diff --git a/README.md b/README.md index 956861d..8e352c4 100644 --- a/README.md +++ b/README.md @@ -24,29 +24,36 @@ homogeneous density.* ## Abstract -We present a new methodology to compute the gravitational fields generated by tesseroids -(spherical prisms) whose density varies with depth according to an arbitrary continuous -function. It approximates the gravitational fields through the Gauss-Legendre Quadrature -along with two discretization algorithms that automatically control its accuracy by -adaptively dividing the tesseroid into smaller ones. The first one is a preexisting two -dimensional adaptive discretization algorithm that reduces the errors due to the -distance between the tesseroid and the computation point. The second is a new -density-based discretization algorithm that decreases the errors introduced by the -variation of the density function with depth. The amount of divisions made by each -algorithm is indirectly controlled by two parameters: the distance-size ratio and the -delta ratio. We have obtained analytical solutions for a spherical shell with radially -variable density and compared them to the results of the numerical model for linear, -exponential, and sinusoidal density functions. These comparisons allowed us to obtain -optimal values for the distance-size and delta ratios that yield an accuracy of 0.1% of -the analytical solutions. The resulting optimal values of distance-size ratio for the -gravitational potential and its gradient are 1 and 2.5, respectively. The density-based -discretization algorithm produces no discretizations in the linear density case, but a -delta ratio of 0.1 is needed for the exponential and the sinusoidal density functions. -These values can be extrapolated to cover most common use cases. However, the -distance-size and delta ratios can be configured by the user to increase the accuracy of -the results at the expense of computational speed. Lastly, we apply this new methodology -to model the Neuquén Basin, a foreland basin in Argentina with a maximum depth of over -5000 m, using an exponential density function. +We present a new methodology to compute the gravitational fields generated by +tesseroids (spherical prisms) whose density varies with depth according to +an arbitrary continuous function. +It approximates the gravitational fields through the Gauss-Legendre Quadrature along +with two discretization algorithms that automatically control its accuracy by adaptively +dividing the tesseroid into smaller ones. +The first one is a preexisting two dimensional adaptive discretization algorithm that +reduces the errors due to the distance between the tesseroid and the computation point. +The second is a new density-based discretization algorithm that +decreases the errors introduced by the variation of the density function with depth. +The amount of divisions made by each algorithm is indirectly controlled +by two parameters: the distance-size ratio and the delta ratio. +We have obtained analytical solutions for a spherical shell with radially variable +density and compared them to the results of the numerical model for linear, +exponential, and sinusoidal density functions. +The heavily oscillating density functions are intended only to test the algorithm to its +limits and not to emulate a real world case. +These comparisons allowed us to obtain optimal values for the distance-size and +delta ratios that yield an accuracy of 0.1% of the analytical solutions. +The resulting optimal values of distance-size ratio for the gravitational potential and +its gradient are 1 and 2.5, respectively. +The density-based discretization algorithm produces no discretizations in the linear +density case, but a delta ratio of 0.1 is needed for the exponential and most sinusoidal +density functions. +These values can be extrapolated to cover most common use cases, which are simpler than +oscillating density profiles. +However, the distance-size and delta ratios can be configured by the user to increase +the accuracy of the results at the expense of computational speed. +Lastly, we apply this new methodology to model the Neuquén Basin, a foreland basin in +Argentina with a maximum depth of over 5000m, using an exponential density function. ## Reproducing the results