engineering have three well-defined properties: a solution exists for the model, solution of the model is unique, and the model solution’s is continuous with the changes in initial conditions (also called parameters or input data). These three properties are also described as existence, uniqueness and stability. It is important to note that the ill-posed problems do not meet all these three well-defined properties. A problem that is not well-posed is known as ill-posed. Several first-order differential equations and inverse problems are ill-posed. If the physical problem or mathematical model is not well-posed, it is required to be reformulated for numerical computation. Typically, ill-posed models or problems require additional assumptions (e.g., smoothness of solution) and the process is usually called as regularization [1]. In geophysical literature, Tikhonov regulation is one of the highly used regularizations for the ill-posed problems.
Continuous mathematical models are often discretized to obtain numerical solutions. These solutions may be continuous with respect to the initial parameters. Furthermore, when these problems are solved with a finite precision, it may suffer from a numerical instability. Even though these problems are well-posed, they may be ill-conditioned. Here, the meaning of ill-conditioned refers to a small error in the initial data resulting in a larger error in the solution. In mathematical literature, an ill-conditioned problem is defined by a large condition number, which is a measure of sensitivity of the model. This gives indication quantitatively how much error is in the output from an error in the input. A physical model is called well-conditioned if it has a low condition number. If the condition number is high, it is called ill-conditioned.
An inverse problem in the field of science and engineering is a process of calculating physical model parameters from a set of real or synthetic observations (in other words, computing the input parameters from the output data/results). Examples are computing images in X-ray computed tomography, source reconstruction, calculating density distributions of the Earth material from the measured gravity potential field, etc. It is known as an inverse problem because it starts with the results of the physical model and computes the physical model parameters (called input to the model). In other words, this can be viewed as the inverse of a forward problem, which starts with the causes and then calculates the effects.
Linear or non-linear Inverse problems are very important mathematical problems in the field of science and engineering. This is due to the fact that these problems give us information about the parameters that cannot be accessed or observed directly. These problems have a wide range of applications in system identification in the field of science and engineering including natural language processing, machine learning, nondestructive testing, and many other domains. This paper is focused on in-depth analysis of ill-posed inverse problems that are usually common in electrical geophysics.
2.2 Fundamentals of Ill-Posed Inverse Problems
These problems attempt to describe a system of coefficient matrix from observed data, which is used to estimate physical model parameters of the forward numerical models. This coefficient matrix usually represents important properties of the media or of the model that is under study. These properties in the field of geophysics are density, electrical conductivity, heat conductivity, magnetic susceptibility, etc., of the Earth materials. In the process of solving such problems, it is possible to delineate many other details such as the structural intrusions, defects, source of contaminations, location, shape, etc. There is a large number of scientific articles and research publications that have dealt with the ill-posed inverse problems directly or indirectly. Since the theory associated with the inverse problems is relatively new, a shortage of textbooks has been felt in this area. This is due to the fact that the new theories, concepts and approaches have been continuously evolving. Kabanikhin [2] has given a very good and detailed overview of the ill-posed inverse problems. In this review paper, definition of ill-posed inverse problems, its types, and several examples of ill-posed inverse problems are described well.
2.3 Brief Historical Development of Resistivity Inversion
According to the literature review in the field of electrical geophysics, interpretation of electrical resistivity data using electrical resistivity inverse methods are commonly done for layered models and geological structures (e.g., groundwater exploration and mapping & monitoring of groundwater). However, in the field of mineral exploration, geothermal exploration, mapping and monitoring of in-situ processes, the layered geologic models are inadequate. With the advent of large computers, two-dimensional (2-D) numerical electrical modelling techniques for surface-to-surface electrode and other electrode configurations are used extensively to interpret electrical data. Integral equation method has a limitation because it allows inhomogeneties only inside the homogeneous sounding host media. Three-dimensional (3-D) numerical modelling methods using finite difference and finite element methods are reported in the geophysical literature. These methods are useful to compute electrical model response over a given 3-D geologic structure. A complete overview about the forward and inverse modelling in electrical geophysics may be found in Narayan [3].
Most of the forward modelling methods have attempted to address some aspect of the design of field experiments. These methods are not very useful for the interpretation of electrical field data on two counts. First, it is based on a trial-and-error mode and second, it is time intensive. Furthermore, they do not yield additional information. Therefore, it is important to introduce a new and effective electrical inversion method to interpret electrical field data in terms of 2-D or 3-D geological models for a variety of electrode configurations. This article emphasizes to use a generalized inverse theory for multi-dimensional structures and an attempt has been made to develop a practical way of inverting the resistivity data for mapping and monitoring two-dimensional (2-D) geologic features using a pair of surface and subsurface electrodes [3]. A brief historical development of resistivity inversion (one-dimensional (1-D) resistivity inversion, two-dimensional (2-D) resistivity inversion, and three-dimensional (3-D) resistivity inversion) used in environmental, engineering and hydrological fields has been reported extensively [4–12].
Electrical impedance tomography (EIT) methods are also known as electrical resistivity imaging methods – another version of resistivity inversion. These methods have proven to work nicely in most of the geophysical settings. EIT methods/imaging methods have been gaining momentum rapidly in recent years. This is due to the fact that they are easy to use and they are non-invasive testing tools. EIT methods or electrical resistivity methods are based on a low-frequency electrical current or DC current (unidirectional flow of electric charge) to probe a medium of the system, and measure its resistance or impedance of electrical current flow. EIT methods are highly sensitive to changes in electrical resistivity. In EIT methods, a known amount of electrical current is injected in the medium of the model and resulting electrical potential field is measured around the boundary points of the medium of the model. From these data, it is possible to perform electrical resistivity inversion of these potential field data to determine the electrical conductivity or resistivity inside the medium of the model that is being probed by the currents. In this way, the internal resistivity or electrical conductivity distribution is reconstructed using electrical measurement from the boundary of the medium of the model under investigation. Such electrical resistivity or conductivity distribution gives valuable information about the interior of the medium. Basically, all EIT methods deal with solving forward problem and inverse problem iteratively. EIT methods are used for various shallow depth archaeological prospecting, geothermal resource prospecting, geo-environmental monitoring, hydrogeological invitations, and geotechnical investigations etc. A good account of EIT methods and electrical resistivity imaging tools may be found in many geophysical literatures [13–20]. In brief, all these inverse problems are solved iteratively until a best possible solution is obtained.
2.4 Overview of Inversion Schemes
In the preceding sections, iterative approaches of electrical inversion methods are described. These approaches consist of a function called “objective function”. It starts an initial parameter guess, updated at the end of every run iteratively and then applied to the “objective function”
