Influence of noise estimation on Electrical Resistivity Tomography Data Inversion

Electrical resistivity tomography is a simple, cost-effective, and highly practical method for surveying near-surface properties. Today, this method is widely used in the discovery and exploitation of water resources, archeology, and environmental and hydro-geophysical studies (such as estimating the hydrogeological parameters of the aquifer). In electrical resistivity imaging, according to the purpose and location of data collection, the electrodes are placed in specific arrays, and data collection is performed. The collected data (potential distribution or apparent resistivity) is then transformed into a distribution of actual electrical resistivity values using inverse modeling methods. Imaging requires defining and solving a nonlinear inverse problem. In this strategy, we optimize the objective function, which consists of fitting field and theoretical data. First, the physics of the problem (forward model) is presented by solving Poisson's equation with the finite difference numerical solution method. An accurate and efficient forward calculation is the basis of most processes of the inversion. Calculation of resistivity forward responses is carried out using simulation of the current flow into the earth’s surface through solving Poisson’s equation. In this contribution, a finite-difference algorithm is applied to discretize the simulated models, restricted by a mixed boundary condition. One of the merits of the finite-difference method over the other methods is its well-known ability to quickly approximate the solutions for any arbitrary and complex structure models. The finite-difference method is relatively fast compared with the finite-element method. However, to include a general topography, the finite-element method becomes a better selection despite being computationally expensive. The partial differential equations governing the resistivity problem are obtained by using the principle of conservation of charge and the continuity equation.The inverse problem is then solved by linearizing the problem in different iterations. A significant part of this research is how to perform inverse modeling of electrical resistivity data. The formulation and solution of the forward and inverse problem in this dissertation have been programmed in MATLAB and part of the program has been written in the C language to increase the computing speed. The field data is noisy due to the non-ideal measuring instruments, improperly filed conditions, operator errors, and geological conditions. Noise values can play a pivotal role in the inversion of electrical resistivity due to the special properties of the inverse problem. A proper estimation of field measurements noise level prevents over- or under-fitting of the calculated data and field data during inversion. Improper fitting (i.e., fitting where the value of the parameter  is much more or less than one) leads to creating an artifact or loss of important details in the final inverted model. In this paper, to deal with the effect of noise level on the ERT inversion results, two methods of reciprocity error method and stacking error method have been used. The results of numerical modeling show that the appropriate estimation of the noise level leads to the estimation of subsurface resistivity models close to the ground reality. We also provide a comparison between the inversion results obtained with the presence of noise level and those derived without including the weighting matrix into the objective function.