Electrical resistivity tomography (ERT) is a widely used for investigating subsurface properties, particularly in near-surface studies. It has found broad application in various fields, such as groundwater exploration, archaeology, environmental monitoring, and hydrogeophysical research, including the evaluation of aquifer parameters. In ERT, electrodes are strategically placed according to the survey goals and site characteristics to gather data. These measurements, which represent the distribution of potential or apparent resistivity, are then analyzed using inverse modeling techniques to obtain the actual resistivity distribution. This process involves solving a nonlinear inverse problem, which aims to minimize discrepancies between field data and theoretical predictions by optimizing an objective function.
    The method is based on forward modeling, which simulates the physical behavior of the system, often by solving Poisson’s equation through a finite difference approach. Accurate forward modeling is crucial for effective inversion. In this study, resistivity responses are derived by simulating the flow of current through the Earth's surface, with Poisson's equation serving as the guide. A finite-difference algorithm is employed to discretize the models, incorporating mixed boundary conditions to enhance precision and reliability. One key advantage of the finite-difference method over other approaches is its established ability to quickly approximate solutions for complex and arbitrary structural models, often providing faster results than the finite-element method. The partial differential equations that describe the resistivity problem are derived using the principles of charge conservation and the continuity equation. To solve the inverse problem, the equations are linearized through iterative processes.
    A central focus of this study is the application of inverse modeling to electrical resistivity data. The forward and inverse problem formulations, along with their respective solutions, have been implemented in MATLAB, with performance improvements achieved through C programming for computational efficiency. Field data are subject to noise, which may arise from factors such as imperfect measuring instruments, suboptimal field conditions, operator errors, and geological influences. These noise components can significantly affect the inversion process, given the inherent challenges of the inverse problem.
    This study investigates the impact of data weighting matrices on the accuracy of geoelectrical data inversion, with focus on electrical resistivity data. The Occam inversion method was utilized as the primary framework for applying various weighting matrices and constraints during the inversion process. Our analysis shows that due to the presence of random noise, variations in the signal-to-noise ratio, the spacing between current and potential electrodes, the different arrays used along a profile, and geological complexities at the data acquisition site, employing data weighting matrices is essential for accurate inversion. Results from synthetic and field models demonstrate that applying a weighting matrix significantly improves the representation of conductive layers and reduces inversion errors. In field studies, validation using agricultural water wells confirmed that inversion results with a weighting matrix closely match geological realities. Additionally, the evaluation of inversion sections using resolution density, upper bounds of the resistivity variation, and sensitivity pattern indicates that the application of weighting matrices produces more reliable results.