Development of a flexible 2D DC Resistivity modelling technique for use in space domain

Geoelectric non-destructive imaging and monitoring of the earth's subsurface requires robust and adaptable numerical methods to solve the governing differential equation. Most of the time, the DC data is acquired along a straight line. Hence, we solve the DC problem for the 2D case. But the source for the DC method exhibits a 3D nature. To account for the source's 3D nature, the 2D DC resistivity modeling is often carried out in the wavenumber domain. There have been studies that suggest ways for the selection of optimum wavenumbers and weights. But, this does not guarantee a universal choice of wavenumbers. The chosen wavenumbers and related weights strongly influence the precision of the resulting solution in the space domain. Many forward modeling studies demonstrate that selecting effective wavenumbers is challenging, especially for complicated models with topography, anisotropy, and significant resistivity differences. Moreover, forward modeling requires many wavenumbers as the models get more complex. 

This study focuses on developing a method that can completely omit wavenumbers for 2D DC resistivity modeling. The present work finds its motivation in a numerical experiment on a simple half-space model. Since the analytical response for such a model can be easily calculated, we match the analytical solution against the responses obtained from various wavenumbers and weights used in the literature. All the responses deviated from the analytical solution after a certain distance, and none of them were found to be accurate for large offsets. It was discovered after thorough testing of the numerical scheme that the wavenumbers selected for the forward modeling significantly impacted how practical the approach is for large offsets. 

To overcome this problem, a new boundary condition is derived and implemented in the existing numerical scheme. The numerical scheme chosen to perform the forward modeling is Mimetic Finite Difference Method (MFDM). We consider that the source is placed on the origin of the coordinate system. This removes the dependency of the source term, expressed in the Fourier domain, on the wavenumber. The solution obtained by solving the resulting equation will be an even function of the wavenumber and be real-valued. This ensures that the potential in the space domain for the 2D model will also be a real-valued even function with a symmetry about a plane perpendicular to the strike direction and passing through the origin. Because the first-order derivative of an even function at the plane of symmetry vanishes, mathematically, it can be expressed as a Neumann boundary condition at the considered plane. Therefore, we propose a scheme to solve the 2D resistivity problem in the space domain using the boundary condition mentioned here.

The developed algorithm is tested on isotropic and anisotropic two-layer models with large contrasts. It is found that the numerical solutions obtained using the modified boundary condition described above show considerable accuracy even for large offsets when compared with the analytical solution. On the other hand, the results obtained using available wavenumbers in the literature are also compared and are found to deviate considerably from the analytical solution at large offsets.