This paper is a chapter of my PhD thesis.
It describes the algorithms used in version 1.2.0 of the open-source
software Tesseroids: gravity forward modeling in spherical coordinates.
The software is a suite of C coded command-line programs that calculate the
gravitational field of a tesseroid (spherical prism) model.
There is also a separate Python implementation of the same algorithm in the
`fatiando.gravmag.tesseroid`

module of the open-source library
Fatiando a Terra: modeling and inversion for geophysics (introduced in version 0.3).

Presentations about the modeling methods and previous versions of the software:

- Optimal forward calculation method of the Marussi tensor due to a geologic structure at GOCE height
- Computation of the gravity gradient tensor due to topographic masses using tesseroids

We present the open-source software Tesseroids, a set of command-line programs to perform the forward modeling of gravitational fields in spherical coordinates. The software is implemented in the C programming language and uses tesseroids (spherical prisms) for the discretization of the subsurface mass distribution. The gravitational fields of tesseroids are calculated numerically using the Gauss-Legendre Quadrature (GLQ). We have improved upon an adaptive discretization algorithm to guarantee the accuracy of the GLQ integration. Our implementation of adaptive discretization uses a "stack" based algorithm instead of recursion to achieve more control over execution errors and corner cases. The algorithm is controlled by a scalar value called the distance-size ratio (D) that determines the accuracy of the integration as well as the computation time. We determined optimal values of D for the gravitational potential, gravitational acceleration, and gravity gradient tensor by comparing the computed tesseroids effects with those of a homogeneous spherical shell. The values required for a maximum relative error of 0.1% of the shell effects are D = 1 for the gravitational potential, D = 1.5 for the gravitational acceleration, and D = 8 for the gravity gradients. Contrary to previous assumptions, our results show that the potential and its first and second derivatives require different values of D to achieve the same accuracy. These values were incorporated as defaults in the software.

Uieda, L., V. Barbosa, and C. Braitenberg (2016), Tesseroids: Forward-modeling gravitational fields in spherical coordinates, GEOPHYSICS, F41–F48, doi:10.1190/geo2015-0204.1.