ROLE OF NO TOPOGRAPHY SPACE IN STOKES-HELMERT SCHEME FOR GEOID DETERMINATION

Transcription

1 LE F N TPGAPY SPACE IN STKES-ELMET SCEME F GEID DETEMINATIN Per Vaníek, ober Tenzer and Jianliang uang Annual meeing of CGU 1

2 We have been formalizing he Sokes soluion of he geodeic boundary value problem in elmer s form (second elmer s condensaion mehod) for almos 10 years. ur firs paper abou his formalizaion was [Vaníek P. and Marinec Z., 1994: The Sokes-elmer scheme for he evaluaion of a precise geoid. Manuscripa Geodaeica, No.19, Springer]. The scheme used in his formalizaion is shown graphically on he nex slide. Molodenskij opined ha i would be impossible o deermine geoid precisely because of he unknown opographical densiy. Ye we are geing good resuls, boh a UNB and a GSD, and he approach seems o make a good physical sense. We ake his as a vindicaion of Sokes s and elmer s ideas.

3 eal Space elmer Space opography ξ g(r,ω) ξ h g h (r,ω) opography elluroid ( ) h SIE elmer elluroid N ( N ) h geoid N N h PIE co-geoid ellipsoid ellipsoid 3

4 eck poined ou in 1993, and recen heoreical work by eck and Novák confirmed ha elmer s anomalies are quie rough due o he presence of he condensaion layer on he geoid. Also, elmer s anomalies are disconinuous on he geoid - see he nex slide. These facs led some people o quesion our approach o he downward coninuaion of elmer s anomalies from he earh surface o he geoid, formulaed as Poisson s soluion o he inverse Dirichle s BVP. Véronneau (GSD) eleced o coninue downward he complee Bouguer anomalies insead of elmer s anomalies, wih good numerical resuls. ence we decided also o have a good look a some such alernaives. 4

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6 Following eck and Novák s heoreical suggesion and Véronneau s experimenal resuls, we have decided o invesigae he inverse Dirichle s problem formulaed in he No Topography space raher han he elmer space. The NT-space is characerized by having he real mass densiy wihin he geoid and zero mass densiy everywhere else cf., Sanos e al. paper a his meeing. The graviy field in he NT-space is smooher han elmer s field and NT-graviy is coninuous everywhere. Applying his approach see he nex slide we ge good numerical resuls cf., Tenzer e al. poser a his meeing wih an addiional bonus: he smooher field makes he ask of Poisson s downward coninuaion for denser graviy daa easier (needed in he fuure?). 6

7 eal Space No Topography Space elmer Space 7

8 g [ r, ] T( r, Ω) Ω = r r The Theory The graviy anomaly g[r (Ω),Ω] on he surface of he earh in he real space is given by he well known formula (he fundamenal gravimeric equaion ): g [ ( Ω ), Ω ] T r= r [ r, ] [ r, Ω] ε [ r, Ω] n ε δ g f sin ϕ T, ε δ Ω ( r Ω ) r r = r ( Ω ) ϕ ε n [ r, Ω] m fcosϕ 1 3 T [ r, Ω] r r 8

9 The ransformaion of g(r,ω) from he real space o he NT-space hen akes place: g[r (Ω),Ω] g NT [r (Ω),Ω]. I is given by he following formula (V (r,ω) is he poenial of opography and V a (r,ω) is he poenial of he amosphere): g NT [ r, Ω] g[ r, Ω] V ( r, Ω) r V ( r, Ω) r= r r= r a r r f sin ϕ r V a [ r, Ω] V [ r, Ω] r V [ r, Ω] 1 V [ r, Ω] r = r m fcos ϕ ϕ 3 r 9

10 where he opographical poenial is given by: V [ r, ] Ω = 4G o r 1 1 3, G o Ω Ω r = G δρ Ω Ω ( Ω ) l 1 ( Ω ) [ r, ψ ( Ω, Ω ), r ] ( ) 1 Ω l [ r, ψ( Ω, Ω ), r ] r = r r dr dω dr dω 10

11 And he opographical aracion is given by: ( r Ω) 1 V, = 4G ( ) o 1 r r Ω = 3 r r. G o Ω Ω r = ( Ω ) l 1 [ r, ψ( Ω, Ω ), r ] r r r dr dω ( Ω ) G δρ Ω Ω r = ( Ω ) l 1 [ r, ψ( Ω, Ω ), r ] r r r dr dω 11

12 Is he NT-anomaly, g NT (r,ω), he same as he sandard complee Bouguer anomaly g CB (r,ω) as we have been using i? N! The g NT (r,ω) is he same as he spherical complee Bouguer anomaly g CB;S (r,ω). This spherical anomaly has been sysemaically invesigaed by Vaníek P., Tenzer., Sjöberg L.E., Marinec Z., Feahersone W.E. [003: New views of he spherical Bouguer graviy anomaly, submied o Geophysical Journal Inernaional]. I has been shown ha i is: defined in he whole 3D space (i.e., i has a well defined disurbing poenial T CB;S (r,ω) associaed wih i) and ha i is indeed harmonic everywhere above he geoid. This is NT he case wih he sandard Bouguer anomaly. 1

13 The spherical complee Bouguer anomaly is defined by he following equaion on he earh surface: Ω Ω 0 TopoC : g B CB; S [ ] = g[ ] 4πGρ [ δρ; ] TC S [ ρ0; ] V T 0 [ ] where g is he generic graviy anomaly in he real space, TopoC B is he correcion for anomalous opographical densiy δρ, TC S is he spherical errain correcion evaluaed for consan opographical densiy ρ 0. 13

14 The difference beween spherical (NT) and planar complee Bouguer anomalies

15 The NT-anomalies coninued downward on he geoid [mgal] 160 [mgal] 140 [mgal] 10 [mgal] 100 [mgal] 80 [mgal] 60 [mgal] 40 [mgal] 0 [mgal] 0 [mgal] -0 [mgal] -40 [mgal] -60 [mgal] -80 [mgal] -100 [mgal] -10 [mgal] -140 [mgal] -160 [mgal] 15

16 nce we have he (NT) anomalies on he surface of he earh in he NT-space, we coninue hem downward in he NT-space: hey are defined everywhere and are harmonic above he geoid. This is done by means of solving he Poisson inegral equaion. Noe ha g NT [r (Ω),Ω] g CB;S [r (Ω),Ω] are coninued downward o he elmer co-geoid o ge g NT [r g (Ω),Ω] where we wan o have hem see slide #7. Thus orhomeric heighs from elmer s space mus be used for he downward coninuaion in he NT-space. These differ from orhomeric heighs in he real space by he geoid - co-geoid separaion. 16

17 The NT-anomalies on elmer s co-geoid, g NT [r g (Ω),Ω], (obained as he Poisson soluion o he downward coninuaion problem), are hen ransformed o elmer s space. We ge g [r g (Ω),Ω], which are hen used as boundary values for he soluion of he BVP of he hird kind in elmer s space as originally envisaged. In spherical approximaion we ge (V c (r,ω) is he poenial of opography condensed on he geoid, V ca (r,ω) is he poenial of he amosphere condensed on he geoid): g 3 3 NT c ca (, Ω) = g (, Ω) V (, Ω) V (, Ω) 17

18 where V c 1 (, Ω) = G σ ( Ω ) l [, ψ ( Ω, Ω ), ] dω Ω Ω σ = ρ r= r dr = ρ V ca ( ) lim a( ) 1, Ω = G ρ r r dr l [, ψ( Ω, Ω ),] Ω Ω r r = ( Ω ) dω and ρ a is he amospheric densiy. 18

19 The final soluion N (Ω) is ransformed ino he geoidal heigh N(Ω) (in he real space) by adding o N (Ω) he primary indirec effec δn (Ω) (in elmer s space). The soluion N (Ω) is again sough in he elmer space in order o minimize he primary indirec opographical effec, i.e., he geoid co-geoid separaion. (The corresponding quaniy δn NT (Ω) in he NT-space is very large and canno be compued accuraely enough.) 19

20 CNCLUSINS 1) This approach seems o work as well as he approach ha uses downward coninuaion in elmer s space. ) I may prove beneficial when denser graviy daa become available for geoid deerminaion. 0

21 Tha s all, folks! 1