Accurate volume integral solutions of direct current resistivity potentials for inhomogeneous conductivities in half space

We develop a novel accurate volume integral formula for solving potentials of three dimensional (3D) direct current resistivity problems with inhomogeneous conductivities in half-space. This new integral formula is composed of the potential, the gradient of Green's function, the gradient of the potential and the anomalous conductivity as the physical variables. First, the unstructured grids are adopted to handle inhomogeneous bodies with complicated shapes. Then, in each anomalous tetrahedron, the potential is represented as its values at vertices and the linear shape functions. Analytical expressions are developed to evaluate singular volume integrals in the final system of linear equations when the observation sites locate in the anomalous body. Finally, two synthetic models are utilized to verify the accuracy and convergence rate of our newly developed volume integral formula and its capability of dealing with complicated models with high conductivity contrasts.