MEAN GRAVITY ALONG PLUMBLINE. University of New Brunswick, Department of Geodesy and Geomatics Engineering, Fredericton, N.B.

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1 MEA GRAVITY ALG PLUMBLIE Beh-Anne Main 1, Chis MacPhee, Rbe Tenze 1, Pe Vaníek 1 and Macel Sans 1 1. Inducin 1 Univesiy f ew Bunswick, Depamen f Gedesy and Gemaics Engineeing, Fedeicn,.B., E3B 5A3, Canada Advanced Cmpuainal Reseach Labay, Faculy f Cmpue Science, Univesiy f ew Bunswick, Fedeicn,.B., E3B 5A3, Canada The mean value f gaviy alng he plumbline beween he geid and he eah suface depends n he mass densiy disibuin wihin he Eah and he shape f he Eah. Since he acual values f gaviy alng he plumbline cann be measued, he mean value f gaviy alng he plumbline has be cmpued fm he gaviy bseved a he suface f he Eah. This can be dne by educing he bseved gaviy accding sme acceped physical mdel.. Mean gaviy alng plumbline Accding he heem f mean inegal values, he mean value g f gaviy alng he plumbline beween he geid and he eah suface eads (eiskanen and Miz, 1967, Eq. 4-0) 1 g g g, Ω d, Ω = g (1) g is he gaviy a a pin f which gecenic psiin is Ω ) φ, λ), and he gecenic Ω f he eah suface is given (wih an accuacy f a ms a few millimees) by he whee adius gecenic adius f he geid g plus he hmeic heigh. The gaviy g can be expessed as (Tenze and Vaníek, 003) g φ ) δg V γ, () whee γ (,φ ) is he nmal gaviy f he gecenic efeence ellipsid, g δ is he gaviy disubance in he Tpgaphy gaviy space (i.e., he gaviy disubance geneaed by he mass wihin he geid iself, Vaníek e al., 003), and V / is he gaviainal aacin f pgaphical masses. g Ω akes he fllwing fm Subsiuing Eq. () back in Eq. (1), he mean gaviy g 1 g V φ ) δg = g 3. Mean nmal gaviy alng plumbline γ d. (3) γ f nmal gaviy alng he plumbline beween he geid and he eah suface 1 g γ γ φ ) d. (4) Ω = g The fis em n he igh-hand-side f Eq. (3) defines he mean value 1

2 eglecing he deflecin f plumbline and he cecin caused by diffeen lenghs f he plumbline (beween he geid and he eah suface) and he ellipsidal nmal (beween he gecenic efeence ellipsid and he elluid), he mean nmal gaviy alng he plumbline can be ewien as (Tenze and Vaníek, 003) γ 1, (5) = ( φ ) n ( φ ) γ φ ) γ, φ d whee ( φ ) is he gecenic adius f he gecenic efeence ellipsid, (Mldensky, 1945), and is he geidal heigh. = ( φ ) is he nmal heigh The mean value f nmal gaviy alng he nmal beween he efeence (gecenic) ellipsid and he elluid is evaluaed by he fllwing fmula (eiskanen and Miz, 1967, Eq. 4-4) 1 Ω a b Ω Ω φ φ ) ( φ ) γ = γ d γ 1 1 f f sin ϕ, (6) = ( φ ) GM a a whee a, b ae he semi-axes and f = ( a - b) / a is he fis numeical flaening f he gecenic efeence ellipsid, is he mean angula velciy f Eah s ain, ϕ is he gedeic laiude, GM is he gecenic gaviainal cnsan, and γ ( φ ) is he nmal gaviy a he suface f he gecenic efeence ellipsid. Fig. 1: Relain beween he mean value γ f nmal gaviy (efeence ellipsid GRS-80) and he nmal heigh.

3 [gal] [gal] 981. [gal] [gal] [gal] [gal] [gal] [gal] The cecin nmal gaviy Fig. : Mean values f nmal gaviy. ε due he geid undulain, i.e., cecin f a shif f he γ inegain ineval fm he efeence ellipsid he geid, is given by (Tenze and Vaníek, 003) γ ( φ ) ( φ ) γ, γ φ ε γ γ g =. (7) n a = ( φ ) Fig. 3: Relain beween he cecin γ and he geidal heigh. ε nmal gaviy due he geid undulain 3

4 55 5. [mgal] 5.1 [mgal] Fig. 4: Cecin nmal gaviy due he geid undulain. 5.0 [mgal] 4.9 [mgal] 4.8 [mgal] 4.7 [mgal] 4.6 [mgal] 4.5 [mgal] 4.4 [mgal] 4.3 [mgal] 4. [mgal] 4.1 [mgal] 4.0 [mgal] 3.9 [mgal] 3.8 [mgal] 3.7 [mgal] 3.6 [mgal] 4. Mean geid-geneaed gaviy disubance The mean value g δ f geid-geneaed gaviy disubance alng he plumbline beween he geid and he eah suface is given by he secnd em n he igh-hand-side f Eq. (3), i.e., R g = R Since he geid-geneaed gaviy disubance g 1 δ δ d. (8) g δ muliplied by a gecenic adial disance is hamnic abve he geid in he Tpgaphy gaviy space, i.e., saisfies he Laplace equain, he gaviy disubance δ g is evaluaed by slving Diichle s bunday value pblem (Kellgg, 199) 1 R δ g = K[, ψ ( Ω, R] δg ( R, Ω ) dω, (9) 4 Ω Ω whee he gaviy disubance δg ( R, δg g is efeed n he c-geid, R is he mean adius f he Eah which appximaes he gecenic adius f he geid ( c-geid) suface, and Pissn s K, ψ Ω, Ω, R is given by inegal kenel [ ] R K[, ψ ( Ω, R ] = R. (10) 3 l [, ψ ( Ω, R] Subsiuing Eqns. (10) and (9) back in Eq. (8) and pefming he adial inegain wih espec, Eq. (8) akes he fllwing fm (Tenze and Vaníek, 003) 4

5 δg = 1 4 R l[, ψ ( Ω, R] Ω Ω R agsinh R sinψ ( Ω, Ω ) anψ ( Ω, Ω ) 1 R 1 = R δg ( R, Ω ) dω. (11) [mgal] 100 [mgal] 80 [mgal] 60 [mgal] 40 [mgal] 0 [mgal] 5 0 [mgal] -0 [mgal] -40 [mgal] -60 [mgal] -80 [mgal] -100 [mgal] -10 [mgal] -140 [mgal] Fig. 5: Mean geid-geneaed gaviy disubances g δ [mgal] 5. Mean pgaphy-geneaed gaviainal aacin The mean value f pgaphy-geneaed gaviainal aacin alng he plumbline beween he geid and he eah suface (given by he hid em n he igh-hand-side f Eq. 3) can be deived as (Tenze and Vaníek, 003) whee V ( R, and ( ) 1 g V 1 R V d = = Ω g = R V ( R, V ( ) = d, (1) V ae he gaviainal penials f pgaphical masses as a efeed n he geid and a he eah suface. V,Ω f he pgaphical masses f a pin Accding Mainec (1998), he gaviainal penial inside he pgaphical mass R R eads 5

6 V ( = G R R 3 R 1 [ ], 3 3 G Ω Ω G Ω Ω R = R R = R ( Ω ) ( Ω ) l 1 [, ψ ( Ω, ] 1 (, Ω ) l [, ψ ( Ω, ] d dω δρ d dω, (13) whee G is ewn s gaviainal cnsan, l [,ψ ( Ω, ] is he spaial disance beween (, (,Ω ), and ψ ( Ω,Ω ) is he spheical disance beween he gecenic diecin Ω and Ω. and The fis em n he igh-hand-side f Eq. (13) is he gaviainal penial f he spheical Bugue shell (f he mean pgaphical densiy and hickness f ) inside he pgaphical masses (Wichienchaen, 198). The secnd em sands f he gaviainal penial f he eain ughness em f densiy (Mainec, 1998), and he hid em epesens he effec f anmalus pgaphical densiy δρ disibuin n he gaviainal penial. Subsiuing Eq. (13) back in Eq. (1), he mean value f pgaphy-geneaed gaviainal aacin alng he adial diecin (which appximaes he plumbline) becmes (Tenze and Vaníek, 003) 1 R = R G V G Ω Ω Ω Ω R d = = R R = R ( Ω ) ( Ω ) 3R G Ω Ω 1 1 ( l [ R, ψ ( Ω, ] l [, ψ ( Ω, ] ) (, Ω ) l [ R, ψ ( Ω, ] l [, ψ ( Ω, ] d dω δρ d dω. (14) b Fig. 6: Relain beween he mean gaviy hmeic heigh g geneaed by he spheical Bugue shell and he. 6

7 55 40 [mgal] 0 [mgal] 00 [mgal] 180 [mgal] 160 [mgal] [mgal] 10 [mgal] 100 [mgal] 80 [mgal] 60 [mgal] [mgal] 0 [mgal] 0 [mgal] Fig. 7: Mean values f he gaviainal aacin caused by he spheical Bugue shell f densiy [mgal] 0 [mgal] 5 10 [mgal] 0 [mgal] -10 [mgal] -0 [mgal] -30 [mgal] -40 [mgal] [mgal] Fig. 8: Mean values f he gaviainal aacin caused by he eain ughness em. 7

8 δρ Fig. 9: Relain beween he mean gaviainal aacin pgaphical densiy disibuin and he hmeic heigh g f he anmalus (laeal) [mgal] 5 0 [mgal] 15 [mgal] 10 [mgal] 5 [mgal] 0 [mgal] -5 [mgal] Cnclusins Fig. 10: Mean values f he gaviainal aacin caused by he anmalus (laeally vaying) pgaphical densiy disibuin. -10 [mgal] The mean value f gaviy (geneaed by he slid Eah wihu he amsphee) alng he plumbline beween he geid and he eah suface (Fig. 11) is descibed as a sum f he mean nmal gaviy beween he ellipsid and he elluid (Eq. 6), he cecin nmal gaviy f geid undulain (Eq. 7), he mean geid-geneaed gaviy disubance (Eq. 11), and he mean pgaphy-geneaed gaviainal aacin (Eq. 14). The mean pgaphy-geneaed gaviainal aacin cnsiss f he 8

9 mean gaviainal aacins caused by he spheical Bugue shell f densiy (fis em in Eq. 14), he eain ughness em (secnd em in Eq. 14), and anmalus pgaphical densiy disibuin (hid em in Eq. 14). Minimum, maximum, and aveage values f hmeic heighs (Fig. 1), geidal heighs (Fig. 13) and he laeal vaiain f he pgaphical densiy (Fig. 14) a he esing aea ϕ, 55, λ 35, 39 in pa f he Canadian Rcky Munains ae in Tab. 1. The esuls f he numeical invesigain f all cmpnens he mean gaviy ae summaized in Tab [gal] [gal] [gal] [gal] [gal] [gal] 981. [gal] 981. [gal] [gal] [gal] [gal] [gal] [gal] [gal] Fig. 11: Mean values f gaviy alng he plumbline beween he geid and he eah suface a a pa f he Canadian Rcky Munains. Tab. 1: hmeic eighs [m] (Fig. 1) Geidal eighs [m] (Fig. 13) Anmalus Laeal Densiy [g.cm -3 ] (Fig. 14) Min. Max. Aveage

10 Tab. : Mean mal Gaviy (Fig. ) Cecin mal Gaviy f Geid Undulain (Fig. 4) Mean Geid-Geneaed Gaviy Disubance (Fig. 5) Mean Gaviain Aacin f Spheical Bugue Shell (Fig. 7) Mean Gaviainal Aacin f Teain Rughness Tem (Fig. 8) Mean Gaviainal Aacin f Anmalus Tpgaphical Densiy (Fig. 10) Mean Gaviy (Fig. 11) Min. [mgal] Max. [mgal] Aveage [mgal] [m] 00 [m] 000 [m] 1800 [m] [m] 1400 [m] 100 [m] 1000 [m] 800 [m] 600 [m] 400 [m] 00 [m] Fig. 1: Teain a a pa f he Canadian Rcky Munains. 0 [m] 10

11 55-1 [m] -13 [m] 5-14 [m] -15 [m] -16 [m] -17 [m] Fig. 13: Geid a a pa f he Canadian Rcky Munains. -18 [m] [g.cm-3] 5 0. [g.cm-3] 0.1 [g.cm-3] 0.0 [g.cm-3] -0.1 [g.cm-3] [g.cm-3] Fig: 14: Laeal vaiain f pgaphical densiy a a pa f he Canadian Rcky Munains. 11

12 Refeences: eiskanen W. A., Miz., 1967: Physical gedesy. W.. Feeman and C., San Fancisc. Kellgg.D., 199: Fundains f penial hey. Spinge. Belin. Mainec Z., 1998: Bunday value pblems f gavimeic deeminain f a pecise. Lecue nes in eah sciences, Vl. 73, Spinge. Mldensky M.S., 1945: Fundamenal pblems f Gedeic Gavimey (in Russian). TRUDY Ts IIGAIK 4, Gedezizda, Mscw. Tenze R., Vaníek P., 003: Rigus definiin f mean gaviy in hey f heigh. Junal f Gedesy, Spinge. (in pepaain) Vaníek P., Tenze R., Sjöbeg L.E., Mainec Z., Feahesne W.E., 003: ew views f he spheical Bugue gaviy anmaly. Gephysical Junal Inenainal. (submied) Wichienchaen C., 198: The indiec effecs n he cmpuain f geid undulains. Dep. f Ged. Sci. Rep.336, hi Sae Univesiy. 1

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