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Geoid, Topography, and the Bouguer Plate Or Shell

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Geoid, Topography, and the Bouguer Plate Or Shell Journal of Geodesy C2001) 75: 210±215 Geoid, topography, and the Bouguer plate or shell P. Vanõ cÆ ek1, P. Nova k1, Z. Martinec2 1 Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, E3B 5A3 Fredericton, Canada e-mail: [email protected]; Tel.: +1-506-4535144; Fax: +1-506-4534943 2 Department of Geophysics, Charles University, V HolesÆ ovicÆ kach 2, Pragues, Czech Republic Received: 24December 1999 / Accepted: 11 December 2000 Abstract. Topography plays an important role in solv- ing many geodetic and geophysical problems. In the evaluation of a topographical eect, a planar model, a 1 Introduction spherical model or an even more sophisticated model can be used. In most applications, the planar model is Periodically, people discover that planar and spherical considered appropriate: recall the evaluation of gravity models of topography give very dierent results for reductions of the free-air, Poincare ±Prey or Bouguer Bouguer anomalies. Similarly, the results for the direct kind. For some applications, such as the evaluation of and indirect topographical eects in the Stokes±Helmert topographical eects in gravimetric geoid computations, technique for geoid computations obtained by means of it is preferable or even necessary to use at least the the planar and spherical models are found to be quite spherical model of topography. In modelling the topo- dierent. Some people claim that the planar model can graphical eect, the bulk of the eect comes from the safely be used for ``local work'' while the spherical Bouguer plate, in the case of the planar model, or from model has to be used for global work. Others still the Bouguer shell, in the case of the spherical model. The maintain that all these questions have already been dierence between the eects of the Bouguer plate and sorted out and that they do not require any more of our the Bouguer shell is studied, while the eect of the rest of attention. So what is going on? topography, the terrain, is discussed elsewhere. It is When looking into this problem CVanõ cÆ ek and Nova k argued that the classical Bouguer plate gravity reduction 1999) we discovered an interesting story, which we will should be considered as a mathematical construction try to recount here. To do so, we focus only on the with unclear physical meaning. It is shown that if the ``in®nite plate'' and the ``spherical shell'' models, leaving reduction is understood to be reducing observed gravity out the terrain eects. The corresponding dierence of onto the geoid through the Bouguer plate/shell then the planar and spherical models of terrain presents an- both models give practically identical answers, as other fascinating story which, in our opinion, requires a associated with Poincare 's and Prey's work. It is shown separate and rather more extensive paper to deal with it why only the spherical model should be used in the adequately. The main point of this ``terrain story'' is the evaluation of topographical eects in the Stokes±Helmert discovery that, contrary to popular belief, the spherical solution of Stokes' boundary-value problem. The reason model terrain eect has to be considered globally. This for this is that the Bouguer plate model does not allow for point has been already discussed by Nova k et al. C1998) a physically acceptable condensation scheme for the and by Nova k and Vanõ cÏ ek C1999). A more formal and topography. complete paper on the subject of spherical terrain model is under review CNova k et al. submitted). Once pub- lished, it will complement the present paper. Key words: Geoid ± Bouguer Reduction of Gravity ± In the present paper, we will focus our attention ®rst Stokes±Helmert's Problem on the dierence between the Bouguer plate and Bou- guer shell eects on Bouguer anomalies. Then, we will tackle another, more or less independent issue, that of the dierence between the two models in the gravimetric geoid determination by means of the Stokes±Helmert technique, where the use of the two models has pro- Correspondence to: P. Vanõ cÆ ek, P. Nova k is currently at: Depart- found consequences. In fact, the planar model, the ment of Geomatics Engineering, The University of Calgary, 2500 Bouguer plate, cannot be used at all if we wish to use a University Drive NW, Calgary, Canada. physically meaningful condensation scheme. 211 2The story of Bouguer plate reduction cally traversing the plate or shell is the same, except for higher-order terms. The situation for the potential is In order to see the pattern, let us show the gravitational naturally dierent: as the potential of the in®nite plate is potential, the gravitational attraction Cnegative ®rst in®nite, we cannot make any direct comparison between vertical derivative of the potential), and the vertical the two models. We can only observe that in the spherical gradient of gravitational attraction Cnegative second model, the potentials at the upside and on the downside of derivative of the potential) of the topographical CBou- the shell dier only by higher-order terms. guer) plate and the topographical CBouguer) shell side by What does it all mean? We wish to address here only side. To keep things simple, let us assume a constant the most interesting question of what this means in the density . Csay, the usual 2670 kg mÀ3) and the same context of the Cincomplete, i.e. without the terrain cor- thickness, H, for both the in®nite plate and the shell of rection) Bouguer gravity anomaly. The incomplete or the inner radius R. This is all shown in Fig. 1. The three simple Bouguer anomaly is computed from the follow- quantities of interest are computed at two points: one on ing formula: the top and one at the bottom of the plate/shell. In B addition, the second derivative, which is discontinuous Dg g A A À c 1 on the top Cand also at the bottom) of the plate/shell, is where g is the observed gravity on the Earth's surface Cat at this point computed in both directions: from above altitude H), A is the ``free-air reduction'' Cto the geoid) and from below. The expressions for the plate are due to the Earth's masses enclosed within the inner derived from Eqs. C3.5) and C3.7) in Heiskanen and radius of the spherical shell Cincluding the latitude and Moritz C1967) by simply extending the ®nite plate to altitude terms), AB is the ``Bouguer reduction'' Cto the in®nity. The expressions for the shell are derived directly geoid) due to the mass of the Bouguer plate, and c is the from the equations for the potential Cof a spherical shell) normal gravity at the reference ellipsoid [Heiskanen and in Wichiencharoen [1982, Eqs. C19), C24) and C25)]. Moritz 1967, Eq. C3.19)]. In this case, the so-called Now, examining Fig. 1, how dierent really are the Bouguer reduction is given by results for the planar and spherical models? Starting from the bottom, with the vertical gradient of attraction, and AB 2pG.H 2 neglecting the higher-order terms Cof the order of H=R and smaller) in the spherical model, the results are identical. For the standard value of mean topographical density of The attraction of the plate at its top is only one half of that 2670 kg mÀ3, the numerical value of the Bouguer of the shell Cat its top and neglecting the higher-order reduction is 0:1119  H mGal. terms), while the attraction at the bottom of the plate is Inspecting again Fig. 1, we can see rather easily that exactly opposite to that at the top. The attraction at the AB is not the dierence between the gravity values at the downside of the shell is zero, as it should be CKellogg top and the bottom of the in®nite plate! It is not the 1929). Note that the change in the attraction when verti- dierence between gravity values on the upside and Fig. 1. The behaviour of gravity and gravity potential induced by Bouguer shell and Bouguer plate. The symbols Oi H=R de- note higher-order terms 212 downside of the spherical shell either! This dierence, in the use of the Cphysically less satisfactory) planar model the planar case, is equal to and that we should therefore obtain the ``correct'' result by using the physically more satisfying spherical model. PP A 4pG.H 3 Rather, the Bouguer gravity reduction is the conse- known in geodesy as the ``Poincare ±Prey gravity reduc- quence of the physically unsatisfactory ``removal of the Bouguer plate'' of an in®nite mass. Thus the Bouguer tion'' [Heiskanen and Moritz 1967, Eq. C3.64)]. In the spherical case the dierence is the same except for the gravity anomaly, useful as it is in many geophysical applications, is really only an arti®cial construction. The higher-order terms, which we will neglect for simplicity. CWe note that we arrive at the stated conclusions, using dierence between using the planar and the spherical either the attraction or the vertical gradient of attraction model is of second order; it is the known spherical Â Ï formulae, as we should.) For the standard topographical correction to the Bouguer plate reduction [Vanõcek and density, the numerical value of the Poincare ±Prey Krakiwsky 1986; Eq. C21.43)] reduction is 0:2238  H mGal. H 2 The discrepancy between the planar and spherical dAB À8pG. 5 S R models was pointed out by Karl C1971). It should be mentioned that planar and spherical terrain models For the standard topographical density of 2670 kg mÀ3, display the same discrepancy.
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