3.02 Potential Theory and Static Gravity Field of the Earth C. Jekeli, The Ohio State University, Columbus, OH, USA

ª 2007 Elsevier B.V. All rights reserved.

3.02.1 Introduction 12 3.02.1.1 Historical Notes 12 3.02.1.2 Coordinate Systems 14 3.02.1.3 Preliminary Definitions and Concepts 14 3.02.2 Newton’s Law of Gravitation 15 3.02.3 Boundary-Value Problems 18 3.02.3.1 Green’s Identities 18 3.02.3.2 Uniqueness Theorems 20 3.02.3.3 Solutions by Integral Equation 21 3.02.4 Solutions to the Spherical BVP 22 3.02.4.1 Spherical Harmonics and Green’s Functions 22 3.02.4.2 Inverse Stokes and Hotine Integrals 25 3.02.4.3 Vening-Meinesz Integral and Its Inverse 26 3.02.4.4 Concluding Remarks 27 3.02.5 Low-Degree Harmonics: Interpretation and Reference 28 3.02.5.1 Low-Degree Harmonics as Density Moments 28 3.02.5.2 Normal Ellipsoidal Field 30 3.02.6 Methods of Determination 32 3.02.6.1 Measurement Systems and Techniques 33 3.02.6.2 Models 37 3.02.7 The Geoid and Heights 38 References 41

Glossary geodetic reference system Normal ellipsoid with deflection of the vertical Angle between direction defined parameters adopted for general geodetic of gravity and direction of normal gravity. and gravimetric referencing. density moment Integral over the volume of a geoid Surface of constant gravity potential that body of the product of its density and integer closely approximates mean sea level. powers of Cartesian coordinates. geoid undulation Vertical distance between the disturbing potential The difference between geoid and the normal ellipsoid, positive if the geoid Earth’s gravity potential and the normal potential. is above the ellipsoid. eccentricity The ratio of the difference of squares geopotential number Difference between gravity of semimajor and semiminor axes to the square of potential on the geoid and gravity potential at a the semimajor axis of an ellipsoid. point. ellipsoid Surface formed by rotating an ellipse gravitation Attractive acceleration due to mass. about its minor axis. gravitational potential Potential due to gravita- equipotential surface Surface of constant tional acceleration. potential. gravity Vector sum of gravitation and centrifugal flattening The ratio of the difference between acceleration due to Earth’s rotation. semimajor and semiminor axes to the semimajor gravity anomaly The difference between Earth’s axis of an ellipsoid. gravity on the geoid and normal gravity on the

11 12 Potential Theory and Static Gravity Field of the Earth

ellipsoid, either as a difference in vectors or a dif- normal gravity Gravity associated with the normal ference in magnitudes. ellipsoid. gravity disturbance The difference between normal gravity potential Gravity potential asso- Earth’s gravity and normal gravity, either as a dif- ciated with the normal ellipsoid. ference in vectors or a difference in magnitudes. orthometric height Distance along the plumb line gravity potential Potential due to gravity from the geoid to a point. acceleration. potential Potential energy per unit mass due to the harmonic function Function that satisfies gravitational field; always positive and zero at Laplace’s field equation. infinity. linear eccentricity The distance from the center of sectorial harmonics Surface spherical harmonics an ellipsoid to either of its foci. that do not change in sign with respect to latitude. mean Earth ellipsoid Normal ellipsoid with para- Stokes coefficients Constants in a series expan- meters closest to actual corresponding parameters sion of the gravitational potential in terms of for the Earth. spherical harmonic functions. mean tide geoid Geoid with all time-varying tidal surface spherical harmonics Basis functions effects removed. defined on the unit sphere, comprising products of multipoles Stokes coefficients. normalized associated Legendre functions and Newtonian potential Harmonic function that sinusoids. approaches the potential of a point mass at infinity. tesseral harmonics Neither zonal nor sectorial non-tide geoid Mean tide geoid with all (direct harmonics. and indirect deformation) mean tide effects zero-tide geoid Mean tide geoid with just the removed. mean direct tidal effect removed (indirect effect due normal ellipsoid Earth-approximating reference to Earth’s permanent deformation is retained). ellipsoid that generates a gravity field in which it is a zonal harmonics Spherical harmonics that do not surface of constant normal gravity potential. depend on longitude.

3.02.1 Introduction (or rise, in the case of some gases), and the more material the greater the tendency to do so. It was Gravitational potential theory has its roots in the late enough of a self-evident explanation that it was not Renaissance period when the position of the Earth in yet to receive the scrutiny of the scientific method, the cosmos was established on modern scientific the beginnings of which, ironically, are credited to (observation-based) grounds. A study of Earth’s grav- Aristotle. Almost 2000 years later, Galileo Galilei itational field is a study of Earth’s mass, its influence (1564–1642) finally took up the challenge to under- on near objects, and lately its redistributing transport stand gravitation through observation and scientific in time. It is also fundamentally a geodetic study of investigation. His experimentally derived law of fall- Earth’s shape, described largely (70%) by the surface ing bodies corrected the Aristotelian view and of the oceans. This initial section provides a historical divorced the effect of gravitation from the mass of backdrop to potential theory and introduces some the falling object – all bodies fall with the same concepts in physical geodesy that set the stage for acceleration. This truly monumental contribution to later formulations. physics was, however, only a local explanation of how bodies behaved under gravitational influence. Johannes Kepler’s (1571–1630) observations of pla- 3.02.1.1 Historical Notes netary orbits pointed to other types of laws, Gravitation is a physical phenomenon so pervasive principally an inverse-square law according to and incidental that humankind generally has taken it which bodies are attracted by forces that vary with for granted with scarcely a second thought. The the inverse square of distance. The genius of Issac Greek philosopher Aristotle (384–322 BC) allowed Newton (1642–1727) brought it all together in his no more than to assert that gravitation is a natural Philosophiae Naturalis Principia Mathematica of 1687 property of material things that causes them to fall with a single and simple all-embracing law that in Potential Theory and Static Gravity Field of the Earth 13 one bold stroke explained the dynamics of the entire needed to be corrected for the irregular direction of universe (today there is more to understanding the gravitation. Today, the modern view of a height dynamics of the cosmos, but Newton’s law remark- reference is changing to that of a geometric, mathe- ably holds its own). The mass of a body was again an matical surface (an ellipsoid) and three-dimensional essential aspect, not as a self-attribute as Aristotle had coordinates (latitude, longitude, and height) of points implied, but as the source of attraction for other on the Earth’s surface are readily obtained geometri- bodies: each material body attracts every other mate- cally by ranging to the satellites of the Global rial body according to a very specific rule (Newton’s Positioning System (GPS). The requirements of law of gravitation; see Section 3.02.2). Newton gravitation for GPS orbit determination within an regretted that he could not explain exactly why Earth-centered coordinate system are now largely mass has this property (as one still yearns to know met with existing models. Improvements in gravita- today within the standard models of particle and tional models are motivated in geodesy primarily for quantum theories). Even Albert Einstein (1879– rapid determination of traditional heights with 1955) in developing his general theory of relativity respect to a level surface. These heights, for example, (i.e., the theory of gravitation) could only improve on the orthometric heights, in this sense then become Newton’s theory by incorporating and explaining derived attributes of points, rather than their cardinal action at a distance (gravitational force acts with the components. speed of light as a fundamental tenet of the theory). Navigation and guidance exemplify a further spe- What actually mediates the gravitational attraction cific niche where gravitation continues to find still intensely occupies modern physicists and important relevance. While GPS also dominates cosmologists. this field, the vehicles requiring completely autono- Gravitation since its early scientific formulation mous, self-contained systems must rely on inertial initially belonged to the domain of astronomers, at instruments (accelerometers and gyroscopes). These least as far as the observable universe was concerned. do not sense gravitation (see Section 3.02.6.1), yet Theory successfully predicted the observed perturba- gravitation contributes to the total definition of the tions of planetary orbits and even the location of vehicle trajectory, and thus the output of inertial previously unknown new planets (Neptune’s discov- navigation systems must be compensated for the ery in 1846 based on calculations motivated by effect of gravitation. By far the greatest emphasis in observed perturbations in Uranus’ orbit was a major gravitation, however, has shifted to the Earth triumph for Newton’s law). However, it was also dis- sciences, where detailed knowledge of the configura- covered that gravitational acceleration varies on tion of masses (the solid, liquid, and atmospheric Earth’s surface, with respect to altitude and latitude. components) and their transport and motion leads Newton’s law of gravitation again provided the back- to improved understanding of the Earth systems (cli- drop for the variations observed with pendulums. An mate, hydrologic cycle, tectonics) and their early achievement for his theory came when he suc- interactions with life. Oceanography, in particular, cessfully predicted the polar flattening in Earth’s shape also requires a detailed knowledge of a level surface on the basis of hydrostatic equilibrium, which was (the geoid) to model surface currents using satellite confirmed finally (after some controversy) with geo- altimetry. Clearly, there is an essential temporal detic measurements of long triangulated arcs in 1736– component in these studies, and, indeed, the tem- 37 by Pierre de Maupertuis and Alexis Clairaut. poral gravitational field holds center stage in many Gravitation thus also played a dominant role in geo- new investigations. Moreover, Earth’s dynamic beha- desy, the science of determining the size and shape of vior influences point coordinates and Earth-fixed the Earth, promulgated in large part by the father of coordinate frames, and we come back to fundamental modern geodesy, Friedrich R. Helmert (1843–1917). geodetic concerns in which the gravitational field Terrestrial gravitation through the twentieth cen- plays an essential role! tury was considered a geodetic area of research, This section deals with the static gravitational although, of course, its geophysical exploits should field. The theory of the potential from the classical not be overlooked. But the advancement in modeling Newtonian standpoint provides the foundation for accuracy and global application was promoted modeling the field and thus deserves the focus of mainly by geodesists who needed a well-defined the exposition. The temporal part is a natural exten- reference for heights (a level surface) and whose sion that is readily achieved with the addition of the astronomic observations of latitude and longitude time variable (no new laws are needed, if we neglect 14 Potential Theory and Static Gravity Field of the Earth general relativistic effects) and will not be expounded z Confocal ellipsoid here. We are primarily concerned with gravitation on Sphere and external to the solid and liquid Earth since this is u δ the domain of most applications. The internal field can also be modeled for specialized purposes (such as submarine navigation), but internal geophysical b modeling, for example, is done usually in terms of the sources (mass density distribution), rather than the resulting field. E x

3.02.1.2 Coordinate Systems Figure 2 Ellipsoidal coordinates. Modeling the Earth’s gravitational field depends on the choice of coordinate system. Customarily, owing to the Earth’s general shape, a spherical polar coor- generated by rotating an ellipse about its minor axis dinate system serves for most applications, and (polar axis). The two focal points of the best-fitting, virtually all global models use these coordinates. Earth-centered ellipsoid (ellipse) are located in the However, the Earth is slightly flattened at the poles, equator about E ¼ 522 km from the center of the and an ellipsoidal coordinate system has also been Earth. A given ellipsoid, with specified semiminor advocated for some near-Earth applications. We note axis, b, and linear eccentricity, E, defines the set of that the geodetic coordinates associated with a geo- ellipsoidal coordinates, as described in Figure 2. The detic datum (based on an ellipsoid) are never used in longitude is the same as in the spherical case. The a foundational sense to model the field since they do colatitude, , is the complement of the so-called not admit to a separation-of-variables solution of reduced latitude; and the distance coordinate, u,is Laplace’ differential equation (Section 3.02.4.1). the semiminor axis of the confocal ellipsoid through Spherical polar coordinates, described with the the point in question. We call ðÞ; ; u ellipsoidal aid of Figure 1, comprise the spherical colatitude, coordinates; they are also known as spheroidal coor- , the longitude, , and the radial distance, r. Their dinates, or Jacobi ellipsoidal coordinates. Their relation to Cartesian coordinates is relation to Cartesian coordinates is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ u2 þ E2 sin cos x ¼ r sin cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ u2 þ E2 sin sin ½2 y ¼ r sin sin ½1 z ¼ u cos z ¼ r cos Points on the given ellipsoid all have u ¼ b; and, all Considering Earth’s polar flattening, a better surfaces, u ¼ constant, are confocal ellipsoids (the approximation, than a sphere, of its (ocean) surface analogy to the spherical case, when E ¼ 0, should is an ellipsoid of revolution. Such a surface is be evident).

3.02.1.3 Preliminary Definitions and z Concepts The gravitational potential, V, of the Earth is gener- θ ated by its total mass density distribution. For r applications on the Earth’s surface it is useful to include the potential, , associated with the centrifu- gal acceleration due to Earth’s rotation. The sum, W ¼ V þ , is then known, in geodetic terminology, y λ as the gravity potential, distinct from gravitational potential. It is further advantageous to define a rela- x tively simple reference potential, or normal potential, Figure 1 Spherical polar coordinates. that accounts for the bulk of the gravity potential Potential Theory and Static Gravity Field of the Earth 15

(Section 3.02.5.2). The normal gravity potential, U,is T N ¼ ½3 defined as a gravity potential associated with a best- fitting ellipsoid, the normal ellipsoid, which rotates This is Bruns’ equation, which is accurate to a with the Earth and is also a surface of constant few millimeters in N, and which can be extended to potential in this field. The difference between the N ¼ T= – ðÞW0 – U0 = for the general case, actual and the normal gravity potentials is known as W0 6¼ U0. The gravity anomaly (on the geoid) is the the disturbing potential: T ¼ W – U; it thus excludes gravity disturbance corrected for the evaluation of the centrifugal potential. The normal gravity poten- normal gravity on the ellipsoid instead of the geoid. tial accounts for approximately 99.999 5% of the total This correction is N q=qh ¼ ðÞq=qh ðÞT= , potential. where h is height along the ellipsoid perpendicular. The gradient of the potential is an acceleration, We have g ¼ – qT=qh, and hence gravity or gravitational acceleration, depending on whether or not it includes the centrifugal accelera- qT 1 q tion. Normal gravity, , comprises 99.995% of the g ¼ – þ T ½4 total gravity, g, although the difference in magni- qh qh tudes, the gravity disturbance, dg, can be as large as The slope of the geoid with respect to the ellip- several parts in 104. A special kind of difference, soid is also the angle between the corresponding called the gravity anomaly, g, is defined as the perpendiculars to these surfaces. This angle is difference between gravity at a point, P, and normal known as the deflection of the vertical, that is, the gravity at a corresponding point, Q, where W ¼ U , P Q deflection of the plumb line (perpendicular to the and P and Q are on the same perpendicular to the geoid) relative to the perpendicular to the normal normal ellipsoid. ellipsoid. The deflection angle has components, , , The surface of constant gravity potential, W ,that 0 respectively, in the north and east directions. The closely approximates mean sea level is known as the spherical approximations to the gravity disturbance, geoid. If the constant normal gravity potential, U ,on 0 anomaly, and deflection of the vertical are given by the normal ellipsoid is equal to the constant gravity potential of the geoid, then the gravity anomaly on the qT qT 2 geoid is the difference between gravity on the geoid g ¼ – ; g ¼ – – T qr qr r ½5 and normal gravity on the ellipsoid at respective points, 1 qT 1 qT ¼ ;¼ – P0, Q0, sharing the same perpendicular to the ellipsoid. rq r sin q The separation between the geoid and the ellipsoid is known as the geoid undulation, N,oralsothegeoid where the signs on the derivatives are a matter of height (Figure 3). A simple Taylor expansion of the convention. normal gravity potential along the ellipsoid perpendi- cular yields the following important formula:

3.02.2 Newton’s Law of Gravitation

In its original form, Newton’s law of gravitation P applies only to idealized point masses. It describes Topographic the force of attraction, F, experienced by two such H, orthometric surface solitary masses as being proportional to the product height of the masses, m1 and m2; inversely proportional to ellipsoidalEllipsoidal Deflection of the distance, ,, between them; and directed along the height,height, h h the vertical W = W0 line joining them:

P0 P0 m m Geoid F ¼ G 1 2 n ½6 N, geoid undulation ,2 g U = U0 Q Q G is a constant, known as Newton’s gravitational 0 0 Ellipsoid constant, that takes care of the units between the Figure 3 Relative geometry of geoid and ellipsoid. left- and right-hand sides of the equation; it can be 16 Potential Theory and Static Gravity Field of the Earth determined by experiment and the current best value force (i.e., it is not an applied force, like friction or is (Groten, 2004): propulsion) – rather, it is known as a kinematic force,

– 11 3 – 1 – 2 that is, one whose action is proportional to the mass G ¼ ðÞ6:67259 0:00030 10 m kg s ½7 on which it acts (like the centrifugal force; see The unit vector n in eqn [6] is directed from either Martin, 1988). Under this precept, the geometry of point mass to the other, and thus the gravitational force space is defined intrinsically by the gravitational is attractive and applies equally to one mass as the fields contained therein. other. Newton’s law of gravitation is universal as far as We continue with the classical Newtonian poten- we know, requiring reformulation only in Einstein’s tial, but interpret gravitation as an acceleration more comprehensive theory of general relativity different from the acceleration induced by real, which describes gravitation as a characteristic curva- applied forces. This becomes especially important ture of the space–time continuum (Newton’s when considering the measurement of gravitation formulation assumes instantaneous action and differs (Section 3.02.6.1). The gravitational potential, V,isa significantly from the general relativistic concept only ‘scalar’ function, and, as defined here, V is derived when very large velocities or masses are involved). directly on the basis of Newton’s law of gravitation. We can ascribe a gravitational acceleration to the To make it completely consistent with this law and gravitational force, which represents the acceleration thus declare it a Newtonian potential, we must that one mass undergoes due to the gravitational impose the following conditions: attraction of the other. Specifically, from the law of lim ,V ¼ Gm and lim V ¼ 0 ½10 gravitation, we have (for point masses) the gravita- ,!1 ,!1 tional acceleration of m1 due to the gravitational Here, m is the attracting mass, and we say that the attraction of m2: potential is regular at infinity. It is easy to show that m the gravitational potential at any point in space g ¼ G 2 n ½8 ,2 due to a point mass, in order to satisfy eqns [8]–[10], must be The vector g is independent of the mass, m1, of the body being accelerated (which Galileo found by Gm V ¼ ½11 experiment). , By the law of superposition, the gravitational where, again, , is the distance between the mass and force, or the gravitational acceleration, due to many the point at which the potential is expressed. Note point masses is the vector sum of the forces or accel- that V for , ¼ 0 does not exist in this case; that is, the erations generated by the individual point masses. field of a point mass has a singularity. We use here Manipulating vectors in this way is certainly feasible, the convention that the potential is always but fortunately a more appropriate concept of grav- positive (in contrast to physics, where it is usually itation as a scalar field simplifies the treatment of defined to be negative, conceptually closer to poten- arbitrary mass distributions. tial energy). This more modern view of gravitation (already Applying the law of superposition, the gravita- adopted by Gauss (1777–1855) and Green tional potential of many point masses is the sum of (1793–1841)) holds that it is a field having a gravita- the potentials of the individual points (see Figure 4): tional potential. Lagrange (1736–1813) fully X developed the concept of a field, and the potential, m V ¼ G j ½12 P , V, of the gravitational field is defined in terms of the j j gravitational acceleration, g, that a test particle would undergo in the field according to the equation And, for infinitely many points in a closed, bounded region with infinitesimally small masses, dm, the V ¼ g ½9 summation in eqn [12] changes to an integration, where is the gradient operator (a vector). Further Z dm elucidation of gravitation as a field grew from V ¼ G ½13 P , Einstein’s attempt to incorporate gravitation into his mass special theory of relativity where no reference frame or, changing variables (i.e., units), dm ¼ dv, where has special significance above all others. It was neces- represents density (mass per volume) and dv is a sary to consider that gravitational force is not a real volume element, we have (Figure 5) Potential Theory and Static Gravity Field of the Earth 17

z spherically symmetric density we may choose the P integration coordinate system so that the polar axis l1 passes through P. Then l j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 2 2 1 l2 , ¼ r9 þ r – 2r9r cos 9; d, ¼ r9r sin 9d9 , mj l3 m2 ½15 y It is easy to show that with this change of variables (from to ,) the integral [14] becomes simply m x 3 GM V ðÞ¼; ; r ; r R ½16 Figure 4 Discrete set of mass points (superposition r principle). where M is the total mass bounded by the sphere. This shows that to a very good approximation the external gravitational potential of a planet such P n as the Earth (with concentrically layered density) is l the same as that of a point mass. Besides volumetric mass (density) distributions, it is of interest to consider surface distributions. Imagine an infinitesimally thin layer of mass on a dv surface, s, where the units of density in this case are those of mass per area. Then, analogous to eqn [14], the potential is ZZ s ρ VP ¼ G ds ½17 s , Figure 5 Continuous density distribution. In this case, V is a continuous function everywhere, but its first derivatives are discontinuous at the sur- ZZ Z face. Or, one can imagine two infinitesimally close V ¼ G dv ½14 P , density layers (double layer, or layer of mass dipoles), volume where the units of density are now those of mass per area times length. It turns out that the potential In eqn [14], , is the distance between the evalua- in this case is given by (see Heiskanen and Moritz, tion point, P, and the point of integration. In spherical 1967, p. 8) polar coordinates (Section 3.02.1.2), these points ZZ  q 1 are ðÞ; ; r and ðÞ9; 9; r9 , respectively. The V ¼ G ds ½18 P qn , volume element in this case is given by s dv ¼ r92 sin 9 d9 d9 dr9. V and its first derivatives where q=qn is the directional derivative along the are continuous everywhere – even in the case that perpendicular to the surface (Figure 5). Now, V itself P is on the bounding surface or inside the mass is discontinuous at the surface, as are all its derivatives. distribution, where there is the apparent singularity In all cases, V is a Newtonian potential, being derived at , ¼ 0. In this case, by changing to a coordinate from the basic formula [11] for a point mass that system whose origin is at P, the volume element follows from Newton’s law of gravitation (eqn [6]). becomes dv ¼ ,2 sin c d dc d, (for some different The following properties of the gravitational colatitude and longitude c and ); and, clearly, the potential are useful for subsequent expositions. singularity disappears – the integral is said to be First, consider Stokes’s theorem, for a vector func- weakly singular. tion, f, defined on a surface, s: ZZ I Suppose the density distribution over the volume depends only on radial distance (from the center of ðÞrf ? n ds ¼ f ? dr ½19 s p mass): ¼ ðÞr9 , and that P is an exterior evaluation point. The surface bounding the masses necessarily is where p is any closed path in the surface, n is the unit a sphere (say, of radius, R) and because of the vector perpendicular to the surface, and dr is a 18 Potential Theory and Static Gravity Field of the Earth differential displacement along the path. From eqn harmonic function in free space. The converse is [9], we find also true: every harmonic function can be repre- sented as a Newtonian potential of a mass rg ¼ 0 ½20 distribution (Section 3.02.3.1). since r r ¼ 0; hence, applying Stokes’s theo- Whether as a volume or a layer density distribu- rem, we find with F ¼ mg that tion, the corresponding potential is the sum or I integral of the source value multiplied by the inverse w ¼ F ? ds ¼ 0 ½21 distance function (or its normal derivative for the dipole layer). This function depends on both the That is, the gravitational field is conservative: the source points and the computation point and is work, w, expended in moving a mass around a closed known as a Green’s function. It is also known as the path in this field vanishes. In contrast, dissipating ‘kernel’ function of the integral representation of the forces (real forces!), like friction, expend work or potential. Functions of this type also play a dominant energy, which shows again the special nature of the role in representing the potential as solutions to gravitational force. boundary-value problems (BVPs), as shown in sub- It can be shown (Kellogg, 1953, p. 156) that the sequent sections. second partial derivatives of a Newtonian potential, V, satisfy the following differential equation, known as Poisson’s equation: 3.02.3 Boundary-Value Problems r2V ¼ – 4G ½22 If the density distribution of the Earth’s interior and where r2 ¼ ? formally is the scalar product of the boundary of the volume were known, then the two gradient operators and is called the Laplacian problem of determining the Earth’s gravitational operator. In Cartesian coordinates, it is given by potential field is solved by the volume integral of eqn [14]. In reality, of course, we do not have access q2 q2 q2 r2 ¼ þ þ ½23 to this information, at least not the density, with qx2 qy2 qz2 sufficient detail. (The Preliminary Reference Earth Note that the Laplacian is a scalar operator. Eqn [22] is Model (PREM), of Dziewonsky and Anderson a local characterization of the potential field, as (1981), still in use today by geophysicists, represents opposed to the global characterization given by eqn a good profile of Earth’s radial density, but does not [14]. Poisson’s equation holds wherever the mass den- attempt to model in detail the lateral density hetero- sity, , satisfies certain conditions similar to continuity geneities.) In this section, we see how the problem of (Ho¨lder conditions; see Kellogg, 1953, pp. 152–153). A determining the exterior gravitational potential can special case of eqn [22] applies for those points where be solved in terms of surface integrals, thus making the density vanishes (i.e., in free space); then Poisson’s exclusive use of accessible measurements on the equation turns into Laplace’ equation, surface.

r2V ¼ 0 ½24 3.02.3.1 Green’s Identities It is easily verified that the point mass potential satisfies eqn [24], that is, Formally, eqn [24] represents a partial differential  equation for V. Solving this equation is the essence 1 r2 ¼ 0 ½25 of the determination of the Earth’s external gravita- , tional potential through potential theory. Like any qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi differential equation, a complete solution is obtained where , ¼ ðÞx – x 0 2þðÞy – y 0 2þðÞz – z 0 2 and the only with the application of boundary conditions, mass point is at ðÞx9; y9; z9 . that is, imposing values on the solution that it must The solutions to Laplace’ equation [24] (that is, assume at a boundary of the region in which it is functions that satisfy Laplace’ equations) are known valid. In our case, the boundary is the Earth’s surface as harmonic functions (here, we also impose the and the exterior space is where eqn [24] holds (the conditions [10] on the solution, if it is a Newtonian atmosphere and other celestial bodies are neglected potential and if the mass-free region includes infi- for the moment). In order to study the solutions to nity). Hence, every Newtonian potential is a these BVPs (to show that solutions exist and are Potential Theory and Static Gravity Field of the Earth 19

unique), we take advantage of some very important n r P theorems and identities. It is noted that only a rather l elementary introduction to BVPs is offered here with ds σ no attempt to address the much larger field of solu- l tions to partial differential equations. n The first, seminal result is Gauss’ divergence the- dv orem (analogous to Stokes’ theorem, eqn [19]), ZZ Z ZZ

r ? f dv ¼ fn ds ½26 Earth v s s where f is an arbitrary (differentiable) vector func- tion and fn ¼ n ? f is the component of f along the outward unit normal vector, n (see Figure 5). The Figure 6 Geometry for special case of Green’s third surface, s, encloses the volume, v. Equation [26] says identity. that the sum of how much f changes throughout the volume, that is, the net effect, ultimately, is equiva- space) that is bounded by the surface, s; P is thus lent to the sum of its values projected orthogonally outside the Earth’s surface. Let V be a solution to eqn with respect to the surface. Conceptually, a volume [24], that is, it is the gravitational potential of the integral thus can be replaced by a surface integral, Earth. From the volume, v, exclude the volume which is important since the gravitational potential is bounded by a small sphere, , centered at P. This due to a volume density distribution that we do not sphere becomes part of the surface that bounds the know, but we do have access to gravitational quan- volume, v. Then, since U, by our definition, is a point tities on a surface by way of measurements. mass potential, r2U ¼ 0 everywhere in v (which Equation [26] applies to general vector functions excludes the interior of the small sphere around P); that have continuous first derivatives. In particular, and, the second identity [29] gives let U and V be two scalar functions having continuous ZZ  second derivatives, and consider the vector function 1 qV q 1 – V ds f ¼ UV. Then, since n ? ¼ q=qn, and , q q , ZZs n n  2 1 qV q 1 r?ðÞUrV ¼rU ? rV þ Ur V ½27 þ – V d ¼ 0 ½30 , qn qn , we can apply Gauss’ divergence theorem to get Green’s first identity, The unit vector, n, represents the perpendicular ZZ Z ZZ ÀÁ pointing away from v. On the small sphere, n is 2 qV rU ? rV þ Ur V dv ¼ U ds ½28 opposite in direction to , ¼ r, and the second inte- v s qn gral becomes Interchanging the roles of U and V in eqn [28], one ZZ  ZZ 1 qV q 1 1 qV obtains a similar formula, which, when subtracted – þ V d ¼ – r 2 d from eqn [28], yields Green’s second identity, r qr qr r ZZ r qr ZZ Z ZZ  – V d ÀÁqV qU U r2V – V r2U dv ¼ U – V ds ZZ qn qn qV v s ¼ – r d – 4V ½29 qr ½31 This is valid for any U and V with continuous deri- vatives up to second order. where d ¼ r 2d, is the solid angle, 4, and V is Now let U ¼ 1=,, where , is the usual distance an average value of V on . Now, in the limit as the between an integration point and an evaluation point. radius of the small sphere shrinks to zero, the right- And, suppose that the volume, v, is the space exterior hand side of eqn [31] approaches 0 – 4VP. Hence, to the Earth (i.e., Gauss’ divergence theorem applies eqn [30] becomes (Kellogg, 1953, p. 219) to any volume, not just volumes containing a mass ZZ  1 1 qV q 1 distribution). With reference to Figure 6, consider V ¼ – V ds ½32 P , q q , the evaluation point, P, to be inside the volume (free 4 s n n 20 Potential Theory and Static Gravity Field of the Earth with n pointing down (away from the space outside uniqueness theorem. That is, we wish to know if a the Earth). This is a special case of Green’s third certain set of boundary values will yield just one identity. A change in sign of the right-hand side potential in space. Before considering such theorems, transforms n to a normal unit vector pointing into v, we classify the BVPs that are typically encountered away from the masses, which conforms more to an when determining the exterior potential from mea- Earth-centered coordinate system. surements on a boundary. In all cases, it is an exterior The right-hand side of eqn [32] is the sum of BVP; that is, the gravitational potential, V, is harmo- single- and double-layer potentials and thus shows nic (r2V ¼ 0) in the space exterior to a closed that every harmonic function (i.e., a function that surface that contains all the masses. The exterior satisfies Laplace’ equation) can be written as a space thus contains infinity. Interior BVPs can be Newtonian potential. Equation [32] is also a solution constructed, as well, but are not applicable to our to a BVP; in this case, the boundary values are inde- objectives. pendent values of V and of its normal derivative, both on s (Cauchy problem). Below and in Section 3.02.4.1, • Dirichlet problem (or, BVP of the first kind ). Solve for we encounter another BVP in which the potential V in the exterior space, given its values every- and its normal derivative are given in a specified where on the boundary. linear combination on s. Using a similar procedure • Neumann problem (or, BVP of the second kind ). Solve and with some extra care, it can be shown (see also for V in the exterior space, given values of its Courant and Hilbert, 1962, vol. II, p. 256 (footnote)) normal derivative everywhere on the boundary. that if P is on the surface, then • Robin problem (mixed BVP, or BVP of the third kind ). ZZ  Solve for V in the exterior space, given a linear 1 1 qV q 1 combination of it and its normal derivative on the VP ¼ – V ds ½33 2 s , qn qn , boundary. where n points into the masses. Comparing this to Using Green’s identities, we prove the following eqn [32], we see that V is discontinuous as one theorems for these exterior problems; similar results approaches the surface; this is due to the double- hold for the interior problems. layer part (see eqn [18]). Theorem 1.IfV is harmonic (hence continuously Equation [32] demonstrates that a solution to a differentiable) in a closed region, v, and if V vanishes particular BVP exists. Specifically, we are able to everywhere on the boundary, s, then V also vanishes measure the potential (up to a constant) and its deri- everywhere in the region, v. vatives (the gravitational acceleration) on the surface Proof. Since V ¼ 0ons, Green’s first identity (eqn and thus have a formula to compute the potential [28]) with U ¼ V gives anywhere in exterior space, provided we also know ZZ Z ZZ the surface, s. Other BVPs also have solutions under qV ðÞrV 2 dv ¼ V ds ¼ 0 ½34 appropriate conditions; a discussion of existence the- s qn orems is beyond the present scope and may be found in (Kellogg, 1953). Equation [33] has deep geodetic The integral on the left side is therefore always significance. One objective of geodesy is to determine zero, and the integrand is always non-negative. the shape of the Earth’s surface. If we have measure- Hence, V ¼ 0 everywhere in v, which implies ments of gravitational quantities on the Earth’s that V ¼ constant in v. Since V is continuous in surface, then conceptually we are able to determine v and V ¼ 0ons, that constant must be zero; and so its shape from eqn [33], where it would be the only V ¼ 0inv. unknown quantity. This is the basis behind the work This theorem solves the Dirichlet problem for the of Molodensky et al. (1962), to which we return trivial case of zero boundary values and it enables the briefly at the end of this section. following uniqueness theorem for the general Dirichlet problem. Theorem 2 (Stokes’ theorem). If V is harmonic (hence continuously differentiable) in a closed region, v, 3.02.3.2 Uniqueness Theorems then V is uniquely determined in v by its values on Often the existence of a solution is proved simply by the boundary, s. finding one (as illustrated above). Whether such as Proof. Suppose the determination is not unique: solution is the only one depends on a corresponding that is, suppose there are V1 and V2, both harmonic Potential Theory and Static Gravity Field of the Earth 21

in v and both having the same boundary values on s. in v; and V2 ¼ V1 on s. By the continuity of V1 and V2, Then the function V ¼ V2 – V1 is harmonic in v with the constant must be zero, and the uniqueness is all boundary values equal to zero. Hence, by proved. Theorem 1, V2 – V1 ¼ 0 identically in v,orV2 ¼ V1 The solution to the Robin problem is unique only everywhere, which implies that any determination is in certain cases. The most famous problem in physi- unique based on the boundary values. cal geodesy is the determination of the disturbing Theorem 3.IfV is harmonic (hence continuously potential, T, from gravity anomalies, g,onthe differentiable) in the exterior region, v, with closed geoid (Section 3.02.1.3). Suppose T is harmonic out- boundary, s, then V is uniquely determined by the side the geoid; the second of eqns [5] provides an values of its normal derivative on s. approximate form of boundary condition, showing Proof. We begin with Green’s first identity, eqn that this is a type of Robin problem. We find that [28], as in the proof of Theorem 1 to show that if the ¼ – 2=r, and, recalling that when v is the exterior normal derivative vanishes everywhere on s, then V is space the unit vector n points inward toward the a constant in v. Now, suppose there are two harmonic masses, that is, q=qn ¼ – q=qr,weget ¼ 1. Thus, functions in v: V1 and V2, with the same normal the condition in Theorem 4 on = is not fulfilled derivative values on s. Then the normal derivative and the uniqueness is not guaranteed. In fact, we will values of their difference are zero; and, by the above see that the solution obtained for the spherical demonstration, V ¼ V2 – V1 ¼ constant in v. Since V boundary is arbitrary with respect to the coordinate is a Newtonian potential in the exterior space, that origin (Section 3.02.4.1). constant is zero, since by eqn [10], lim,!1 V ¼ 0. Thus, V ¼ V , and the boundary values determine 2 1 3.02.3.3 Solutions by Integral Equation the potential uniquely. This is a uniqueness theorem for the exterior Green’s identities show how a solution to Laplace’s Neumann BVP. Solutions to the interior problem equation can be transformed from a volume integral, are unique only up to an arbitrary constant. that is, an integral of source points, to a surface Theorem 4. Suppose V is harmonic (hence continu- integral of BVPs, as demonstrated by eqn [32]. The ously differentiable) in the closed region, v, with uniqueness theorems for the BVPs suggest that the boundary, s; and, suppose the boundary values are potential due to a volume density distribution can given by also be represented as due to a generalized density layer on the bounding surface, as long as the result is q V g ¼ V js þ ½35 harmonic in exterior space, satisfies the boundary q s n conditions, and is regular at infinity like a Then V is uniquely determined by these values if Newtonian potential. Molodensky et al. (1962) sup- = > 0. posed that the disturbing potential is expressible as Proof. Suppose there are two harmonic functions, ZZ V1 and V2, with the same boundary values, g,ons. T ¼ ds ½39 s , Then V ¼ V2 – V1 is harmonic with boundary values  where is a surface density to be solved using the q q V2 V1 boundary condition. With the spherical approxima- ðÞjV2 – V1 s þ – ¼ 0 ½36 qn qn s tion for the gravity anomaly, eqn [5], one arrives at

With U ¼ V ¼ V2 – V1, Green’s first identity, eqn the following integral equation [28], gives ZZ  q 1 2 ZZZ ZZ 2 cos – þ ds ¼ g ½40 2 – qr , r, ðÞrðÞV – V dv ¼ ðÞV – V ðÞV – V ds s 2 1 2 1 2 1 v s The first term accounts for the discontinuity at the ½37 surface of the derivative of the potential of a density Then layer, where is the deflection angle between the ZZ Z ZZ normal to the surface and the direction of the (radial) ðÞrðÞV – V 2 dv þ ðÞV – V 2 dS ¼ 0 ½38 derivative (Heiskanen and Moritz, 1967, p. 6; Gu¨nter, 2 1 2 1 s 1967, p. 69). This Fredholm integral equation of the Since = > 0, eqn [38] implies that rðÞ¼V2 – V1 0 second kind can be simplified with further approx- in v; and V2 – V1 ¼ 0ons. Hence V2 – V1 ¼ constant imations, and, a solution for the density, , ultimately 22 Potential Theory and Static Gravity Field of the Earth leads to the solution for the disturbing potential approximation based on an ellipsoid of revolution is (Moritz, 1980). briefly examined in Section 3.02.5.2 for the normal Other forms of the initial representation have also potential. In spherical polar coordinates, ðÞ; ; r ,the been investigated, where Green’s functions other Laplacian operator is given by Hobson (1965, p. 9) than 1=, lead to simplifications of the integral equa-   1 q q 1 1 q q tion (e.g., Petrovskaya, 1979). Nevertheless, most 2 2 r ¼ 2 r þ 2 sin practical solutions rely on approximations, such as r qr qr r sin q q 1 q2 the spherical approximation for the boundary condi- þ ½41 2 2 2 tion, and even the formulated solutions are not r sin q strictly guaranteed to converge to the true solutions A solution to r2V ¼ 0 in the space outside a sphere (Moritz, 1980). Further treatments of the BVP in a of radius, R, with center at the coordinate origin can geodetic/mathematical setting may be found in the be found by the method of separation of variables, volume edited by Sanso` and Rummel (1997). whereby one postulates the form of the solution, V,as In the next section, we consider solutions for T as surface integrals of boundary values with appropriate V ðÞ¼; ; r f ðÞ gðÞ hrðÞ ½42 Green’s functions. In other words, the boundary values (whether of the first, second, or third kind) Substituting this and the Laplacian above into eqn may be thought of as sources, and the consequent [24], the multivariate partial differential equation potential is again the sum (integral) of the product of separates into three univariate ordinary differential a boundary value and an appropriate Green’s func- equations (Hobson, 1965, p. 9; Morse and Feshbach, tion (i.e., a function that depends on both the source 1953, p. 1264). Their solutions are well-known func- point and the computation point in some form of tions, for example, inverse distance in accordance with Newtonian 1 V ðÞ¼; ; r P ðcos Þ sin m ½43a potential theory). Such solutions are readily obtained nm r nþ1 if the boundary is a sphere. or

1 V ðÞ¼; ; r P ðcos Þcos m ½43b 3.02.4 Solutions to the Spherical BVP nm r nþ1

This section develops two types of solutions to stan- where PnmðÞt is the associated Legendre function of dard BVPs when the boundary is a sphere: the the first kind and n, m are integers such that spherical harmonic series and an integral with a 0 m n, n 0. Other solutions are also possible Green’s function. All three types of problems are (e.g., gðÞ¼ ea ðÞaPR and hðrÞ¼r n), but only solved, but emphasis is put on the third BVP since eqns [43] are consistent with the problem at hand: gravity anomalies are the most prevalent boundary to find a real-valued Newtonian potential for the values (on land, at least). In addition, it is shown how exterior space of the Earth (regular at infinity and the Green’s function integrals can be inverted to 2-periodic in longitude). The general solution is a obtain, for example, gravity anomalies from values linear combination of solutions of the forms given by of the potential, now considered as boundary values. eqns [43] for all possible integers, n and m, and can be Not all possible inverse relationships are given, but it written compactly as should be clear at the end that, in principle, virtually  X1 Xn nþ1 any gravitational quantity can be obtained in space R V ðÞ¼; ; r vnmYnmðÞ; ½44 from any other quantity on the spherical boundary. n¼0 m¼ – n r

where the Ynm are surface spherical harmonic func- 3.02.4.1 Spherical Harmonics and Green’s tions defined as Functions ( For simple boundaries, Laplace’s equation [24] is rela- cos m; m 0 YnmðÞ¼; PnmjjðÞcos ½45 tively easy to solve provided there is an appropriate sinjjm ; m < 0 coordinate system. For the Earth, the solutions com- monlyrelyonapproximatingtheboundarybyasphere. and Pnm is a normalization of Pnm so that the ortho- This case is described in detail and a more accurate gonality of the spherical harmonics is simply Potential Theory and Static Gravity Field of the Earth 23

ZZ 1 longitudinal sectors; and tesseral (the zeros of Ynm tes- YnmðÞ; Yn9m9ðÞ; d 4 ( sellate the sphere) (Figure 7). 1; n ¼ n9 and m ¼ m9 While the spherical harmonic series has its ¼ ½46 0; n 6¼ n9 or m 6¼ m9 advantages in global representations and spectral inter- pretations of the field, a Green’s function representation and where ¼ fgðÞj; 0 ; 0 2 provides a more local characterization of the field. represents the unit sphere, with d ¼ sin d d. Changing a boundary value anywhere on the globe For a complete mathematical treatment of spheri- changes all coefficients, vnm, according to eqn [47], cal harmonics, one may refer to Mu¨ller (1966). The which poses both a numerical challenge in applications, bounding spherical radius, R, is introduced so that as well as in keeping a standard model up to date. all the constant coefficients, vnm, also known as However, since the Green’s function essentially depends Stokes constants, have identical units of measure. on the inverse distance (or higher power thereof), a Applying the orthogonality to the general solution remote change in boundary value generally does not [44], these coefficients can be determined if the appreciably affect the local determination of the field. function, V, is known on the bounding sphere When the boundary is a sphere, the solutions to (boundary condition): the BVPs using a Green’s function are easily derived ZZ 1 from the spherical harmonic series representation. vnm ¼ V ðÞ; ; R YnmðÞ; d ½47 4 Moreover, it is possible to derive additional integral relationships (with appropriate Green’s functions) Equation [44] is known as a spherical harmonic among all the derivatives of the potential. To forma- expansion of V and with eqn [47] it represents a lize and simultaneously simplify these derivations, solution to the Dirichlet BVP if the boundary is a consider harmonic functions, f and h, where h sphere. The solution thus exists and is unique in the depends only on and r, and function g, defined on sense that these boundary values generate no other the sphere of radius, R. Thus let potential. We will, however, find another equivalent  X1 Xn nþ1 form of the solution. R f ðÞ¼; ; r fnmYnmðÞ½; 48 In a more formal mathematical setting, the solution n¼0 m¼ – n r [46] is an infinite linear combination of orthogonal basis  X1 nþ1 functions (eigenfunctions) and the coefficients, v ,are R nm hðÞ¼; r ðÞ2n þ 1 hnPnðÞ½cos 49 the corresponding eigenvalues. One may also interpret n¼0 r the set of coefficients as the spectrum (Legendre spec- X1 Xn trum) of the potential on the sphere of radius, R gðÞ; ; R ¼ g nmYnmðÞ; ½50 (analogous to the Fourier spectrum of a function on n¼0 m¼ – n pffiffiffiffiffiffiffiffiffiffiffiffiffi the plane or line). The integers, n, m, correspond to where PnðÞ¼cos Pn0ðÞcos = 2n þ 1 is the nth wave numbers, and are called degree (n) and order (m), degree Legendre polynomial. Constants fnm and gnm respectively. The spherical harmonics are further clas- are the respective harmonic coefficients of f and g sified as zonal (m ¼ 0), meaning that the zeros of Yn0 when these functions are restricted to the sphere of divide the sphere into latitudinal zones; sectorial radius, R. Then, using the decomposition formula for (m ¼ n), where the zeros of Ynn divide the sphere into Legendre polynomials,

θ θ λ θ λ P80 (cos ) P88 (cos ) cos 8 P87 (cos ) cos 7

Figure 7 Examples of zonal, sectorial, and tesseral harmonics on the sphere. 24 Potential Theory and Static Gravity Field of the Earth

Xn 1 corresponding Green’s function, also known as PnðÞ¼cos c YnmðÞ; YnmðÞ½9;9 51 2n þ 1 m¼ – n Poisson’s kernel. For convenience, one separates Earth’s gravita- where tional potential into a reference potential (Section cos c ¼ cos cos 9 þ sin sin 9cosðÞ½ – 9 52 3.02.1.3) and the disturbing potential, T. The disturb- ing potential is harmonic in free space and satisfies it is easy to prove the following theorem. the Poisson integral if the boundary is a sphere. In Theorem (convolution theorem in spectral analysis on the deference to physical geodesy where relationships sphere). ZZ between the disturbing potential and its derivatives 1 are routinely applied, the following derivations are f ðÞ¼; ; r gðÞ9;9; R hðÞc; r d 4 ½53 developed in terms of T, but hold equally for any if and only if fnm ¼ g nmhn exterior Newtonian potential. Let  Here, and in the following, d ¼ sin 9d9d9. The X1 Xn nþ1 GM R angle, c, is the distance on the unit sphere between TðÞ¼; ; r CnmYnmðÞ½; 59 R r points ðÞ; and ðÞ9; 9 . n¼0 m¼ – n Proof. The forward statement [53] follows directly by where M is the total mass (including the atmosphere) substituting eqns [51] and [49] into the first equation of the Earth and the dCnm are unitless harmonic [53], together with the spherical harmonic expansion coefficients, being also the difference between coeffi- [50] for g. A comparison with the spherical harmonic cients for the total and reference gravitational expansion for f yields the result. All steps in this proof potentials (Section 3.02.5.2). The coefficient, C00,is are reversible, and so the reverse statement also holds. zero under the assumption that the reference field Consider now f to be the potential, V,outsidethe accounts completely for the central part of the total sphere of radius, R, and its restriction to the sphere to be field. Also note that these coefficients specifically the function, g : gðÞ; ¼ V ðÞ; ; R . Then, clearly, refer to the sphere of radius, R. hn ¼ 1, for all n;bythetheoremabove,wehave The gravity disturbance is defined (in spherical ZZ approximation) to be the negative radial derivative of 1 V ðÞ¼; ; r V ðÞ9;9; R UðÞc; r d ½54 T, the first of eqns [5]. From eqn [59], we have 4 q gðÞ¼; ; r – TðÞ; ; r where qr  X1 Xn nþ2  GM R X1 nþ1 R ¼ ðÞn þ 1 CnmYnmðÞ; UðÞ¼c; r ðÞ2n þ 1 P ðÞ½cos c 55 R2 r r n n¼0 m¼ – n n¼0 ½60 For the distance and, applying the convolution theorem [53], we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi obtain , ¼ r 2 þ R2 – 2rR cos c ½56 ZZ R between points ðÞ; ; and ðÞ9; 9; , with r R, TðÞ¼; ; r gðÞ9;9; R HðÞc; r d ½61 r R 4 the identity (the Coulomb expansion; Cushing, 1975, p. 155), where with gnm ¼ ðÞn þ 1 dCnm=R and fnm ¼ dCnm,we  have hn ¼ fnm=gnm ¼ R=ðÞn þ 1 , and hence (taking X1 nþ1 1 1 R ¼ P ðÞcos c ½57 care to keep the Green’s function unitless) , n R n¼0 r  X1 nþ1 2n þ 1 R yields, after some arithmetic (based on taking the HðÞ¼c; r PnðÞ½cos c 62 n þ 1 r derivative on both sides with respect to r), n¼0 The integral in [61] is known as the Hotine integral, RrðÞ2 – R2 UðÞ¼c; r 3 ½58 the Green’s function, H, is called the Hotine kernel, , and with a derivation based on equation [57], it is Solutions [44] and [54] to the Dirichlet BVP for a given by (Hotine 1969, p. 311) spherical boundary are identical (in view of the con-  2R , volution theorem [53]). The integral in [54] is known HðÞ¼c; r – ln 1 þ ½63 as the Poisson integral and the function U is the , 2R sin2c=2 Potential Theory and Static Gravity Field of the Earth 25

Equation [61] solves the Neumann BVP when the more common expression for the disturbing poten- boundary is a sphere. tial, but it embodies hidden constraints. The gravity anomaly (again, in spherical approx- We note that gravity anomalies also serve as imation) is defined by eqn [5] boundary values in the harmonic series form of the  solution for the disturbing potential. Applying the q 2 gðÞ¼; ; r – – TðÞ½; ; r 64 orthogonality of the spherical harmonics to eqn [65] qr r yields immediately ZZ or, also, 2  R X1 Xn nþ2 Cnm ¼ gðÞ9;9; R YnmðÞ9;9 d; GM R 4ðÞn – 1 GM ; ; – ; gðÞ¼r 2 ðÞn 1 CnmYnmðÞ R n¼0 m¼ – n r n 2 ½69 ½65 A similar formula holds when gravity disturbances In this case, we have gnm ¼ ðÞn – 1 dCnm=R and are the boundary values (n – 1 in the denominator hn ¼ R=ðÞn – 1 . The convolution theorem in this changes to n þ 1). In either case, the boundary values case leads to the geodetically famous Stokes integral, formally are assumed to reside on a sphere of radius, ZZ R. An approximation results if they are given on the R TðÞ¼; ; r gðÞ9;9; R SðÞc; r d ½66 geoid, as is usually the case. 4 where we define Green’s function to be  3.02.4.2 Inverse Stokes and Hotine X1 nþ1 2n þ 1 R SðÞ¼c; r P ðÞcos c Integrals n – 1 r n n¼2 The convolution integrals above can easily be R R R, R2 ¼ 2 þ – 3 – 5 cos c inverted by considering again the spectral , r r 2 r 2 relationships. For the gravity anomaly, we note that R2 , þ r – R cos c – 3 cos c ln ½67 f ¼ rg is harmonic with coefficients, r 2 2r fnm ¼ GMðÞ n – 1 dCnm=R. Letting gnm ¼ GM dCnm=R, more commonly called the Stokes kernel. Equation we find that hn ¼ n – 1; from the convolution theo- [66] solves the Robin BVP if the boundary is a rem, we can write sphere, but it includes specific constraints that ensure ZZ 1 the solution’s uniqueness – the solution by itself is gðÞ¼; ; r TðÞ9;9; R ZˆðÞc; r d ½70 not unique, in this case, as proved in Section 3.02.3.2. 4R Indeed, eqn [65] shows that the gravity anomaly has where no first-degree harmonics for the disturbing poten-  X1 nþ2 tial; therefore, they cannot be determined from the R ZˆðÞ¼c; r ðÞ2n þ 1 ðÞn – 1 PnðÞ½cos c 71 boundary values. Conventionally, the Stokes kernel n¼0 r also excludes the zero-degree harmonic, and thus the The zero- and first-degree terms are included provi- complete solution for the disturbing potential is sionally. Note that given by  q 2 GM ZˆðÞ¼c; r – R þ UðÞc; r ½72 TðÞ¼; ; r C00 q r  r r X1 2 GM R that is, we could have simply used the Dirichlet þ C1mY1mðÞ; R ZZm¼ – 1 r solution [54] to obtain the gravity anomaly, as given R þ gðÞ9;9; R SðÞc; r d ½68 by [70], from the disturbing potential. It is convenient 4 to separate the kernel function as follows:  The central term, C00, is proportional to the differ- X1 nþ2 R ence in GM of the Earth and reference ellipsoid and is ZˆðÞ¼c; r ZðÞc; r – ðÞ2n þ 1 P ðÞ½cos c 73 r n zero to high accuracy. The first-degree harmonic n¼0 coefficients, C1m, are proportional to the center-of- where mass coordinates and can also be set to zero with  X1 nþ2 appropriate definition of the coordinate system (see R ZðÞ¼c; r ðÞ2n þ 1 n PnðÞ½cos c 74 Section 3.02.5.1). Thus, the Stokes integral [66] is the n¼1 r 26 Potential Theory and Static Gravity Field of the Earth

8 9 ()> 1 q > We find that <> => ;;ðÞr 1 r q 1 ¼ > >TðÞ; ; r gðÞ¼; ; r – TðÞ; ; r ;;ðÞr ;ðÞr :> 1 q ;> r ZZ – 1 r sin q þ TðÞ9;9; R ZðÞc; r d ½75 ½78 4R where ; are the north and east deflection compo- Now, since Z has no zero-degree harmonics, its inte- nents, respectively, and is the normal gravity. gral over the sphere vanishes, and one can write the Clearly, the derivatives can be taken directly inside numerically more convenient formula: the Stokes integral, and we find ZZ 1 1 () ZZ gðÞ¼; ; r – TðÞþ; ; r ðTðÞ9;9; R ;; r 4R ðÞr R ¼ gðÞ9;9; R – TðÞÞ; ; R ZðÞc; r d ½76 ;; 4r ;ðÞr ðÞr () q cos This is the inverse Stokes formula. Given T on the SðÞc; r d ½79 sphere of radius R (e.g., in the form of geoid undula- qc sin tions, T ¼ N), this form is useful when the gravity anomaly is also desired on this sphere. It is one way to where determine gravity anomalies on the ocean surface 8 9 8 9 > 1 q > > qc > from satellite altimetry, where the ocean surface is <> => <> => approximated as a sphere. Analogously, from eqns r q 1 q q ¼ ½80 [60] and [64], it is readily seen that the inverse > 1 q > r > 1 qc > qc : – ; : – ; Hotine formula is given by r sin q sin q ZZ 1 1 and gðÞ¼; ; r TðÞþ; ; r ðTðÞ9;9; R r 4R qc 1 – TðÞÞ; ; R ZðÞc; r d ½77 ¼ ðÞ¼sin cos 9 – cos sin 9cosðÞ9 – cos q sinc Note that the difference of eqns [76] and [77] yields 1 qc 1 – ¼ sin 9sinðÞ¼9 – sin ½81 the approximate relationship between the gravity sin q sin c disturbance and the gravity anomaly inferred from 9; 9 ; eqns [5]. The angle, , is the azimuth of ðÞat ðÞon Finally, we realize that, for r ¼ R, the series for Z the unit sphere. The integrals [79] are known as the Vening-Meinesz integrals. Analogous integrals for ðÞc; R is not uniformly convergent and special numerical procedures (that are outside the present the deflections arise when the boundary values are scope) are required to approximate the correspond- the gravity disturbances (the Green’s functions are ing integrals. then derivatives of the Hotine kernel). For the inverse Vening-Meinesz integrals, we need to make use of Green’s first identity for surface functions, f and g: 3.02.4.3 Vening-Meinesz Integral and Its ZZ ZZ Z Inverse f ðÞg ds þ rf ? rg ds ¼ f rg ? n db Other derivatives of the disturbing potential may also s s b ½82 be determined from boundary values. We consider here only gravity anomalies, being the most preva- where b is the boundary (a line) of surface, s; r and lent data type on land areas. The solution is either in are the gradient and Laplace–Beltrami operators, the form of a series – simply the derivative of the which for the spherical surface are given by series [59] with coefficients given by eqn [69], or an 0 1 q integral with appropriate derivative of the Green’s B C 2 2 B q C q q 1 q function. The horizontal derivatives of the disturbing r¼@ A; ¼ þ cot þ q q2 q sin2 q2 potential are often interpreted as the deflections of 1 the vertical to which they are proportional in sphe- sin q ½83 rical approximation (eqn [5]): Potential Theory and Static Gravity Field of the Earth 27 and where n is the unit vector normal to b. For a Clearly, following the same procedure for f ¼ T, closed surface such as the sphere, the line integral we also have the following relationship: vanishes, and we have ZZ R ZZ ZZ TðÞ¼; ; r 0 ððÞ9;9; R cos 4 f ðÞg d ¼ – rf ? rg d ½84 q þ ðÞ9;9; R sin Þ BðÞc; r d ½90 qc The surface spherical harmonics, Y ðÞ; , satisfy nm where the following differential equation:  X1 nþ1 ; ; ðÞ2n þ 1 R YnmðÞþnnðÞþ 1 YnmðÞ¼0 ½85 BðÞ¼c; r PnðÞ½cos c 91 n¼2 nnðÞþ 1 r Therefore, the harmonic coefficients of T ðÞ; ; r on the sphere of radius, R, are It is interesting to note that instead of an integral over the sphere, the inverse relationship between the dis- GM turbing potential on the sphere and the deflection of ½ TðÞ; ; r nm¼ – nnðÞþ 1 Cnm ½86 R the vertical on the same sphere is also more straight- Hence, by the convolution theorem (again, consider- forward in terms of a line integral: ing the harmonic function, f ¼ rg), Z ð; Þ ZZ 0 1 TðÞ¼; ; R TðÞþ0;0; R ððÞ9;9; R ds gðÞ¼; ; r – TðÞ9;9; R W ðÞc; r d 4 ð0; 0Þ 4R – 9;9; ½87 ðÞR dsÞ½92 where where  X1 nþ2 ðÞ2n þ 1 ðÞn – 1 R ds ¼ R d; ds ¼ R sin d ½93 W ðÞ¼c; r PnðÞ½cos c 88 n¼2 nnðÞþ 1 r and where the zero-degree term of the gravity anom- aly must be treated separately (e.g., it is set to zero in this case). Using Green’s identity [84] and eqns [80] 3.02.4.4 Concluding Remarks and [81], we have ZZ The spherical harmonic series, [59], represents the 1 general solution to the exterior potential, regardless gðÞ¼; ; r rTðÞ9;9; R ? rW ðÞc; r d 4R ZZ of the way the coefficients are determined. We know R how to compute those coefficients exactly on the ¼ 0 ððÞ9;9; R cos 4R basis of a BVP, if the boundary is a sphere. More q complicated boundaries would require corrections þ ðÞ9;9; R sin Þ W ðÞc; r d ½89 qc or, if these are omitted, would imply an approxima- tion. If the coefficients are determined accurately where normal gravity on the sphere of radius, R, (e.g., from satellite observations (Section 3.02.6.1), is approximated as a constant: ;ðÞR .0. Equation but not according to eqn [69]), then the spherical [89] represents a second way to compute harmonic series model for the potential is not a gravity anomalies from satellite altimetry, where spherical approximation. The spherical approxima- the along-track and cross-track altimetric differences tion enters when approximate relations such as eqns are used to approximate the deflection components [5] are used and when the boundary is approximated (with appropriate rotation to north and east as a sphere. However determined, the infinite series components). Employing differences in altimetric converges uniformly for all r > Rc, where Rc is the measurements benefits the estimation since systema- radius of the sphere that encloses all terrestrial tic errors, such as orbit error, cancel out. To speed masses. It may also converge below this sphere, but up the computations, the problem is reformulated would represent the true potential only in free space in the spectral domain (see, e.g., Sandwell and (above the Earth’s surface, where Laplace’s equation Smith, 1996). holds). In practice, though, convergence is not an 28 Potential Theory and Static Gravity Field of the Earth issue since the series must be truncated at some finite of the Earth’s density, which also leads to fundamen- degree. Any trend toward divergence is then part of tal geometric characterizations, particularly for the the overall model error, due mostly to truncation. second-degree harmonics. Let Model errors exist in all the Green’s function  nþ1 integrals as they depend on spherical approximations ðÞa a R Cnm ¼ vnm ½94 in the boundary condition. In addition, the surface of GM a integration in these formulas is formally assumed be unitless coefficients that refer to a sphere of to be the geoid (where normal derivatives of the radius, a. (Recall that coefficients, vnm, eqn [44], potential coincide with gravity magnitude), but it is refer to a sphere of radius, R.) Relative to the approximated as a sphere. The spherical approxima- ðÞa central harmonic coefficient, C00 ¼ 1, the next sig- tion results, in the first place, from a neglect of the ðÞa nificant harmonic, C20 , is more than 3 orders of ellipsoid flattening, which is about 0.3% for the magnitude smaller; the remaining harmonic coeffi- Earth. When working with the disturbing potential, cients are at least 2–3 orders of magnitude smaller this level of error was easily tolerated in the past (e.g., than that. The attenuation after degree 2 is much if the geoid undulation is 30 m, the spherical approx- more gradual (Table 1), indicating that the bulk of imation accounts for about 10 cm error), but today it the potential can be described by an ellipsoidal field. requires attention as geoid undulation accuracy of The normal gravitational field is such a field, but it 1 cm is pursued. also adheres to a geodetic definition that requires the The Green’s functions all have singularities when underlying ellipsoid to be an equipotential surface in the evaluation point is on the sphere of radius, R. For the corresponding normal gravity field. This section points above this sphere, it is easily verified that all examines the low-degree harmonics from these two the Legendre series for the Green’s functions con- perspectives of interpretation and reference. verge uniformly, since jjPnðÞcos c 1. When r ¼ R, the corresponding singularities of the integrals are either weak (Stokes integral) or strong (e.g., Poisson integral), requiring special definition of the integral as Cauchy principal value. 3.02.5.1 Low-Degree Harmonics as Finally, it is noted that the BVP solutions Density Moments also require that no masses reside above the geoid Returning to the general expression for the gravita- (the boundary approximated as a sphere). To satisfy tional potential in terms of the Newtonian density this condition, the topographic masses must be integral (eqn [14]), and substituting the spherical redistributed appropriately by mathematical reduc- harmonic series for the reciprocal distance (eqn [57] tion and the gravity anomalies, or disturbances with [51]), measured on the Earth’s surface must be reduced to the geoid (see Chapter 10.05). The mass redistribution must then be undone (mathematically) in order to obtain the correct potential on or above the geoid. Table 1 Spherical harmonic coefficients of the total gravitational potentiala Details of these procedures are found in Heiskanen and Moritz (1967, chapter 3). In addition, the atmo- Degree Order, (a) (a) sphere, having significant mass, affects gravity (n) (m) Cnm Cn,m anomalies at different elevations. These effects 204.841 70E 04 0.0 may also be removed prior to using them as boundary 212.398 32E 10 1.424 89E 09 values in the integral formulas. 2 2 2.439 32E 06 1.400 28E 06 3 0 9.571 89E 07 0.0 3 1 2.030 48E 06 2.481 72E 07 3 2 9.048 02E 07 6.190 06E 07 3 3 7.212 94E 07 1.414 37E 06 3.02.5 Low-Degree Harmonics: 4 0 5.399 92E 07 0.0 Interpretation and Reference 415.361 67E 07 4.735 73E 07 4 2 3.505 12E 07 6.624 45E 07 The low-degree spherical harmonics of the Earth’s 4 3 9.908 68E 07 2.009 76E 07 441.884 72E 07 3.088 27E 07 gravitational potential lend themselves to interpreta- tion with respect to the most elemental distribution aGRACE model GGM02S (Tapley et al., 2005). Potential Theory and Static Gravity Field of the Earth 29

ZZZ  8 9 X1 nþ1 1 r9 > r9sin 9sin 9; m ¼ – 1 > V ðÞ¼; ; r G ZZ Z <> => r 0 r G volume n¼0 !v ¼ pffiffiffi r9cos 9; m ¼ 0 dv 1m 2 > > Xn 3R > > 1 volume :> ;> Y ðÞ; Y ðÞ9;9 dv 9 9 9; nm nm 8 r sin9 cos m ¼ 1 2n þ 1 m¼ – n  > ycm; m ¼ – 1 > X1 Xn nþ1 > > R G < = ¼ GMffiffiffi r Rnþ1ðÞ2n þ 1 ¼ p zcm; m ¼ 0 ½99 n¼0 m¼ – n 3R2 > > ZZ Z ! :> ;> x ; m ¼ 1 n m ðÞr9 YnmðÞ9;9 dv volume Nowadays, by tracking satellites, we have access to YnmðÞ; ½95 the center of mass of the Earth (including its atmo- sphere), since it defines the center of their orbits. yields a multipole expansion (so called from electro- Ignoring the small motion of the center of mass statics) of the potential. The spherical harmonic (annual amplitude of several millimeters) due to the (Stokes) coefficients are multipoles of the density temporal variations in the mass distribution, we may distribution (cf. eqn [44]), choose the coordinate origin for the geopotential ZZ Z model to coincide with the center of mass, thus G 9 n 9;9 annihilating the first-degree coefficients. vnm ¼ nþ1 ðÞr YnmðÞdv ½96 R ðÞ2n þ 1 v The second-order density moments likewise are One may also consider the nth-order moments of related to the second-degree harmonic coefficients density (from the statistics of distributions) defined (quadrupoles). They also define the inertia tensor of by the body. The inertia tensor is the proportionality ZZ Z factor in the equation that relates the angular ðÞn 0 0 0 momentum vector, H, and the angular velocity, w, ¼ ðÞx ðÞy ðÞz dv; n ¼ þ þ ½97 v of a body, like the Earth:

The multipoles of degree n and the moments of order H ¼ I! ½100 n are related, though not all ðÞn þ 1 ðÞn þ 2 =2 and is given by moments of order n can be determined from the 0 1 2nþ1 multipoles of degree n, when n 2 (clearly Ixx Ixy Ixz B C risking confusion, we defer to the common nomen- B C I ¼ @ Iyx Iyy Iyz A ½101 clature of order for moments and degree for spherical harmonics). This indeterminacy is directly con- Izx Izy Izz nected to the inability to determine the density It comprises the moments of inertia on the diagonal distribution uniquely from external measurements ZZ Z of the potential (Chao, 2005), which is the classic ÀÁ 92 92 ; geophysical inverse problem. Ixx ¼ y þ z dv ZZ Zvolume The zero-degree Stokes coefficient is coordinate ÀÁ 92 92 ; invariant and is proportional to the total mass of the Iyy ¼ z þ x dv ½102 ZZ Zvolume ÀÁ Earth: 2 2 Izz ¼ x9 þ y9 dv ZZZ volume G GM v00 ¼ dv ¼ ½98 R volume R and the products of inertia off the diagonal It also represents a mass monopole, and it is propor- ZZ Z tional to the zeroth moment of the density, M. Ixy ¼ Iyx ¼ – x9y9 dv; The first-degree harmonic coefficients (represent- ZZ Zvolume ing dipoles) are proportional to the coordinates of the Ixz ¼ Izx ¼ – x9z9 dv; ½103 ZZ Zvolume center of mass, ðÞxcm; ycm; zcm , which are proportional to the first-order moments of the density, as verified Iyz ¼ Izy ¼ – y9z9 dv by recalling the definition of the first-degree spheri- volume cal harmonics: Note that 30 Potential Theory and Static Gravity Field of the Earth

0 1 ðÞ2 ðÞ2 ðÞ2 ; – ; cos 0 – sin 0 0 Ixx ¼ 020 þ 002 Ixy ¼ 110 etc ½104 B C B C and there are as many (six) independent tensor com- R ¼ @ sin 0 cos 0 0 A ½107 ponents as second-order density moments. Using the 001 explicit expressions for the second-degree spherical and with eqns [105], it is straightforward to show that harmonics, we have from eqn [96] with n ¼ 2: pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 15G 15 G 15 G – – v2; – 2 ¼ 3 ðÞIuu Ivv sin 2 0 v ; – ¼ – I ; v ; – ¼ – I 10R 2 2 5R3 xy 2 1 5R3 yz pffiffiffiffiffi ½108 pffiffiffiffiffi pffiffiffi 15G 15 G 5 G ÀÁ v ¼ – ðÞI – I cos 2 – – ½105 2; 2 3 uu vv 0 v2; 1 ¼ 3 Ixz v2; 0 ¼ 3 Ixx þ Iyy 2Izz 10R pffiffiffiffiffi5R 10R 15 G ÀÁ Hence, we have v ¼ I –I 2; 2 10R3 yy xx 1 – 1 v2; – 2 0 ¼ tan ½109 These are also known as MacCullagh’s formulas. Not 2 v2; 2 all density moments (or, moments of inertia) can be where the quadrant is determined by the signs of the determined from the Stokes coefficients. harmonic coefficients. From Table 1, we find that If the coordinate axes are chosen so as to diago- ¼ – 14:929; that is, the u-figure axis is in the nalize the inertia tensor (products of inertia are then 0 mid-Atlantic between South America and Africa. equal to zero), then they are known as principal axes The second-degree, second-order harmonic coef- of inertia, or also ‘figure’ axes. For the Earth, the ficient, v , indicates the asymmetry of the Earth’s z-figure axis is very close to the spin axis (within 2; 2 mass distribution with respect to the equator. Since several meters at the pole); both axes move with v > 0 (for the Earth), equations [108] show that respect to each other and the Earth’s surface, with 2; 2 I > I and thus the equator ‘bulges’ more in the combinations of various periods (daily, monthly, vv uu direction of the u-figure axis; conversely, the equator annually, etc.), as well as secularly in a wandering is flattened in the direction of the v-figure axis. This fashion. Because of these motions, the figure axis is flattening is relatively small: 1.1 105. not useful as a coordinate axis that defines a frame Finally, consider the most important second- fixed to the (surface of the) Earth. However, because degree harmonic coefficient, the second zonal har- of the proximity of the figure axis to the defined monic, v . Irrespective of the x-axis definition, it is reference z-axis, the second-degree, first-order har- 2; 0 proportional to the difference between the moment monic coefficients of the geopotential are relatively of inertia, I , and the average of the equatorial small (Table 1). ÀÁzz moments, I þ I =2. Again, since v < 0, the The arbitrary choice of the x-axis of our Earth- xx yy 2; 0 Earth bulges more around the equator and is flat- fixed reference coordinate system certainly did not tened at the poles. The second zonal harmonic attempt to eliminate the product of inertia, I (the xy coefficient is roughly 1000 times larger than the x-axis is defined by the intersection of the Greenwich other second-degree coefficients and thus indicates meridian with the equator, and the y-axis completes a a substantial polar flattening (owing to the Earth’s right-handed mutually orthogonal triad). However, it early more-fluid state). This flattening is approxi- is possible to determine where the x-figure axis is mately 0.003. located by combining values of the second-degree, second-order harmonic coefficients. Let u, v, w be the axes that define a coordinate system in which the 3.02.5.2 Normal Ellipsoidal Field inertia tensor is diagonal, and assume that Iww ¼ Izz. A rotation by the angle, – 0, about the w- (also z-) Because of Earth’s dominant polar flattening and the figure axis brings this ideal coordinate system back to near symmetry of the equator, any meridional section the conventional one in which we calculate the har- of the Earth is closer to an ellipse than a circle. For monic coefficients. Tensors transform under rotation, this reason, ellipsoidal coordinates have often been defined by matrix, R , according to advocated in place of the usual spherical coordinates. In fact, for geodetic positioning and geographic map- I ¼ R I R T ½106 xyz uvw ping, because of this flattening, the conventional With the rotation about the w-axis given by the (geodetic) latitude and longitude are coordinates matrix, that define the direction of the perpendicular to an Potential Theory and Static Gravity Field of the Earth 31 ellipsoid, not the radial direction from the origin. integrals have also been developed and applied in These geodetic coordinates are not the ellipsoidal practice (e.g., Fei and Sideris, 2000). coordinates defined in Section 3.02.1 (eqn [2]) and The simplicity of the boundary values of the would be rather useless in potential modeling normal gravitational potential allows its extension because they do not separate Laplace’ differential into exterior space to be expressed in closed analytic equation. form. Analogous to the geoid in the actual gravity Harmonic series in terms of the ellipsoidal coor- field, the normal ellipsoid is defined to be a level dinates, ðÞ; ; u , however, can be developed easily. surface in the normal gravity field. In other words, They have not been adopted in most applications, the sum of the normal gravitational potential and the perhaps in part because of the nonintuitive nature of centrifugal potential due to Earth’s rotation is a con- the coordinates. Nevertheless, it is advantageous to stant on the ellipsoid: model the normal (or reference) gravity field in terms e ; ; ; of these ellipsoidal coordinates since it is based on an V ðÞþb ðÞ¼b U0 ½112 ellipsoid. Laplace’ equation in these ellipsoidal coor- Hence, the normal gravitational potential on the dinates can be separated, analogous to spherical ellipsoid, V e ðÞ; ; b , depends only on latitude and coordinates, and the solution is obtained by succes- is symmetric with respect to the equator. sively solving three ordinary differential equations. Consequently, it consists of only even zonal harmo- Applied to the exterior gravitational potential, V, the nics, and because the centrifugal potential has only solution is similar to the spherical harmonic series zero- and second-degree zonals, the corresponding (eqn [44]) and is given by ellipsoidal series is finite (up to degree 2). The solu- tion to this Dirichlet problem is given in ellipsoidal X1 Xn Q ðÞiu=E coordinates by V ðÞ¼; ; u nmjj ve Y ðÞ½; 110 Q ðÞib=E nm nm  n¼0 m¼ – n nmjj GM E 1 q 1 V eðÞ¼; ; u tan – 1 þ !2a2 cos2 – E u 2 e q 3 where E is the linear eccentricity associated with the 0 ½113 coordinate system and Qnm is the associated Legendre function of the second kind. The coefficients of the where a is the semimajor axis of the ellipsoid, !e is e series, vnm, refer to an ellipsoid of semiminor axis, b; Earth’s rate of rotation, and and, with the series written in this way, they are all  2 real numbers with the same units as V. An exact 1 u – 1 E u q ¼ 1 þ 3 tan – 3 ; q0 ¼ qj ½114 relationship between these and the spherical harmo- 2 E2 u E u¼b nic coefficients, vnm, was given by Hotine (1969) and Jekeli (1988). Heiskanen and Moritz (1967) and Hofmann- With this formulation of the potential, Dirichlet’s Wellenhof and Moritz (2005) provide details of the BVP is solved for an ellipsoidal boundary using the straightforward derivation of these and the following orthogonality of the spherical harmonics: expressions. The equivalent form of V e in spherical harmonics ZZ is given by e 1 vnm ¼ V ðÞ; ; b YnmðÞ; d ½111 ! 4  X1 2n e GM a V ðÞ¼; ; r 1 – J2n P2nðÞcos ½115 where d ¼ sin dd and ¼ fðÞjd; 0 d ; r n¼1 r 0 2g. Note that while the limits of integra- where tion and the differential element, d, are the same as  for the unit sphere, the boundary values are on the 3e2n 5n J ¼ ðÞ– 1 nþ1 1 – n þ J ; n 1 ellipsoid. Unfortunately, integral solutions with ana- 2n ðÞ2n þ 1 ðÞ2n þ 3 e2 2 lytic forms of a Green’s function do not exist in this ½116 case, because the inverse distance now depends on two surface coordinates and there is no correspond- and e ¼ E=a is the first eccentricity of the ellipsoid. ing convolution theorem. However, approximations The second zonal coefficient is given by have been formulated for all three types of BVPs (see  e2 2 !2a2E I e – I e Yu et al., 2002, and references therein). Forms of – e zz xx J2 ¼ 1 ¼ 2 ½117 ellipsoidal corrections to the classic spherical 3 15q0 GM Ma 32 Potential Theory and Static Gravity Field of the Earth where the second equality comes directly from the the factor, e2n, and only harmonics up to degree 10 last of eqns [105] and the ellipsoid’s rotational sym- are significant. Normal gravity, being the gradient of e e metry (Ixx ¼ Iyy). This equation also provides a direct the normal gravity potential, is used only in applica- relationship between the geometry (the eccentricity tions tied to an Earth-fixed coordinate system. or flattening) and the mass distribution (difference of second-order moments) of the ellipsoid. Therefore, J2 is also known as the dynamic form factor – the flattening of the ellipsoid can be described either 3.02.6 Methods of Determination geometrically or dynamically in terms of a difference in density moments. In this section, we briefly explore the basic technol- The normal gravitational potential depends solely ogies that yield measurements of the gravitational on four adopted parameters: Earth’s rotation rate, !e ; field. Even though we have reduced the problem of the size and shape of the normal ellipsoid, for exam- determining the exterior potential from a volume ple, a, J2; and a potential scale, for example, GM. The integral to a surface integral (e.g., either eqns [66] mean Earth ellipsoid is the normal ellipsoid with or [69]), it is clear that in theory we can never parameters closest to actual corresponding para- determine the entire field from a finite number of meters for the Earth. GM and J2 are determined by measurements. The integrals will always need to be observing satellite orbits, a can be calculated by fit- approximated numerically, and/or the infinite series ting the ellipsoid to mean sea level using satellite of spherical harmonics needs to be truncated. altimetry, and Earth’s rotation rate comes from astro- However, with enough effort and within the limits nomical observations. Table 2 gives current best of computational capabilities, one can approach the values (Groten, 2004) and adopted constants for the ideal continuum of boundary values as closely as Geodetic Reference Systems of 1967 and 1980 desired, or make the number of coefficients in the (GRS67, GRS80) and the World Geodetic System series representation as large as possible. The 1984 (WGS84). expended computational and measurement effort In modeling the disturbing potential in terms of has to be balanced with the ability to account for spherical harmonics, one naturally uses the form of inherent model errors (such as the spherical approx- the normal gravitational potential given by eqn [115]. imation) and the noise of the measuring device. To Here, we have assumed that all harmonic coefficients be useful for geodetic and geodynamic purposes, the refer to a sphere of radius, a. Corresponding coeffi- instruments must possess a sensitivity of at least a few – e cients for the series of T ¼ V V are, therefore, parts per million, and, in fact, many have a sensitivity 8 > of parts per billion. These sensitivities often come at > 0; n ¼ 0 <> – J the expense of prolonging the measurements (inte- ðÞa CðÞa – pffiffiffiffiffiffiffiffiffiffiffiffiffin ; n ¼ 2; 4; 6; ... Cnm ¼ > n0 ½118 gration time) in order to average out random noise, > 2n þ 1 :> thus reducing the achievable temporal resolution. ðÞa ; Cnm otherwise This is particularly critical for moving-base instru- For coefficients referring to the sphere of radius, R,as mentation such as on an aircraft or satellite where n ðÞa in eqn [59], we have Cnm ¼ ðÞa=R Cnm . The temporal resolution translates into spatial resolution harmonic coefficients, J2n, attenuate rapidly due to through the velocity of the vehicle.

Table 2 Defining parameters for normal ellipsoids of geodetic reference systems

Reference 3 2 2 system a (m) J2 GM (m s ) !e (rad s )

GRS67 6378160 1.0827E 03 3.98603E14 7.2921151467E 05 GRS80 6378137 1.08263E 03 3.986005E14 7.292115E 05 WGS84 6378137 1.08262982131E 03 3.986004418E14 7.2921151467E 05 Best 6378136.7 0.1 (1.0826359 0.0000001)E 03 (3.986004418 0.000000008)E14 7.292115E 5 current (mean-tide system) (zero-tide system) (includes atmosphere) (mean value) valuesa aGroten E (2004) Fundamental parameters and current (2004) best estimates of the parameters of common relevance to astronomy, geodesy, and geodynamics. Journal of Geodesy 77: (10–11) (The Geodesist’s Handbook pp. 724–731). Potential Theory and Static Gravity Field of the Earth 33

3.02.6.1 Measurement Systems and Gravimeters, especially static instruments, are Techniques designed to measure acceleration at very low fre- quencies (i.e., averaged over longer periods of time), Determining the gravitational field through classical whereas accelerometers typically are used in naviga- measurements relies on three fundamental laws. The tion or other motion-sensing applications, where first two are Newton’s second law of motion and his accelerations change rapidly. As such, gravimeters law of gravitation. Newton’s law of motion states that generally are more accurate. Earth-fixed gravimeters the rate of change of linear momentum of a particle is actually measure gravity (Section 3.02.1.3), the dif- equal to the totality of forces, F, acting on it. Given 2 2 ference between gravitation and centrifugal more familiarly as mi d x=dt ¼ F, it involves the acceleration due to Earth’s spin (the frame is not inertial mass, mi ; and, conceptually, the forces, F, inertial in this case). The simplest, though not the should be interpreted as action forces (like propul- first invented, gravimeter utilizes a vertically, freely sion or friction). The gravitational field, which is part falling mass, measuring the time it takes to fall a of the space we occupy, is due to the presence of given distance. Applying eqn [119] to the falling masses like the Earth, Sun, Moon, and planets, and mass in the frame of the device, one can solve for g induces a different kind of force, the gravitational (assuming it is constant): xtðÞ¼0:5gt 2. This free-fall force. It is related to gravitational acceleration, g, gravimeter is a special case of a more general gravi- through the gravitational mass, mg , according to the meter that constrains the fall using an attached spring law of gravitation, abbreviated here as mgg¼Fg. or the arm of a pendulum, where other (action) forces Newton’s law of motion must be modified to include (the tension in the arm or the spring) thus enter into Fg separately. Through the third fundamental law, the equation. Einstein’s equivalence principle, which states that The first gravimeter, in fact, was the pendulum, inertial and gravitational masses are indistinguish- systematically used for gravimetry as early as the able, we finally get 1730s and 1740s by P. Bouguer on a geodetic expedi- tion to measure the size and shape of the Earth d2x (meridian arc measurement in Peru). The pendulum ¼ a þ g ½119 dt 2 served well into the twentieth century (until the early 1970s) both as an absolute device, measuring where a is the specific force (F=mi ), or also the the total gravity at a point, or as a relative device inertial acceleration, due to action forces. This equa- indicating the difference in gravity between two tion holds in a nonrotating, freely falling frame (that points (Torge, 1989). Today, absolute gravimeters is, an inertial frame), and variants of it can be derived exclusively rely on a freely falling mass, where in more complicated frames that rotate or have their exquisitely accurate measurements of distance and own dynamic motion. However, one can always time are achieved with laser interferometers and assume the existence of an inertial frame and proceed atomic clocks (Zumberge et al., 1982; Niebauer et al., on that basis. 1995). Accurate relative gravimeters are much less There exists a variety of devices that measure the expensive, requiring a measurement of distance motion of an inertial mass with respect to the frame change only, and because many errors that cancel of the device, and thus technically they sense a; such between measurements need not be addressed. devices are called accelerometers. Consider the spe- They rely almost exclusively on a spring-suspended cial case that an accelerometer is resting on the test mass (Nettleton, 1976; Torge, 1989). Developed Earth’s surface with its sensitive axis aligned along early in the twentieth century in response to oil- the vertical. In an Earth-centered frame (inertial, if exploration requirements, the relative gravimeter we ignore Earth’s rotation), the free-fall motion of has changed little since then. Modern instruments the accelerometer is impeded by the reaction force of include electronic recording capability, as well as the Earth’s surface acting on the accelerometer. In specialized stabilization and damping for deployment this case, the left-hand side of eqn [119] applied to on moving vehicles such as ships and aircraft. The the motion of the accelerometer is zero, and the accuracy of absolute gravimeters is typically of the accelerometer, sensing the reaction force, indirectly order of parts per billion, and relative devices in field measures (the negative of) gravitational acceleration. deployments may be as good but more typically are This accelerometer is given the special name, at least 1 order of magnitude less precise. Laboratory gravimeter. relative (in time) gravimeters, based on cryogenic 34 Potential Theory and Static Gravity Field of the Earth instruments that monitor the virtual motion of a test orbits (falls to) the Earth. Ever since Sputnik, the first mass that is electromagnetically suspended using artificial satellite, launched into the Earth orbit in superconducting persistent currents (Goodkind, 1957, Earth’s gravitational field could be determined 1999), are as accurate as portable absolute devices by tracking satellites from precisely known ground (or more), owing to the stability of the currents and stations. Equation [119], with gravitational accelera- the controlled laboratory environment (see Chapters tion expressed as a truncated series of spherical 3.03 and 3.04). harmonics (gradient of eqn [44]), becomes On a moving vehicle, particularly an aircraft, the  ! Xn Xn nþ1 d2x max R relative gravitational acceleration can be determined ; ! 2 ¼ vnmr YnmðÞþ e t þ R only from a combination of gravimeter and kinematic dt n¼0 m¼ – n r positioning system. The latter is needed to derive the ½120 (vertical) kinematic acceleration (the left-hand side of eqn [119]). Today, the GPS best serves that func- where !e is Earth’s rate of rotation and dR represents tion, yielding centimeter-level precision in relative residual accelerations due to action forces (solar three-dimensional position. Such combined GPS/ radiation pressure, atmospheric drag, Earth’s albedo, gravimeter systems have been used successfully to etc.), gravitational tidal accelerations due to other determine the vertical component of gravitation over bodies (Moon, Sun, planets), and all other subsequent large, otherwise-inaccessible areas such as the Arctic indirect effects. The x left-handÀÁ side of eqn [120] is Ocean (Kenyon and Forsberg, 2000) and over other more explicitly xtðÞ¼x ðÞt ; ðÞþt !et; rtðÞ , and areas that are more economically surveyed from the the spatial coordinates on the right-hand side are also air for oil-exploration purposes (Hammer, 1982; functions of time. This makes the equation more Gumert, 1998). The airborne gravimeter is specially conceptual than practical since it is numerically designed to damp out high-frequency noise and is more convenient to transform the satellite position usually stabilized on a level platform. Three-dimen- and velocity into Keplerian orbital elements (semi- sional moving-base gravimetry has also been major axis of the orbital ellipse, its eccentricity, its demonstrated using the triad of accelerometers of a inclination to the equator, the angle of perigee, the high-accuracy inertial navigation system (the type right ascension of the node of the orbit, and the mean that are fixed to the aircraft without special stabiliz- motion) all of which also change in time, but most ing platforms). The orientation of all accelerometers much more slowly. This transformation was derived on the vehicle must be known with respect to inertial by Kaula (1966) (see also Seeber, 1993). space, which is accomplished with precision gyro- In the most general case (nmax > 2 and dR 6¼ 0), scopes. Again, the total acceleration vector of the there is no analytic solution to eqn [120] or its trans- vehicle, d2x=dt 2, can be ascertained by time differ- formations to other types of coordinates. The entiation of the kinematic positions (from GPS). One positions of the satellite are observed by ranging of the most critical errors is due to the cross-coupling techniques and the unknowns to be solved are the of the horizontal orientation error, c, with the large coefficients, vnm. Numerical integration algorithms vertical acceleration (the lift of the aircraft, essen- have been specifically adapted to this problem and tially equal to – g). This is a first-order effect extremely sophisticated models for dR are employed (g sin dc) in the estimation of the horizontal gravita- with additional unknown parameters to be solved in tion components, but only a second-order effect order to estimate as accurately as possible the grav- (gðÞ1 – cos dc ) on the vertical component. Details of itational coefficients (e.g., Cappelari et al., 1976; moving-base vector gravimetry may be found in Pavlis et al., 1999). The entire procedure falls under Jekeli (2000a, chapter 10) and Kwon and Jekeli the broad category of dynamic orbit determination, (2001). and the corresponding gravitational field modeling The ultimate global gravimeter is the satellite in may be classified as the ‘timewise’ approach. The free fall (i.e., in orbit due to sufficient forward velo- partial derivatives of eqn [120] with respect to city) – the satellite is the inertial mass and the unknown parameters, p ¼ fg...; vnm; ... , are inte- ‘device’ is a set of reference points with known coor- grated numerically in time, yielding estimates for dinates (e.g., on the Earth’s surface, or another H ¼ qx=qp ¼ fg...; qx=qvnm; ... . These are then satellite whose orbit is known; see Chapter 3.05). used in a least-squares adjustment of the linearized The measuring technology is an accurate ranging model relating observed positions (e.g., via ranges) to system (radar or laser) that tracks the satellite as it parameters Potential Theory and Static Gravity Field of the Earth 35

x ¼ Hp þ e ½121 independent, disconnected platforms moving in simi- where dx and dp are differences with respect to lar orbits, the gravitational difference depends also on previous estimates, and e represents errors. (Tapley, their relative motion in inertial space. This is the 1973). concept for satellite-to-satellite tracking, by which A gravimeter (or accelerometer) on a satellite does one satellite precisely tracks the other and the change not sense the presence of a gravitational field. This is in the range rate between them is a consequence of a evident from the fact that the satellite is in free fall gravitational difference, a difference in action forces, (apart from small accelerations due to action forces and a centrifugal acceleration due to the rotation of such as atmospheric drag) and the inertial test mass the baseline of the satellite pair. It can be shown that and the gravimeter, itself, are equally affected by the line-of-sight acceleration (the measurement) is gravitation (i.e., they are all in free fall). However, given by two accelerometers fixed on a satellite yield, through 2 d T T the difference in their outputs, a gradient in accel- ¼ e ðÞþgxðÞ2 – gxðÞ1 e ðÞa2 – a1 dt 2 ! 2 2 eration that includes the gradient of gravitation. On a 1 d d þ x – ½125 nonrotating satellite, the acceleration at an arbitrary dt dt point, b, of the satellite is given by where e is the unit vector along the instantaneous d2x a ¼ – gbðÞ ½122 baseline connecting the two satellites, is the base- b dt 2 line length (the range), and x ¼ x2 – x1 is the in a coordinate system with origin at the center of difference in position vectors of the two satellites. mass of the satellite. Taking the difference (differen- Clearly, only the gravitational difference projected tial) of two accelerations in ratio to their separation, along the baseline can be determined (similar to a we obtain single-axis gradiometer), and then only if both satel- lites carry accelerometers that sense the a g b ¼ – ½123 nongravitational accelerations. Also, the orbits of b b both satellites need to be known in order to account where the ratios represent tensors of derivatives in for the centrifugal term. the local satellite coordinate frame. For a rotating Two such satellite systems were launched satellite, this equation generalizes to recently to determine the global gravitational field. One is CHAMP (Challenging Mini-Satellite Payload)in qa qg d b ¼ – b þ 2 þ ½124 2000 (Reigber et al., 2002) and the other is GRACE q q b b dt (Gravity Recovery and Climate Experiment) in 2002 where is a skew-symmetric matrix whose off-diag- (Tapley et al., 2004a). CHAMP is a single low-orbiting onal elements are the components of the vector that satellite (400–450 km altitude) being tracked by the defines the rotation rate of the satellite with respect high-altitude GPS satellites, and it also carries a to the inertial frame. Thus, a gradiometer on a satel- magnetometer to map the Earth’s magnetic field. lite (or any moving vehicle) senses a combination of GRACE was more specifically dedicated to determin- gravitational gradient and angular acceleration ing with extremely high accuracy the long to (including a centrifugal type). Such a device is sched- medium wavelengths of the gravitational field and uled to launch for the first time in 2007 as part of the their temporal variations. With two satellites in vir- mission GOCE (Gravity Field and Steady-State Ocean tually identical low Earth orbits, one following the Circulation Explorer ; Rummel et al., 2002). If the entire other, the primary data consist of intersatellite ranges tensor of gradients is measured, then, because of the observed with K-band radar. The objective is to sense symmetry of the gravitational gradient tensor and of changes in the gravitational field due to mass transfer 2 , and the antisymmetry of d=dt, the sum dab=db on the Earth within and among the atmosphere, the T þðÞdab=db eliminates the latter, while the differ- hydrosphere/cryosphere, and the oceans (Tapley T ence dab=db – ðÞdab=db can be used to infer , et al., 2004b). subject to initial conditions. An Earth-orbiting satellite is the ideal platform on When the two ends of the gradiometer are fixed to which to measure the gravitational field when seek- one frame, the common linear acceleration, d2x=dt 2, ing global coverage in relatively short time. One cancels, as shown above; but if the two ends are simply designs the orbit to be polar and circular; 36 Potential Theory and Static Gravity Field of the Earth and, as the satellite orbits, the Earth spins under- linear relationship between observations and neath, offering a different section of its surface on unknown harmonic coefficients: each satellite revolution. There are also limitations.

2 Xnmax Xn  À Á First, the satellite must have low altitude to achieve d T ¼ vnme rUnm ; ; r j high sensitivity, since the nth-degree harmonics of 2 x2 dt n¼0 m¼ – n n þ x1; x2 the field attenuate as ðÞR=r 1. On the other hand,  the lower the altitude, the shorter the life of the – rU ðÞj; ; r þ a þ c ½126 nm x1 satellite due to atmospheric drag, which can only be nþ1 countered with onboard propulsion systems. Second, where UnmðÞ¼; ; r ðÞR=r YnmðÞ; and da, dc because of the inherent speed of lower-orbit satellites are the last two terms in eqn [125]. Given the latter (about 7 km s1), the resolution of its measurements and a set of line-of-sight accelerations, a theoretically is limited by the integration (averaging) time of the straightforward linear least-squares adjustment sensor (typically 1–10 s). Higher resolution comes solves for the coefficients. A similar procedure can only with shorter integration time, which may reduce be used for gradients observed on an orbiting the accuracy if this depends on averaging out random satellite: noise. Figure 8 shows the corresponding achievable

Xnmax Xn resolution on the Earth’s surface for different satellite qa ¼ v rrTU ðÞ; ; r þ c ½127 instrumentation parameters, length of time in polar nm nm x qb n¼0 m¼ – n orbit and along-orbit integration time, or smoothing x (Jekeli, 2004). In each case, the indicated level of where c comprises the rotational acceleration terms resolution is warranted only if the noise of the sensor in eqn [124]. (after smoothing) does not overpower the signal at In situ measurements of line-of-site acceleration this resolution. or of more local gradients would need to be reduced Both CHAMP and GRACE have yielded global from the satellite orbit to a well-defined surface (such gravitational models by utilizing traditional satellite- as a sphere) in order to serve as boundary values in a tracking methods and incorporating the range rate solution to a BVP. However, the model for the field is appropriately as a tracking observation (timewise already in place, rooted in potential theory (trun- approach). However, the immediate application of cated series solution to Laplace’ equation), and one eqn [125] suggests that gravitational differences can may think of the problem more in terms of fitting a be determined in situ and used to determine a model three-dimensional model to a discrete set of observa- for the global field directly. This is classified as the tions. This operational approach can readily, at least spacewise approach. In fact, if the orbits are known conceptually, be expanded to include observations with sufficient accuracy (from kinematic orbit deter- from many different satellite systems, even airborne mination, e.g., by GPS), this procedure utilizes a and ground-based observations. Recently, a rather different theory has been con- sidered by several investigators to model the

10 000 gravitational field from satellite-to-satellite tracking observations. The method, first proposed by Wolff

1000 Measurement (1969), makes use of yet another fundamental law: the integration law of conservation of energy. Simply, the range rate time 100 between two satellites implies an along-track velocity 10 s 5 s difference, or a difference in kinetic energy.

Resolution (km) 10 Observing this difference leads directly, by the con- 1 s servation law, to the difference in potential energy, 1 that is, the gravitational potential and other potential 1 day 1 mo. 6 mo.1 yr. 5 yr. energies associated with action forces and Earth’s Mission duration rotation. Neglecting the latter two, conservation of Figure 8 Spatial resolution of satellite measurements energy implies vs mission duration and integration time. The satellite altitude is 450 km (after Jekeli, 2004) mo., month. 2 1 d Reproduced by permission of American Geophysical V ¼ x1 – E0 ½128 Union. 2 dt Potential Theory and Static Gravity Field of the Earth 37

where E0 is a constant. Taking the along-track differ- For a global harmonic analysis, the number 2 ential, we have approximately of coefficients, ðÞnmax þ 1 , must not be greater than the number of data. A general, conservative

d d rule of thumb for the maximum resolution (half- V ðÞx2 – V ðÞ¼x1 x1 ½129 dt dt wavelength) of a truncated spherical harmonic series is, in angular degrees on the unit sphere, wherejj dx2=dt – dx1=dt d=dt.Thisveryrough conceptual relationship between the potential differ- 180 ence and the range rate applies to two satellites closely ¼ nmax following each other in similar orbits. The precise formulation is given by (Jekeli, 1999) and holds for Thus, data on a 1 1 angular grid of latitudes and any pair of satellites, not just two low orbiters. longitudes would imply nmax ¼ 180. The number of Mapping range rates between two polar orbiting satel- data (64 800) is amply larger than the number of lites (such as GRACE) yields a global distribution of coefficients (32 761). This majority suggests a least- potential difference observations related again linearly squares adjustment of the coefficients to the data, in to a set of harmonic coefficients: either method, especially because the data have errors (Rapp, 1969). As nmax increases, a rigorous,  Xnmax Xn À Á optimal adjustment usually is feasible, for a given V ðÞx2 – V ðÞ¼x1 vnm Unm ; ; r x computational capability, only under restrictive n¼0 m¼ – n  2 À Á assumptions on the correlations among the errors of – Unm ; ; r ½130 x1 the data. Also, the obvious should nevertheless be noted that the accuracy of the model in any area The energy-based model holds for any two vehi- depends on the quality of the data in that area. cles in motion and equipped with the appropriate Furthermore, considering that a measurement con- ranging and accelerometry instrumentation. For tains all harmonics (up to the level of measurement example, the energy conservation principle could error), the estimation of a finite number of harmonics also be used to determine geopotential differences from boundary data on a given grid is corrupted by between an aircraft and a satellite, such as a GPS those harmonics that are in the data but are not satellite (at which location the geopotential is known estimated. This phenomenon is called aliasing in quite well). The aircraft would require only a GPS spectral analysis and can be mitigated by appropriate receiver (to do the ranging) and a set of acceler- filtering of the data (Jekeli, 1996). ometers (and gyros for orientation) to measure the The optimal spherical harmonic model combines action forces (the same system components as in air- both satellite and terrestrial data. The currently best borne accelerometry discussed above). Resulting known model is EGM96 complete to degree and potential differences could be used directly to order nmax ¼ 360 (Lemoine et al., 1998). It is an model the geoid using Bruns’ equation. updated model for WGS84 based on all available satellite tracking, satellite altimetry, and land gravity (and topographic) data up to the mid-1990s. 3.02.6.2 Models Scheduled to be revised again for 2006 using more recent data, as well as results from the satellite mis- We have already noted the standard solution options sions, CHAMP and GRACE, it will boast a maximum to the BVP using terrestrial gravimetry in the form of degree and order of 2160 (5’ resolution). In construct- gravity anomalies: the Green’s function approach, ing combination solutions of this type, great effort is Stokes integral, eqn [66], and the harmonic analysis expended to ensure the proper weighting of observa- of surface data, either using the integrals [69] or tions in order to extract the most appropriate solving a linear system of equations (eqn [65] trun- information from the diverse data, pertaining to dif- cated to finite degree) to obtain the coefficients, dCnm. ferent parts of the spatial gravitational spectrum. The The integrals must be evaluated using quadratures, satellite tracking data dominate the estimation of the and very fast numerical techniques have been devel- lower-degree harmonics, whereas the fine resolution oped when the data occupy a regular grid of of the terrestrial data is most amenable to modeling coordinates on the sphere or ellipsoid (Rapp and the higher degrees. It is beyond the present scope to Pavlis, 1990). Similar algorithms enable the fast solu- delve into the numerical methodology of combina- tion of the linear system of eqns [65]. tion methods. Furthermore, it is an unfinished story 38 Potential Theory and Static Gravity Field of the Earth as new in situ satellite measurements from GRACE integral in eqn [132] thus approximates the geoid by and GOCE will affect the combination methods of a sphere. Furthermore, it is assumed that no masses the future. exist external to the geoid. Part of the reduction to Stokes integral is used in practice only to take the geoid of data measured on the Earth’s surface advantage of local or regional data with higher reso- involves redistributing the topographic masses on or lution than was used to construct the global models. below the geoid. This redistribution is undone out- Even though the integral is a global integral, it can be side the solution to the BVP (i.e., Stokes integral) in truncated to a neighborhood of the computation order to regain the disturbing potential for the actual point since the Stokes kernel attenuates rapidly as Earth. Conceptually, we may write the reciprocal distance. Moreover, the corresponding ZZ  R ðÞnmax truncation error may be reduced if the boundary TP ¼ g – g – c SP; P 0 d 4 P 0 values exclude the longer-wavelength features of ðÞnmax the field. The latter constitute an adequate represen- þ TP þ TP ½133 tation of the remote zone contribution and can be where dc is the gravity reduction that brings the ðÞn included separately as follows. Let g max denote gravity anomaly to a geoid with no external masses, the gravity anomaly implied by a spherical harmonic and dTP is the effect (called indirect effect) on the model, such as given by eqn [65], truncated to disturbing potential due to the inverse of this reduc- degree, nmax. From the orthogonality of spherical tion. This formula holds for T anywhere on or above harmonics, eqn [46], it is easy to show that the geoid and thus can also be used to determine the  Xnmax Xn nþ1 geoid undulation according to Bruns’ formula [3]. ðÞnmax GM R T ðÞ¼; ; r CnmYnmðÞ; R ZZn¼2 m¼ – n r R ¼ gðÞnmax ðÞ9;9; R SðÞc; r d 4 3.02.7 The Geoid and Heights ½131 Thus, given a spherical harmonic model The traditional reference surface, or datum, for fgCnmj2 n nmax; – n m n , we first remove heights is the geoid (Section 3.02.1.3). A point at the model in terms of the gravity anomaly and then mean sea level usually serves as starting point restore it in terms of the disturbing potential, chan- (datum origin) and this defines the datum for vertical ging Stokes formula [66] to control over a region or country. The datum (or ZZ geoid) is the level continuation of the reference sur- R TðÞ¼; ; r gðÞ9;9; R – gðÞnmax ðÞ9;9; R face under the continents, and the determination of 4 gravity potential differences from the initial point to ðÞnmax SðÞc; r d þ T ðÞ; ; r ½132 other points on the Earth’s suface, obtained by level- In theory, if gðÞnmax has no errors, then the residual ing and gravity measurements, yields heights with g – gðÞnmax excludes all harmonics of degree respect to that reference (or in that datum). The n nmax, and orthogonality would also allow the gravity potential difference, known as the geopoten- exclusion of these harmonics from S. Once the inte- tial number, at a point, P, relative to the datum origin, gration is limited to a neighborhood of ðÞ; ; R ,asit P0, is given by (since gravity is the negative vertical must be in practice, there are a number of ways to derivative of the gravity potential) Z modify the kernel so as to minimize the resulting P truncation error (Sjo¨berg, 1991, and references CP ¼ W0 – WP ¼ g dn ½134 therein). The removal and restoration of a global P0 model, however, is the key aspect in all these where g is gravity (magnitude), dn is a leveling incre- methods. ment along the vertical direction, and W0 is the In practical applications, the boundary values are gravity potential at P0. By the conservative nature on the geoid, being the surface that satisfies the of the gravity potential, whatever path is taken for the boundary condition of the Robin BVP (i.e., we integral yields a unique geopotential number for P. require the normal derivative on the boundary and From these potential differences, one can define var- measured gravity is indeed the derivative of the ious types of height, for example, the orthometric potential along the perpendicular to the geoid). The height (Figure 3): Potential Theory and Static Gravity Field of the Earth 39

CP computation of T on the geoid can be done with HP ¼ ½135 gP assumptions on the density of the topographic masses where above the geoid and a proper reduction to the geoid, Z using only an approximate height. Indeed, since the 1 P vertical gradient of the disturbing potential is the g P ¼ g dH ½136 4 2 HP P0 gravity disturbance, of the order of 5 10 ms ,a height error of 20 m leads to an error of 102 m2 s2 is the average value of gravity along the plumb line in T, or just 1 mm in the geoid undulation. It should be from the geoid at P to P. Other height systems are 0 noted that a model for the disturbing potential as a also in use (such as the normal and dynamic heights), series of spherical harmonics, for example, derived but they all rely on the geopotential number (see from satellite observations, satisfies Laplace’s equation Heiskanen and Moritz, 1967, chapter 4, for details). and, therefore, does not give the correct disturbing For a particular height datum, there is theoreti- potential at the geoid (if it lies below the Earth’s sur- cally only one datum surface (the geoid). But access face where Laplace’s equation does not hold). to this surface is far from straightforward (other than The ability to derive orthometric heights (or other at the defined datum origin). If P is defined at mean 0 geopotential-related heights) from GPS has great sea level, other points at mean sea level are not on the economical advantage over the laborious leveling same level (datum) surface, since mean sea level, in procedure. This has put great emphasis on obtaining fact, is not level. Erroneously assuming that mean sea an accurate geoid undulation model for land areas. level is an equipotential surface can cause significant Section 3.02.6 briefly outlined the essential methods distortions in the vertical control network of larger to determine the geoid undulation from a combina- regions, as much as several decimeters. This was the tion of spherical harmonics and an integral of local case, for example, for the National Geodetic Vertical gravity anomalies. When dealing with a height datum Datum of 1929 in the US for which 26 mean sea level or the geoid, the constant N0 ¼ – ðÞW0 – U0 = points on the east and west coasts were assumed to lie requires careful attention. It can be determined by on the same level surface. Accessibility to the geoid comparing the geoid undulation computed according (once defined) at any point is achieved either with to a model that excludes this term (such as eqn [133]) precise leveling and gravity (according to eqns [134] with at least one geoid undulation (usually many) and [135]), or with precise geometric vertical posi- determined from leveling and GPS, according to tioning and knowledge of the gravity potential. eqn [137]. Vertical control and the choice of height Geometric vertical positioning, today, is obtained datum are specific to each country or continent, very accurately (centimeter accuracy or even better) where a local mean sea level was the adopted with differential GPS. Suppose that an accurate grav- datum origin. Thus, height datums around the ity potential model is also available in the same world are ‘local geoids’ that have significant differ- coordinate system as used for GPS. Then, determin- ences between them. Investigations and efforts have ing the GPS position at P0 allows the evaluation of been under way for more than two decades to define the gravity potential, W0, of the datum. Access to the a global vertical datum; however, it is still in the geoid at any other point, P, or equivalently, deter- future, awaiting a more accurate global gravity mining the orthometric height, HP, can be done by potential model and, perhaps more crucially, a con- first determining the ellipsoidal height, h, from GPS. sensus on what level surface the global geoid should Then, as shown in Figure 3, be. On the oceans, the situation is somewhat less H ¼ h – N ½137 P P complicated. Oceanographers who compute sea-sur- where, with T ¼ W – U evaluated on the geoid, face topography from satellite altimetry on the basis Bruns’ extended equation (Section 3.02.1) yields of eqn [137] depend critically on an accurate geoid undulation, or equivalently on an accurate model of N ¼ T= – ðÞW – U = ½138 0 0 T. However, no reduction of the disturbing potential where U0 is the normal gravity potential of the nor- from mean sea level to the geoid is necessary, the mal ellipsoid. deviation being at most 2 m and causing an error in In a sense, the latter is a circular problem: deter- geoid undulation of less than 0.1 mm. Thus, a sphe- mining N requires N in order to locate the point on rical harmonic model of T is entirely appropriate. the geoid where to compute T. However, the Furthermore, it is reasonable to ensure that the 40 Potential Theory and Static Gravity Field of the Earth

constant, W0 – U0, vanishes over the oceans. That is, The mean tidal potential is inherent in all our one may choose the geoid such that it best fits mean terrestrial observations (the boundary values) and sea level and choose an ellipsoid that best fits this cannot be averaged away; yet, the solutions to the geoid. It means that the global average value of the BVP assume no external masses. Therefore, in prin- geoid undulation should be zero (according to eqn ciple, the effect of the tide potential including its [138]). The latter can be achieved with satellite alti- mean, or permanent, part should be removed from metry and oceanographic models of sea surface the observations prior to applying the BVP solutions. topography (Bursa et al., 1997). On the other hand, the permanent indirect effect is Several interesting and important distinctions not that well modeled and arguably should not be should be made in regard to the tidal effects on the removed; after all, it contributes to the Earth’s shape geoid. The Sun and the Moon generate an appreci- as it actually is in the mean. Three types of tidal able gravitational potential near the Earth (the other systems have been defined to distinguish between planets may be neglected). In an Earth-fixed coordi- these corrections. A mean quantity refers to the quan- nate system, this extraterrestrial potential varies in tity with the mean tide potential retained (but time- time with different periods due to the relative varying parts removed); a nontidal quantity implies motions of the Moon and Sun, and because of that all tidal effects (time-varying, permanent, direct Earth’s rotation (Torge, 2001, p. 88). There is also a and indirect effects) have been removed computa- constant part, the permanent tidal potential, repre- tionally; and the zero-tide quantity excludes all time- senting the average over time. It is not zero, because varying parts and the permanent direct effect, but it the Earth–Sun–Moon system is approximately retains the indirect permanent effect. coplanar. For each body with mass, MB, and distance, If the geoid (an equipotential surface) is defined rB, from the Earth’s center, this permanent part is solely by its potential, W0, then a change in the poten- given by tial due to the tidal potential, V tide (time-varying and constant parts, direct and indirect effects), implies that  3 r 2 ÀÁ1 1 the W0-equipotential surface has been displaced. The V BðÞ¼; r GM 3 cos2 – 1 sin2" – ½139 c B 2 geoid is now a different surface with the same W0. 4 rB 2 3 This displacement is equivalent to a change in geoid where " is the angle of the ecliptic relative to the undulation, dN ¼ V tide=, with respect to some pre- equator (" 23 449). Using nominal parameter defined ellipsoid. The permanent tidal effect (direct values for the Sun and the Moon, we obtain at and indirect) on the geoid is given by ÀÁ mean Earth radius, R ¼ 6371km, 2 NðÞ¼ – 0:099ðÞ 1 þ k2 3 cos – 1 m ½142 ÀÁ sþm 2 2 – 2 Vc ðÞ; R ¼ – 0:97 3 cos – 1 m s ½140 If N represents the instantaneous geoid, then the The gravitational potential from the Sun and geoid without any tidal effects, that is, the nontidal Moon also deforms the quasi-elastic Earth’s masses geoid, is given by with the same periods and similarly includes a con- Nnt ¼ N – N ½143 stant part. These mass displacements (both ocean and solid Earth) give rise to an additional indirect change The mean geoid is defined as the geoid with all but the mean tidal effects removed: in potential, the tidal deformation potential (there are also secondary indirect effects due to loading of the N ¼ N – ðÞN – N ½144 ocean on the solid Earth, which can be neglected in this discussion; see Chapter 3.06). The indirect effect This is the geoid that could be directly observed, for is modeled as a fraction of the direct effect (Lambeck, example, using satellite altimetry averaged over time. 1988, p. 254), so that the permanent part of the tidal The zero-tide geoid retains the permanent indirect potential including the indirect effect is given by effect, but no other tidal effects: ÀÁÀÁ N ¼ N – N þ 0:099k 3 cos2 – 1 m ½145 sþm sþm z 2 Vc ðÞ¼; R ðÞ1 þ k2 Vc ðÞ; R ½141 The difference between the mean and zero-tide where k2 ¼ 0:29 is Love’s number (an empirical geoids is, therefore, the permanent component of number based on observation). This is also called the direct tidal potential. We note that, in principle, the mean tide potential. each of the geoids defined above, has the same Potential Theory and Static Gravity Field of the Earth 41

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