Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7
The followig is a bief eview of the g tems i Moitz s aalytic cotiuatio method to solve the simple Molodesky poblem. This eview should complemet, ot eplace, the expositio i (Moitz 98. Fo the gavity aomaly, g, o the telluoid ad the fictitious, aalytically cotiued aomaly, g ', o the level though the evaluatio poit,, we have ( g = U g ', ( whee the upwad cotiuatio opeato, U, is epeseted by U U. ( = = z = =! z We assume this opeato is ivetible, ca also be epeseted as D = U, whee the dowwad cotiuatio opeato, D, D = D. ( q= q We itoduce the Molodesky shikig facto, k, such that h = kh, k. (4 k The the upwad cotiuatio opeato fo each diffeet topogaphic suface, defied by h k, becomes U ( k k U. (5 = = k z = =! z Similaly, we defie the ivese opeato fo each of these topogaphies: D ( k q= q k Dq =. (6 ( k ( k Now, sice U D = I, whee I is the idetity opeato, we have k k + q = q = =. (7 = q= = = U D k U D k U D I
This holds fo abitay k; hece, the coefficiets of k o the left side of the last equality must equal the coefficiets of k o the ight side: = : U D = I = = : U D = U D + U D = = = : U D = (8 Hece, sice U = I, we have D = I ; (9 ad, i geeal, D = U D. ( = Now, let g D ( g = ; the ' k = k = = ; ( = = g D g k D g k g ad, let k =, so that. ( g D g D g g ' = = = whee, with the defiitio of = = U, we have ( h h z = =! z. (! h = = g g g
Fo example, we have ( g = D g = g ; (4 g = ( h h g h ; (5 g = ( h h g ( h h g h h. (6 It is stogly emphasized that the fuctios, g, ae ot fuctios i thee dimesios eve though they cotai a height. These fuctios ae defied o level sufaces ad ca ot be diffeetiated with espect to the vetical by applyig a chai ule to thei defiitio i tems of height. Istead, we eed to defie a vetical deivative opeato, which we ca do if we thik of the aalytic cotiuatio of each g above the level suface. Hofma-Wellehof ad Moitz (5, Sec.-4 povide such a opeato: f f R f f = L( f = + d, (7 R π l fo a fuctio, f, defied o the sphee (ad which is aalytically cotiued ito the exteio space accodig to Laplace s equatio Diichlet bouday-value poblem!. We ca eglect the fist tem because it is small, ad wite L f R f f = d. (8 π l The fuctios, g, ae the give by whee ( h h! = g = L g, (9 L ( = L L. (
Note that Moitz (98 defies the successive opeato, L = = L L L! h ; that is, L! L. Fially, we expess the distubig potetial at as follows: ( ( ψ, ( R T = g + g + g + S d 4π whee we ecogize that the g, i themselves, ae ested global itegals because of the L- opeato. Each g is a fuctio of the poit,, o the level suface though. Fo example, whee =, R g g g h h L g h h d ', ( ' = = π l' ( ( ( l ' is the distace betwee the two poits, ad ( g = ( h h L( g ( h h L( L ( g ', o the level suface. Also, fo ( R ( g ( g R L( g L( g = h h d h h d ' ' ' ' ( π l' π l' whee each of the itegals fo g ivolve itegals. Specifically, the fist itegal is R π ( g ( g ' l ' R = π d ' R g g R g g h h d " h h d " " ' " ' π l' " π l " l' ( ( The fist tem fo g ca also be witte as d ' (4
( h h L( g ( h h L( ( h h L( g = ( h h L( hl ( g L ( h L( g ( h h L( hl ( g ( h h hl ( L( g = = (5 showig that cae must be execised whe dealig with these opeatos ote that fom (8 ( ( R h ' L g h L g L( hl( g = d '. (6 π l The complete tem, g, the is ' ' ( g = ( h h L( hl ( g ( h h hl ( L( g ( h h L( L( g = ( h h L( hl ( g ( h h L( L ( g (7 I ay case, it quickly becomes itactable umeically, uless oe uses Fouie tasfoms (as developed by Sideis 987. We also ote that all g deped o the poit, ; so that fo each diffeet evaluatio poit fo the distubig potetial, the g have to be computed idividually, if based o the ecusio fomula (9. Howeve, we see that the equivalet fomulatio (7, just i tems of suface data, g ad h, cicumvets this difficulty. Sideis (987 makes this moe geeal. Coside agai the basic aalytic cotiuatio, give by ( ad (9, epeated hee fo coveiece, ( g ' ( g =, (8 = ( h h = ((, ( g g L g! = = g. (9 This epesets the aalytic cotiuatio of a fuctio, g, o some suface (the telluoid to the fuctio, g ', o a level suface though, cotaiig the poits,. If that level suface is the ellipsoid, the h =, ad we get fom (9 h (,, ( g = L g g = g! =
ad, fom (8, the aalytically cotiued gavity aomaly o the ellipsoid, g = g. ( = The, Sideis (987 makes the statemet that it is obvious that ( g is the upwad cotiuatio of g, claimig that this upwad cotiuatio effect is give by h (,, ( u u g = L g g =! such that that is, = ( g = g + g ; ( = u ( g g g L ( g u h = + =. (4! = The poof of this is ot obvious, ad Sideis (987 does ot give it, but veifies the esult fo =,,, 4. Fo example, h! = ( g = L ( g = g + h L( g = h L ( g + h L( g = ( h h L( g which is the same as equatio (; ad, h h, (6! = ( g = L ( g = g + hl ( g + L ( g which, the eade is asked to veify, is the same as (7. I this way, the g ca be computed oce ad fo all, as ca thei opeatios, ( eeded i the ecusio (. Havig these, the compute fo ay. Fo example, up to =,,(5 L g, as g eeded i Stokes s fomula ae easy to
( g g g g g h L g g h L g h L L g + + = + + + + +, (7 whee h is fixed ad both sides ae fuctios of the suface poit,. Refeeces Hofma-Wellehof, B., Moitz, H. (5: hysical Geodesy. Spige-Velag, Beli. Moitz, H. (98: Advaced hysical Geodesy. Hebet Wichma Velag, Kaluhe. Sideis, M.G. (987: Spectal Methods fo the Numeical Solutio of Molodesky s oblem. hd Dissetatio, Depatmet of Suveyig Egieeig, Uivesity of Calgay, Calgay, Albeta.