Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid

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1 JOUNAL OF GEOPHYSICAL ESEACH: SOLID EATH, VOL. 118, , doi: /2013jb010457, 2013 Ellipsoidal topogaphic potntial: Nw solutions fo spctal fowad gavity modling of topogaphy with spct to a fnc llipsoid S. J. Classns 1 and C. Hit 1 civd 19 Jun 2013; visd 14 Octob 2013; accptd 16 Octob 2013; publishd 8 Novmb [1] Fowad gavity modling in th spctal domain taditionally lis on sphical appoximation. Howv, this lvl of appoximation is insufficint fo som psnt day high-accuacy applications. H w psnt two solutions that avoid th taditional sphical appoximation in spctal fowad gavity modling. Th fist solution (th xtndd intgation mthod) applis intgation ov masss fom a fnc sph to th topogaphy and applis a coction fo th masss btwn llipsoid and sph. Th scond solution (th hamonic combination mthod) computs topogaphic potntial cofficints fom a combination of sufac sphical hamonic cofficints of topogaphic hights abov th llipsoid, basd on a lation among sphical hamonic functions intoducd by Classns (2005). Using a dg 2160 sphical hamonic modl of th topogaphic masss, both mthods a applid to div th Eath s llipsoidal topogaphic potntial in sphical hamonics. Th hamonic combination mthod convgs fastst and akin to th EGM2008 gopotntial modl gnats additional sphical hamonic cofficints in spctal band 2161 to 2190 which a found cucial fo accuat valuation of th llipsoidal topogaphic potntial at high dgs. Thfo, w commnd us of th hamonic combination mthod to modl llipticity in spctal-domain fowad modling. Th mthod yilds llipsoidal topogaphic potntial cofficints which a compatibl with global Eath gopotntial modls constuctd in llipsoidal appoximation, such as EGM2008. It shows that th sphical appoximation significantly undstimats dg colation cofficints among gopotntial and topogaphic potntial. Th topogaphic potntial modl is, foxampl, of immdiat valu fo th calculation of Bougu gavity anomalis in fully llipsoidal appoximation. Citation: Classns, S. J., and C. Hit (2013), Ellipsoidal topogaphic potntial: Nw solutions fo spctal fowad gavity modling of topogaphy with spct to a fnc llipsoid, J. Gophys. s. Solid Eath, 118, , doi: /2013jb Intoduction [2] Modling of th gavitational potntial gnatd by th topogaphy of th Eath and oth clstial objcts has long bn an activ fild of sach. Knowldg of th topogaphic potntial is usful mainly bcaus th shot-wavlngth signal of obsvd gavity-latd quantitis is stongly dominatd by th contibution fom th topogaphy. It can thfo b usd to pdict a dtaild gavity fild wh no o only fw obsvations a availabl. This is impotant fo th constuction of high-solution Eath gavity modls [.g., Pavlis and app, 1990; Pavlis t al., 2012], modling of th gavity fild of clstial objcts such as th Moon, Mas, and Vnus [.g., Wiczok, 2007; Hit t al., 2012a], and th cation of 1 Wstn Austalian Cnt fo Godsy and Th Institut fo Goscinc sach, Cutin Univsity, Pth, Wstn Austalia, Austalia. Cosponding autho: S. J. Classns, Wstn Austalian Cnt fo Godsy and Th Institut fo Goscinc sach, Cutin Univsity, GPO Box U1987, Pth, WA 6845, Austalia. (s.classns@cutin.du.au) Amican Gophysical Union. All ights svd /13/ /2013JB synthtic Eath gavity modls [.g., Haagmans, 2000; Classns, 2003; Baghbandi and Sjöbg, 2012]. [3] A scond majo ang of applications uss th diffncs btwn a modl of th topogaphic potntial and th contibution of topogaphy fom obsvd gavity-latd quantitis. Most impotantly, this givs insight into mass igulaitis within th plant s intio [.g., Völgysi and Toth, 1992; Wiczok and Phillips, 1998]. Th sulting signal is much smooth than th actual gavity fild, which also facilitats data pdiction and downwad continuation of satllit obsvations [.g., Hck and Wild, 2005]. A latd subjct is that of tain coctions in goid dtmination accoding to Stoks s thoy, which quis th moval of all masss outsid th goid [.g., Sjöbg, 1998; Sun, 2002]. [4] Gnation of a topogaphic potntial modl quis fowad modling of mass contibutions though Nwton s intgal, ith in th spac domain o in th spctal domain. Topogaphy can b ith uncompnsatd [.g., Hit t al., 2012b], o an isostatic compnsation can b assumd [.g., umml t al., 1988; Gafand and Engls, 1993]. S Tsoulis [2001] and Göttl and umml [2009] fo a futh discussion on isostatic compnsation mchanisms and th topogaphic-isostatic potntial. 5991

2 [5] Many diffnt mthods fo fowad modling in th spac domain hav bn dvlopd; an ovviw of and compaisons btwn th diffnt mthods a povidd in Hck and Sitz [2007] and Tnzt al. [2010]. Compaisons of fowad modling in th spac and spctal domain a povidd by Kuhn and Sitz [2005], Wild-Pfiff and Hck [2007], and Balmino t al. [2012]. [6] Fowad modling in th spctal domain is computationally mo fficint and has bn widly usd fo sval dcads. Th solution of topogaphic potntial modls has incasd fom sphical hamonic dg and od 180 in th 1980s [app, 1982; umml t al., 1988] to 10,800 cntly [Balmino t al., 2012]. Th incas in solution dmands mo pcis modling mthodologis. [7] A common tchniqu mployd in spctal fowad modling is th us of a sis xpansion of pows of th topogaphic hight and sufac sphical hamonic cofficints (SHCs) of ths pows of topogaphic hight to gnat solid SHCs of th topogaphic potntial. Ealy contibutions hav usd a lina appoximation [.g., Lambck, 1979; app, 1982]. umml t al. [1988] xtndd this to thid-od pows and Balmino [1994] gnalizd it to high-od pows. Convgnc of th sis xpansion was studid by Sun and Sjöbg [2001], Novák [2010], and Hit and Kuhn [2012]. [8] On subjct that has civd littl attntion thus fa is th valuation of os intoducd by th sphical appoximation that is usd almost univsally. In spctal fowad modling, a mass sph is usd as a fnc, and th plant s topogaphy is assumd to sid on this sphical sufac. It is wll known that th Eath is to a much high dg of accuacy modld by an oblat llipsoid of volution. This is commonly accountd fo in th cation of global gavity modls [.g., Pavlis t al., 2012], but not in spctal fowad modling, which maks topogaphic potntial modls incompatibl with global gavity modls. Th topogaphic potntial gnatd taking into account th plant s llipticity is hin calld th llipsoidal topogaphic potntial (ETP). [9] Sjöbg [2004] divs llipsoidal coctions to topogaphic ffcts in goid modling, but this wok dos not povid a mthodology fo gnating th ETP. Futhmo, th coctions divd w limitd to th od of th squad fist numical ccnticity of th llipsoid 2, which is insufficint fo high dg and od SHCs. To ou knowldg, spctal fowad modling of th ETP has bn studid only by Novák and Gafand [2005], Balmino t al. [2012], and Wang and Yang [2013]. [10] Novák and Gafand [2005] modl th ETP and its vtical gadint by a sis of bas functions that a othonomal on th llipsoid [Gafand and Engls, 1992], using godtic coodinats. Ths bas functions a diffnt fom th sphical hamonic functions usd in global gavity modls, so th sulting xpansion of th ETP is not dictly compatibl with global gavity modls. Th appoach has also not bn applid globally, and th convgnc of th sis xpansions has not bn studid. Balmino t al. [2012] povid a mthod to comput th ETP using sufac sphical hamonic xpansions, but thy us th sphical appoximation fo thi numical computations (up to dg and od (d/o) 10,800). Thy did comput llipsoidal coctions but only fo long wavlngths (up to d/o 120). Wang and Yang [2013] us two mthods to comput th ETP: a sphical hamonic solution that quis a global intgation fovy dg n, and a solution using llipsoidal hamonics which is implmntd up to dg and od 180 only. [11] In this pap, two mthods that avoid th classical sphical appoximation in spctal domain fowad modling a intoducd. Both mthods us sufac sphical hamonic xpansions with spct to a fnc llipsoid. Us of only sphical hamonics has sval advantags ov llipsoidal hamonics: it is simpl, dos not qui th us of llipsoidal coodinats, and th sulting xpansion of th ETP is dictly compatibl with global gavity modls. It also avoids numical issus in th computation of llipsoidal hamonic functions [.g., Sona, 1995], although much impovmnt in this fild has bn mad cntly [.g., Sba t al., 2012; Fukushima, 2013]. [12] Th fist of ou two mthods is simila to on suggstd by Balmino t al. [2012]; it is also simila to th sphical hamonic solution by Wang and Yang [2013], but it uss binomial xpansions instad of but-foc computations that includ a global intgation fovy dg n. Th scond mthod is a nw, diffnt mthod which will pov to hav significant advantags. [13] Th two mthods a divd in sction 2. In sction 3, thy a compad to th sphical appoximation and to on anoth, both thotically and numically, and th sulting pow spctum of th ETP is compad to that of th Eath Gavitational Modl 2008 (EGM2008) global gavity modl [Pavlis t al., 2012]. Som xampls of applications a povidd in sction 4, and th final sction contains a discussion of th sults. 2. Mthods 2.1. Topogaphic Potntial [14] Th sphical hamonic xpansion of th gavitational potntial of a body is [.g., umml t al., 1988] VðPÞ ¼ GM n;m nþ1 V nm Y nmðpþ (1) P wh V(P) is th gavitational potntial in point P, G is th univsal gavitational constant, M is th mass of th body, is a fnc sph adius, P is th distanc btwn point P and th coodinat systm oigin, n, m a th sphical hamonic dg and od, Y nm a fully nomalizd (4π nomalizd) sphical hamonic functions, and th SHCs V nm a [umml t al., 1988] V nm ¼ 1 Mð2n þ 1Þ Q nρ ðqþy nm ðqþd Q (2) wh th intgation is ov th whol body (domain ) and ρ (Q) is th dnsity of th body in valuation point Q. In sphical coodinats, quation (2) ads V π 2π nm ¼ 1 ðθ;λþ Mð2n þ 1Þ Q nρ θ¼0 λ¼0 ¼0 ðqþy nm ðqþ 2 Q sinθ dθdλd (3) wh θ is th sphical pola colatitud, λ is th longitud, is th distanc fom th oigin and (θ,λ) is th distanc btwn th oigin and th body sufac. 5992

3 [15] Th topogaphic potntial is commonly dfind as th potntial gnatd by topogaphic masss, ith with spct to th goid [.g., Sjöbg, 1998] o th fnc llipsoid [.g., Novák and Gafand, 2005; Vajda t al., 2007]. A futh altnativ, lss common in godsy, is to dfin th topogaphy with spct to a sphical sufac, using topogaphic hights abov a sph [.g., Wiczok and Phillips, 1998]. Balmino t al. [2012] discuss th diffncs btwn ths dfinitions. H w us a dfinition with spct to th llipsoid. [16] W dfin th topogaphic potntial as th diffnc btwn potntials gnatd by (a) a body with igula topogaphy and dnsity distibution ρ (Q) (quation (3)) and (b) a fnc llipsoid with dnsity distibution ρ (Q), wh ρ (Q)=ρ (Q) fo all points Q that fall insid both th body () and th llipsoid. As a sult, it contains th combind ffct of topogaphic masss abov th llipsoid (wh tain hight is positiv) and th lack of topogaphic mass und th llipsoid (wh tain hight is ngativ). [17] Th SHCs of th topogaphic potntial a thn wh V nm ¼ 2 π Mð2n þ 1Þ V T ðθ; λþ ¼ 8 >< >: ¼ 2π θ¼0 λ¼0 ¼ V T ðθ; λþy nm ðθ; λþsinθ dθdλ (4) Q nþ2ρ Q Q nþ2ρ Q ð Þd fo > ð Þd fo < and is th distanc fom th oigin to an llipsoidal fnc sufac (th llipsoidal adius). Not that th squa of th fnc adius has bn movd outsid th intgals in quation (4) fo mathmatical convninc. To allow analytical intgation ov in quation (5), th dnsity is usually assumd adially invaiant. An altnativ that assums a vaiabl dnsity function as a pow sis of th adial distanc is povidd in amillin [2002]. In th cas of adial invaianc, th intgal in quation (5) is simpl, and idntical fo both cass V T ðθ; λ Þ ¼ ρθ; ð λþ n þ 3 wh ρ(θ, λ)=ρ fo > and ρ(θ, λ)=ρ fo <. [18] Wh infomation about adial vaiations in dnsity within th topogaphy is availabl, th topogaphy can b placd by a lay of constant dnsity and th sam mass as th oiginal lay: th quivalnt ock topogaphy/ockquivalnt topogaphy (ET/ET) [.g., Balmino t al., 1973; Tsoulis, 1999; Hit t al., 2012b]. Th hight of this lay, th ock-quivalnt hight, can b computd in plana appoximation [.g., Balmino t al., 1973; umml t al., 1988; Hit t al., 2012b] o in sphical appoximation [.g., Classns, 2003; Mladk, 2006]. It is customay to plac ocan wat, fsh lak wat, and ic by quivalnt ock lays, sulting in ngativ ET hights ov all of Eath s ocans [.g., Hit t al., 2012b]. [19] Latal vaiations in dnsity can b accommodatd by using sufac dnsity functions [Kuhn and Fathston, 2003], by using diffnt sufac hamonic analyss ov vaious domains [Balmino t al., 2012], o by including th (5) (6) dnsity function in th global intgation within th sphical hamonic analyss [Eshagh, 2009; Tnzt al., 2012]. In th maind of this pap, w assum constant dnsity of ockquivalnt topogaphy (ρ(θ, λ)=ρ) fo th sak of simplicity, but ou sults can b xtndd to accommodat latally vaiant dnsity using on of th abov mntiond mthods. An isostatic compnsation mchanism can also b applid to gnat th so-calld topogaphic-isostatic potntial [.g., app, 1982; Sünkl, 1986; umml t al., 1988]. H w only consid th uncompnsatd topogaphic potntial, but ou sults can asily b xtndd to also includ an isostatic compnsation pat Ellipsoidal Topogaphic Potntial [20] In pactical applications of spctal fowad modling of th topogaphic potntial, a sphical appoximation is commonly applid to simplify quation (6). Th appoximations mad a and ¼ (7) ¼ þ H (8) wh H is th othomtic hight of th topogaphy. Howv, this sphical appoximation is no long sufficint, spcially fo spctal analysis of high-dg and high-od SHCs. [21] Instad of th sphical appoximations in quations (7) and (8), w us quation (6) in unaltd fom. Th sphical hamonic synthsis in quation (4), with quation (6), could b pfomd numically [Wang and Yang, 2013], but this is computationally infficint bcaus V T is dpndnt on sphical hamonic dg n. To mak th computations mo fficint, a binomial xpansion can b applid to th tms in quation (6) that a dpndnt on n. This is commonly don in sphical appoximation [.g., umml t al., 1988; Wiczok and Phillips, 1998], and can also b applid in th cunt llipsoidal appoximation. In sphical appoximation, th scond tm btwn th squa backts in quation (6) cancls, but it nds to b takn into account in llipsoidal appoximation. Blow, w div two diffnt mthods to do this Mthod 1: Extndd Intgation (EI) Mthod [22] Th fist tm within th squa backts in quation (6) can b xpandd into a binomial sis [cf. Classns, 2006] wh n þ 3 ¼ k k¼0 ¼ 1 þ 1 k¼1 k! k j¼1 l k ðn þ 4 jþ l k (9) l ¼ (10) [23] Not that th fnc adius of th sphical hamonic xpansion is commonly st qual to th smimajo axis of th godtic fnc llipsoid. Givn th Eath s 5993

4 flattning, l achs valus with an absolut magnitud in xcss of 20 km na th pols on Eath. [24] A simila binomial sis can b applid to th scond tm within th squa backts in quation (6) wh ¼ k¼0 n þ 3 k k l (11) l ¼ (12) [25] Insting quations (9) and (11) into quation (6) givs V T ðqþ ¼ ρ " k n þ 3 n þ 3 l l # k (13) k k¼1 [26] Th summation uns fom k = 1, bcaus th tm with k = 0 vanishs. Insting quation (13) into quation (4) and aanging th od of summation and intgation givs V nm ¼ ρ 3 Mð2n þ 1Þðn þ 3Þ n þ 3 k¼1 k " l k # l k σ Y nm ðqþdσ Y nm ðqþdσ σ ð14þ [27] Th two intgations ov th unit sph can b combind into on, but w spaat thm h, as it povids a usful intptation of th pocss whn compad to th sphical appoximation. Equation (14) can b simplifid to V nm ¼ 4πρ 3 n þ 3 l ðþ k nm l ðþ k nm (15) Mð2n þ 1Þðn þ 3Þ k k¼1 wh w hav intoducd th following fully nomalizd sufac sphical hamonic sis l k ¼ l ðþ k nm Y nm (16) n;m with and with l ðþ k nm ¼ 1 4π l k Y nmdσ (17) σ l k ¼ l ðþ k nm Y nm (18) n;m l ðþ k nm ¼ 1 4π l k Y nmdσ (19) σ [28] This shows that it is possibl to modl th topogaphic potntial with spct to th llipsoid using only sphical hamonics. [29] Compaing quation (15) to th solutions in sphical appoximation of umml t al. [1988] and Wiczok and Phillips [1998], it is obvious that this mthod ssntially computs th contibution fom th sph to th topogaphy (takn with spct to th llipsoid, sulting in intgation ov a gnally xtndd ang) and thn subtacts th contibution fom th mass btwn th sph and th llipsoid. Balmino t al. [2012] hav divd a solution simila to this but appa not to hav implmntd it. [30] Bcaus th sis in quation (15) convgs, not all n + 3 tms nd to b takn into account but th sis can b tuncatd aft sufficint pcision has bn obtaind. If applid to th Eath, sis convgnc is slow than in sphical appoximation, bcaus l and l ach significantly lag magnituds than th ock-quivalnt hights. Th at of convgnc is shown in sction Mthod 2: Hamonic Combination (HC) Mthod [31] A scond, nw mthod avoids th us of l and l, which a lag ov much of th Eath s sufac. It is basd on a diffnt binomial xpansion of th scond tm in quation (6), taking into account that th llipsoidal sufac is asily dscibd mathmatically as a function of latitud [.g., Classns, 2006]. It also lis on a lation among sphical hamonic functions divd by Classns [2005]. [32] Fist, quation (6) is wittn as follows: " V T ðqþ ¼ ρ # 1 (20) n þ 3 [33] W now apply a binomial sis xpansion to th tm btwn squa backts 1 ¼ n þ 3 d k k wh k¼1 ¼ 1 k¼1 k! k j¼1 ðn þ 4 jþ d k (21) d ¼ (22) [34] Th distanc d closly appoximats th llipsoidal hight of th ock-quivalnt topogaphy, but is masud along th diction to th llipsoid s oigin. Insting quations (20) and (21) into quation (4) givs, aft changing th od of intgation and summation V nm ¼ 2 ρ Mð2n þ 1Þn þ 3 n þ 3 k k¼1 σ d k Y nm ðqþdσ (23) [35] W now apply a scond binomial sis to th fist tm within th intgal [cf. Classns, 2006] 1 2 sin 2 θ b ¼ ¼ b j¼0 2 0 ð 1Þ j n þ 3 2 A 2j sin 2j θ (24) j [36] As is common in godsy, w hav h assumd that th fnc llipsoid is an oblat llipsoid of volution 5994

5 among sphical hamonic functions of qual od m [Classns, 2005, quation (27)] sin 2j θy nm ¼ j i¼ j nm Y nþ2i;m (25) wh nm a fully nomalizd sinusoidal Lgnd wight functions [Classns, 2005, 2006], which can b computd though th cusion lations in Appndix A. Insting quations (24) and (25) into quation (23) givs Figu 1. EI mthod: Potntial contibutions of th fist 25 intg pows of th topogaphy. Contibution of 1st pow (blu), contibution of 25th pow (d). Total contibution (black lin). Shown a dimnsionlss potntial dg vaiancs of th diffncs (l l ). dfind by its smimajo axis a and smimino axis b o squad fist numical ccnticity 2. Not th diffnc with th binomials sis usd in mthod 1 (quation (11)). Th sis in quation (24) is infinit, but Classns [2006] has shown that it always convgs. Convgnc is most apid fo low dgs n. W also apply th following lation V nm ¼ 2 ρ b Mð2n þ 1Þn þ 3! 0 n þ 3 ð 1Þ j n þ 3 2 A 2j j k¼1 k j¼0 j nm i¼ j σ d k Y nþ2i;m Q ð Þdσ (26) [37] Intoducing th following fully nomalizd sufac sphical hamonic sis wh d k ¼ d ðþ k nm Y nm (27) n;m d ðþ k nm ¼ 1 4π d k Y nmdσ (28) σ givs th final xpssion fo th solid SHCs of th ETP Figu 2. EI mthod: Potntial contibutions of th fist th intg pows of th topogaphy. Shown a dimnsionlss potntial dg vaiancs of l, l, and (l l ). 5995

6 Figu 3. HC mthod: Potntial contibutions of th fist 10 intg pows of th topogaphy. Contibution of 1st pow (blu), contibution of 10th pow (d). Shown a dimnsionlss potntial dg vaiancs in spctal band of dgs 0 to V nm ¼ 4πρb 3 b n n þ 3 Mð2n þ 1Þðn þ 3Þ k¼1 k 0 ð 1Þ j n þ 3 2 A 2j j nm d ðþ k j¼0 i¼ j j nþ2i;m ð29þ [38] This mthod thus lis on a combination of sufac SHCs of qual od m. Th summations ov k and j can b tuncatd; th at of convgnc is shown in sction 3.3. Whn a sphical fnc sufac is slctd, th solutions of both mthods (quations (15) and (29)) dgnat into th wll-known sphical appoximation [.g., umml t al., 1988; Wiczok and Phillips, 1998] V nm ¼ 4πρ 3 Mð2n þ 1Þðn þ 3Þ n þ 3 H ðþ k nm (30) k wh k¼1 HC mthod (cf. sction 2.3 and 2.4) spaatly, and (ii) compa th mthods to gain insight into similaitis and diffncs. In all tsts, w us th ET2012 ock-quivalnt topogaphy modl dvlopd at Cutin Univsity. ET2012 is a sphical hamonic modl of Eath s uncompnsatd topogaphic masss complt to dg and od 2160, which cosponds to 5 ac min spatial solution. It dscibs th masss of (i) Eath s visibl topogaphy, (ii) ocan wat, (iii) majo ic shts of Gnland and Antactica, and (iv) majo inland laks (of Noth Amica and Asia) using a singl constant mass dnsity of 2670 kg m 3.Thcompssionof wat and ic masss was accomplishd as dscibd in Hit t al. [2012b, sction 3.2] fo a dg 360 pdcsso of th dg 2160 ET2012 modl. Full dtails on data sts and mthodsusdisinhit [2013, Appndix A]. Th SHCs of ET2012 a publicly availabl via du.au/sach/modls/eath2012/, fil Eath2012.ET2012. SHCto2160.dat. [41] Ou numical tsts us th gomtical and physical paamts of th Godtic fnc Systm 1980 (GS80) fnc llipsoid: smimajo axis a = 6,378,137 m, smimino axis b = 6,356, m, and GM = m 3 s 2 [Moitz, 2000]. With th CODATA (Committ on Data fo Scinc and Tchnology) numical valu fo G = m 3 kg 1 s 2 [Moht al., 2012, p. 72], it follows fo Eath s mass: M = kg. Fo all sphical appoximations tstd in this study, w us th GS80 smimajo axis a as th fnc sph adius. [42] Tsting of th two mthods dscibd in sction 2 quis gocntic adii of th topogaphy which w obtaind fom xpanding th ET2012 topogaphy to dg and od Th quantitis l (quation (10)), l (quation (12)), and d (quation (22)) and th topogaphic hight functions (THF) l /, l /, and d / w ppad in tms of 2 ac min global gids (howv, du to th limitd solution of th ET2012 modl ths gids do not contain infomation at spatial scals small than 5 ac min). Th THFs w thn aisd to intg pows k (anging fom 1 up to 25) and th sulting (l /) k, (l /) k, and (d / ) k hamonically analyzd to giv sts of SHCs l ðþ k nm, l ðþ k nm, and d ðþ k nm. H ðþ k nm ¼ 1 4π H k Y nmdσ (31) σ [39] Balmino t al. [2012] div an llipsoidal coction to th sphical appoximation which, lik ou solution, involvs a summation ov sufac SHCs of qual od m. Howv, thi coctions us an xpansion of th llipsoidal adius to th fist od of th llipsoid s flattning. This is akin to tuncating quation (24) aft j = 1, which is insufficint fo high dg SHCs du to th appaanc of dg n in th binomial cofficint. 3. Numical Study 3.1. Gnal maks [40] Th pimay pupos of th numical study is to (i) analyz th convgnc bhavio of th EI mthod and th Figu 4. HC mthod: As Figu 3, but focus on spctal band of dgs 2150 to

7 Figu 5. Compaison among th mthods in th spctal domain, (a) potntial dg vaiancs of th topogaphic potntial in sphical and llipsoidal appoximation (mthods EI and HC), and of EGM2008, all in spctal band 0 to 2220 and 2150 to 2200 (clos-up), (b) as bfo, but in spctal band 0 to 300, (c) ETP dg vaiancs of th EI and HC mthods, and thi diffncs in spctal band 0 to Not that sufac sphical hamonic xpansions a not stictd to data bing on a sph [.g., Jkli, 1988]. All sphical hamonic analyss w caid out to dg and od 2699 with th algoithm of Discoll and Haly [1994] as implmntd in th SHTools packag ( ipgp.f/). Not that th sulting xpansions lack pow in th highst dgs du to th limitd solution of th ET2012 modl. Howv, ths xpansions w only usd to dg 2160 (EI mthod) and 2220 (HC mthod) Mthod 1: Extndd Intgation (EI) Mthod [43] W invstigatd th convgnc of th llipsoidal topogaphic potntial fom th EI mthod by valuating quation (18) spaatly fo intg pows of th THF fom k = 1 to k = 2, to k = 25. Th (dimnsionlss) potntial dg vaiancs of th sulting contibutions V ðþ k nm a shown in Figu 1 togth with th total (accumulativ) V nm sulting fom addition of th fist 25 contibutions. [44] Fom Figu 1, intg contibutions a quid up to k = 22 to sufficintly convg at dg and od This is substantially slow than in th sphical cas wh convgnc is achd with k = 7 [cf. Hit and Kuhn, 2012, Figu 1]. Th dg vaiancs of th singl contibutions xhibit numous intsctions in spctal band of dgs ~700 to 2160, showing that much of th high-dg spctal ngy is dlivd by th high-od pows. In spctal band ~1000 to 2160, pows k = 5 to 15 of th THFs mak lag contibution than th low-intg pows 1 to 4. This bhavio is vy diffnt to th sphical cas, wh in spctal band of 0 to 2160 ach intg pow of th THF maks a contibution small than th pvious on [cf. Hit and Kuhn, 2012, Figu 1], with th fist intsction obsvd only aound dg ~3000 [cf. Balmino t al., 2012, Figu 7]. [45] Figu 1 also shows that ov most pats of th spctum th a always singl contibutions V ðþ k nm that hav high spctal ngy than th total contibution V nm, and 5997

8 btwn th llipsoid and sph a of vy long wavlngth chaact (Figu 2). Akin to th potntial cofficints of a nomal gavity fild implid by a fnc lvl llipsoid (.g., GS80), th spctal pow of l/ is stictd to th vn low-dg zonal hamonics [.g., Moitz, 2000, p. 130], and ngligibl fo hamonic dgs of ~12 and lag (Figu 2). [47] A dtaild inspction of th l/, l/, and (l/ l/) contibutions vals that at vn, low hamonic dgs th l/ contibution is always lag than that of th diffnc (l/ l/); hnc, l/ ducs th ngy of l/ (not th duction of spik-lik ffcts in l/ in Figu 2). lativ to th total contibution shown in Figu 1, th spctal ngy of th l/ contibution is at last 10 ods of magnituds small fo k 4, so can b safly nglctd fo all high intg pows. Figu 6. Compaison among th mthods in th spatial domain, (a) Ellipsoidal ffct: diffncs among gavity distubancs fom HC mthod in llipsoidal appoximation (band 0 to 2190) and in sphical appoximation (band 0 to 2160), min/max/man/ms = 2.9/4.7/0.6/1.2 mgal, (b) Diffncs among gavity distubancs fom th HC mthod (band 0 to 2190) and th EI mthod (band 0 to 2160), min/ max/man/ms = 180/193/0/17 mgal. this ffct bcoms mo ponouncd th shot th spatial scals. At dg 2160, th spctal pow of th fist 16 intg contibutions is lag than that of th total contibution. Hnc, addition of succssiv contibutions has som canclation ffct on th total contibution. Notwithstanding this obsvation, with k = 22 th EI mthod quis a lag numb of intg pow contibutions to convg, and this is owing to th fact that th THFs a much lag than in th sphical cas. [46] Fo th fist fou intg pows (k = 1 to k = 4), w analyzd th potntial contibution mad by th two THFs l/ and l/ usd as input in th EI mthod (quation (15)). As xpctd, th l/ contibutions thos of th masss 3.3. Mthod 2: Hamonic Combination (HC) Mthod [48] Th convgnc bhavio of mthod 2 (HC mthod) was invstigatd by valuating quation (29) fo all indics k = 1 to k = 10 spaatly. Th inn summations (ov j) w valuatd to jmax = 30 which nsus convgnc of ths tms [cf. Classns, 2006, p. 140; Classns and Fathston, 2005, Figu 2]. Figu 3 shows th singl contibutions mad by th fist 10 intg pows of th THF d/. In contast to th EI mthod, sufficint convgnc is alady achd fo k = 7, which is compaabl to th topogaphic potntial contibutions in sphical appoximation [Hit and Kuhn, 2012, Figu 1]. In a lativ sns, th bhavio of th contibutions shown in Figu 3 is compaabl to th sphical cas, and th a no intsctions in th spctal band of dgs 0 to Du to this fast convgnc, th HC mthod is computationally mo fficint than th EI mthod. [49] Th most impotant obsvation is mad aound hamonic dg 2160 wh all contibutions xpinc a dop in spctal ngy. Figu 4 povids a dtail plot of all contibutions in spctal band 2150 to 2195, showing that ðk Þ th tms V nm byond dg 2160 mak som notabl contibution to about 2175, whil diminishing aound dg This flcts an impotant attibut of th HC mthod. ðk Þ ðk Þ Each cofficint V nm dpnds on a goup of SHCs d nþ2i;m within a spctal backt of 2 jmax (60 in th psnt cas) to ith sid of sphical hamonic dg n, sulting in additional SHCs of up to dg 2220 in th psnt cas. Figu 7. Gavity distubancs fom th HC mthod ov Euop in spctal (a) band 721 to 2160, (b) band 2161 to 2190, and (c) band 721 to 2190, units in mgal. 5998

9 Figu 8. Maximum diffncs btwn gavity distubancs fom th EI and HC mthods along paallls fo fou diffnt spctal bands (0 to 720, 0 to 1800, 0 to 2100, and 0 to 2160), units in mgal. Howv, bcaus of th convgnc of th summation ov j in quation (29), th cofficints V ðþ k nm bcom ngligibl byond hamonic dg ~2180. [50] Th obsvd bhavio is a ky chaactistic of llipsoidal potntial modling [Classns, 2006] and also sn in high-dg gopotntial modls such as EGM2008 [Pavlis t al., 2012] that a basd on llipsoidal appoximation. EGM2008 was dvlopd in llipsoidal hamonics to dg and od 2160 and tansfomd to sphical hamonics using th tansfomation dscibd in Jkli [1988]. In th cas of EGM2008, Jkli s tansfomation givs is to additional SHCs in th spctal band of dgs 2161 to 2190 as discussd in dtail by Holms and Pavlis [2007]. In dict analogy to EGM2008, considation of ths additional SHCs is cucially impotant to accuatly psnt th ETP, as will b dmonstatd in sction Compaisons in th Spctal Domain [51] Figu 5a compas dg vaiancs of th (total contibutions fom th) EI and HC mthods with ach oth, with thos fom (convntional) topogaphic potntial modling in sphical appoximation (quation (30)), and with dg vaiancs fom th EGM2008 global gavity modl. Fo asons of consistncy, th latt w computd fom th SHCs of EGM2008, not fom llipsoidal hamonic cofficints which a also availabl. [52] Th dg vaiancs fom th two llipsoidal mthods (EI and HC) a in clos agmnt ov most of th spctum. Th spctum of th topogaphic potntial in sphical appoximation has smingly mo pow as th dg incass, with diffncs of about on od of magnitud at n = Ths diffncs a as xpctd, givn diffnt fnc sufacs (sufac of sph vsus sufac of llipsoid) w usd in th cation of th SHCs in sphical and llipsoidal appoximation. Th fnc sph adius in sphical appoximation was st qual to th smimajo axis a (th customay valu), which placs th topogaphy futh fom th Eath s oigin compad to th llipsoidal solution, sulting in mo pow at high dgs. [53] Figu 5b shows that th signals fom th two ETP mthods a commnsuat with EGM2008 fom hamonic dg of ~250 and high, whil th topogaphic potntial has significantly high pow at low hamonic dgs. This wll-known bhavio is causd by isostatic compnsation masss at mdium and long wavlngths, which a not modld by th (uncompnsatd) ET2012 topogaphy and divd potntial cofficints, but a constitunt of Eath s obsvd gavity fild, s also umml t al. [1988], Watts [2001], Wiczok [2007], and Hit t al. [2012a]. [54] Figu 5a (insid panl) povids a dtail viw on th spcta of th fou potntial modls in spctal band 2150 to 2200, xmplifying th simila chaactistics of EGM2008 and th ETP fom th HC mthod (sction 3.3). Both modls povid additional SHCs byond dg 2160, which apidly loos spctal pow and ach th lvl of (this is 10 ods of magnitud small than th signal) na dg 2190 fo EGM2008 and na dg 2180 fo th ETP. [55] Figu 5c compas th dg vaiancs of th two ETP mthods, and thos of thi cofficint diffncs. Th dg vaiancs of th cofficint diffncs (i.., th diffnc spctum) a found to b 5 to 7 ods of magnitud small than th signal of th topogaphic potntial itslf. This indicats a asonabl agmnt among th mthods ov most of th spctum. Impotantly, th spcta of th HC and EI mthods incasingly dviat fom ach oth at high spatial dgs, as is indicatd by th diffnc spctum. At dg 2160, th diffnc spctum is lss than 1 od of magnitud blow th signal cuv, which points at a significant discpancy among th two mthods vy clos to th maximum dg Compaisons in th Spac Domain [56] In od to futh invstigat th discpancis among th two mthods, adial divativs of th topogaphic potntial (also known as gavity distubancs, shot: gavity) w calculatd at th sufac of th GS80 llipsoid (HC and EI mthods), and at th sufac of th sph with adius (fom th topogaphic potntial modl in sphical appoximation). Fom Figu 6a, llipsoidal topogaphic gavity fom th HC mthod and gavity in sphical appoximation a in clos agmnt, with th diffncs (MS 1 mgal, maximum Figu 9. Ellipsoidal ffct: diffncs among hight anomalis fom HC mthod in llipsoidal appoximation (band 0 to 2190) and in sphical appoximation (band 0 to 2160), units in m. 5999

10 shown in Figu 9. Ths diffncs ach a magnitud of ~15 m, and a pdominantly of a long-wavlngth natu. Figu 10. EGM2008 Bougu gavity distubancs at th Eath s sufac in fully llipsoidal appoximation in spctal band 0 to 2190, topogaphic gavity distubancs fom th HC mthod, min/max/man/ms = 964/455/-34/201 mgal. diffnc 4.7 mgal) likly flcting th ffct of diffnt mass aangmnt in th two appoximations. Diffncs in gavity fom th two ETP mthods xhibit lag latitud-dpndnt discpancis that incas towad th pols (Figu 6b) to magnituds as lag as ~150 mgal. Ths discpancis a causd by th lack of cofficints byond dg and od 2160 in th EI mthod, as xmplifid in th nxt paagaph. W not that th ETP fom th HC mthod was valuatd in ou tsts to dg 2190, and not to dg 2160 (.g., Figu 6a). [57] Figu 7 shows th impotanc of taking into account th SHCs byond dg 2160 fo th accuat valuation of ETP at high dgs. sticting th valuation to dg 2160 poducs latitud-dpndnt pattns in high latituds, which incas towad th pols and ach ~100 mgal amplituds (Figu 7a). Similaffcts w potd by Holms and Pavlis [2007] fo a pdcsso modl of EGM2008 if tuncatd to dg Evaluation of th SHCs byond hamonic dg 2160 poducs almost idntical pattns, howv, with opposit sign (Figu 7b), which is why valuation to dg 2190 is f of any latitud-dpndnt pattns (Figu 7c). [58] Ths compaisons povid vidnc that (i) th EI and HC mthods a not igoously compatibl, and fom Figus 6 and 7 (ii) th latitud-dpndnt os a unambiguously attibutabl to th EI mthod. W finally attmptd to naow th discpancis among th HC and EI mthods, by valuating gavity distubancs fom both mthods in spctal bands of hamonic dgs 0 to 720, 0 to 1800, 0 to 2100, and 0 to 2160, and analyzing thi diffncs along latitud bands (simila to Holms and Pavlis [2007]). Figu 8 shows th maximum diffnc as a function of th latitud, and spctal bands. Th agmnt among gavity fom both appoachs is btt than 0.1 mgal (xpandd to dg 720) and btt than 0.5 mgal (to dg 1800) anywh on Eath (cf. Figu 8), which is satisfactoy. Howv, th maximum discpancis incas to ~5 mgal (whn valuating to dg 2100) and dtioat to ~150 mgal (dg 2160). Togth with Figu 6, this shows that th poblms with th EI mthod chifly sid in th high dgs and high latituds, whil th HC mthod is f of thos ffcts (s Figus 6a and 7). [59] Th diffncs in tms of hight anomalis btwn th topogaphic potntial in sphical appoximation and th llipsoidal topogaphic potntial (using th HC mthod) a 4. Application Exampls [60] W applid th HC mthod (sction 2.4) along with th ET2012 topogaphy modl (sction 3.1) fo computation of th fist dg 2190 EGM2008 Bougu gavity map in fully llipsoidal appoximation. W computd gavity distubancs fom (i) EGM2008 and (ii) ET2012/HC in full solution, i.., fom dg 2 to 2190 at th Eath s sufac in tms of 5 ac min solution gids. This was accomplishd by calculating gavity distubancs and thi fist fiv adial divativs fom both modls at a fnc hight of 4000 m abov th GS80 fnc llipsoid and continuation of gavity distubancs fom th fnc hight to th Eath s sufac using Tayloxpansions as dscibd in Hit [2012] fo EGM2008 and Hit and Kuhn [2012] fo th topogaphic potntial. [61] Th Eath s sufac was psntd by th Eath2012 sufac modl ( au/sach/modls/eath2012/, fil Eath2012.topo_ai. SHCto2160.dat). EGM2008 Bougu gavity distubancs, obtaind as diffnc btwn EGM2008 and ET2012 implid gavity ffcts in llipsoidal appoximation, a shown in Figu 10. Th map concptually impovs on th pviously publishd map by Balmino t al. [2012], which is basd on a mixtu of appoximation lvls (topogaphy-implid gavity ffcts in sphical appoximation with only low-dg llipsoidal coctions, combind with EGM2008 in full llipsoidal appoximation). Fom Figu 6a, th llipsoidal ffct (i.., diffncs among sphical and llipsoidal appoximation) on th topogaphy-implid gavity is at th mgal lvl, so compaativly small but nonngligibl fo accuat applications. [62] As a scond application xampl, w computd dg colation cofficints among EGM2008, and th ET2012 topogaphic potntial modl in llipsoidal (using th HC appoach) and sphical appoximation (Figu 11). Th colation among EGM2008 and th topogaphic potntial in sphical appoximation incass at low and mdium dgs, achs a maximum of about aound dg 500 bfo dcasing to +0.6 at dg Howv, a mo alistic pictu of th EGM2008 quality is obtaind fom Figu 11. Dg colation cofficints among SHCs of EGM2008 and of th implid topogaphic potntial in sphical and llipsoidal appoximation (HC mthod). 6000

11 th llipsoidal topogaphic potntial, with colation cofficints found to b as lag as aound dg 1000, and at dg To ou knowldg, this high colation btwn gopotntial and topogaphic potntial cofficints has not bn obsvd bfo. It is obvious that topogaphic potntial SHCs in sphical appoximation considably undstimat th colation, indicating poo modl quality at shot spatial scals, which maks thm of littl us fovaluation of high-dg gopotntial modls such as EGM2008 which a dvlopd in llipsoidal appoximation. K 2;2 nm K 2;2 nm ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i uðn 2 m 2 Þ ðn þ 1Þ 2 m 2 ¼ t ð2n 3Þð2n 1Þ 2 ð2n þ 1Þ K 0;2 nm ¼ 2 ð n2 þ m 2 þ n 1Þ ð2n 1Þð2n þ 3Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ih i u t ðn þ 1Þ 2 m 2 ðn þ 2Þ 2 m 2 ð2n þ 1Þð2n þ 3Þ 2 ð2n þ 5Þ (A5) (A6) (A7) 5. Discussion, Conclusions, and commndations [63] Th ffct of th sphical appoximation in fowad gavity modling has bn shown to b significant in both th spatial and th spctal domain, spcially affcting th pow of high-dg topogaphic potntial SHCs. It is thfo most cucial fo quantitis with substantial pow in th high dgs, and fo computation of global gavity modls o any typ of spctal analysis. Th two mthods intoducd h fo modling th llipsoidal topogaphic potntial, though distinctly diffnt in thi appoach, show good agmnt acoss almost th nti spctum. It can b concludd that of th two mthods, th hamonic combination mthod is supio, bcaus (i) it povids fast convgnc and hnc quis lss pows of th THFs, and mo impotantly (ii) it povids additional cofficints byond dg 2160 that a vital fo accuat valuation of th ETP. [64] Th colation btwn th ETP and EGM2008 cofficints was found to b much gat fo th llipsoidal appoximation than fo th sphical appoximation. Not only do th dg vaianc spcta of th ETP and EGM2008 xhibit simila pow fom dg ~250 onwad, th dg colation cofficints a also much high than fo th sphical appoximation. Ths numical sults claly show that th solution in llipsoidal appoximation dlivs a significant impovmnt ov th sphical appoximation. W commnd that th hamonic combination mthod b usd fo spctal fowad gavity modling of any clstial objct that can closly b appoximatd by an oblat llipsoid of volution. Appndix A: Lgnd Wight Functions [65] Th fully nomalizd sinusoidal Lgnd wight functions nm in quation (25) can b computd via vaious cusiv schms [Classns, 2005] nm ¼ 1 K 2i 2k;2j 2 nm K 2k;2 nþ2i 2k;m k¼ 1 nm ¼ 1 K 2k;2 nm K2i 2k;2j 2 nþ2k;m k¼ 1 nm ¼ 1 K 2iþ2k;2j 2 nm K 2k;2 nþ2i;m k¼ 1 wh (A3) follows fom (A1) and th lation nm ¼ K 2i;2j nþ2i;m (A1) (A2) (A3) (A4) [66] Equations (A1), (A2), (A3) can all b usd to comput th function nm fo any pai of i and j fom th initial valus [67] Th initial valus shown h only hold fo th fully nomalizd (4π nomalizd) functions. Any oth fom of nomalization will not affct th cusion lations, but will sult in diffnt initial valus, which can asily b divd. Dtails on th pactical and numical diffncs btwn th vaious cusiv schms can b found in Classns [2005]. [68] Acknowldgmnts. 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