the spheroidal fixed free two-boundary-value problem for geoid

The spheroidal ®xed±free two-boundary-value problemfor geoid determination (the spheroidal Bruns' transform)

E. W. Grafarend1, A. Ardalan1, M. G. Sideris2

1Department of Geodesy and Geo Informatics, Stuttgart University, Geschwister-Scholl-Str. 24 D, Stuttgart, Germanye-mail: [email protected]; Tel.: +49 711 1213390; Fax: +49 711 12132852Department of Geomatics Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4

Received: 4 May 1999 /Accepted: 21 May 1999

Abstract. The target of the spheroidal Gauss±Listinggeoid determination is presented as a solution of thespheroidal ®xed±free two-boundary value problembased on a spheroidal Bruns' transformation (``spheroi-dal Bruns' formula''). The nonlinear spheroidal Bruns'transform (nonlinear spheroidal Bruns' formula), thespheroidal ®xed part and the spheroidal free part of thetwo-boundary value problem are derived. Four di�erentspheroidal gravity models are treated, in particular todetermine whether they pass the test to ®t to thepostulate of a level ellipsoidal gravity ®eld, namely ofSomigliana±Pizzetti type.

Key words. Two-boundary value problem á Spheroidalboundary value problem á Spheroidal Stokes' operator áSpheroidal Bruns' formula á Geoid determination

1 Introduction

The long road to the Gauss±Listing geoid, namely itsphysical and mathematical foundation, took us tothe formulation of the ®xed±free two-boundary valueproblem.

Functionals of the gravity potential, for instance

� the zero derivative of the gravity potential by meansof gravimetric levelling

� the vertical derivative of the gravity potential bymeans of relative or absolute gravimetry

� the horizontal derivative of the gravity potential bymeans of astrogeodetic or GPS±LPS levelling,

are determined by measurements on the Earth's topo-graphic surface, M2

h. In contrast, the Gauss±Listinggeoid as an equipotential surface M2

g close to mean sea

level is partially within the topography, accordingly notaccessible to direct measurement. The problem of geoiddetermination as an inverse problem of potential theoryis constituted as a free boundary problem with respect toa properly chosen reference ®gure.

If there were no topographic masses but a ``quiet sea-level surface'', the theory of equilibrium ®gures accordingto Newton would teach us that an ellipsoid of revolutionE2

a;a;b would characterize to ®rst order the surface ofthe Earth, for instance the Maclaurin equilibrium ®gure.The great geodetic expeditions (see Kakkuri et al. 1986;Smith 1986, TobeÂ 1986)

� to Peru by L. Godin, P. Bouguer, C.M. de LaCondamine, J.J. Sartacilia, A. de Ulloa, in the period1735±1744

� to Lapland by P.L. de Maupertuis, A.C. Clairaut,A. Celsius, R. Outhier, P.C. Le Monnier, C.E.L.Comus, A. Hellant, between 1736 and 1737

have indeed revealed the truth that globally the Earth isnot a sphere, but an oblate ellipsoid-of-revolution E2

a;a;b.Accordingly, a properly chosen reference ®gure for theGauss±Listing geoid is E2

a;a;b, nowadays given as a WorldGeodetic Datum (see e.g. Eitschberger and Grafarend1974; Grafarend and Ardalan 1998). Geoidal undula-tions are referred to a level ellipsoid E2

a;a;b, which hasbeen developed by Pizzetti (1894) and Somigliana (1930)and extensively analysed by Grafarend et al. (1977) andGrafarend and Ardalan (1999) in functional analyticalterms. The cartographic community has for a long timeadopted the concept to develop geodetic map projec-tions of the ellipsoid-of-revolution E2

a;a;b, namely

� the Universal Stereographic Projection (UPS), e.g.Grafarend and You (1995)

� the Universal Mercator Projection (UMP), e.g.E. Grafarend and R. Sy�us (1998)

� the Universal Transverse Mercator Projection(UTM), e.g. Grafarend (1995)

� the Hotine recti®ed skew orthomorphic projection(oblique Mercator projection HOM), e.g. Engels andGrafarend (1995)Correspondence to: E.W. Grafarend

Journal of Geodesy (1999) 73: 513±533

� the Universal Lambert Projection, e.g. Grafarendand You (1995),

all of conformal type.The target of the spheroidal Gauss±Listing geoid

determination is presented here as a solution of thespheroidal ®xed±free two-boundary-value problembased on a spheroidal Bruns' transformation (``spheroi-dal Bruns' formula''). Such a two-boundary-valueproblem has already been developed by Mihelcic (1972)and Grafarend and Sanso (1984), though in di�erentcontext, and recently by Martinec (1998a, pp. 6±7). In-deed, we go through six de®nitions, ®ve lemmas, ®vecorollaries, seven tables and eight ®gures before we ®-nally arrive at the nonlinear spheroidal Bruns' transform(nonlinear spheroidal Bruns' formula), the spheroidal®xed part and the spheroidal free part of the two-boundary-value problem. Four di�erent spheroidalgravity models are treated, in particular to determinewhether they pass the test to ®t to the postulate of a levelellipsoidal gravity ®eld, namely of Somigliana±Pizzettitype. In this way, the spheroidal free boundary valuepart which coincides with the spheroidal Stokes' prob-lem has already been solved by Martinec and Grafarend(1997a) and Ritter (1998a, b).

Finally, the Appendix gives an introduction to thetheory of a directional derivative.

2 Formulation of the nonlinear ®xed±freetwo-boundary value problem and its linearization

De®nition 1 focuses on the formulation of the nonlinear®xed±free two-boundary value problem of physicalgeodesy. (1) represents the Laplace±Poisson equationin the external space of the planet Earth and in termsof the gravity potential w�x� in a reference framefe1; e2; e3j 0g, which rotates around the 3-axis withrotational speed x. In contrast, (2) summarizes theLaplace±Poisson equation in the massive interior andthe physical boundary of the planet Earth, again in arotating frame of reference and in a massive body of amass density ®eld q�x�. With respect to a proper gaugethe modulus of gravity, the l2-norm of the gradient ofthe gravity potential ®eld w�x�, is observable on theboundary oGe of the Earth, the topographic surface M2

h.Here, the index refers to the ellipsoidal height H whereits height function H�L;B� of ellipsoidal longitude andlatitude has been represented in orthonormal ellipsoidalfunctions (spheroidal functions) by Grafarend andEngels (1992a, b), also numerically. (3) de®nes the ®xednonlinear boundary value problem on M2

h. The keyproblem of physical geodesy, the analysis of the Gauss±Listing Geoid, the geodetic reference surface close tomean sea level, has been formulated by (4) as a freeboundary value problem w�x� � w0 with respect to thefundamental geodetic parameter w0 which has recentlybeen analysed by Grafarend and Ardalan (1997), amongothers. Indeed, in the functional equation w�x� � w0 theplacement vector x is unknown. Finally, (5) summarizesthe regularity condition of the gravity potential ®eld atin®nity.

De®nition 1. (The non-linear ®xed±free two-boundary-value problem)

1. div grad w�x� � 2x2 8 x 2 R3=D [ oG�e(external space of the planet Earth)

2. div grad w�x� � ÿ4pgq�x� � 2x2 8x 2 D [ oGÿe(internal space plus boundary of the planet Earth)

3. kgrad w�x�k2 � c�x� 8 x 2 oGe �: M2h

(boundary value data of type modulus of gravity)4. w�x� � w0 8 x 2 oGi �: M2

g(equipotential value at the level of the geoid close tomean sea level)

5. w�x� � 12x2 xÿ x j exh iexk k22 � gm

xk k2 � Ow1xk k32

� �for xk k2!1(regularity condition at in®nity)

In order to linearize the nonlinear ®xed and the freetwo-boundary value problem, as the ®xed±free two-boundary value problem, we apply the Euler d-pertur-bation theory. Indeed, we take advantage of priorinformation of the terrestrial gravity ®eld by means ofa normal gravity ®eld (reference gravity ®eld) outlinedin Lemma 1.

Lemma 1. (The ®xed±free two-boundary-value problemfunctionally linearized in gravity space; Euler d-pertur-

Fig. 1. Fibering of the point set R3 equipped with the Euclideanmetric, D domain of the massive Earth, oG�e approach to the Earthsurface from the outside, oGÿe approach to the Earth surface from theinside, (oGe external boundary, two-dimensional Riemann manifoldM2

h), oGi internal boundary representing the geoid, the equipotentialsurface of gravity close to Mean Sea Level(two-dimensional Riemannmanifold M2

g)

Fig. 2. Geometric ®bering of the Euclidian space fR3; gklg, M2h

topographic surface of the Earth, M2g geoidal surface of the Earth,

shaded topographic masses

514

bation theory; M2h topographic surface versus M2

ggeoidal surface.)

Euler d-increments:

w � W � dW

c � C� dC

� Ck k2�C jdCh iCk k2

� 1

2 Ck k2dC2 ÿ 1

2 Ck k32C jdCh i2

ÿ 1

2 Ck k32C jdCh idC2 ÿ 1

8 Ck k32dC4

� Oc�

C jdCh i2dC2

q � P � dP

x2 � X2 � 2 X jdXh i � dX2

1. The Laplace±Poisson equation in the external spaceR3=D [ oG�e :

Div Grad W �x� � 2X2 versus Div Grad dW �x�� 4 X dXjh i � Ox�dX2�

8x 2 R3=D [ oG�e

2. The Laplace±Poisson equation in the internal spacebounded by the external topographic surface oGe:

Div Grad W �x� versus Div Grad dW �x�� ÿ4pGq�x� � 2X2 � ÿ4pdq�x� � 4 XjdXh i

�Ox�dX2�8x 2 D [ oGÿe

3. Upper ®xed boundary-value problem (linearizedoblique boundary-value problem) modulus of gravitydata on a given regularized surface M2

h of the planetEarth:

kC�x�k2 � GradW �x�k k2versus

�fixed BVPh� dC�x�

� 1

kCk2hCjdCi � 1

2kCk2dC2

ÿ 1

2 Ck k32hCjdCi ÿ 1

2 Ck k32CjdCh idC2

ÿ 1

8 Ck k32dC4 � Oc CjdCh i2dC2

n o� hGradW �x�jGrad dW �x�i

kGradW �x�k2

� hGrad dW �x�jGrad dW �x�i2kGradW �x�k2

ÿ 1

2kGradW �x�k32hGradW �x�jGrad dW �x�i2

ÿ 1

2 GradW �x�k k32hGradW �x�jGrad dW �x�i

� hGrad dW �x�jGrad dW �x�iÿ 1

8kGradW �x�k32hGrad dW �x�jGrad dW �x�i2

� Oc�hGjdGi2 dG2

4. Lower free boundary-value problem (linearized freeDirichlet boundary-value problem) potential data on anunknown geoidal surface M2

g of the planet Earth:

(free BVPg� w0 � W �x� � dW �x� 8x 2 oGi �: M2g

5. Regularity condition at in®nity:

W �x� � 1

2kxÿ hxjexiexk22 �

GMkxk

� Ow1

kxk32

!for kxk2 !1

dW �x� � hXjdXikxÿ hxjexik22

� Odw1

kxk32

!for kxk2 !1

Lemma 1 makes at ®rst a statement about thedecomposition of the actual gravity potential and theactual modulus of gravity in terms of a normal partW ;C (reference part) written in capital letters and aperturbation part (disturbing part) written as Eulerincrements dW ; dC. In addition, the mass density q andthe squared rotational speed x are decomposed in thesame way. It is worth mentioning that the Taylorexpansion of the modulus of gravity (the length of thegravity vector) appears here for the ®rst time up toorder ®ve in dC. The quadratic term has already beenanalysed by Martinec (1998a, p. 11), among others.Under (1) we have reviewed the Laplace±Poissonequation in the external space, both in terms of thenormal gravity ®eld and in terms of the perturbedgravity ®eld. In contrast, (2) is a formulation of theLaplace±Poisson equation in the internal space bound-ed by the external topographic surface oGe, both interms of the normal gravity ®eld and in terms of theperturbed gravity ®eld. (3) is the highlight of the upper®xed boundary-value problem, an expansion of the

Fig. 3. Decomposition of the scalar valued gravity potential w�x�,of the vector-valued gravity c�x� and of the scalar-valued modulusof gravity c�x�k k2� c

515

nonlinear boundary operator up to order ®ve in dC. Incontrast, (4) is a formulation of the lower free boundaryvalue problem with potential data on an unknowngeoidal surface M2

g. The index g refers to ``ground'' or``geoid''. Finally, (5) summarizes the regularity con-ditions at in®nity for the normal part and perturbedpart of the gravity potential.

Since the geoidal surface M2g is not known a priori,

but by prior information only approximately, by meansof Lemma 2 we aim at a Taylorization of all functionalswith respect to an approximate geoidal surface, in par-ticular to the geodetic reference ellipsoid E2

a;a;b. Indeed,we apply Lagrange perturbation by introducing La-grange increments DW , DC.

Lemma 2 collects the basic results in Taylorizing the®xed±free two-boundary value problem both in gravityspace and in geometry space. The Lagrange D-pertur-bation theory leading to two-point function D incre-ments for the gravity potential DW �x;X� as well as themodulus of gravity DC�x;X� is applied on the basis ofseries inversion (see e.g. Grafarend et al. 1996), in orderto derive the nonlinear Bruns' transformation (nonlinearBruns' formula). The Taylor series are based on the di-rectional derivative rYX :� hrXjYi, namely the pro-jection of the gradient rX on the vector Y. h�j�i denotesthe standard Euclidean inner product (scalar product).For instance, the projection of gradients rW ;rC, re-spectively, on the normal vector of an equipotentialsurface W �X� � W0 (a constant), in particular to a levelellipsoid E2

a;a;b of Somigliana±Pizzetti type has beenchosen. All expansions have been worked out up to thirdorder. Due to our choice of the directional derivative theprojected gradients are isozenithal. It is for this reasonthat we have called the Taylor increments in geometryspace ``normal isozenithal height h''. The isozenithalexpansion w0 � w�x� � W �x� � dW �x�, namely of W �x�,has led us to power series in terms of isozenithal heighth, namely to the nonlinear Bruns' transformation(nonlinear Bruns' formula) we are highlighting in de®-nition 2.

Lemma 2. (The ®xed±free two-boundary value problemTaylorized in gravity space as well as in geometry space;Lagrange D-perturbation theory; directional derivativerYX :� hrXjYi; forward Bruns' transformation.)

Laplace D increments (two-point functions):

DW �x;X� :� w�x� ÿ W �X�DC�x;X� :� c�x� ÿ C�X�DX :� xÿ X 8X(DW potential anomaly, DC gravity anomaly, DX place-ment anomaly)

Series inversion (Grafarend et al. 1996) y�x� 2 HOMPOLY as well as x�y� 2 HOM POLY

y�x� � a11x� a12x2 � � � � � a1nxn

versus

x�y� � b11y � b12y2 � � � � � b1nyn

if a11; . . . ; a1n are given, then

b11 � aÿ111

b12 � aÿ311 a12

b13 � 2aÿ611 a212 ÿ aÿ411 a13

The Bruns' transformation:

c�x� � C�x� � dC�x�

C�x� � C�X� � 1

1!�rNC�X��h

� 1

2!�rNrNC�X��h2 � OC�h3� �B1�

dC�x� � dC�X� � 1

1!�rNdC�X��h

� 1

2!�rNrNdC�X��h2 � OdC�h3� �B2�

w�x� � W �x� � dW �x�

W �x� � W �X� � 1

1!�rN W �X��h

� 1

2!�rNrN W �X��h2 � OW �h3� �B3�

dW �x� � dW �X� � 1

1!�rNdW �X��h

� 1

2!�rNrNdW �X��h2 � OdW �h3� �B4�

8 x 2 oGi �: M2g and X 2M2

G e.g. E2a;a;b

N 2NM2G e.g. NE2

a;a;b

(the normal space of the model equipotential surfaceW � W0, e.g. the ellipsoid of revolution E2

a;a;b)h is called normal isozenithal height.

Forward solution:

w0 � w�x� � W �x� � dW �x�

� W �X� �X1n�1

a1nhn � dW �x�

� W0 �X1n�1

a1nhn � dW �x�

()ÿ dW �x� � w0 ÿ W0 � ÿdW �x� � dw �

X1n�1

a1nhn

8 x 2 oGi :�M2g and X 2M2

G e.g. E2a;a;b

subject to:

a11 :� 1

1!rN W �X�; a12 � 1

2!rNrN W �X�; . . . ;

a1n :� 1

n!rN � � � rN|��{z��}

n times

W �X�

h �X1n�1

b1n�ÿdW �x� � DW �x;X��n

516

subject to:

b11 :� 1

rN W �X�

b12 :� ÿ 1

�rN W �X��31

2!rNrN W �X�

b13 :� rNrN W �X��rN W �X��3 ÿ

rNrNrN W �X�6�rN W �X��4

De®nition 2. (Bruns' transformation)

Let the directional derivative of the ®rst, second,. . .,nth order of the normal gravity potential W �X� be givenat a point X of the normal geoid M2

G. e.g. E2a;a;b the

level ellipsoid of revolution. Then the normal ellipsoidalheight h is given by

h � b11�ÿdW � DW � � b12�ÿdW � DW �2

� b13�ÿdW � DW �3 � � � � � b1n�ÿdW � DW �n� Ohf�ÿdW � DW �ng �B5�

subject to:

b11�rN W �; b12�rN W ;rNrN W �;b13�rN W ;rNrN W ; rNrNrN W �; . . . ;

b1n�rN W ;rNrN W ; . . . ; rN � � � rN|��{z��}n times

W �

a transformation which is called nonlinear Bruns'transformation (Grafarend and Niemeier 1971).

Let us back-substitute the nonlinear Bruns' trans-formation (Bruns' formula) following Lemma 3 intothe series expansion of the modulus of gravity c�x� �C�x� � dC�x�, namely C�x�, in order to derive the Eulerincrement dC�x� of the modulus of gravity, also calledthe modulus of gravity disturbance, as power series ofthe directional, derivative rNdW �X� of the gravity po-tential disturbance dW �X� relative to a given placementvector X 2 E2

a;a;b which extends to a point on the geo-detic reference ellipsoid E2

a;a;b, the level ellipsoid ofSomigliana±Pizzetti type.

Lemma 3. (The ®xed±free two-boundary-value problemtaylorized in gravity space as well as in geometry space;Lagrange D-perturbation theory; directional derivativercX :� hrXjYi; backward Bruns' transformation.)

Backward solution:

Transplant the Bruns' transformation (B5) to (B1):

c�x� � C�x� � dC�x� � C�x� �X1n�1

c1nhn � dC�x�

()ÿ dC�x� � c�x� ÿ C�X� � ÿdC�x� � DC�x;X�

�X1n�1

c1nhn

8 x 2 oGi :�M2g and X 2M2

G e.g. E2a;a;b (level ellipsoid of

revolution)

subject to:

c11 :� 1

1!rN C�X�; c12 :� 1

2!rNrNC�X�; . . . ;

c1n :� rN � � � rN|��{z��}n times

C�X�

h �X1n�1

b1n�ÿdW � DW �n

ÿ dC� Dc�x;X� �X1n�1

c1nhn

� c11h� c12h2 � � � � � c1nhn � OC�hn�1�� c11b11�ÿdW � DW � � c11b12�ÿdW � DW �2

� c11b13�ÿdW � DW �3 � � � �� c12b211�ÿdW � DW �2

� 2c12b11b12�ÿdW � DW �3 � � � �� c13b311�ÿdW � DW �3 � 3c13b211b12�ÿdW � DW �3

� OCf�ÿdW � DW �4gÿdC� DC�x;X� � d11�ÿdW � DW � � d12�ÿdW � DW �2

� d13�ÿdW � DW �3

� OCf�ÿdW � DW �4gsubject to:

d11 :� c11b11 � rNC�X�=rN W �X�d12 :� c11b12 � c12b2

11

� ÿ 1

2rN C�X��rNrN W �X��=�rN W �X��3

� 1

2�rNrNC�X��=�rN W �X��3

d13 :� c11b13 � 2c12b11b12 � c13b311 � 3c13b211b12

� rN C�X��rNrN W �X��=�rN W �X��6

ÿrNC�X��rNrNrN W �X��=6�rN W �X��4

ÿ 1

2rNrNC�X��rNrN W �X��=�rN W �X��4

� 1

6rNrNrNC�X�=�rN W �X��3

ÿ 1

4rNrNrNC�X��rNrN W �X��=�rN W �X��4 � � �

dC�x� � dC�X� � 1

1!�rdC�X��h� OdC�h2�

dW �x� � dW �X� � 1

1!�rdW �X��h� OdW �h2�

dC�x� � rNdW �X� � 1

2rN W �X� �rNdW �X��2

� OdCfh; �rN W �X�rNdW �X��2g

517

dW �x� � dW �X� � OdW �h�Finally, we present the general form of the previouslyde®ned free boundary value problem Taylorized ingravity space as well as in geometry space relative toM2

G, in particular E2a;a;b, in Lemma 4. Here we have

written the potential anomalies DW �x;X� and themodulus of gravity anomaly as given quantities onthe left-hand side, and the polynomial in terms ofdW �X�, rNdW �X� etc. as functionals of the gravitypotential disturbance dW �X�, its directional derivativesrNdW �X� etc., on the right-hand side. All functionalsof dW �X� have to be taken at a point X of a normalequipotential surface M2

G, e.g. the level ellipsoid ofrevolution E2

a;a;b.

Lemma 4. (Free boundary-value problem Taylorized ingravity space as well as in geometry space of referenceM2

G, e.g. E2a;a;b.)

Let the directional derivatives of ®rst, second, third,etc. order of the normal gravity potential W �X� and themodulus of normal gravity C�X� be given at a point Xof a normal equipotential surface M2

G, e.g. the levelellipsoid of revolution E2

a;a;b. Thus, the nonlinear Robinboundary value problem of the geoid M2

g is given by

ÿ d11DW �x;X� ÿ d12DW �x;X�2 � DC�x;X�� ÿd11dW �X� � rNdW �X� � d12dW �X�2

� OfdW DW ; dW 3g8X 2M2

G; e.g. E2a;a;b �level ellipsoid of revolution�

3 The spheroidal ®xed±free two-boundary value problem

In order to implement the standard representation ofthe surface of the Earth in ellipsoidal coordinates whichhas been popularized by the Navstar global positioningsystem (GPS) or GLONASS, we introduce the Gaussspheroidal geometry space in De®nition 3 and the Jacobispheroidal geometry space in De®nition 4. While theGauss ellipsoidal coordinates fL;B;Hg are based on theGauss map of the ellipsoidal surface normal given inspherical form (also called minimal distance mappingof the Earths topographic surface), the Jacobi ellipsoi-dal coordinates (special form of Jacobi elliptic coordi-nates called spheroidal) have been introduced asorthogonal coordinates fk;/; ug in which the Laplaceequation separates. Such a separation of the Laplaceequation is not enjoyed by Gauss spheroidal coordi-nates, as was shown by Grafarend (1988). Bothreference coordinates are given by direct equations ofthe embedding into R3 as well as by the inverseequations, their Jacobi maps, and their tangent spacesin R3 and ``on'' E2

a;a;b. The metric of R3 is ®nallycomputed in terms of the ellipsoidal coordinatesfL;B;Hg, fk;/; ug, respectively.

De®nition 3. [The Gauss spheroidal geometry space(Gauss map of the ellipsoid of revolution E2

a;a;b)]

appr oGi � appr M2g � E2

a;a;b

� fX 2 Rjf �X ; Y ; Z� :� �X 2 � Y 2�=a2 � Z2=b2ÿ 1 � 0

and R3 3 a > b 2 R3goGe :�M2

h

� fX 2 R3jf �X ; Y ; Z� � 0

and X �L;B;H�; Y �L;B;H�; Z�L;B;H�;0 � L < 2p; ÿp=2 < B < �p=2g

Direct equations: e2 :� �a2 ÿ b2�=a2

X � a��1ÿ e2 sin2 B

p � H�L;B�" #

cosB cos L

Y � a��1ÿ e2 sin2 B

p � H�L;B�" #

cosB sin L

Z � a�1ÿ e2��1ÿ e2 sin2 B

p � H�L;B�" #

sinB

Inverse equations:

L � arctanY =X

sgn X � �; sgn Y � � : 0 � L < p=2

sgn X � ÿ; sgn Y � � : p=2 � L < p

sgn X � ÿ; sgn Y � ÿ : p � L < 3p=2

sgn X � �; sgn Y � ÿ : 3p=2 � L < 2p

26664B;H are obtained by Newton iteration or by solving asystem of algebraic equations (Grafarend and Lohse1991) or by using closed formulae of Heikkinen (1982).H�L;B� is given as a set of orthonormal functions onE2

a;a;b by Grafarend and Engels (1992a, b).

Jacobi map:

J :�XL XB XH

YL YB YH

ZL ZB ZH

24 35subject to:

XL � DLX � ÿ a��1ÿ e2 sin2 B

p � H

" #cosB sin L

� HL cosB cos L

XB � DBX � ÿ a��1ÿ e2 sin2 B

p � H

" #sinB cos L

� ae2 sinB cosB

�1ÿ e2 sin2 B�3=2cosB cosL

� HB cosB cosL

XH � DH X � cosB cosL

YL � DLY � a��1ÿ e2 sin2 B

p �H

" #cosB cosL

�HL cosB sinL

518

YB � DBY � ÿ a��1ÿ e2 sin2 B

p � H

" #sinB sin L

� ae2 sinB cosB

�1ÿ e2 sin2 B�3=2cosB sin L

� HB cosB sin L

YH � DH Y � cosB sin L

ZL � DLZ � HL sinB

ZB � DBZ � a�1ÿ e2��1ÿ e2 sin2 B

p � H�L;B�" #

cosB

� a�1ÿ e2�e2 sinB cosB

�1ÿ e2 sin2 B�3=2sinB� HB sinB

ZH � DH Z � sinB

Tangent space of oGe :�M2g:

l1 �oXoL� E1J11 � E2J21 � E3J31

l2 �oXoB� E1J12 � E2J22 � E3J32

R3 � spanfE1;E2;E3gwhere fE1; E2; E3g are orthonormal base vectors whichspan fR3; gklg.Tangent space of fR3; gklg:

lL �oXoL�H fixed� � E1J11 � E2J21 � E3J31;

lB �oXoB�H fixed� � E1J12 � E2J22 � E3J32

lH �oXoH� E1 cosB cosL� E2 cosB sin L� E3 sinB

Metric of fR3; gklg:

dS2 � �dL; dB; dH �J �JdLdBdH

24 35G :� J �J

�

�a��

1ÿe2 sin2 Bp �H

�2cos2 B 0 0

0 a�1ÿe2��1ÿe2 sin2 B�3=2�H� �2

0

0 0 1

266664377775

De®nition 4. [The Jacobi spheroidal geometry space(cover of R3 by a triply orthogonal spheroidal coordi-nate system).]

appr oGi � appr M2g � E2

a;a;b

� fX 2 R3j�X 2 � Y 2�=a2 � Z2=b2 � 1;

�a 2 R�� > �b 2 R��g

oGe :�M2h

� fX 2 R3 j f �X ; Y ; Z� � 0

X �k;/; u�; Y �k;/; u�; Z�k;/; u�;0 � k < 2p;ÿp=2 < / < �p=2g

Direct equations: e2 :� a2 ÿ b2

X ��u2 � e2

pcos/ cos k

Y ��u2 � e2

pcos/ sin k

Z � u sin/

Inverse equations:

k� arctan Y =X

sgn X ��; sgn Y �� : 0� k< p=2

sgn X �ÿ; sgn Y �� : p=2� k< p

sgn X �ÿ; sgn Y �ÿ : p� k< 3p=2

sgn X ��; sgn Y �ÿ : 3p=2� k< 2p

26664/ � �sgn Z�arcsin

�1

2e2

�e2 ÿ �X 2 � Y 2 � Z2�

��X 2 � Y 2 � Z2 ÿ e2�2 � 4e2Z2

q ��1=2

u ��1

2

�X 2 � Y 2 � Z2 ÿ e2

��X 2 � Y 2 � Z2 ÿ e2�2 � 4e2Z2

q ��1=2

[Thong and Grafarend 1989, formula 1(66)].

Jacobi map:

J :�Xk X/ Xu

Yk Y/ Yu

Zk Z/ Zu

24 35subject to:

Xk � DkX

� ÿ��u2 � e2

pcos/ sin k� u uk��

u2 � e2p cos/ cos k

Yk � DkY

��u2 � e2

pcos/ cos k� u uk��

u2 � e2p cos/ sin k

Zk � DkZ � uk sin/

X/ � D/X

� ÿ��u2 � e2

psin/ cos k� u u/��


Y/ � D/Y

� ÿ��u2 � e2

psin/ sin k� u u/��

u2 � e2p cos/ sin k

Z/ � D/Z � u cos/� u/ sin/

Xu � DuX � u��u2 � e2p cos/ cos k

519

Yu � DuY � u��u2 � e2p cos/ sin k

Zu � DuZ � sin/

Tangent space of oGe :�M2h:

l1 �oXok� E1J11 � E2J21 � E3J31

l2 �oXo/� E1J12 � E2J22 � E3J32

Tangent space of fR3; gklg:

lk �oXok�u fixed� � E1J11 � E2J21 � E3J31

l/ �oXo/�u fixed� � E1J12 � E2J22 � E3J32

lu �oXou

� E1u��


� E2u��

u2 � e2p cos/ sin k� E3 sin/

Metric of fR3; gklg:

dS2 � �dk; d/; du�J�Jdkd/du

24 35

G :� J �J ��u2 � e2� cos2 / 0 0

0 u2 � e2 sin2 / 0

0 0 u2�e2 sin2 /u2�e2

24 35[Thong and Grafarend 1989, formula 1(73)]

Cartan orthonormal frame of fR3; gklg:Ek :� lk � lkk k2� ÿE1 sin k� E2 cos k

E/ :� l/ � l/

2

� ÿE1

��u2 � e2p��

u2 � e2 sin2 /q sin/ cos k

ÿ E2

��u2 � e2p��

u2 � e2 sin2 /q sin/ cos k

� E3u��

u2 � e2 sin2 /q cos/

Eu :� lu � kluk2� �E1

1��u2 � e2 sin/

p cos/ cos k

� E2u��

u2 � e2 sin/p cos/ sin k

� E3

��u2 � e2p��

u2 � e2 sin2 /q sin/

`ìf X 2 E2a;a;b or u � b applies, then Eu � N is the surface

normal vector of E2a;a;b''.

Corollary 1. (Direct transformation of Gauss ellipsoidalcoordinates into Jacobi spheroidal coordinates andvise versa.)

Direct Equations:

k � L

/ � arctan��1ÿ e2p

tanB�

u � 1��1ÿ e2p cosB

a�1ÿ e2��1ÿ e2 sin2 B�1=2

� H

" #� 1� �1ÿ e2� tan2 B� �1=2

Inverse equations:

L � k

B � arctan1��

1ÿ e2p tan/

� �

H ��1ÿ e2p

u cos�/� 1� 1

1ÿ e2tan2 /

� �1=2ÿ a 1ÿ e2ÿ �

� 1ÿ e2tan2 /

1ÿ e2 � tan2 /

� �ÿ1=2In order to extend the range of the Laplace equation-down to the approximate geoid, namely the normalequipotential surface M2

G, we have to eliminate thetopographic masses which are `òn top'' of M2

G from thegravity potential w�x�. Such a procedure, called ``removestep'', establishes the topographic reduction or the terraine�ect. Indeed, the Newton potential generated by thetopographic masses ``between M2

G and M2h'' has to be

computed in Jacobi spheroidal coordinates as hasalready been done by Feistritzer (1998). It is an openquestion as to how well the ellipsoidal topographicpotential can be represented in a ¯at approximation andby means of Fast Fourier algorithms as has successfullybeen done for the spherical topographic potential bySideris (1985, 1990, 1995), Sideris and Tziavos (1988),Li and Sideris (1992, 1994, 1995), Forsberg and Sideris(1993), Sideris and Li (1993), and Peng et al. (1995).From experience in computing the spherical topographicpotential (terrain e�ect), we already know its largeimpact on potential values. It is for this reason thatphysical geodesists have developed a balancing proce-dure by means of Helmert's topographic mass condensa-tion favoured by VanõÂ cÏ ek and SjoÈ berg (1991), Martinecet al. (1993), VanõÂ cÏ ek et al. (1995), Featherstone et al.(1998), Nahavandchi and SjoÈ berg (1998) and Sun and P.VanõÂ cÏ ek (1998), by means of isostatic compensationfavoured by SjoÈ berg (1998), Arabelos and Tziavos(1998) or by means of impact of those masses whichare `ùnder'' M2

G (subtop masses), namely the topo-graphic sandwiches M2

G, Konrad discontinuity, Mohodiscontinuity etc., in summary the crustal gravitational

520

®eld favoured by Grafarend and Engels (1993), Engelset al. (1995, 1996) and Kakkuri and Wang (1998).

As soon as we have established the ``remove step'' fortopographic masses (``terrain e�ect''), we have to orga-nize the step of downward continuation of topographicgravity dc�x�, x 2M2

h, to dc�X�, X 2M2G. Nowadays

such a downward continuation is based on the Abel±Poisson integral, here the ellipsoidal Abel±Poisson inte-gral as it is derived by Martinec and Grafarend (1997b)and applied for the downward continuation of gravitypotential disturbance by Feistritzer (1998) probing var-ious regularization procedures (Tykhonov±Philipps andothers) for the inverse ellipsoidal Abel±Poisson integral.So far, the problem of downward continuation ofgravity disturbance based upon the inverse ellipsoidalAbel±Poisson integral has not been tackled.

The remove step and the downward continuationstep are ®nally followed by the restore step as follows.On the level of the approximate geoid M2

G the gravita-tional impact of

(i) the topographic masses and/or(ii) the balancing masses of type

(ii1) simple layer condensation of Helmert type or(ii2) isostatic compensation or(ii3) crust (seismology: top±Konrad, Konrad±Moho

topographic sandwiches)

is restored. Nothing has been done in Gauss or Jacobispheroidal geometry space. Only Feistritzer (1998)implemented the (i)±(ii2) restore step for the potentialdisturbance in Jacobi spheroidal geometry.

Finally, De®nition 5 gives a summary of additive de-composition of the gravity potential for both the removeand the restore step; Fig. 4, in contrast, illustrates theremove step, downward continuation, and the restorestep. However, we are left with the problem of decidingupon a proper normal gravity potential (reference po-tential) with respect to spheroidal geometry. In actualfact there are basically two possibilities, which are re-

viewed brie¯y in Table 1 for spherical coordinates andin Table 2 for spheroidal coordinates.

De®nition 5. (Additive decomposition of the normalgravity potential ®eld.)

The remove step:

w�x� � W �x� � dW �x�

W �x� � U�top; x� � U�subtop; x�� U�normal; x� � V �x� 8 x 2M2

h

versus

The restore step:

w�X� � W �X� � dW �X�

W �X� � U�top; X� � U�subtop; X�� U�normal; X� � V �X� 8 X 2M2

G; e.g. E2a;a;b

Thong and Grafarend (1989) have reviewed the funda-mental solutions of the three-dimensional Laplace

Fig. 4. Removal of the topographic and sub-topographic massesfrom w�x� and c�x� � c�x�k k2 data at x 2M2

h versus restorationof topographic and sub-topographic masses for W �X� and C�X� �C�X�k k2 data at X 2M2

G, e.g. E2a;a;b; downward continuation of

dW �x� and dC�x� from x 2M2h to X 2M2

G, e.g. E2a;a;b, by means of

the spheroidal Abel±Poisson integral

Table 1. Spherical normalgravity ®eld Type of gravity potential Type of equipotential surface Author

gmr

Sphere Stokes (1849)

gmr� 1

2x2r2 cos2 / Algebraic surface of order 6 Bode and Grafarend

(1981, 1982)

gmr� r30

r3e2m�/; k�u2m � 1

2x2r2 cos2 /

Algebraic surface of order 14 Bruns (1878)

gmr� r30

r3e2m�k;/�u2m

� r40r4

e3m�k;/�u3m � r50r5

e4m�k;/�u4m

� 1

2x2r2 cos2 /

Algebraic surface of order 22 Helmert (1884)

XJ

j�0

X�j

ÿj

�r0r

�j�1ejm�k;/� ujm

� 1

2x2r2 cos2 /

Algebraic surface of order 4J � 6 Grafarend et al. (1985)

521

equation in spheroidal coordinates/elliptic coordinatesof various types. One choice is the spheroidal harmonicexpansion in the coordinates fk;/; ug which we havefavoured here. Its ®rst term (Table 2, model 1) containsthe gravitational mass (GM), the absolute eccentricitye �

��a2 ÿ b2p

, and the zero-degree/order Legendre func-tion Q00 of the second kind. Q00 has been expressed incolumn two. As an equipotential surface of the gravi-tational potential it generates an ellipsoid of revolution.If we add the centrifugal potential to the ®rst-orderspheroidal potential term and transform its representa-tion into spheroidal coordinates fk;/; ug and spheroidalharmonic functions, we arrive at the normal gravita-tional potential of type model 2. As an equipotentialsurface it can be interpreted as an algebraic surface oforder 6. In contrast, model 3 relates the normal gravitypotential to the Somigliana±Pizzetti ®eld which isgenerated by the postulate that at least one referencesurface in the spheroidal expansion of the gravitypotential (including the centrifugal potential) should bean ellipsoid of revolution. Column one contains theSomigliana±Pizzetti ®eld in modern functional analyticnotation, namely the fully normalized associated Leg-endre function P �lm�u=e� of the ®rst kind as well asQ�lm�u=e� of the second kind appear; column two is anexplicit write-up of the Somigliana±Pizzetti level ellip-soid reference ®eld. For the special case u � b, where bdenotes the semi-minor axis of an ellipsoid of revolution(for example the reference ellipsoid), the level surface isellipsoidal. Finally, model 4 contains the full spheroidalharmonic expansion of the normal gravity potentialincluding centrifugal potential. As an equipotentialsurface it is illustrated as an algebraic surface of order4J � 6, where J is the order of the spheroidal harmonicexpansion.

For a more familiar, conventional reference we havelisted the four models for a spherical harmonic expan-sion in Table 1. However, there is one essential di�er-ence: there is no spherical equipotential surface for agravity potential which includes the centrifugal poten-tial. The ®rst term of a spherical harmonic expansion(Table 1, ®rst case) according to Stokes (1849) generatesan equipotential surface of type sphere. For solving thespherical Stokes boundary value problem, Bode andGrafarend (1981, 1982) introduced a geodetic referencesurface as an equipotential surface (an algebraic surfaceof order 6) which is generated by the additive decom-position of a gravitational potential of type gm=r and thecentrifugal potential x2r2 cos2 /=2 (Table 1, secondcase). Bruns (1878) recommended as a reference equi-potential surface the gravity ®eld expansion up to order2 and showed its behaviour as an algebraic surface oforder 14 (Table 1, third case). Helmert (1884) went astep further to take reference to a gravity ®eld expansionup to order 4 and recommended such an algebraic sur-face of order 22 as a reference equipotential case (Table1, fourth case). Finally, we have added the gravity ®eldexpanded up to degree and order J used by Grafarendet al. (1985) for solving the non-spherical Stokes'boundary-value problem. In Table 1, ®fth case, it isdocumented as an algebraic surface of order 4J � 6.T

able

2.Spheroidalnorm

algravitypotential®eld

W�k;/;u�.

e�

��

a2ÿ

b2p

Typeofgravitypotential

U(functionalspace

notation)

Typeofgravitypotential

U(explicitnotation)

Typeofequipotential

surface

Author

Model

1G

M eQ

00

u e�� e 00

GM e

arccot

u e��Spheroid

(ellipsoid

ofrevolution)

Grafarend

Model

2G

M eQ� 00

u e�� e 00�2 9

X2e2

1�

P� 2

u e��h

i e 00

ÿ2

9�� 5pX

2e2

1�

P� 2

u e��h

i e 20�/�

GM e

arccot

u e��1 2X�u

2�

e2�cos2

/Algebraic

surface

oforder

6Grafarend

Model

3G

M eQ� 00

u e�� e 00�1 3

X2a2

2P� 2�u e�Q

� 00�u e�ÿ3P� 1�u e�

2P� 2�b e�Q

� 00�b e�ÿ3P� 1�b e�

�1 �� 5p

e 20�/��

2 9X

2e2

1�

P� 2

u e��h

i e 00

ÿ2

9�� 5pX

2e2

1�

P� 2

u e��h

i e 20�/�

GM e

arccot

u e�1 6

X2a2�3

u2 e2�1�arccot�u e�ÿ3

u e

�3b2 e2�1�arccot�b e�ÿ3

b e

��3

sin2/ÿ1��

1 2X�u

2�

e2�cos2

/

Spheroid

(ellipsoid

of

revolutionfor

u�

b)Somigliana±Pizzetti

Model

4XJ j�

1

X�j m�ÿ

j

Q� jm�u e�

Q� jm�b e�

e jm�k;/�u

jm2 9X

2e2

1�

P� 2

u e��h

i e 00

ÿ2

9�� 5pX

2e2

1�

P� 2

u e��h

i e 20�/�

XJ j�1

X�j m�ÿ

j

Q� jm�u e�

Q� jm�b e�

e jm�k;/�u

jm�1 2X

2�u

2�

e2�cos2

/Algebraic

surface

of

theorder

4J�6

522

Considerable work was involved in the representationof the normal gravitational potential, its gradient, themodulus of gravity, and most extensively its directionalderivatives in terms of spheroidal harmonics, namelychoosing model 1±4 for comparison. Lemma 5 containsthe general representation of the gradient and direc-tional derivatives of the normal gravity potentialU�k;/; u� in terms of orthogonal coordinates. Here thedirectional derivative grad U j euh i � reu U has beenchosen. Corollaries 2±4 and Tables 3, 4 and 5 show indetail the explicit computations of the modulus ofgravity C and the directional derivatives grad U j euh iand grad C j euh i for the spheroidal normal gravity po-tential U�k;/; u� of type ®rst model, second model andthird model (Somigliana±Pizzetti). Unfortunately, wehad to cut short the lengthy derivations for space rea-sons. It should be noted that all derivations are basedupon the monomial expansion��1� xp

� 1� 1

2xÿ 1 � 1

2 � 4 x2 � 1 � 1 � 32 � 4 � 6 x3

ÿ 1 � 1 � 3 � 52 � 4 � 6 � 8 x4 � O�x5�

for jxj � 1. In order to obtain some numerical insightinto the order of magnitude of the various terms we havecomputed

1. D/U��g//p

, as a function of / 2 �ÿp=2; �p=2� at al-titude u � b� 104 m (Fig. 6)

2. D/U��g//p

as a function of u 2 � b; b� 104 m� for/ � p=4 (Fig. 7)

3. �D/U�2=�2�u2 � e2�jDuU j � as a function of/ 2 �ÿp=2;�p=2� at altitude u � b� 104 m (Fig. 8).

Figure 8 shows that the second-order term �D/U�2=�2�u2 � e2� jDuU j� contains only 7:35� 10ÿ9% of the

spheroidal gravity e�ect, a good reason to neglect it inCorollary 4.

Lemma 5. (Gradient of gravity potential, norm ofgravity vector.)

Gradient of potential:

gradU � llDlU � glvlvDlU

� glv lvk k ecDlU � glv ��gvvp

evDuU

� Ek1��gkkp DkU � E/

1��g//p D/U � Eu

1��guup DkU

�)gradU � Ek

1��u2 � e2� cos2 /

p DkU

� E/1��

u2 � e2 sin2 /q D/U � Eu

��u2 � e2p��

u2 � e2 sin2 /q DkU

Directional derivative along the surface normal of E2a;a;b:

gradU jEuh i � reu U

��u2 � e2p��

u2 � e2 sin2 /q DuU � 1��

guup ouU

Modulus of gravity:

C ��gradU j gradUh i

p� 1

�u2 � e2� cos2 /�DkU�2 � 1

u2 � e2 sin2 /�D/U�2

�� u2 � e2

u2 � e2 sin2 /�DkU�2

�1=2

Table 3. Spheroidal Stokes' operator DC�x;X� :� c�x� ÿ C�X� � �rEu C�X��h�rEu dW �X�Spheroidal Stokes' operator

Model 1 Dc�x; X� � GM u

��u2 � e2p

�u2 � e2 sin2 /�2 �1

�u2 � e2 sin2 /� ��u2 � e2p

!h� u2 � e2��

u2 � e2 sin2 /q o

oudW �X�

Model 2 Dc�x; X� �(

GM u

��u2 � e2p

�u2 � e2 sin2 /�2 �1

�u2 � e2 sin2 /� ��u2 � e2p

" #

� u1

2

X4�u2 � e2��4u2 � e2 � 3e2 sin2 /�GM�u2 � e2 sin2 /�

e2 cos2 / X2

�u2 � e2 sin2 /�2" #

cos2 /

)h�

��u2 � e2p��

u2 � e2 sin2 /q o

oudW �X�

Model 3 Dc�x; X� �(

u e2�ÿ1� sin2 /��u2 � e2p �u2 � e2 sin2 /�3=2

C� 2GM u

�u2 � e2��u2 � e2 sin2 /�

� 1

3a2X2 �ÿ3u3 ÿ 5ue2�e� �3u4 � �6u2 � 3e2�e2�arccot�ue�

�u2 � e2��u2 � e2 sin2 /�� arccot�be� e2 � �ÿ3e� 3 arccot�be� b�b��3 sin2 /ÿ 1� � X2 u2 � e2

u2 � e2 sin2 /cos2 /

)h

��u2 � e2p��

u2 � e2 sin2 /q o

oudW �X� � O�e8�

Model 4 Dc�x; X� � 1��guup

XJ

j�1

X�j

m�ÿj

Du�Q�jm�ue��Q�jm�be�

ejm�k;/� ujm �Q�jm�ue�Q�jm�be�

Du�ejm�k;/��ujm � X2u cos2 /

( )" #h�

��u2 � e2 sin2 /

q��u2 � e2p o

oudW �X�

523

Table

4.SpheroidalStokes'operator

DC�x;X�:�

c�x�ÿ

C�X��r

EuC�X��h�r E

udW�X�(

spheroidalBruns'transform

ationinserted)

SpheroidalStokes'operator

Model

1Dc�x

;X��

GM

u

��

u2�

e2p

�u2�

e2sin2/�2�

1

�u2�

e2sin2/��

��u2�

e2p

!,

(1 ��

u2�

e2p

GM

��

��u2�

e2sin2/

q) dW

�X��

u2�

e2��

��

u2�

e2sin2/

qo ou

dW�X�

Model

2Dc�x

;X��ÿ( G

Mu

��

u2�

e2p

�u2�

e2sin2/�2�

1

�u2�

e2sin2/��

��u2�

e2p

"# �

u1 2

X4�u

2�

e2��4

u2�

e2�3e2sin2/�

GM�u

2�

e2sin2/�

e2cos2

/X

2

�u2�

e2sin2/�2

"# co

s2/

),

GM

��

u2�

e2p

��

��u2�

e2sin2/

q�

X2u��

��u2�

e2p

cos2

/��

��

u2�

e2sin2/

q8 > < > :

9 > = > ;dW�X��

��

u2�

e2p ��

��u2�

e2sin2/

qo ou

dW�X�

Model

3Dc�x

;X��ÿ(

ue2�ÿ

1�sin2/�

��

u2�

e2p

�u2�

e2sin2/�3=

2C�

2G

Mu

�u2�

e2��u

2�

e2sin2/��

1 3a2

X2

�ÿ3

u3ÿ5ue

2�e��3

u4��6

u2�3e2�e2�arccot�u e�

�u2�

e2��u

2�

e2sin2/��a

rccot�b e�e

2��ÿ

3e�3arccot�b e�b

�b��3

sin2/ÿ1�

�X

2u2�

e2

u2�

e2sin2/cos2

/

),(

��

u2�

e2p ��

��u2�

e2sin2/

q" ÿ

GM

u2�

e2ÿ1 3X

2a2

e�3u2�2e2��ÿ

3u3ÿ3

ue2�arccot�u e�

�u2�

e2��a

rccot�b e�e

2��ÿ

3e�3arccot�b e�b

�b��3

sin2/ÿ1��

X2ucos2

/

#) dW�X�

��

��u2�

e2p ��

��u2�

e2sin2/

qo ou

dW�X��

O�e8�

Model

4Dc�x

;X��ÿ

1 �� g uu

pXJ j�

1

X�j m�ÿ

j

Du�Q� jm�u e��

Q� jm�b e�

e jm�k;/�u

jm�

Q� jm�u e�

Q� jm�b e�D

u�e j

m�k;/��u

jm�

X2ucos2

/

()

"#

,"1 �� g u

up

( XJ j�1

X�j m�ÿ

j


Q� jm�b e�

e jm�k;/�u

jm�k;/��

Q� jm�u e�

Q� jm�b e�D

u�e j

m�k;/��

u jm�

X2ucos2

/

)# dW�x��

��

��u2�

e2sin2/

q��

��u2�

e2p

o oudW�X�

524

Corollary 2. (First model.)

U�u� � GMe

arccotue)

C�/; u� � 1��u2 � e2p GM��

u2 � e2 sin2 /q

DuU�u� � GMe�ÿ1� 1

1� u2e2

1

e� ÿGM

1

u2 � e2

gradU j Euh i ��u2 � e2p��

u2� e2 sin2 /q DuU � 1��

guup DuU �rN U

�ÿ GM��u2� e2p ��

u2� e2 sin2 /q

gradC j Euh i

��u2 � e2p��

u2 � e2 sin2 /q DuC � 1��

guup DuC � rNC

� GM u

��u2 � e2p

�u2 � e2 sin2 /�2 �1

�u2 � e2 sin2 /� ��u2 � e2p

" #

Corollary 3. (Second model.)

U�/; u� � GMe

arccotue

� �� 1

2X2�u2 � e2� cos2 /

D/U � ÿX2�u2 � e2� sin/ cos/

D/U � ÿ GMu2 � e2

� X2u cos2 /

�)C2 � sin2 / cos2 /

u2 � e2 sin2 /X4�u2 � e2�2

� u2 � e2

u2 � e2 sin2 /ÿ GM

u2 � e2� X2u cos2 /

� �2

C � GM��u2 � e2p ÿ 1��

u2 � e2 sin2 /q

�(1� 1

2

X4

�GM�2 �u2 � e2�2�u2 � e2 sin2 /� cos2 /

ÿ 2X2

GM�u2 � e2� cos2 /

)1=2

� GM��u2 � e2p ÿ 1��

u2 � e2 sin2 /q

�(1� 1

2

X4

�GM�2 �u2 � e2�2�u2 � e2 sin2 /� cos2 /

ÿ X2

GM�u2 � e2� cos2 /� O2

)or

C ��u2 � e2p��

u2 � e2 sin2 /q GM

u2 � e2� 1

2

X4

GM

�

��u2 � e2��u2 � e2 sin2 /� cos2 /ÿ X2 cos2 /� O2

�where O2 is of the order of centrifugal accelerationsquared.

gradU j Euh i

��u2 � e2p��

u2 � e2 sin2 /q DuU � 1��

guup DuU � rN U

� ÿ GM��u2 � e2p ��

u2 � e2 sin2 /q � X2u

��u2 � e2p

cos2 /��u2 � e2p

sin2 /

gradC j Euh i

��u2 � e2p��

u2 � e2 sin2 /q DuC � 1��

guup DuC � rN C

� GM u

��u2 � e2p

�u2 � e2 sin2 /�2 �1

�u2 � e2 sin2 /� ��u2 � e2p

" #

� u1

2

X4�u2 � e2��4u2 � e2 � 3e2 sin2 /�GM�u2 � e2 sin2 /�

"

� e2 cos2 /X2

�u2 � e2 sin2 /�

#cos2 /

Corollary 4. (Third model.)

U�/; u� � GMe

arccotue

� �� 1

6X2a2

3 u2e2 � 1

� �arccot u

e

ÿ �ÿ 3 ue

3 b2e2 � 1

ÿ �arccot b

e

ÿ �ÿ 3 be

�3 sin2 /ÿ 1�

� 1

2X2�u2 � e2� cos2 /

Table 5. Spheroidal Stokes' operator of the harmonic type up toorder �e3� (identical for all models)

Spheroidal Stokes' operator��u2 � e2 sin2 /

q��u2 � e2p Dc�x; X� � 2

udW �X� � o

oudW �X � � O�e3�

Fig. 5. Directional derivative of the normal gravity potential ®eldW �X� and of the modulus C�X� � C�X�k k2� gradW �X�k k2 of thegravity vector C�X� with respect to E2

a;a;b

525

D/U � a2X2 �3u2 � e2�arccot ue

ÿ �ÿ 3ue

�3b2 � e2�arccot be

ÿ �ÿ be� � sin/ cos/

ÿ X2�u2 � e2� sin/ cos/

DuU � ÿ GMu2 � e2

ÿ 1

3X2a2

� e�3u2 � 2e2� � �ÿ3u3 ÿ 3ue2�arccot ue

ÿ ��u2 � e2� arccot b

e

ÿ �e2 � �ÿ3e� 3arccot b

e

ÿ �b�b� �

� �3 sin2 /ÿ 1� � X2u cos2 /

grad U j Euh i

��u2 � e2p��

u2 � e2 sin2 /q DuU � 1��

guup DuU � ruU

��u2 � e2p��

u2 � e2 sin2 /q h

ÿ GMu2 � e2

ÿ 1

3X2a2

� e�3u2 � 2e2� � �ÿ3u3 ÿ 3ue2�arccot�ue��u2 � e2� arccot�be�e2 � �ÿ3e� 3 arccot�be�b�b

� �� 3 sin2 /ÿ 1� � X2u cos2 /

iC �

��gradU j gradUh i

p�

��D/U�2

u2 � e2 sin2 /� u2 � e2

u2 � e2 sin2 /�DuU�2

s

� 1��u2 � e2 sin2 /

q ��D/U�2 � �u2 � e2��DuU�2

q

C ��u2 � e2p��

u2 � e2 sin2 /q jDuU j

��1� �D/U�2�DuU�2

1

u2 � e2

s

C ��u2 � e2p��

u2 � e2 sin2 /q jDuU j 1� 1

2

�D/U�2�DuU�2

1

u2 � e2� O2

!

C ��u2 � e2p��

u2 � e2 sin2 /q jDuU j � 1

2

�D/U�2jDuU j

1

u2 � e2� O2

( )

if �D/U�=��DuU�2�u2 � e2�� < 1

�)

C��u2� e2p��

u2� e2 sin2/q (��ÿ GM

u2� e2ÿ 1

3X2a2

� e�3u2� 2e2�� ÿ3u3ÿ 3ue2�arccot�ue��u2� e2��arccot�be�e2��ÿ3e� 3arccot�be�b�b�

� �3sin2/ÿ 1��X2ucos2/��

� 1

2

�a2X2 �3u2� e2�arccot u

e

ÿ �ÿ 3ue

�3b2� e2�arccot be

ÿ �ÿ be� � sin/cos/

ÿ X2�u2 � e2� sin/ cos/

�.�ÿ GM

u2 � e2

ÿ 1

3X2a2

e�3u2 � 2e2� � �ÿ3u3 ÿ 3ue2�arccot ue

ÿ ��u2 � e2��arccot b

e

ÿ �e2 � �ÿ3e� 3 arccot b

e

ÿ �b�b�

� �3 sin2 /ÿ 1� � X2u cos2 /�� O2

)According to Fig. 8, �D/U�2=f2 �u2 � e2� DuUj jg con-tains only 7:35� 10ÿ9% of the whole gravity acceler-ation signal. The value of 7:35� 10ÿ9% is smaller thanthe order of magnitude of e8�e8 is the ®rst eccentricityof the reference ellipsoid). Accordingly, C can bewritten as

C ��u2 � e2p��

u2 � e2 sin2 /q ��ÿ GM

u2 � e2ÿ 1

3X2a2

� e�3u2 � 2e2� � �ÿ3u3 ÿ 3ue2�arccot ue

ÿ ��u2 � e2� arccot b

e

ÿ �e2 � �ÿ3e� 3 arccot b

e

ÿ �b�b� �

� �3 sin2 /ÿ 1� � X2u cos2 /�� O�e8�

�

Fig. 6. Variations of D/U=��g//p

versus / 2 �ÿ p2; � p

2� at the altitude

u � b� 104 m

Fig. 7. Variations of D/U=��g//p

versus u � b� 104m for / � ÿ p4

526

gradC j euh i

��u2� e2p��

u2� e2 sin2/q DuC� 1��

guup DuC�ruC

� ue2�ÿ1� sin2/��u2� e2�1=2�u2� e2 sin2/�3=2

C

� 2GMu

�u2� e2��u2� e2 sin2/��1

3a2X2

� �ÿ3u3ÿ 5ue2�e��3u4��6u2� 3e2�e2�arccot�ue��u2� e2��u2� e2 sin2/��arccot�be�e2��ÿ3e� 3arccot�be�b�b�

� �3sin2/ÿ 1��X2 u2� e2

u2� e2 sin2/cos2/�O�e8�

The computation in the various tables has been basedupon the fully normalized associated Legendre functionsof ®rst kind and second kind summarized in Table 9 aswell as their derivatives in Table 10.

Fig. 8. Variations of �D/U�2=f2 �u2 � e2� DuUj jg versus / 2 �ÿ p2 ;

p2�

at the altitude u � b� 104 m

Table 6. Spheroidal normal modulus of gravity ®eld C�k;/; u� � gradU�k;/; u�k k2Type of normal modulus of gravity C(functional space notation)

Type of the normal modulus of gravity C(explicit notation)

Model 1 1��guup GM

eDu Q00

ue

� �� e00 GM��u2 � e2p ��

u2 � e2 sin2 /q

Model 2(

1

guu

�GMe

Du Q�00ue

� �� e00 � 2

9X2e2Du P �2

ue

� �� e00

ÿ 2

9��5p X2e2Du P �2

ue

� �� e20�/�

�2� 1

g//�ÿ 2

9��5p X2e2 1� P �2

ue

� �� D/�e20�/� �

�2�1=2

��u2 � e2p��

u2 � e2 sin2 /q (

ÿ GMu2 � e2

� X2u cos2 /

� 1

2

X4�u2 � e2� sin2 / cos2 /

ÿ GMu2�e2 � X2u cos2 /

� O2

)

Model 31��guup GM

e

�� Du Q�00ue

� �� e00

� 1

3X2a2

�2Du P �2

ue

� �� Q�00

ue

� �� 2 P �2

ue

� �Du Q�00

ue

� �� ÿ 3Du P �1

ue

� ��

2 P �2be

� �Q�00

be

� �ÿ 3 P �1

be

� �� 2

9X2e2Du P �2

ue

� �� e00

ÿ 2

9��5p X2e2Du P �2

ue

� �� e20�/�

�� O�e8�

��u2 � e2p��

u2 � e2 sin2 /q ��ÿ GM

u2 � e2ÿ 1

3X2a2

� e�3u2 � e2� � �ÿ3u3 ÿ 3ue2�arccot�ue��u2 � e2��arccot�be�e2 � �ÿ3e� 3 arccot�be�b�b�

� �3 sin2 /ÿ 1� � X2u cos2 /� O�e8��

Model 4 1��guup

��XJ

j�1

X�j

m�ÿj

Du�Q�jm�ue� �Q�jm�be�

ejm�k;/� ujm

� Q�jm�ue�Q�jm�be�

Du�ejm�k;/� � ujm � 2

9X2e2Du P �2

ue

� �� e00

ÿ 2

9��5p X2e2Du P �2

ue

� �� e20�/�

�� O2

1��guup

��XJ

j�1

X�j

m�ÿj

Du�Q�jm�ue��Q�jm�be�

ejm�k;/� ujm

� Q�jm�ue�Q�jm�be�

Du�ejm�k;/�� ujm � X2u cos2 /

�� O2

527

Table

7.Directionalderivativeofthenorm

algravitypotential®eldrU�k;/;u��

grad

WjE

uh

i�D

uW=�� g u

up

Typeofnorm

almodulusofgravity

potential(functionalspace

notation)

Typeofnorm

almodulusofgravity

potential(explicitnotation)

Model

11 �� g u

up

GM e

Du

Q00

u e��

� e 00

ÿ1 ��

u2�

e2p

GM

��

��u2�

e2sin2/

qModel

21 �� g u

up

� GM e

Du

Q� 00

u e��

� e 00�2 9X

2e2

Du

P� 2

u e�� e 0

0

ÿ2

9�� 5pX

2e2

Du

P� 2

u e�� e 2

0�/��

ÿG

M��

��u2�

e2p

��

��u2�

e2sin2/

q�

X2u��

��u2�

e2p

cos2

/��

��

u2�

e2sin2/

q

Model

31 �� g u

up

� GM e

Du

Q� 00

u e��

� e 00�1 3

X2a2� 2

Du

P� 2

u e�� Q

� 00

u e��2

P� 2

u e�� Du

Q� 00

u e��

� ÿ3D

uP� 1

u e��

, � 2P� 2

b e��Q� 00

b e��ÿ3

P� 1

b e�� 2 9

X2e2

Du

P� 2

u e�� e 0

0ÿ

2

9�� 5p

X2e2

Du

P� 2

u e�� e 2

0�/��

��

u2�

e2p ��

��u2�

e2sin2/

q� ÿ

GM

u2�

e2ÿ1 3X

2a2

�e�3

u2ÿ2e2��ÿ

3u3ÿ3ue

2�arccot�u e�

�u�

e2��a

rccot�b e�e

2��ÿ

3e�3arccot�b e�b�b�

��3

sin2/ÿ1��

X2acos2

/

�

Model

41 �� g u

up

� XJ j�1

X�j m�ÿ

j


Q� jm�b e�

e jm�k;/�u

jm�

Q� jm�u e�

Q� jm�b e�

�D

u�e j

m�k;/��

u jm�2 9X

2e2

Du

P� 2

u e�� e 0

0ÿ

2

9�� 5pX

2e2

Du

P� 2

u e�� e 2

0�/��

1 �� g uu

p� XJ j�

1

X�j m�ÿ

j


Q� jm�b e�

e jm�k;/�u

jm�

Q� jm�u e�

Q� jm�b e�

�D

u�e j

m�k;/��

u jm�

X2ucos2

/

�

528

Table

8.Directionalderivativeofthenorm

almodulusofgravity®eldr E

uC�k;/;u��

grad

CjE

uh

i�D

uC=�� g u

up

Typeofnorm

almodulusofgravity

C(functionalspace

notation)

Typeofnorm

almodulusof

gravity(explicitnotation)

Model

11 �� g u

up

� Du

1 �� g uu

p�� G

M eD

uQ

00

u e��

��

� � �e 00�

1 �� g uu

pG

M eD

2 uQ

00

u e��

��

� � �e 00�G

Mu

��

u2�

e2p

�u2�

e2sin2/�2�

1

�u2�

e2sin2/��

��u2�

e2p

"#

Model

21 g u

u

1 2

� Du

1 g uu�� G

M eD

uQ� 00

u e��

� e 00�2 9X

2e2

Du

P� 2

u e�� e 0

0

ÿ2

9�� 5pX

2e2

Du

P� 2

u e�� e 2

0�/�� 2 �

2 g uu

Duu�U�

�D

u1 g /

/�� ÿ

2

9�� 5pX

2e2

1�

P� 2

u e��

� D/�e 2

0�/�� 2 �

2 g //

Du/�U��

�� 1 g u

u

� GM e

Du

Q� 00

u e��

� e 00�2 9X

2e2

Du

P� 2

u e�� e 0

0ÿ

2

9�� 5p

X2e2

Du

P� 2

u e�� e 2

0�/�� 2

�1 g /

/

� ÿ2

9�� 5pX

2e2

1�

P� 2

u e��

� D/�e 2

0�/�� 2� ÿ

1=2

GM

u

��

u2�

e2p

�u2�

e2sin2/�2�

1

�u2�

e2sin2/��

��u2�

e2p

"#

�u

1 2

X4�u

2�

e2��4

u2�

e2�3e2sin2/�

GM�u

2�

e2sin2�/�

"�

e2cos2

/X

2

�u2�

e2sin2/�2# co

s2/

Model

31 �� g u

up

� Du

1 �� g uu

p�� G

M e� � � �D

uQ� 00

u e��

� e 00�1 3

X2a2� 2

Du

P� 2

u e�� Q

� 00

u e��2

P� 2

u e�� Du

Q� 00

u e��

� ÿ3D

uP� 1

u e��

� 2P� 2

b e��Q� 00

b e��ÿ3

P� 1

b e��

�2 9X

2e2

Du

P� 2

u e�� e 0

0ÿ

2

9�� 5pX

2e2

Du

P� 2

u e�� e 2

0�/��

1 �� g uu

pD

uuU� �

O�e8�

ue2�ÿ

1��ÿ

1�sin2/�

�u2�

e2�1=

2�u

2�

e2sin2/�3=

2C�

2G

Mu

�u2�

e2��u

2�

e2sin2/��

1 3a2

X2

��ÿ

3u2ÿ5ue

2�e��3

u4��6

u2�3e2�e2

arccot�u e�

�u2�

e2��u

2�

e2sin2/��a

rccot�b e�e

2��ÿ

3e�3arccot�b e�b�b�

��3

sin2/ÿ1��

X2

u2�

e2

t2�

e2sin2/cos2

/�O�e��

Model

41 �� g u

up

� Du

1 �� g uu

p�� XJ j�

1

X�j m�ÿ

j


Q� jm�b e�

e jm�k;/�u

jm�

Q� jm�u e�

Q� jm�b e�D

u�e j

m�k;/��u

jm

�2 9X

2e2

Du

P� 2

u e�� e 0

0ÿ

2

9�� 5pX

2e2

Du

P� 2

u e�� e 2

0�/��

1 �� g uu

pjD

uu�W�j�O2

�1 �� g u

up

� Du

1 �� g uu

p�� XJ j�

1

X�j m�ÿ

j

Du�Q� jm

u eÿ� �Q� jm�b e�

e jm�k;/�u

jm�

Q� jm

u eÿ�Q� jm�b e�

�D

u�e j

m�k;/��

u jm�

X2ucos2

/

� � � ��1 �� g u

up

Duu�U��

O2

�

529

Table 9. Normalized associated Legendre functions of the ®rst kind P �nm�sin/�, P �nm�ue�; and the associated Legendre functions of the secondkind Q�nm�ue�n m P �nm�sin/� P �nm�ue� Q�nm�ue�0 0 1 1 arccot�ue�1 0

��3p

sin/ ue

1ÿ uearccot

ue

� �2 0

��5p

2�3 sin2 /ÿ 1� 1

23

u2

e2� 1

� �1

23

u2

e2� 1

� �arccot

ue

� �ÿ 3

ue

� �n m

��4n2 ÿ 1p��

n2 ÿ m2p sin/ P �nÿ1;m

ÿ��2n� 1��n� mÿ 1��nÿ mÿ 1�p ��n2 ÿ m2��2nÿ 3�p

� P �nÿ2;m 8 n 2 �3; 1� and m 2 �0; nÿ 2�

�n� m�!p n!

�Z p

0

ue�

��u2

e2� 1

rcos s

!n

� cosms ds

�ÿ1�m 2n�n� m�!m!

�nÿ m�!�2m�!

�Z 10

sinh2m s ds

�ue ��u2e2 � 1

qcosh s�n�m�1

Table 10. Derivatives of normalized associated Legendre functions of the ®rst kind D/�P �nm�sin/� � Du�P �nm�ue� �, and the associated Legendrefunction of the second kind Du�Q�nm�ue� �n m D/�P �nm�sin/� � Du�P �nm�ue� � Du�Q�nm�ue� �0 0 0 0 ÿ e

u2 � e2

1 0��3p

cos/ 1

eÿ 1

earccot

ue

� �� u

u2 � e2

2 0 3��5p

sin/ cos/ 3ue2 3

ue2arccot

ue

� �ÿ 1

2

3 u2 � e2

e �u2 � e2� ÿ3

2

1

e

n m��n�n� 1� ÿ m�mÿ 1��

4ÿ 2dlmp

sP �n;mÿ1

ÿ��n�n� 1� ÿ m�mÿ 1�p ��

2p P �n;m�1

subject to

dlm �1 for m � 1

0 for m 6� 1

�

�n� m�!p n!

Z p

0

�n�

u� ��u2 � e2p

cos se

�n

��u2 � e2

p� u cos s

� ��.n

u��u2 � e2

p� e2 cos s� u2 cos s

o� cosmsds

�ÿ1�m2n�n� m�!m!

�nÿ m�!�2m�!Z 10

ÿ�n� m� 1�

� sinh2m s�ue�

��u2

e2

r� 1 cosh s

�n�m�1

��u2 � e2

p� u cosh s

h i.h ��

u2 � e2p ��

u2 � e2 sin2 /q

cosh sids

Table 11. Spheroidal Bruns' transformation h � ÿdW �x�=rEu W �X�Spheroidal Bruns' transformation

Model 1 h � dW �x�,(

1��u2 � e2p GM��

u2 � e2 sin2 /q )

Model 2 h � ÿdW �x�,(

GM��u2 � e2p ��

u2 � e2 sin2 /q � X2u

��u2 � e2p

cos2 /��u2 � e2 sin2 /

q )

Model 3 h � ÿdW �x�,( ��

u2 � e2p��

u2 � e2 sin2 /q "

ÿ GMu2 � e2

ÿ 1

3X2a2

�e�3u2 � 2e2� � �ÿ3u3 ÿ 3ue2�arccot u

e

� ��u2 � e2� arccot

be

� �e2 � ÿ3e� 3 arccot

be

� �b

� �b

� � �3 sin2 /ÿ 1� � X2u cos2 /

#)

Model 4 h � ÿdW �x�,"

1��guup

(XJ

j�1

X�j

m�ÿj

Du Q�jmue

� �� Q�jm

be

� � ejm�k;/� ujm�k;/� �Q�jm

ue

� �Q�jm

be

� �Du�ejm�k;/�� ujm � X2u cos2 /

)#

530

Finally, we have succeeded in computing the sphe-roidal Bruns' formula (spheroidal Bruns' transforma-tion) in its linear form by means of Table 8, namely forall spheroidal models. Correspondingly, Table 9 con-tains the spheroidal Stokes' boundary operator (freeboundary value problem, linearized in spheroidal ge-ometry and gravity space) for all four models beforesubstitution of the spheroidal Bruns' formula. As soonas we implement the spheroidal Bruns' transformationwe arrive at the ®nal spheroidal Stokes' boundaryoperator in Table 10 which was the starting pointof Martinec and Grafarend (1997a, b) and Ritter(1998a, b) in solving the spheroidal Stokes' boundaryvalue problem.

Corollary 5. (Spheroidal Stokes' operator, harmonicStokes' operator.)

The free part of the ®xed±free two-boundary-valueproblem in its linearized version agrees with thespheroidal Stokes' boundary-value problem. Table 9reviews the spheroidal Stokes' operator before theimplementation of the Bruns' transformation (``Bruns'formula'') for the four di�erent normal gravity models,while Table 10 indicates the spheroidal Stokes' operatorafter the implementation of the Bruns' transformation(``Bruns' formula''). In contrast, Table 11 summarizesthe spheroidal Stokes' operator up to order e3 when it istransformed into its harmonic form, namely by multi-plication with

��guup

. Up to the order e3 all four modelsagree with each other.

Acknowledgements. This paper was prepared while the thirdauthor was an Alexander von Humboldt visiting scholar at theGeodetic Institute of the University of Stuttgart. This support,as well as Prof. Grafarend's hospitality, are gratefully acknowl-edged.

Appendix

Directional derivativeConsider a vector space V over the ®eld R of realnumbers equipped with a canonical di�erential struc-ture. We identify the tangent space TuV at u 2 V withthe vector space itself. We can make the followingstatements.

1. The directional derivative of a function w 2 C1�V� inthe direction of the vector v is denoted by

rvw :� hdw j vi � hgradw j vi dx;

2. the following rules apply:

(1) rv�w1 � w2� � rvw1 �rvw2

(2) rv�sw� � srvw� �rvs�w(3) rv1�v2w � rv1w�rv2w(4) rsvw � srvw;

3. in the language of di�erential geometry the directionalderivative is a ¯at a�ne connection on V withouttorsion;

4. Fig. 9 is an illustration of the various rules of thedirectional derivative.

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