# Computation of Gravity and Magnetic Anomalies of Non-homogeneous
and Arbitrarily Shaped Bodies Using Finite Element Techniques

*Mello, Ulisses T.*

IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598

## Abstract

In this study, I discuss alternative techniques to calculate gravity and magnetic
anomalies of non-homogeneous, arbitrarily shaped geological bodies by using 2D
and 3D finite element discretization. To compute magnetic and gravity anomalies it
is common to represent geological bodies by irregular polygons or polyhedral
shapes. However with advances in discretization techniques, such as finite element,
it is convenient to develop algorithms that take advantage of these advances. These
algorithms can be especially useful in cases where gravity and magnetic anomalies
need to be computed in association with other calculations using Finite Element
Methods such as monitoring of reservoir production, integrated basin modeling and
tectonic modeling. Although magnetic and gravity anomalies can be calculated by
using traditional finite element solutions for boundary value problems (Laplace and
Poisson equations), I suggest the use of finite element Gaussian quadrature as an
alternative method. This numerical integration method was extensively tested and is
widely available in practically any finite element implementation to compute
“stiffness” matrices. This method can also be used as a first step to define the
necessary boundary conditions in the boundary value approach instead of large-
distance approximations or infinite finite elements currently used on the calculation
of domain boundaries. The finite element Gaussian quadrature approach is
accurate, fast, easily implemented and parallelized. In addition, it allows an arbitrary
representation of complex geological bodies and non-homogeneous property
distributions, such as density contrast and magnetic susceptibility, with fewer
elements when higher order interpolation are utilized.