DC resistivity inversion using conjugate gradient and maximum likelihood techniques with hydrogeological applications

This study introduces a DC 2D inversion algorithm that employs conjugate gradients relaxation to solve the maximum likelihood inverse equations. The adoption of the maximum likelihood algorithm was motivated by its advantage of not requiring the calculation of electrical field derivatives. While the inversion algorithm based on the maximum likelihood inverse theory has been extensively described for 3D DC inversion using finite differences modelling, its application in the 2D finite element method has received limited attention. A significant difference between 3D finite difference modelling and 2D finite element methods lies in the integration variable lambda. In our 2D case, the electrical potential is initially calculated in the Laplace and Fourier domains, which include the stiffness matrix. However, to obtain the stiffness matrix in the Cartesian domain, we had to develop a suitable transformation method since no existing resources in the literature addressed this specific condition. In this study, we successfully transformed the stiffness matrix using a similar approach to the potential calculation. The results obtained from synthetic and real models demonstrate the method’s potential for various applications, as exemplified by the hydrogeological study presented in this work.