The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution

Abstract

When topography is represented by a simple regular grid digital elevation model, the analytical rectangular prism approach is often used for a precise gravity field modelling at the vicinity of the computation point. However, when the topographical surface is represented more realistically, for instance by a triangular irregular network (TIN) model, the analytical integration using arbitrary polyhedral bodies (the analytical line integral approach) can be implemented directly without additional data pre-processing (gridding or interpolation). The analytical line integral approach can also facilitate 3-D density models created for complex geometrical bodies. For the forward modelling of the gravitational field generated by the geological structures with variable densities, the analytical integration can be carried out using polyhedral bodies with a varying density. The optimal expression for the gravitational attraction vector generated by an arbitrary polyhedral body having a linearly varying density is known. In this article, the corresponding optimal expression for the gravitational potential is derived by means of line integrals after applying the Gauss divergence theorem.

Gravity effects of polyhedral bodies with linearly varying density

“…Due to the different approach exploited thus far, the previous expression has a rather different form from the analogous quantity considered by Pohanka (1998), Holstein (2003 and Hamayun et al (2009). Thus, the equivalence of their expression with (17) can be proved only numerically, see e.g.…”

“…This simple example has been considered for comparing the results obtained by means of formulas (51), (54) and (60) with those obtained by using the approach first contributed by Pohanka (1998), Holstein (2003 and Hamayun et al (2009). This has already been described in D'Urso (2014) for the potential so that we limit hereafter to address the case of the first and second-order derivatives.…”

“…However, as shown in the sequel, the expression (17) is well defined for every position of P with respect to the polyhedral body in the sense that it does not need to be supplemented with the small positive number ε, introduced in the pionering papers on the subject by Pohanka (1988Pohanka ( , 1998, and used as well by Hamayun et al (2009), to avoid undefined operations when the computation point is near to Fr(Ω).…”

“…The evaluation of (7) has been already pursued in the literature, see e.g. Pohanka (1998), Hansen (1999, Holstein (2003) and Hamayun et al (2009), by observing that…”

“…More recently Hamayun et al (2009) have derived the expression for the potential of a body with a linearly varying density, by extending the original approach by Pohanka (1988) limited to the constant density case, and proved that the computational algorithm exploited in Pohanka (1998) for the gravity attraction can be applied as well to the potential.…”