedo, etc.), gavitational tidal acceleations due to othe bodies (Moon, Sun, planets), and all othe subsequent indiect effects. The x left-hand side of eqn [20] is moe explicitly xt ðþ¼x ðþ; t ðþþ! t e t; t ðþ, and the spatial coodinates on the ight-hand side ae also functions of time. This makes the equation moe conceptual than pactical since it is numeically moe convenient to tansfom the satellite position and velocity into Kepleian obital elements (semimajo axis of the obital ellipse, its eccenticity, its inclination to the equato, the angle of peigee, the ight ascension of the node of the obit, and the mean motion) all of which also change in time, but most much moe slowly. This tansfomation was deived by Kaula (966) (see also Seebe, 993). In the most geneal case (n max > 2 and dr 6¼ 0), thee is no analytic solution to eqn [20] o its tansfomations to othe types of coodinates. The positions of the satellite ae obseved by anging techniques and the unknowns to be solved ae the coefficients, v nm. Numeical integation algoithms have been specifically adapted to this poblem and extemely sophisticated models fo dr ae employed with additional unknown paametes to be solved in ode to estimate as accuately as possible the gavitational coefficients (e.g., Cappelai et al., 976; Pavlis et al., 999). The entie pocedue falls unde the boad categoy of dynamic obit detemination, and the coesponding gavitational field modeling may be classified as the timewise appoach. The patial deivatives of eqn [20] with espect to unknown paametes, p ¼ f...; v nm ;... g, ae integated numeically in time, yielding estimates fo H ¼ qx=qp ¼ f...; qx=qv nm ;... g. These ae then used in a least-squaes adjustment of the lineaized model elating obseved positions (e.g., via anges) to paametes

25 Potential Theoy and Static Gavity Field of the Eath 35 x ¼ Hp þ e ½2Š whee dx and dp ae diffeences with espect to pevious estimates, and e epesents eos. (Tapley, 973). A gavimete (o acceleomete) on a satellite does not sense the pesence of a gavitational field. This is evident fom the fact that the satellite is in fee fall (apat fom small acceleations due to action foces such as atmospheic dag) and the inetial test mass and the gavimete, itself, ae equally affected by gavitation (i.e., they ae all in fee fall). Howeve, two acceleometes fixed on a satellite yield, though the diffeence in thei outputs, a gadient in acceleation that includes the gadient of gavitation. On a nonotating satellite, the acceleation at an abitay point, b, of the satellite is given by a b ¼ d2 x dt 2 gb ð Þ ½22Š in a coodinate system with oigin at the cente of mass of the satellite. Taking the diffeence (diffeential) of two acceleations in atio to thei sepaation, we obtain a b b ¼ g b ½23Š whee the atios epesent tensos of deivatives in the local satellite coodinate fame. Fo a otating satellite, this equation genealizes to qa b qb ¼ qg b qb þ 2 þ d dt ½24Š whee is a skew-symmetic matix whose off-diagonal elements ae the components of the vecto that defines the otation ate of the satellite with espect to the inetial fame. Thus, a gadiomete on a satellite (o any moving vehicle) senses a combination of gavitational gadient and angula acceleation (including a centifugal type). Such a device is scheduled to launch fo the fist time in 2007 as pat of the mission GOCE (Gavity Field and Steady-State Ocean Ciculation Exploe ; Rummel et al., 2002). If the entie tenso of gadients is measued, then, because of the symmety of the gavitational gadient tenso and of 2, and the antisymmety of d=dt, the sum da b =db þðda b =dbþ T eliminates the latte, while the diffeence da b =db ðda b =dbþ T can be used to infe, subject to initial conditions. When the two ends of the gadiomete ae fixed to one fame, the common linea acceleation, d 2 x=dt 2, cancels, as shown above; but if the two ends ae independent, disconnected platfoms moving in simila obits, the gavitational diffeence depends also on thei elative motion in inetial space. This is the concept fo satellite-to-satellite tacking, by which one satellite pecisely tacks the othe and the change in the ange ate between them is a consequence of a gavitational diffeence, a diffeence in action foces, and a centifugal acceleation due to the otation of the baseline of the satellite pai. It can be shown that the line-of-sight acceleation (the measuement) is given by d 2 dt 2 ¼ et ð gx ð 2Þ gx ð ÞÞþe T ð a 2 a Þ þ d dt x 2 d! 2 dt ½25Š whee e is the unit vecto along the instantaneous baseline connecting the two satellites, is the baseline length (the ange), and x ¼ x 2 x is the diffeence in position vectos of the two satellites. Clealy, only the gavitational diffeence pojected along the baseline can be detemined (simila to a single-axis gadiomete), and then only if both satellites cay acceleometes that sense the nongavitational acceleations. Also, the obits of both satellites need to be known in ode to account fo the centifugal tem. Two such satellite systems wee launched ecently to detemine the global gavitational field. One is CHAMP (Challenging Mini-Satellite Payload) in 2000 (Reigbe et al., 2002) and the othe is GRACE (Gavity Recovey and Climate Expeiment) in 2002 (Tapley et al., 2004a). CHAMP is a single low-obiting satellite ( km altitude) being tacked by the high-altitude GPS satellites, and it also caies a magnetomete to map the Eath s magnetic field. GRACE was moe specifically dedicated to detemining with extemely high accuacy the long to medium wavelengths of the gavitational field and thei tempoal vaiations. With two satellites in vitually identical low Eath obits, one following the othe, the pimay data consist of intesatellite anges obseved with K-band ada. The objective is to sense changes in the gavitational field due to mass tansfe on the Eath within and among the atmosphee, the hydosphee/cyosphee, and the oceans (Tapley et al., 2004b). An Eath-obiting satellite is the ideal platfom on which to measue the gavitational field when seeking global coveage in elatively shot time. One simply designs the obit to be pola and cicula;

26 36 Potential Theoy and Static Gavity Field of the Eath and, as the satellite obits, the Eath spins undeneath, offeing a diffeent section of its suface on each satellite evolution. Thee ae also limitations. Fist, the satellite must have low altitude to achieve high sensitivity, since the nth-degee hamonics of the field attenuate as ðr=þ n þ. On the othe hand, the lowe the altitude, the shote the life of the satellite due to atmospheic dag, which can only be counteed with onboad populsion systems. Second, because of the inheent speed of lowe-obit satellites (about 7 km s ), the esolution of its measuements is limited by the integation (aveaging) time of the senso (typically 0 s). Highe esolution comes only with shote integation time, which may educe the accuacy if this depends on aveaging out andom noise. Figue 8 shows the coesponding achievable esolution on the Eath s suface fo diffeent satellite instumentation paametes, length of time in pola obit and along-obit integation time, o smoothing (Jekeli, 2004). In each case, the indicated level of esolution is waanted only if the noise of the senso (afte smoothing) does not ovepowe the signal at this esolution. Both CHAMP and GRACE have yielded global gavitational models by utilizing taditional satellitetacking methods and incopoating the ange ate appopiately as a tacking obsevation (timewise appoach). Howeve, the immediate application of eqn [25] suggests that gavitational diffeences can be detemined in situ and used to detemine a model fo the global field diectly. This is classified as the spacewise appoach. In fact, if the obits ae known with sufficient accuacy (fom kinematic obit detemination, e.g., by GPS), this pocedue utilizes a Resolution (km) day mo. 6 mo. y. 5 y. Mission duation Measuement integation time 0 s 5 s Figue 8 Spatial esolution of satellite measuements vs mission duation and integation time. The satellite altitude is 450 km (afte Jekeli, 2004) mo., month. Repoduced by pemission of Ameican Geophysical Union. s linea elationship between obsevations and unknown hamonic coefficients: d 2 dt 2 x; x 2 ¼ Xnmax X n n¼0 m¼ n v nm e T U nm ; ; jx2 U nm ð; ; Þj x þ a þ c ½26Š whee U nm ð; ; Þ ¼ ðr=þ nþ Y nm ð; Þ and da, dc ae the last two tems in eqn [25]. Given the latte and a set of line-of-sight acceleations, a theoetically staightfowad linea least-squaes adjustment solves fo the coefficients. A simila pocedue can be used fo gadients obseved on an obiting satellite: qa Xnmax ¼ qb x X n n¼0 m¼ n v nm T U nm ð; ; Þ x þ c ½27Š whee c compises the otational acceleation tems in eqn [24]. In situ measuements of line-of-site acceleation o of moe local gadients would need to be educed fom the satellite obit to a well-defined suface (such as a sphee) in ode to seve as bounday values in a solution to a BVP. Howeve, the model fo the field is aleady in place, ooted in potential theoy (tuncated seies solution to Laplace equation), and one may think of the poblem moe in tems of fitting a thee-dimensional model to a discete set of obsevations. This opeational appoach can eadily, at least conceptually, be expanded to include obsevations fom many diffeent satellite systems, even aibone and gound-based obsevations. Recently, a athe diffeent theoy has been consideed by seveal investigatos to model the gavitational field fom satellite-to-satellite tacking obsevations. The method, fist poposed by Wolff (969), makes use of yet anothe fundamental law: the law of consevation of enegy. Simply, the ange ate between two satellites implies an along-tack velocity diffeence, o a diffeence in kinetic enegy. Obseving this diffeence leads diectly, by the consevation law, to the diffeence in potential enegy, that is, the gavitational potential and othe potential enegies associated with action foces and Eath s otation. Neglecting the latte two, consevation of enegy implies V ¼ d 2 dt x 2 E 0 ½28Š

27 Potential Theoy and Static Gavity Field of the Eath 37 whee E 0 is a constant. Taking the along-tack diffeential, we have appoximately V ðx 2 Þ V ðx Þ ¼ d dt x d dt ½29Š whee jdx 2 =dt dx =dtj d=dt. This vey ough conceptual elationship between the potential diffeence and the ange ate applies to two satellites closely following each othe in simila obits. The pecise fomulation is given by (Jekeli, 999) and holds fo any pai of satellites, not just two low obites. Mapping ange ates between two pola obiting satellites (such as GRACE) yields a global distibution of potential diffeence obsevations elated again linealy to a set of hamonic coefficients: Xnmax V ðx 2 Þ V ðx Þ ¼ X n n¼0 m¼ n v nm U nm ; ; x2 U nm ; ; x ½30Š The enegy-based model holds fo any two vehicles in motion and equipped with the appopiate anging and acceleomety instumentation. Fo example, the enegy consevation pinciple could also be used to detemine geopotential diffeences between an aicaft and a satellite, such as a GPS satellite (at which location the geopotential is known quite well). The aicaft would equie only a GPS eceive (to do the anging) and a set of acceleometes (and gyos fo oientation) to measue the action foces (the same system components as in aibone acceleomety discussed above). Resulting potential diffeences could be used diectly to model the geoid using Buns equation Models We have aleady noted the standad solution options to the BVP using teestial gavimety in the fom of gavity anomalies: the Geen s function appoach, Stokes integal, eqn [66], and the hamonic analysis of suface data, eithe using the integals [69] o solving a linea system of equations (eqn [65] tuncated to finite degee) to obtain the coefficients, dc nm. The integals must be evaluated using quadatues, and vey fast numeical techniques have been developed when the data occupy a egula gid of coodinates on the sphee o ellipsoid (Rapp and Pavlis, 990). Simila algoithms enable the fast solution of the linea system of eqns [65]. Fo a global hamonic analysis, the numbe of coefficients, ðn max þ Þ 2, must not be geate than the numbe of data. A geneal, consevative ule of thumb fo the maximum esolution (halfwavelength) of a tuncated spheical hamonic seies is, in angula degees on the unit sphee, ¼ 80 n max Thus, data on a angula gid of latitudes and longitudes would imply n max ¼ 80. The numbe of data (64 800) is amply lage than the numbe of coefficients (32 76). This majoity suggests a leastsquaes adjustment of the coefficients to the data, in eithe method, especially because the data have eos (Rapp, 969). As n max inceases, a igoous, optimal adjustment usually is feasible, fo a given computational capability, only unde estictive assumptions on the coelations among the eos of the data. Also, the obvious should nevetheless be noted that the accuacy of the model in any aea depends on the quality of the data in that aea. Futhemoe, consideing that a measuement contains all hamonics (up to the level of measuement eo), the estimation of a finite numbe of hamonics fom bounday data on a given gid is coupted by those hamonics that ae in the data but ae not estimated. This phenomenon is called aliasing in spectal analysis and can be mitigated by appopiate filteing of the data (Jekeli, 996). The optimal spheical hamonic model combines both satellite and teestial data. The cuently best known model is EGM96 complete to degee and ode n max ¼ 360 (Lemoine et al., 998). It is an updated model fo WGS84 based on all available satellite tacking, satellite altimety, and land gavity (and topogaphic) data up to the mid-990s. Scheduled to be evised again fo 2006 using moe ecent data, as well as esults fom the satellite missions, CHAMP and GRACE, it will boast a maximum degee and ode of 260 (5 esolution). In constucting combination solutions of this type, geat effot is expended to ensue the pope weighting of obsevations in ode to extact the most appopiate infomation fom the divese data, petaining to diffeent pats of the spatial gavitational spectum. The satellite tacking data dominate the estimation of the lowe-degee hamonics, wheeas the fine esolution of the teestial data is most amenable to modeling the highe degees. It is beyond the pesent scope to delve into the numeical methodology of combination methods. Futhemoe, it is an unfinished stoy

28 38 Potential Theoy and Static Gavity Field of the Eath as new in situ satellite measuements fom GRACE and GOCE will affect the combination methods of the futue. Stokes integal is used in pactice only to take advantage of local o egional data with highe esolution than was used to constuct the global models. Even though the integal is a global integal, it can be tuncated to a neighbohood of the computation point since the Stokes kenel attenuates apidly as the ecipocal distance. Moeove, the coesponding tuncation eo may be educed if the bounday values exclude the longe-wavelength featues of the field. The latte constitute an adequate epesentation of the emote zone contibution and can be ð included sepaately as follows. Let g n max Þ denote the gavity anomaly implied by a spheical hamonic model, such as given by eqn [65], tuncated to degee, n max. Fom the othogonality of spheical hamonics, eqn [46], it is easy to show that T ðnmax X nmax Þ ð; ; Þ ¼ GM R ¼ R 4 X n n¼2 m¼ n g ðnmax R nþ C nm Y nm ð; Þ Þ ð9; 9; R ÞSðc; Þd ½3Š Thus, given a spheical hamonic model fc nm j2 n n max ; n m ng, we fist emove the model in tems of the gavity anomaly and then estoe it in tems of the distubing potential, changing Stokes fomula [66] to Tð; ; Þ ¼ R gð9; 9; RÞ g ðnmaxþ ð9; 9; RÞ 4 Sðc; Þd þ T ðnmaxþ ð; ; Þ ½32Š ð In theoy, if g n max Þ has no eos, then the esidual ð g g n maxþ excludes all hamonics of degee n n max, and othogonality would also allow the exclusion of these hamonics fom S. Once the integation is limited to a neighbohood of ð; ; RÞ,asit must be in pactice, thee ae a numbe of ways to modify the kenel so as to minimize the esulting tuncation eo (Sjöbeg, 99, and efeences theein). The emoval and estoation of a global model, howeve, is the key aspect in all these methods. In pactical applications, the bounday values ae on the geoid, being the suface that satisfies the bounday condition of the Robin BVP (i.e., we equie the nomal deivative on the bounday and measued gavity is indeed the deivative of the potential along the pependicula to the geoid). The integal in eqn [32] thus appoximates the geoid by a sphee. Futhemoe, it is assumed that no masses exist extenal to the geoid. Pat of the eduction to the geoid of data measued on the Eath s suface involves edistibuting the topogaphic masses on o below the geoid. This edistibution is undone outside the solution to the BVP (i.e., Stokes integal) in ode to egain the distubing potential fo the actual Eath. Conceptually, we may wite T P ¼ R g g ðnmaxþ c 4 P 0S P; P 0d þ T ðnmaxþ P þ T P ½33Š whee dc is the gavity eduction that bings the gavity anomaly to a geoid with no extenal masses, and dt P is the effect (called indiect effect) on the distubing potential due to the invese of this eduction. This fomula holds fo T anywhee on o above the geoid and thus can also be used to detemine the geoid undulation accoding to Buns fomula [3] The Geoid and Heights The taditional efeence suface, o datum, fo heights is the geoid (Section ). A point at mean sea level usually seves as stating point (datum oigin) and this defines the datum fo vetical contol ove a egion o county. The datum (o geoid) is the level continuation of the efeence suface unde the continents, and the detemination of gavity potential diffeences fom the initial point to othe points on the Eath s suface, obtained by leveling and gavity measuements, yields heights with espect to that efeence (o in that datum). The gavity potential diffeence, known as the geopotential numbe, at a point, P, elative to the datum oigin, P 0, is given by (since gavity is the negative vetical deivative of the gavity potential) C P ¼ W 0 W P ¼ Z P P 0 g dn ½34Š whee g is gavity (magnitude), dn is a leveling incement along the vetical diection, and W 0 is the gavity potential at P 0. By the consevative natue of the gavity potential, whateve path is taken fo the integal yields a unique geopotential numbe fo P. Fom these potential diffeences, one can define vaious types of height, fo example, the othometic height (Figue 3):

29 Potential Theoy and Static Gavity Field of the Eath 39 whee H P ¼ C P g P ½35Š g P ¼ Z P g dh ½36Š H P P 0 is the aveage value of gavity along the plumb line fom the geoid at P 0 to P. Othe height systems ae also in use (such as the nomal and dynamic heights), but they all ely on the geopotential numbe (see Heiskanen and Moitz, 967, chapte 4, fo details). Fo a paticula height datum, thee is theoetically only one datum suface (the geoid). But access to this suface is fa fom staightfowad (othe than at the defined datum oigin). If P 0 is defined at mean sea level, othe points at mean sea level ae not on the same level (datum) suface, since mean sea level, in fact, is not level. Eoneously assuming that mean sea level is an equipotential suface can cause significant distotions in the vetical contol netwok of lage egions, as much as seveal decimetes. This was the case, fo example, fo the National Geodetic Vetical Datum of 929 in the US fo which 26 mean sea level points on the east and west coasts wee assumed to lie on the same level suface. Accessibility to the geoid (once defined) at any point is achieved eithe with pecise leveling and gavity (accoding to eqns [34] and [35]), o with pecise geometic vetical positioning and knowledge of the gavity potential. Geometic vetical positioning, today, is obtained vey accuately (centimete accuacy o even bette) with diffeential GPS. Suppose that an accuate gavity potential model is also available in the same coodinate system as used fo GPS. Then, detemining the GPS position at P 0 allows the evaluation of the gavity potential, W 0, of the datum. Access to the geoid at any othe point, P, o equivalently, detemining the othometic height, H P, can be done by fist detemining the ellipsoidal height, h, fom GPS. Then, as shown in Figue 3, H P ¼ h P N ½37Š whee, with T ¼ W U evaluated on the geoid, Buns extended equation (Section 3.02.) yields N ¼ T= ðw 0 U 0 Þ= ½38Š whee U 0 is the nomal gavity potential of the nomal ellipsoid. In a sense, the latte is a cicula poblem: detemining N equies N in ode to locate the point on the geoid whee to compute T. Howeve, the computation of T on the geoid can be done with assumptions on the density of the topogaphic masses above the geoid and a pope eduction to the geoid, using only an appoximate height. Indeed, since the vetical gadient of the distubing potential is the gavity distubance, of the ode of ms 2,a height eo of 20 m leads to an eo of 0 2 m 2 s 2 in T, o just mm in the geoid undulation. It should be noted that a model fo the distubing potential as a seies of spheical hamonics, fo example, deived fom satellite obsevations, satisfies Laplace s equation and, theefoe, does not give the coect distubing potential at the geoid (if it lies below the Eath s suface whee Laplace s equation does not hold). The ability to deive othometic heights (o othe geopotential-elated heights) fom GPS has geat economical advantage ove the laboious leveling pocedue. This has put geat emphasis on obtaining an accuate geoid undulation model fo land aeas. Section biefly outlined the essential methods to detemine the geoid undulation fom a combination of spheical hamonics and an integal of local gavity anomalies. When dealing with a height datum o the geoid, the constant N 0 ¼ ðw 0 U 0 Þ= equies caeful attention. It can be detemined by compaing the geoid undulation computed accoding to a model that excludes this tem (such as eqn [33]) with at least one geoid undulation (usually many) detemined fom leveling and GPS, accoding to eqn [37]. Vetical contol and the choice of height datum ae specific to each county o continent, whee a local mean sea level was the adopted datum oigin. Thus, height datums aound the wold ae local geoids that have significant diffeences between them. Investigations and effots have been unde way fo moe than two decades to define a global vetical datum; howeve, it is still in the futue, awaiting a moe accuate global gavity potential model and, pehaps moe cucially, a consensus on what level suface the global geoid should be. On the oceans, the situation is somewhat less complicated. Oceanogaphes who compute sea-suface topogaphy fom satellite altimety on the basis of eqn [37] depend citically on an accuate geoid undulation, o equivalently on an accuate model of T. Howeve, no eduction of the distubing potential fom mean sea level to the geoid is necessay, the deviation being at most 2 m and causing an eo in geoid undulation of less than 0. mm. Thus, a spheical hamonic model of T is entiely appopiate. Futhemoe, it is easonable to ensue that the

30 40 Potential Theoy and Static Gavity Field of the Eath constant, W 0 U 0, vanishes ove the oceans. That is, one may choose the geoid such that it best fits mean sea level and choose an ellipsoid that best fits this geoid. It means that the global aveage value of the geoid undulation should be zeo (accoding to eqn [38]). The latte can be achieved with satellite altimety and oceanogaphic models of sea suface topogaphy (Busa et al., 997). Seveal inteesting and impotant distinctions should be made in egad to the tidal effects on the geoid. The Sun and the Moon geneate an appeciable gavitational potential nea the Eath (the othe planets may be neglected). In an Eath-fixed coodinate system, this extateestial potential vaies in time with diffeent peiods due to the elative motions of the Moon and Sun, and because of Eath s otation (Toge, 200, p. 88). Thee is also a constant pat, the pemanent tidal potential, epesenting the aveage ove time. It is not zeo, because the Eath Sun Moon system is appoximately coplana. Fo each body with mass, M B, and distance, B, fom the Eath s cente, this pemanent pat is given by V B c ð; Þ ¼ 3 4 GM B 2 B 2 3 cos 2 2 sin2 " 3 ½39Š whee " is the angle of the ecliptic elative to the equato (" ). Using nominal paamete values fo the Sun and the Moon, we obtain at mean Eath adius, R ¼ 637km, Vc sþm ð; RÞ ¼ 0:97 3 cos 2 m 2 s 2 ½40Š The gavitational potential fom the Sun and Moon also defoms the quasi-elastic Eath s masses with the same peiods and similaly includes a constant pat. These mass displacements (both ocean and solid Eath) give ise to an additional indiect change in potential, the tidal defomation potential (thee ae also seconday indiect effects due to loading of the ocean on the solid Eath, which can be neglected in this discussion; see Chapte 3.06). The indiect effect is modeled as a faction of the diect effect (Lambeck, 988, p. 254), so that the pemanent pat of the tidal potential including the indiect effect is given by V c sþm ð; RÞ ¼ ð þ k 2 ÞVc sþm ð; RÞ ½4Š whee k 2 ¼ 0:29 is Love s numbe (an empiical numbe based on obsevation). This is also called the mean tide potential. The mean tidal potential is inheent in all ou teestial obsevations (the bounday values) and cannot be aveaged away; yet, the solutions to the BVP assume no extenal masses. Theefoe, in pinciple, the effect of the tide potential including its mean, o pemanent, pat should be emoved fom the obsevations pio to applying the BVP solutions. On the othe hand, the pemanent indiect effect is not that well modeled and aguably should not be emoved; afte all, it contibutes to the Eath s shape as it actually is in the mean. Thee types of tidal systems have been defined to distinguish between these coections. A mean quantity efes to the quantity with the mean tide potential etained (but timevaying pats emoved); a nontidal quantity implies that all tidal effects (time-vaying, pemanent, diect and indiect effects) have been emoved computationally; and the zeo-tide quantity excludes all timevaying pats and the pemanent diect effect, but it etains the indiect pemanent effect. If the geoid (an equipotential suface) is defined solely by its potential, W 0, then a change in the potential due to the tidal potential, V tide (time-vaying and constant pats, diect and indiect effects), implies that the W 0 -equipotential suface has been displaced. The geoid is now a diffeent suface with the same W 0. This displacement is equivalent to a change in geoid undulation, dn ¼ V tide =, with espect to some pedefined ellipsoid. The pemanent tidal effect (diect and indiect) on the geoid is given by N ðþ¼ 0:099ð þ k 2 Þ 3 cos 2 m ½42Š If N epesents the instantaneous geoid, then the geoid without any tidal effects, that is, the nontidal geoid, is given by N nt ¼ N N ½43Š The mean geoid is defined as the geoid with all but the mean tidal effects emoved: N ¼ N ðn N Þ ½44Š This is the geoid that could be diectly obseved, fo example, using satellite altimety aveaged ove time. The zeo-tide geoid etains the pemanent indiect effect, but no othe tidal effects: N z ¼ N N þ 0:099k 2 3 cos 2 m ½45Š The diffeence between the mean and zeo-tide geoids is, theefoe, the pemanent component of the diect tidal potential. We note that, in pinciple, each of the geoids defined above, has the same

31 Potential Theoy and Static Gavity Field of the Eath 4 potential value, W 0, in its own field. That is, with each coection, we define a new gavity field and the coesponding geoid undulation defines the equipotential suface in that field with potential value given by W 0. This is fundamentally diffeent than what happens in the case when the geoid is defined as a vetical datum with a specified datum oigin point. In this case one needs to conside also the vetical displacement of the datum point due to the tidal defomation of the Eath s suface. The potential of the datum then changes because of the diect tidal potential, the indiect effect due to mass changes, and the indiect effect due to the vetical displacement of the datum (fo additional details, see Jekeli (2000b)). Refeences Busa M, Radej K, Sima Z, Tue S, and Vatt V (997) Detemination of the geopotential scale facto fom TOPEX/Poseidon satellite altimety. 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