Gravitational field calculation in spherical coordinates using variable densities in depth

Abstract

SUMMARY 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 2-D 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 per cent 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. Finally, 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.

Gravity Field Modeling Using Tesseroids with Variable Density in the Vertical Direction

“…However, its accuracy decreases rapidly toward the polar regions due to the significant change of the tesseroid surface from the equator (with an almost rectangular horizontal shape) to poles (with nearly a triangular horizontal shape). In the quadrature method, the approximate solutions of Newton's volume integrals are directly solved by using numerical quadrature rules, in which the Gauss-Legendre quadrature (GLQ) is popularly applied, leading to the so-called GLQ method (e.g., Asgharzadeh et al 2007;Wild-Pfeiffer 2008;Li et al 2011;Uieda et al 2016;Deng and Shen 2018;Soler et al 2019). In some approaches, the volume integral is analytically integrated along the radial direction first.…”

“…Fukushima (2017Fukushima ( , 2018) developed a novel method, which first computes the GP of the tesseroids at an arbitrary point with very high precision by the powerful DEQ rule and then approximates the GV and GGT by numerical partial differentiation of the computed GP. In Li et al (2011), Grombein et al (2013), Uieda et al (2016), Lin and Denker (2019), Soler et al (2019), andZhong et al (2019), the improvement of the approximation is achieved by regularly or adaptively subdividing the tesseroids close to the computation point into smaller tesseroid elements along both horizontal and vertical dimensions or only in the horizontal dimension first, and then summing all effects of the subdivided tesseroid elements computed by the GLQ rule. Among the above-mentioned approaches, the last one is easy to implement and further requires no elementary body conversion, flat Earth approximation, coordinate transformation, tesseroid rotation, and numerical differentiation.…”

“…However, these formulas are limited to the computation of the GP and GV. Most recently, Soler et al (2019) proposed a new method, which is able to compute the gravitational effects of the tesseroids whose density varies with depth according to an arbitrary continuous function by the GLQ method. In the computation, the continuous density function for the tesseroid is in fact discretized into a set of smaller tesseroid elements along the radial direction by an adaptive density-based discretization algorithm, while each tesseroid element has a linear density function of the radius.…”

An Improved Method to Moho Depth Recovery From Gravity Disturbance and Its Application in the South China Sea

“…$$(r,\,\varphi ,\,\lambda )$$ is the computation point position defined in the spherical coordinate system, and $$\,\rho \left({r}^{\prime }\right)$$ is the radially variable density function. Since Equation 1 has no analytical solution, 3DGLQ is employed to estimate its approximate solution (Soler et al., 2019; Uieda et al., 2016) $$${\int }_{{\lambda }_{1}}^{{\lambda }_{2}}{\int }_{{\varphi }_{1}}^{{\varphi }_{2}}{\int }_{{r}_{1}}^{{r}_{2}}\rho \left({r}^{\prime }\right)\,f\left({r}^{\prime },\,{\varphi }^{\prime },\,{\lambda }^{\prime }\right)\mathrm{d}{r}^{\prime }\mathrm{d}{\varphi }^{\prime }\mathrm{d}{\lambda }^{\prime }\,\approx \,A\sum\limits _{i\,=\,1}^{{N}^{r}}\sum\limits _{j\,=\,1}^{{N}^{\varphi }}\sum\limits _{k\,=\,1}^{{N}^{\lambda }}{W}_{i}^{r}{W}_{j}^{\varphi }{W}_{k}^{\lambda }\,\rho \left({r}_{i}\right)\,f\left({r}_{i},\,{\varphi }_{j},\,{\lambda }_{k}\right)$$$ where …”

Gravitational field calculation in spherical coordinates using variable densities in depth

“…This new code is freely available under the BSD 3-clause opensource license. All source code, Python scripts, data, and model results are made available through an online repository doi.org/10.6084/m9.figshare.8239622 (Soler et al, 2019) or github.com/pinga-lab/tesseroid-variable-density. The repository also contains instructions for replicating all results presented here.…”