Gravitational Attraction of a Rectangular Parallelepiped

Abstract

In the inverse solution of a gravity problem when s2 Y2 1 z* the direct method of interpretation is not effective, the g2 = YP II{ k dy SI Y1 I I dx' + y2 + 22 ZI interpretation is done by assuming a geometric model of the subsurface mass distribution based on geologic SZ YZ 22 or some other considerations, calculating the gravity =YP I r In (y + 4x2 + y2 + z*) s1 I I I dr. Y, 21 effect of the model, and finally matching with the Let observed anomaly curve. The model is repeatedly changed until there is an agreement of the theoretically I = .f In (y + dx" + y* + z*) dx calculated and observed anomaly curve. Again for =xln(y+~x*+y*+z*)-Il. estimation of terrain effect, it is necessary to com-where pute the gravity from the known mass distribution. So the importance of direct potential problems in gravity I, = X2 should not be overlooked. I ( y + 4x2 + y2 + 2) 4x2 + y2 + 22 dr Talwani and Ewing (1960) developed a method of computation of gravity attraction of 3-D bodies of = I x*(tix2+y2+z*-y) arbitrary shape. An arbitrarily shaped body may be (x' + 2)4x" + y2 + 22 0!x considered to consist of a number of right rectangular X2 prisms, and for calculation of the effect of the whole = I xz + zz dr block, it is sufficient if the gravity effect of an individual prism is evaluated. Nagy (1966) considered the gravitational attraction of a right rectangular -y I X2 (x' + 2)4x* + y2 + 12 ak.prism. Nagy' s formula has certain limitations which we discuss below, and some suggestions are put The first term will eventually become zero when forth to overcome them. limits of y are put. So I, may be written as THEORY X2The vertical component of gravitational attraction at the origin of a right rectangular parallelepiped bounded by planes x = x,, x = x2; y = y,, y = y, and z = zi, z = z2 (Figure 1) is given by 1,=-y I (x' + z*) dx' + yz + zz dx _ dx Y 4x2 + y2 + z= g,(O,O,O) -g, = YP zdxdydz (x' + y* + 2)3' 2 '(1) where y is the universal constant of gravitation, and p is the density of the prism. ,Neglecting sign of g,, we have from (1) where z2 dx (x2' + 22) dx" + yz + 22 1 y [In (x + \/x2 + y2 + z2) -121.

Terrain modeling in forward gravimetric problems: a case study on local terrain effects

“…Special cases of Eqs. (3) and (4) describe the gravity field of other prismatic mass distributions very popular to exploration geophysicists, such as the right rectangular prism or parallelepiped (MacMillan, 1930;Mader, 1951;Nagy, 1966;Waldvogel, 1976;Banerjee and Gupta, 1977) or the prism with polygonal cross section (Plouf, 1976;Cady, 1980). The right rectangular prism model has been widely used in terrain correction computations, especially after the advent of digital computers (Kane, 1962), while the use of table methods based on the gravitational attraction of hole cylinders was an indispensable part of terrain effect computations to the community for decades (Hammer, 1939).…”

Gravity effects of polyhedral bodies with linearly varying density

“…This has motivated, after the first fundamental contribution by Mader (1951), additional papers on the same issue aiming at improving the numerical efficiency and the generality of the resulting formulas, Koch (1965), Nagy (1966), Banerjee and DasGupta (1977), Nagy et al (2000), Smith (2000), Tsoulis (2000), Jiancheng and Wenbin (2010), Tsoulis et al (2003) and D'Urso (2012).…”

The Integration of 3D Modeling and Simulation to Determine the Energy Potential of Low-Temperature Geothermal Systems in the Pisa (Italy) Sedimentary Plain

“…The Bouguer correction, with the Bullard B term, reduces the infinite Bouguer slab to a spherical cap. Finally, the terrain was corrected, up to 30 km, with curvature correction beyond 10 km, using a right rectangular prism algorithm [47]. The corrections were computed using a mass density of 2300 kg·m −3 .…”