*2.2. Inversions*

The MT inversion problem is an optimization problem in which the objective is to predict a model that is close to the real geoelectric structure from the observed response (Figure 1b). The optimization process update the geoelectric model iteratively to find the minimum objective function. The objective function of this optimization problem can be divided into two terms, one corresponding to data fitting and one corresponding to the model smoothness [18]. The data fitting term measures the difference between the observed response and the predicted response (Figure 1b). The smoothness term measures the change in the magnitude of the resistivity of each stratum [21]. The objective function can be expressed as follows:

$$\min \Phi(\mathbf{m}) = \min \left( \lambda \left\| \mathbf{C}\_{\text{W}} \mathbf{m} \right\|^2 + \left\| \mathbf{C}\_{d} (\mathbf{F}[\mathbf{m}] - \mathbf{d}) \right\|^2 \right) \tag{3}$$

where <sup>Φ</sup>(**m**) is the objective function; **F** is the forward modeling operator; **d** represents the observed data; **C***m* and **C***d* are the covariance matrices of the model vector and the observed data vector, respectively; and *λ* is the Lagrange multiplier weighting the model smoothness term relative to the total norm. The objective function is updated with the predicted model **m**, and its value gradually decreases in each iteration.

To minimize the objective function, several iterative methods of linear inversions have been proposed [22,23]. The occam's inversion is a popular and stable inversion algorithm based on an iterative method in which the model is directly updated in each iteration, causing the value of the objective function to decrease steadily [24–26]. The model is updated as follows:

$$\mathbf{m}\_{k+1} = \begin{bmatrix} \frac{1}{\lambda} \mathbf{C}\_{m}^{T} \mathbf{C}\_{m} + \left(\mathbf{C}\_{d} \mathbf{J}\_{k}\right)^{T} \mathbf{C}\_{d} \mathbf{J}\_{k} \end{bmatrix}^{-1} \left(\mathbf{C}\_{d} \mathbf{J}\_{k}\right)^{T} \mathbf{C}\_{d} \mathbf{d}\_{\mathcal{S}} \tag{4}$$
 
$$\mathbf{d}\_{\mathcal{S}} = \mathbf{d} - \mathbf{F}[\mathbf{m}\_{k}] + \mathbf{J}\_{k} \mathbf{m}\_{k}$$

The iteration process begins with an initial model guess **m**0, and the model is updated to **<sup>m</sup>***k* in the *k*th iteration. The optimal model is considered to be found when a maximum number of iterations, a convergence threshold for the objective function or some other termination criterion is reached. In addition, it is important to note that the model parameters are typically expressed in terms of the logarithm of the resistivity in order to reduce the variations in the gradient.

The linear inversion methods can easily become trapped in local minima and require considerable computational effort to calculate the gradient of the objective function [24]. Moreover, they are critically dependent on the initial model [27]. However, global optimization methods based on heuristic algorithms overcome these shortcomings [8,28].
