Suppementa Notes to ysica Geodesy GS6776 Cristoper Jekei Geodetic Science Scoo of Eart Sciences Oio State University 016
I. Terrain eduction (or Correction): Te terrain correction is a correction appied to te Bouguer reduction to account for te fact tat by removing te Bouguer pate (se), te topograpic mass above te pate as not been removed and te mass witin te vaeys beow te upper eve of te pate (se) was removed by mistake. For te atter, we fi in te vaeys wit negative density and compute te corresponding effect. We consider te sperica case and ten make a panar approximation. First, define te residua terrain reative to a point,, on te Eart s surface as te difference between te actua terrain and te spere troug te point (Figure 1). Te potentia due to te masses above and te negative masses beow is given by Newton s aw of gravitation: terrain + + dr dr + + = ( + ) ( + ) δv Gρ d Gρ d + dr = Gρ ( + ) d + (1) were te fina integra incudes eigts bot ess tan and greater tan. Te integra approximates te differentia voume eement using a constant radius, +. Te gravitationa effect (te residua terrain effect) is defined by te negative radia derivative of te potentia, + δvterrain 1 δ gterrain = = Gρ + drd. () r r + For te meaning of te symbos, see Figure 1, and aso note tat te distance,, is given by = r + r r r cosψ, (3) were r is te radius of te evauation point, and r is te radius of te integration point. Te tota terrain effect is te combination of te residua terrain effect and te effect of te Bouguer pate (or se). Te negative of te residua terrain effect is te terrain correction. Now, it is easiy verified tat 1 1 r =. (4) r r r Hence, equation () becomes
+ ( + ) r gterrain = G drd r r + δ ρ + + Gρ r + = d (5) Te ast equaity ods if r = + (i.e., te computation point is on te topograpic surface). From Figure 1, it is easiy sown tat 1 = + +, (6) r from wic one can write te series 1 1 1 = 1+ + r 1 1 1 = 1 + r (7) provided tat < 1. (8) Te atter condition is generay assumed to be satisfied if te terrain incination is ess tan 45 degrees. Now, we make te usua panar approximation, wic means tat terms of te order of are negected. Tis means, for exampe, tat + = 1+. (9) Ten, wit tis panar approximation and retaining just te second-order term of te type (8) in equation (7), te integrand in equation (5) becomes + 1 1 +. (10) - 3 -
Finay, te residua terrain effect is ten approximated by ( ) δ g Gρ d. (11) terrain 3 Te terrain correction is te remova of te terrain reative to te computation point and is, tus, te negative of te residua terrain effect: ( ) At = δ g Gρ d terrain 3 (1) Te terrain correction is aways positive! Figure 1: Geometry for terrain effect integra. - 4 -
II. Hemert s (Second) Condensation Metod One metod to re-distribute te removed masses above te geoid is known as Hemert s (Second) Condensation Metod, in wic te topograpic masses are simpy condensed or paced on te geoid as an infinitey tin ayer. Tis satisfies te condition of te geodetic boundary vaue probem tat no mass soud exist above te geoid. In can be sown tat tis is a specia case of te ratt-hayford metod of isostatic reduction (zero dept of compensation). Tere are two versions of te gravity reduction according to Hemert s condensation. In te first, te masses are removed (causing an effect at te Eart s surface point were gravity is measured), te resuting anomay is downward continued (in free air) to te geoid, and te masses are subsequenty restored on te geoid as a ayer, causing an effect at te corresponding point on te geoid. In te second case, te remova of masses and te restoration as a ayer are considered as one process tat affects te vaue of measured gravity at te Eart s surface point. Subsequenty te gravity anomay is downward continued (in free air) to te geoid. Wic metod to use as been te subject of muc controversy in te past (and peraps te present, as we). Jekei and Serpas (003) ave done anayses tat sow bot metods are teoreticay correct and neiter metod is better numericay in moderate topograpy, but in roug topograpy, te first approac seems generay better. Tis approac is discussed beow. In eiter case te effect on gravity due to te remova and restoration of a terrestria masses (Bouguer pate pus residua terrain) above te geoid is caed te direct topograpic effect. In Hemert s condensation te topograpic masses are restored on te geoid as a ayer wit varying density given by (Martinec 1998) + ( θ, λ) 1 (, ) = (,, r) r dr c(, ), (13) κ θ λ ρ θ λ ρ θ λ were ρ c is te mean density of te crust, assumed constant (but, κ ceary is not constant). Te formuation in equation (13) is suc tat te tota mass of te Eart is preserved. Martinec (1998) aso considers an aternative condensation tat eaves te center of mass uncanged (equation (13) canges te center of mass sigty and tus affects te first-degree armonics). We ave aready determined (wit severa approximations) te effect of removing te topograpic masses above te geoid. If we take te more accurate mode for te Bouguer pate reduction as tat due to a sperica se of tickness,, ten removing tis causes a cange in gravity given by A se s.bouguer r Gδ M =, (14) were, negecting terms of O( ), δ M se 4πρ. (15) Tus te tota remova of topograpic masses affects te vaue of surface gravity by te sum of - 5 -
te sperica Bouguer pate effect and te terrain correction, given by ( ) AT 4π Gρ Gρ d. (16) + Now, te gravitationa potentia of te ayer is given by ayer 0 3 δv = Gρ d, (17) were 0 is aso sown in te Figure. If = 0 (a point on te geoid), ten te derivative of tis potentia is discontinuous and must be evauated in te imit as 0. erforming te imit from te outside of te geoid, Heiskanen and Moritz (1967, p.6) give: 1 δ gayer ( 0 ) = π Gρ Gρ d. (18) r Now from equations (3) and (4), wit r 0 r = =, we ave r r r r 1 1 = 3 0 0 0 ; (19) so tat r 1 1 1 = = 0 r = 0 4 sin. (0) ψ Separating a constant-eigt term and inserting equation (0) simpifies equation (18) to Gρ 1 Gρ δ gayer ( 0 ) = π Gρ + d + d. (1) 4 ψ sin 0 Te first integra in equation (1) is easiy evauated to be π Gρ by noting tat ψ ψ d = sin cos dψ dα, () were 0 α π and 0 ψ π. Sigty re-writing te second integra, we tus ave - 6 -
( ) Gρ δ π ρ, (3) gayer 0 = 4 G + d 0 were te integra, compared to simiar integras, as in equation (5), can be negected under te panar approximation (i.e., negecting terms of O( ) ): δ g πgρ. (4) ayer 0 4 Now, combining te effects of removing te topograpic masses according to equation (16) and restoring tem according to equation (4), we find tat te direct topograpica effect is given by ( ) δ g A δ g Gρ d, (5) = T + ayer 0 3 wic is just te terrain correction. Tus, te Bouguer reduction of a tick se and te restoration of tat se as a tin ayer wit corresponding density togeter yied equa and opposite effects. (Te same woud od if te infinitey extended tick pate were condensed onto an infinitey extended and infinitey tin ayer.) Terefore, often ony te terrain correction is appied to free-air gravity anomaies and tese are ten caed Faye anomaies. In te case were te terrain correction is negigiby sma, one coud even just use te free-air anomaies in Stokes s integra and te conditions of no mass outside te geoid, as we as te anomay being ocated on te geoid woud bot be satisfied ocay! eferences Heiskanen WA, Moritz H (1967): ysica Geodesy. W.H. Freeman and Co., San Francisco. Jekei, C., Serpas, J.G. (003): eview and numerica assessment of te direct topograpica reduction in geoid determination. Journa of Geodesy, 77, 6-39. Martinec, Z. (1998): Boundary-Vaue robems for Gravimetric Determination of a recise Geoid. Springer-Verag, Berin. - 7 -