On the Construction of a Synthetic Earth Gravity Model

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1 O the Costructio of a Sythetic Earth Gravity Model M. Kuh, W.E. Featherstoe Wester Australia Cetre for Geodesy, Curti Uiversity of Techology, GPO Box U987, Perth WA 6845, Australia Abstract. A sythetic Earth gravity model (SEGM), based o realistic-as-possible iformatio ad assumptios about the mass distributio of the Earth, geerates self-cosistet gravity field quatities. Therefore, such a model is well suited to validate the theories ad methods used i practical gravity field determiatio. This paper describes the theoretical ad practical methods required to costruct a SEGM usig recetly available data o the Earth s mass distributio. The approach preseted is a first attempt to costruct a SEGM purely by forward gravity field modellig. Numerical results will be give for global sythetic geoid heights ad gravity aomalies, which show commo features with EGM96. Keywords. Sythetic Earth gravity model, forward modellig, Earth s mass-desity distributio Itroductio A sythetic Earth gravity model (SEGM) geerates self-cosistet gravity field parameters based o a realistic-as-possible represetatio of the Earth. Usig Newto s law of gravitatio, for istace, the gravitatioal potetial ad its derivatives i arbitrary directios ca be uiquely determied from a give mass distributio. These self-cosistet, sythetic (simulated) gravity field quatities ca be used to study the differet theories ad methods used i practice to aalyse the Earth s gravity field without relyig o observatios, which are subject to measuremet errors. The importace of a SEGM has bee already recogized by the IAG via special study group SSG 3.77 Sythetic Modellig of the Earth s Gravity Field (see The cocept of a SEGM ca be divided ito a source model ad a effect model (e.g., Pail 999). Source models use reasoably realistic mass desity iformatio to produce the gravity field by umerical, or eve aalytical (Allasia 22), itegratio of Newto s law of gravitatio. I this regard, poit-mass models are simple ad very useful for geeratig the local ad global gravity field, which have bee developed by, e.g., Barthelmes ad Dietrich (99), Lehma (993), Vermeer (995), Ihde et al. (998) ad Claesses et al. (2). Effect models represet the gravity field based o observatios, so as to make them realistic, but the parameters are assumed error-free. Differet degrees of spherical harmoic effects model have bee used by, e.g., Tziavos (996), Featherstoe (22) ad Novak et al. (2). A sythetic spherical harmoic effects model up to degree 54 (to ±45 latitude) ad degree 27 (over the whole Earth) is available at the aforemetioed website. These maximum degrees of expasio are costraied by computer uderflow ad overflow errors (Holmes ad Featherstoe 22). Global models based upo the spherical harmoic represetatio of Newto s itegral have bee developed by, e.g., Rummel et al. (988), Pail (999), Haagmas (2) ad Claesses (22). The latter three models are classified here as hybrid source/effects sythetic models because they combie both approaches. Most importatly, the aforemetioed studies usig source ad/or effects models are defied for or focus oly upo the exteral gravity field. However, from a give mass distributio, the gravity field ca also be described iside the Earth s masses usig Newto s law of gravitatio, which completes a SGEM. Therefore, this paper cocetrates o the costructio of a SEGM usig oly mass-desity iformatio as pure source model. I geeral, two steps have to be performed: () Costruct a reasoably realistic mass distributio of the Earth, called here the sythetic Earth mass model (SEMM). (2) Geerate the differet gravity field parameters iduced by these masses. The theoretical ad practical aspects of these two steps will be discussed. A overview o the geeratio of two gravity field parameters (geoid

2 heights ad gravity aomalies) usig the spherical harmoic approach ad the direct evaluatio of Newto s itegral will the be give. Prelimiary SEGM results will the be preseted usig the spherical harmoic represetatio of Newto s volume itegral, ad compared with EGM96. 2 Sythetic Earth Mass Model Startig with a simple referece model of the Earth s mass distributio, the [realistic] simulated mass distributio ca be represeted by its deviatio from this referece model. Accordigly, there will be mass excesses ad deficiecies with respect to the referece model. Oe example of a referece model is PREM (Dziewoski ad Aderso 98), but this oly uses a radially symmetric desity variatio. Istead, a referece ellipsoid icludig the complete mass of the Earth (e.g. GRS8 with GM = 39865x 8 m 3 s -2 ; Moritz 98) is used because it accouts for the ellipsoidal shape of the Earth. Also, istead of usig a mea desity of the whole Earth (~555 kg/m 3 ), the mass distributio ca be adapted to follow a geophysical model (e.g. PREM) for the radial variatio. All additioal mass distributios ca the be cosidered as deviatios (or aomalies) with respect to this referece model. These masses are described geometrically with the 3D mass-desity fuctio ρ = ρ ref ρ, which is the differece betwee the desity ρ ref of the referece model ad the actual desity ρ. The vertical variatio of such masses ca be described by the geocetric radius or the ellipsoidal height. This ca be achieved either by discrete values associated with a fixed mass colum (e.g., by a digital elevatio model; DEM) or by aalytical fuctios such as spherical harmoics. Crust R Cotiet ρ(θ,λ) t (+) (-) c c (+) H (θ,λ) () H (θ,λ) mass layer Matle Ocea m o RL geoid Fig. : Represetatio of a mass layer by deviatios above (+) ad below (-) the mea sphere of radius R L. L To a spherical approximatio, the vertical mass variatio ca be described by the spherical height ( + ) (measured alog the radial) H ( Ω) above or ( ) H ( Ω) below a mea sphere of radius R L = R L (Fig. ). Here Ω = ( λ, θ ) deotes the coordiate pair of spherical logitude λ ad co-latitude (polar distace) θ. The height fuctios H ( Ω) ad H ( Ω) are give i terms of spherical harmoics ( + ) ( ) by Nmax H ( Ω) = H m Y m ( Ω) () = m= with the coefficiets H m ad the surface spherical harmoics Pm (cosθ ) cos mλ m Ym ( Ω) = (4) Pm (cosθ ) cos mλ m < for the associated Legedre fuctios of the first kid P (cosθ m ) (e.g., Heiskae ad Moritz 967). The over-bars i Eqs. () ad (2) idicate that the correspodig parameters are fully ormalized. If the Earth s gravity field is oly required outside the masses, the variable mass-desity ca be replaced with a mea value by itroducig equivalet rock heights (e.g., Rummel et al. 988). Here, the mass of a specific colum with the actual desity ρ is replaced by a colum of equal mass usig the mea desity ρ. The correspodig height is the give by the equivalet rock height (ibid.) H er ρ ( Ω) = H ( Ω). (3) ρ Whe usig Eq. (3), oly the costat desity differece ρ ref ρ with respect to the referece model desity, together with the equivalet rock height, eed oly be cosidered i further computatios. Importatly however, the gravity field outside the masses will be differet whe usig the equivalet rock heights because the exact mass distributio has bee chaged by this approximatio. Further work is uderway to study the effect of this. The additioal mass distributios with respect to the referece ellipsoid ca be cosidered as: Topographic ad ocea water masses Mass aomalies i the crust Mass aomalies i the matle, or deeper The accumulatio (superpositio) of all this iformatio the represets the sythetic Earth mass model (SEMM). However, a costrait must be itroduced such that, after addig ad subtractig the additioal masses, the total mass of the Earth remais uchaged; otherwise large biases will be itroduced.

3 3 Gravity Field Parameters from the SEMM Usig Newto s law of gravitatio, ay kid of gravity field parameter ca be uiquely derived form a SEMM. Here, two differet approaches to derive a SEGM from a SEMM will be discussed. For global applicatios, the gravitatioal potetial ad its derivatives ca be expressed i terms of spherical harmoic coefficiets of the correspodig heights ad/or surface desities (called here the spherical harmoic approach). O the other had, the gravity field parameters ca be derived directly by umerically evaluatig Newto s itegral (called here the direct itegratio approach). 3. Spherical Harmoic Approach The gravitatioal potetial caused by the equivalet mass layer is give by + Nmax R V ( Ω, r) = Vm Ym( Ω). (4) = r m= where the correspodig fully ormalized spherical harmoic coefficiets are (e.g., Rummel et al. 988, Wieczorek ad Phillips 998, Ramille 22, Kuh ad Featherstoe 22, this issue) + 2 RL * Vm = µ κpm (5) R p= with * p Γ( + 3) + κ pm = κpm, p N (6) p! Γ( + 4 p) for the fully ormalized spherical harmoic coefficiets κ of the surface desity (±) fuctios p m p H + κ p = ρ, p N (7) p RL ad the factor 4πGR µ =. (8) 2 + I Eqs. (4) to (7), the superscript + idicates masses above the mea sphere R L ad - masses below it. Positive ad egative sigs are also carried though the calculatios to represet mass excesses ad deficiecies, respectively, with respect to the referece model. If equivalet rock heights are used i Eq. (7), H ( Ω) has to be replaced by er + 2 H (Ω). The upward cotiuatio factor ( R L / R) i Eq. (5) shows that the effect of the deeper mass distributios is atteuated for higher degrees, due to simply to their greater distace from the geoid or Earth s surface. The superpositio of all gravitatioal potetial mi effects V iduced by the regarded mass distributios (also icludig that of the referece model ref V ) the represets the total gravitatioal potetial of the SEGM, give by SEGM m ref V = V + V, (9) where m V m V = I i m i= () ad I idicates the umber of differet mass distributios used about each R L. The gravitatioal potetial of the referece ellipsoid is give by (e.g., Moritz 98) = 2 ref GM a V J 2 P2 (cosθ ) () r = r where the eve zoal harmoics J 2 = V2 ad associated Legedre fuctios P 2 (cosθ ) are ot ormalised. Itroducig the cetrifugal potetial Φ, which is assumed to be the same for the actual ad referece gravity fields, the total potetial W SEGM ad disturbig potetial T SEGM of the SEGM are give by SGEM SGEM ref W = V + V + Φ (2) SEGM SEGM ref m T = V V = V. (3) Usig Eq. (3), the spectral represetatio of other parameters of the aomalous gravity field ca be give by applyig Meissl s spectral scheme (Rummel ad va Geldere 995). Icludig the relatio give by Eq. (6), a exteded Meissl scheme for forward gravity field modellig ca be defied (Kuh ad Featherstoe 22, this issue). As well as allowig the computatio of differet gravity field parameters from mass distributios, this exteded Meissl scheme ca be used to study the spectral sesitivity of the SEGM to differet mass distributios (ibid.). 3.2 Direct Itegratio Approach Newto s itegral ca be evaluated by discretised umerical itegratio usig the gravitatioal potetial (or its derivatives) caused by regularly shaped bodies, such as prisms or tesseroids (i.e., spherical volume elemets). The major advatage of this approach is that it allows the straightforward computatio of gravity field parameters iside the masses. It also allows the computatio of highfrequecy source models i local regios that host very detailed iformatio o the mass distributio, which ca the be superimposed o the SEGM geerated by the spherical harmoic approach.

4 -5-5 Formulae for the gravitatioal potetial ad its derivatives caused by prisms ca be foud i Nagy et al. (2; also see the correctios i Deis ad Featherstoe, 22; this issue), whereas the gravitatioal potetial ad acceleratio caused by tesseroids are give, to a first-order approximatio, by Kuh (22). Followig the approach give i Eq. (9), the gravitatioal potetial of the SEGM V SEGM is the give by umerical itegratio of Newto s itegral over all mass distributios residual to the referece model. The gravitatioal potetial of the referece model ca be evaluated outside the masses either by Eq. () or by a simple poit-mass model alog the mior axis (e.g., Barthelmes ad Dietrich 99). For evaluatios iside the masses, however, the actual mass distributio of the SEMM has to be used. Fially, the gravitatioal potetial of the referece model, as well as its cetrifugal potetial, is oly eeded if absolute SEGM values, such as gravity at the Earth s surface, are eeded. For parameters of the aomalous gravity field, the represetatio of T = V is sufficiet [cf. Eq. (2)], which SEGM m has bee used to produce the prelimiary results preseted here. For the testig of geoid computatio theories, for example, (cf. Novak et al., 2), the aomalous SEGM field is always sufficiet. 4 Prelimiary Source SEGM I this Sectio, a first attempt will be made to costruct a SEGM by forward gravity field modellig oly. This prelimiary model is based o the followig realistic data. Topography ad Bathymetry Topographic heights ad bathymetric depths are take from the global 5 x 5 JGP95E digital elevatio model (DEM; Lemoie et al. 998, chapter 2). For the topographic masses ad ocea water mass deficiecies with respect to the referece mass distributio (described below), mass-desities of t 3 3 ρ = 267kg / m ad ρ w = 634kg / m have bee used, respectively. Crust The crustal mass distributio is take from the global 2 o x 2 o CRUST 2. model, which (oddly) is a update of the CRUST 5. model (Mooey et al. 998). I this model, the crust is defied by the depth ad mass-desity of five differet layers (soft ad hard sedimets, upper, middle ad lower crust). The bottom of the lowest layer defies the Mohorovicic discotiuity, which is used to model the mass aomalies associated with the udulatig crust/matle trasitio zoe. The global mea Moho depth i this model is 22 km (38 km beeath cotiets, 2.6 km beeath oceas). Matle The laterally variable desity of the upper matle has also bee take from CRUST 2.. The deeper matle masses are ot icluded i this study. Istead, they are assumed costat ad equal to the desity of the referece model. However, they will be icluded i the first release of the SEGM, because the results preseted later show that crustal ad topographic masses aloe do ot adequately represet all low frequecies of the Earth s gravity field. Referece Model The referece model used for the SEGM is assumed to be the GRS8 ellipsoid (Moritz 984) with its idirectly itroduced mass give by the geocetric gravitatioal costat GM. However, the [assumed] homogeeous mass distributio iside GRS8 has bee chaged regardig the outer shell of the Earth. Istead, this shell is take as a homogeeous mass layer of costat thickess (34 km) with a mea desity of 286 kg/m 3, i accordace with the mea cotietal crust. Beeath this, a mea upper matle desity of 3365 kg/m 3 has bee itroduced. The mea desity of the lower matle ad core the has to be chose such that the referece model represets the remaiig mass of the Earth. The geoid height ad gravity aomalies iduced by the above mass distributios (for degrees = 2 36) are illustrated i Figs. 2 ad 3. By way of compariso, the geoid height ad gravity aomalies from EGM96 are illustrated i Figs. 4 ad Fig. 2: Sythetic geoid heights derived from the JGP95E DEM ad CRUST 2. crustal model. Cotour iterval 5 m

9 6 3-3 -6-9 6 2 8 24 3 36 (9E, N) ad ear the orth-west coast of America (2W, 3N), are ot that well represeted by the sythetic data. Therefore, these ca be attributed to deeper sources.

Further studies will be used to determie if cosiderig the additioal mass distributios i the matle ca chage these magitudes. -2-6 -2-8 -4 4 8 2 6 2 Fig.

N from EGM96 N from SEGM N from JGP95 (topo./bathy.) N from CRUST 2. (oly) -9 6 2 8 24 3 36 Fig. 4: EGM96 geoid heights. Cotour iterval 2 m. 9 6. 2 6 2 8 24 3 36 degree Fig.

5 (9E, N) ad ear the orth-west coast of America (2W, 3N), are ot that well represeted by the sythetic data. Therefore, these ca be attributed to deeper sources. Also, the sythetic geoid heights ad gravity aomalies geerally show a greater rage of magitudes with a maily log wavelegth structure, which is attributed to the eglect of the deeper mass sources. Further studies will be used to determie if cosiderig the additioal mass distributios i the matle ca chage these magitudes Fig. 3: Sythetic gravity aomalies derived from the JGP95E DEM ad CRUST 2. crustal model. Uits i mgal rms of degree variace [m]. N from EGM96 N from SEGM N from JGP95 (topo./bathy.) N from CRUST 2. (oly) Fig. 4: EGM96 geoid heights. Cotour iterval 2 m degree Fig. 6: Degree variaces of the geoid height from the SEGM, EGM96, JGP95E ad CRUST Fig. 5: EGM96 gravity aomalies. Uits i mgal. I Figs. 2 ad 4, several commo structures ca be idetified. For example, a large geoid highs over New Guiea (4E, 5N) ad ear Icelad (W, 65N) ad large geoid lows over the Himalayas (9E, 35N) or Siberia plai (8E, 6N) ca be see both i the sythetic data ad i EGM96. This idicates that these structures ca be attributed to crustal mass distributios oly. Other structures, however, such as the lows i the Bay of Begal The misrepresetatio of the low frequecy part of the SEGM spectrum ca also be see i the degree variaces of the geoid heights (Fig. 6). This shows more eergy i the low degrees of the SEGM, compared with EGM96. However, for higher degrees ( > ), the degree variaces of the SEGM ad the EGM96 show early the same rate of power decay. This adds evidece to the ability of the SEGM to geerate a realistic gravity field i the medium to high frequecies. The low frequecy effects will have to be refied, either usig poit masses (cf. Hipki, 2) or refied matle desity models from seismic tomography. 5 Coclusios A prelimiary SEGM has bee preseted, based upo forward gravity field modellig usig recetly released geodetic ad geophysical iformatio about the Earth s topography ad crust. Eve with such a simple model, it is possible to obtai a very

6 reasoable represetatio of the Earth s gravity field i the medium ad high frequecies. This has bee show umerically by a rather good agreemet of the SEGM with EGM96 i may regios. It also shows that several features of the Earth s gravity field ca be attributed partly or completely to the crustal mass distributio. However, there are also several regios that do ot agree with EGM96, which is attributed to the omissio of deeper mass sources i the SEGM. Therefore, further studies will cosider deeper mass sources to determie if they ca explai the higher power see i the lowfrequecy SEGM with respect to EGM96. The umerical results preseted have bee limited to degree 36, which is sufficiet for most crustal mass aomalies. Iformatio o higher frequecies comes maily from the topographic masses, which have bee developed up to degree 44. However, the effect of these masses has bee trucated to degree 36 so as to compare them directly with EGM96. Future extesios to this SEGM will also iclude this high frequecy iformatio. To iclude the very high frequecy iformatio available i local DEMs, the direct evaluatio of Newto s volume itegral must be used for these mass distributios. Ackowledgemets This study was fuded by the Australia Research Coucil through a grat A27 o costructig a sythetic Earth s gravity field. We would gratefully thak Dr.-Ig. K. Seitz (Uiversity of Karlsruhe, Germay) for software o spherical harmoic aalysis ad sythesis. Refereces Allasia G (22) Approximatig potetial itegrals by cardial basis iterpolats o multivariate scattered data. Comp & Math Applic 43(3-5): Barthelmes L, Dietrich R (99) Use of poit masses o optimised positios for the approximatio of the gravity field. I: Rapp RH, Sasó F (Eds) Determiatio of the Geoid, Spriger, Berli, pp Claesses SJ, Featherstoe WE, Barthelmes F (2) Experieces with poit-mass gravity field modellig i the Perth regio, Wester Australia, Geom Res Aust 75: Claesses SJ (22) A sythetic Earth model aalysis, implemetatio, validatio ad applicatio, MSc thesis, Delft Uiversity of Techology, Delft, 75 pp. Deis ML, Featherstoe WE (22 this issue) Evaluatio of orthometric correctio algorithms to spirit levellig usig a simulated moutai, Proc GG22, Thessaloiki. Dziewoski AM, Aderso DL (98) Prelimiary referece Earth model. Phys Earth Plaet Iter 25: Featherstoe, W.E. (22) Tests of two forms of Stokes s itegral usig a sythetic gravity field based o spherical harmoics. I: Grafared EW, Krumm FW, Schwarze VS (Eds), Geodesy - The Challege for the Third Milleium, Spriger, Berli, pp Haagmas R (2) A sythetic Earth for use i geodesy. J Geod 74: Hipki RG (2) The statistics of pik oise o the sphere: applicatios to matle desity aomalies, Geophys J It 44: Holmes SA, Featherstoe WE (22) A uified approach to the Cleshaw summatio ad the recursive computatio of very high degree ad order ormalised associated Legedre fuctios, J Geod 76: Ihde J, Schirmer U, Stefai F, Töppe F (998) Geoid modellig with poit masses, Proc. 2d Cotietal Workshop o the Geoid i Europe, Budapest, pp Kuh M (22) Geoid determiatio with desity hypotheses from isostatic models ad geological iformatio, J Geod (submitted). Kuh M, Featherstoe WE (22 this issue) O the optimal spatial resolutio of crustal mass distributios for forward gravity field modellig. Proc GG22, Thessaloiki. Lehma R (993) The method of free-positioed poit masses geoid studies o the Golf of Bothia, Bull Geod 67: 3-4. Lemoie FG ad 4 other authors (998) The developmet of the NASA GSFC ad Natioal Imagiary ad Mappig Agecy (NIMA) geopotetial model EGM96. Rep. NASA/TP , Natioal Aeroautics ad Space Admiistratio, Marylad, 575 pp. Moritz H (98) Advaced physical geodesy. Herbert Wichma, Karlsruhe, 5 pp. Moritz H (984) Geodetic Referece System 98, Bull Geod 54: Mooey WD, Laske G, Masters TG (998) CRUST 5.: a global crustal model at 5x5 degrees, J Geophys Res 3: Nagy D, Papp G, Beedek J (2) The gravitatioal potetial ad its derivatives for the prism, J Geod 74: Novak P, Vaicek P, Veroeau M, Holmes SA, Featherstoe WE (2) O the accuracy of modified Stokes s itegratio i high-frequecy gravimetric geoid determiatio, J Geod 74: Ramille G (22) Gravity/magetic potetial of ueve shell topography, J Geod 76: Rummel R, Rapp RH, Sükel H. (988) Comparisos of global topographic/isostatic models to the Earth s observed gravity field, Rep 388, Dept Geod Sci ad Surv, Ohio State Uiv, Columbus. Rummel R, va Geldere M (995) Meissl scheme spectral characteristics of physical geodesy, mauscr geod 2: Pail R (999) Sythetic global gravity model for plaetary bodies ad applicatios i satellite gradiometry, PhD thesis, Techical Uiversity Graz, Austria, 38 pp. Tziavos IN (996) Compariso of spectral techiques for geoid computatios over large regios, J Geod 7: Vermeer M (995) Mass poit geopotetial modellig usig fast spectral techiques; historical overview, toolbox descriptio, umerical experimet, mauscr geod 2: Wieczorek MA, Phillips RJ (998): Potetial aomalies o a sphere: applicatio to the thickess of the luar crust. J Geophys Res 3(E):

Gravity An Introduction

Gravity An Introduction Gravity A Itroductio Reier Rummel Istitute of Advaced Study (IAS) Techische Uiversität Müche Lecture Oe 5th ESA Earth Observatio Summer School 2-13 August 2010, ESA-ESRIN, Frascati / Italy gravitatio ad