Supplemental Notes. Determination of Spherical Harmonic Models GS ADVANCED PHYSICAL GEODESY. Christopher Jekeli

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1 Suppleetal Notes o Deteriatio of Spherial Haroi Models GS887 - ADVANED PHYSIAL GEODESY hristopher Jeeli Divisio of Geodesti Siee Shool of Earth Siees The Ohio State Uiversity Marh 07

2 The followig osiders basi ethods of deteriig the spherial haroi oeffiiets of the disturbig potetial fro terrestrial gravity aoaly data. We do ot osider their deteriatio usig satellite-traig data, satellite altietry, other satellite data, or the obiatio of satellite ad terrestrial data, whih are topis beyod the preset sope of these short otes. Let the disturbig potetial be represeted by a spherial haroi series, + R, () = = r GM T r = Y R B (, θ, λ ) ( θ, λ ) B where the defiitios of all quatities have bee give i other otes ad are thus assued to be well ow. This is a exat represetatio for poits outside the Brilloui sphere (the sphere whose radius, R B, is the greatest distae to a poit o the Earth s surfae fro its eter), sie the series overges uiforly to the true disturbig potetial at these poits i spae. For poits below the Brilloui sphere, but still above the Earth s surfae, the series ay ot overge. However, a truated series would possibly still be a legitiate odel for the disturbig potetial eve at these poits sie the (truated) series is still haroi (although, the effet of evetual divergee ay ause sigifiat odel error). O the other had, the series (truated or ot) is ot a theoretially orret odel at poits below the Earth s surfae (suh as, at the geoid o otiets) sie the disturbig potetial is ot haroi there. We wish to deterie the oeffiiets,, for the odel () of the disturbig potetial fro gravity data o Earth s surfae. There are at least two distit ethods to aoplish this, based, i the first plae, o the spherial approxiatio of the boudary oditio: T g = T r R r R = r = R r= R, () where g is forally the gravity aoaly o the geoid, approxiated as a sphere of radius, R. Thus, eve though we started with a exat odel, already we ae soe approxiatios. We will osider a ore rigorous proedure later. Usig the orthogoality of the surfae spherial harois, Y, ad RB R, we have (usig Rapp s (969) teriology) Method A: = g ( R, θ ', λ ') Y, ( θ ', λ ') d 4πγ ( ), ; (3) Method B: g ( R, θ, λ ) = γ ( ) Y ( θ, λ ). (4) = = where γ GM R. These ethods use gravity aoalies as observatios to deterie the paraeters,. Method A, give by equatio (3), is also ow as the quadratures ethod sie it ivolves the uerial itegratio of the gravity aoalies. Method B, give by equatio (4), is also ow as the least-squares ethod. However, both ethods a be forulated i ters of a least-squares adjustet, where Method A uses a set of oditio

3 equatios ad Method B uses a set of observatio equatios. There is a third ethod, leastsquares olloatio to solve for the oeffiiets usig gravity aoalies as observatios. This has bee studied to soe extet i the past, but was foud ot to yield partiularly good results, ostly due to the poor owledge of the ovariae betwee the gravity aoalies ad the oeffiiets (if we would ow this orrelatio well eough, the we would probably ot eed to solve for the oeffiiets). Thus we will liit our disussio to Methods A ad B, ad variatios thereof. Followig Rapp (969), the disretizatio of the itegrals (3) is give by R = g ( R, θ j ', λ ') Y, ( θ ', λ ') d 4π GM ( ), (5) j, j, where eah sall area eleet, j,, orrespods to a observatio g j,, usually o soe regular grid o the sphere, ad usually the ea gravity aoaly over the eleet, j,. learly, we have a fiite uber of observatios ad we a oly solve for a fiite uber of oeffiiets. Equatio (5) a be writte i vetor/atrix for, = B g, (6) where = (,, ) T is a fiite vetor of the oeffiiets to be solved, g (,, ) T, is a fiite vetor of the observatios, ad the atrix B has eleets lie ( ) j, = g j, R Y, ( θ, λ ) d. (7) 4π GM Exat expressios a be derived for the itegrals of the spherial haroi futios. Typially, we ay thi of the gravity aoaly observatios as distributed o a spherial grid of ells defied by ostat latitude ad logitude lies. If there are J = π θ suh ells i latitude ad K = π λ ells i logitude, the we have JK observatios. If ax is the axiu degree of the oeffiiets we see to solve, the there are L = ( ) ax + 4 oeffiiets i total (exludig the zero ad first-degree harois). There is a fudaetal rule of thub that the axiu degree of the spherial haroi odel is related to the resolutio of the gravity aoaly data as follows: ax π 80 = =. (8) θ [rad] θ [deg] Thus, if θ = λ, we have JK = π θ = ax > L. That is, we have ore observatios tha paraeters ad we ay wish to pursue a least-squares solutio. Without goig ito the derivatio we state oly that the solutio for Method A is give by

4 ( P ) M M ( B ) ˆ = + g, (9) 0 0 ( ) D = ˆ M + P, (0) where 0 is a vetor of iitial values for the oeffiiets, P is the orrespodig weight atrix (the iitial values ad weight atrix ight oe fro a previous satellite solutio), ad M = BP B, () T g where P g is the observatio weight atrix (obtaied fro the dispersio atrix of the observatio errors). Equatio (0) gives the dispersio atrix of the errors i the estiated paraeters, where is the variae of uit weight. We see that eve for sall ax the solutio is oputatioally foridable. The atrix, M, has diesios, L L ; ad so, for ax = 80, we eed to ivert a 3,757 3,757 atrix. learly, soe approxiatios ust be ade. For exaple, suppose that the iverse weight atrix for the observatios is P = diag siθ, whih iludes a way of opesatig for the irease i the uber of ( ) g g j data as the observatio poits approah the poles, that is, by dereasig the data weights. Furtherore, if we assue for the iitial oeffiiets that P = 0, ad if we approxiate the eleets of B as j, ( ) 4πγ ( θ λ ) Y, j,, () the with, = siθ θ λ, a eleet of the atrix, M, beig the produt of a row of B, j j orrespodig to degree ad order, ( ), ad a olu of order, ( ', '), is give by T B, orrespodig to degree ad g θ λ ( 4πγ ) ( )( ' ) (, ) Y (, ) θ λ θ λ Y, j ', ' j j,. (3) j, By the orthogoality of the spherial haroi futios, the su approxiately ollapses to either zero or 4π. Assuig uity for the variae of uit weight, we fid that the variae of the estiated oeffiiets is give by = θ λ g 4 πγ ( ). (4) 3

5 For Method B, we ust truate the series (4) ad the a write the observatio equatios i vetor/atrix for g = A, (5) where eleets of the atrix, A, are of the for ( ) Y, ( j, ) γ θ λ. (6) The solutio for the oeffiiets is give by T T ( A P g A P ) A P g ( A ) ˆ = + g. (7) 0 0 With the sae assuptio o the weight atries as i Method A, we fid the sae variae for the estiated oeffiiets, equatio (4) (it is left to the studet to verify this). Note that ulie the quadratures ethod, i priiple, Method B does ot require gravity aoaly data over the whole sphere; o the other had, a better solutio for the oeffiiets results for evely distributed data. Both Methods A ad B ivolve the iversio of huge atries i the ost geeral ase of a full variae-ovariae atrix for the observatios ad for large observatio or oeffiiet sets. Rapp (969) tested the ethods for ax = 4 ad 5 5 ea gravity aoalies, ad with diagoal weight atries, whih siplified the solutios to soe extet. He oluded that either ethod had a sigifiat advatage, but did ote that the oeffiiet solutio of Method B yielded saller observatio residuals; that is, the aoalies iplied by the estiated oeffiiets were loser to the observed aoalies tha for Method A. I fat, Method B is the ethod that was used for EGM96 (Leoie et al. 998) ad EGM08 (Pavlis et al. 0). O the other had, several of the previous high-degree solutios by Method A siply applied the quadratures forulas (uerial itegratio) rather tha perforig a least-squares solutio. Equatio (4) a be used to obtai a estiate of the error i the disturbig potetial (or ay of its futioals) give a ertai resolutio ad auray i the gravity aoaly data. The resolutio is defied by θ (ad λ ) ad the auray is give by g. If the errors i the oeffiiets are idepedet (whih ould be laied o the basis of the orthogoality of the spherial haroi futios, if the gravity aoaly observatio errors are also idepedet), the it is easy to show that the oissio error (the error i the estiate of T due to errors i the gravity aoaly observatios) has variae GM oissio, ax ( T ( R,, )) Y (, ) θ λ = θ λ. (8), R = = osiderig all possible values of T o the sphere, the average variae of the oissio error exludes the square of the spherial haroi (sie Y averages to uity). Substitutig (4), the average variae of the oissio error beoes 4

6 R θ λ ax +. (9) 4 g oissio ( T ) = π = ( ) Iediately, we a deterie the average stadard deviatio of the orrespodig geoid udulatio error: oissio = oissio. (0) γ ( N ) ( T ) A ote of autio is eeded here. The oissio error thus oputed depeds o the idepedee of the errors i the gravity aoalies (ad hee the idepedee of the errors i the haroi oeffiiets). That is, it assues that the oise is uorrelated, or white. It is ow fro statistis that uorrelated oise is redued (with respet to its ea) whe it is averaged. Whe we opute the disturbig potetial fro gravity aoalies, we are itegratig (i.e., suig, or, i essee, averagig) gravity aoalies via Stoes s itegral, ad so we are averagig observatio errors. Therefore, the estiate (9) teds to be rather low, perhaps optiistially low; that is, it oly represets a lower boud o the stadard deviatio. The true variae (or, stadard deviatio) ould easily be higher if these siplified assuptios are ot fulfilled. For exaple, ay orrelatios or systeati opoets i the aoaly errors ivalidate this siple forula. Furtherore, it represets a global average; stadard deviatios ay be higher i soe regios ad lower i others. Despite all these aveats, it is a reasoable first assesset of the oissio error, ad we a tur the proble aroud to as: what iiu auray i the gravity aoaly is eessary to obtai a ertai auray i the disturbig potetial (or, geoid udulatio)? Returig to the proble of deteriig aurate spherial haroi oeffiiets, we ade two sigifiat approxiatios, aog several less sigifiat oes, that yielded a rather siple relatioship betwee the gravity aoalies ad the oeffiiets (equatio (3) or (4)). The ore sigifiat approxiatios ilude the spherial approxiatio i the boudary oditio ad the approxiatio of the boudary by a sphere. We ay proeed to eliiate these approxiatios by first defiig a speial gravity aoaly, all it g, for whih equatios (3) ad (4) are exat. We the deterie the redutios eessary to trasfor our observed (Molodesy) free-air aoalies to g. We have already foud forulas that aout for the liear approxiatio; we will osider oe ethod of dealig with the spherial approxiatios. Other redutios aout for the effet of the atosphere (we require that the disturbig potetial be haroi i free spae), ad for possible approxiatios ade i oputig oral gravity o the telluroid. Aalytially reduig the gravity aoaly to a sphere, however, is ot pratial eve though it is required for equatio (3) to be exat. Oe would have to perfor aalytial otiuatio of the aoaly by a vertial distae of up to te to twelve iloeters. For Method B, we a avoid this requireet by iludig a radial depedee: GM g r Y R r = = ax + R. () (, θ, λ ) ( ) ( θ, λ ) = 5

7 This oly hages the idividual eleets of the A -atrix i the odel (5), ot its essetial validity. We aot do the sae i Method A sie we eed to itegrate over a surfae o whih the spherial haroi futios are orthogoal. Istead, we re-forulate the proble i ellipsoidal oordiates whih allows us to use a ellipsoid as boudary ad express the disturbig potetial as a series of ellipsoidal harois. What aes this approah feasible is a exat relatioship betwee spherial harois ad ellipsoidal harois. This relatioship was developed by Hotie (969) ad further ivestigated by Jeeli (98). We do ot attept to go ito the details of this relatioship, whih is atheatially ot too diffiult, but legthy. By this approah we oly eed to perfor a aalyti otiuatio of the Molodesy free-air aoalies to the ellipsoid, whih ivolves a redutio over a vertial distae roughly equal to the topography, typially less (ofte uh less) tha a few iloeters, plus the geoid udulatio (obtaied fro soe a priori odel). We proeed as follows. Defie (without approxiatio!) T g = T. () r r The it is easily show that r g is a haroi futio (it satisfies Laplae s equatio i free spae). Trasforig to ellipsoidal oordiates (u = sei-ior axis of the ofoal ellipsoid through the oputatio poit; δ = redued o-latitude; λ = logitude), we ay expad r g i solid ellipsoidal haroi futios: = = ( u E) ( b E) ax S e r g ( u, δ, λ ) = a g Y, ( δ, λ ), (3) S where S is a fully oralized assoiated Legedre futio of the seod id, where a, b are, respetively, sei-ajor ad sei-ior axes of the oral ellipsoid that has E for its e liear eetriity; ad where g is a ellipsoidal haroi oeffiiet with the sae uits as δ λ, ow the gravity aoaly. By the orthogoality of the spherial haroi futios, Y, (, ) with oordiates, ( δ, λ ), projeted oto the uit sphere, we have (,, ) (, ) si, (4) e g = re g b δ λ Y, δ λ δ dδ dλ 4π a where r e is the radius to a poit o the oral ellipsoid.. The spherial haroi oeffiiet is related to its ellipsoidal ousis aordig to s e,, S, ( b E) = 0 g = L g, (5) 6

8 where the fators, L,, are ow ad deped o the itegers,,. Therefore, oe the ellipsoidal oeffiiets are deteried by uerial quadratures (iludig the possibility of least-squares estiatio) o the basis of equatio (4), the spherial haroi oeffiiets a be s deteried fro equatio (5). The spherial oeffiiets, g, also satisfy r g r a g Y ax + a, (6) s (, θ, λ ) ( θ, λ ) = r = = assuig the ellipsoid with sei-ajor axis, a, is a Brilloui ellipsoid, fro whih we fid, usig (), a a s g = GM R ( ). (7) It reais to deterie g o the oral ellipsoid fro Molodesy free-air aoalies o the Earth s surfae. Forally, this is ahieved as follows g g ε ellipsoid Earth's surfae spherial approxiatio ε other approxiatios D = + + +, (8) where D represet the dowward otiuatio of the free-air aoaly to the ellipsoid. The details ay be foud i Rapp ad Pavlis (990) ad i Leoie et al. (998). Eve though Method B, i theory, does ot deped o a regular boudary surfae, it is advatageous uerially if all data are o a surfae of revolutio. That is why equatio (8) is also used to develop EGM96 ad EGM08, whih both are based o Method B. Referees Hotie M (969): Matheatial Geodesy. ESSA Moograph No., US departet of oere, Washigto D. Jeeli (988): The exat trasforatio betwee ellipsoidal ad spherial haroi expasios. Mausripta Geodaetia, 3, Leoie FG, Keyo S, Fator JK, Trier RG, Pavlis NK, hi DS, ox M, Kloso SM, Luthe SB, Torree MH, Wag YM, Williaso RG, Pavlis E, Rapp RH, Olso TR (998) The developet of the joit NASA GSF ad the Natioal Iagery ad Mappig Agey (NIMA) geopotetial odel EGM96, NASA Tehial Paper NASA/TP , Goddard Spae Flight eter, Greebelt Pavlis NK, Holes SA, Keyo S, Fator JF (0): The developet ad evaluatio of Earth Gravitatioal Model (EGM008). Joural of Geophysial Researh, 7, B04406, doi: 0.09/0JB Rapp RH (969): Aalytial ad uerial differees betwee two ethods for the obiatio of gravietri ad satellite data. Bollettio di Geofisia Teoria ed Appliata, XI(4-4), Rapp RH, Pavlis NK (990): The developet ad aalysis of geopotetial oeffiiet odels to spherial haroi degree 360. Joural of Geophysial Researh, 95(B3),