3-D Dc Resistivity Modelling Based On Spectral Element Method With Unstructured Tetrahedral Grids

In this paper, we propose a spectral element method (SEM) based on unstructured tetrahedral grids for direct current (dc) resistivity modelling. Unlike the tensor-product of 1-D Gauss- Lobatto-Legendre (GLL) quadrature in conventional SEM, we use Proriol-Kooniwinder- Dubiner (PKD) polynomials to form the high-order basis polynomials on tetrahedral grids. The final basis functions are established by using Vandermonde matrix. Compared to traditional SEM, our method truly takes into account the high precision of spectral method and the flexibility of finite element method with unstructured grids for modelling the complex underground structures. After addressing the theory on the construction of basis functions and interpolation and integration nodes, we validate our algorithm using the analytical solutions for a layered earth model and the results from other methods for multiple geoclectrical models. We further investigate a dual-track scheme for improving the accuracy of our SEM by increasing the order of interpolation polynomials or by refining the grids.