## 2 Method
\begin{split}\phi=[\mathbf{d}-\mathbf{F}(\mathbf{m})]& \mathbf{W}_{d}[\mathbf{d}-\mathbf{F}(\mathbf{m})]+a_{H}\Big{(} \mathbf{m}^{T}\mathbf{S}_{x}^{T}\mathbf{W}_{x}\mathbf{S}_{x}\mathbf{m}+\mathbf{ m}^{T}\mathbf{S}_{y}^{T}\mathbf{W}_{y}\mathbf{S}_{y}\mathbf{m}\Big{)}\\ +& a_{V}\Big{(}\mathbf{m}^{T}\mathbf{S}_{z}^{T} \mathbf{W}_{z}\mathbf{S}_{z}\mathbf{m}\Big{)}+a_{p}(\mathbf{P}\mathbf{m}- \mathbf{m}_{p})^{T}W_{p}(\mathbf{P}\mathbf{m}-\mathbf{m}_{p}).\end{split}  \tag{3}\]  
The regularization parameters \(a\) set the relative weighting for the different constraints in this objective function. \(\mathbf{W}_{d}\) is a diagonal matrix whose elements depend on the standard deviations associated with the measurements. The diagonal weighting matrices \(\mathbf{W}_{x}\), \(\mathbf{W}_{y}\), \(\mathbf{W}_{z}\), and \(\mathbf{W}_{p}\) can be used to assign different degrees of constraining to different regions of the interpretive models by ascribing different weights to each constraining relation between pairs of parameters, i.e., each row in the matrix Equation (1). The inversion is an implementation of the Gauss-Newton method, with Levenberg-Marquardt iterations [13; 14]. In each iteration, the Marquardt's damping parameter \(\mu\) is multiplied or divided by a constant factor, based on the success of the generated model in decreasing the value of the objective functional \(\phi\). In the examples shown here, the multiplicative factor is \(2\). In the \((k+1)^{th}\) iteration, the regularized estimator is, then, written as  
\[\mathbf{m}_{k+1}=\mathbf{m}_{k}+\left[\mathbf{J}_{k}^{T}\mathbf{W}_{ d}\mathbf{J}_{k}+a_{H}(\mathbf{S}_{x}^{T}\mathbf{W}_{x}\mathbf{S}_{x}+\mathbf{S}_{y}^{T} \mathbf{W}_{y}\mathbf{S}_{y})+a_{V}(\mathbf{S}_{z}^{T}\mathbf{W}_{z}\mathbf{S}_ {z})+\right.\\ a_{p}\mathbf{W}_{p}+\mu\mathbf{I}\right]^{-1}\left[(\mathbf{W}_{ d}\mathbf{J}_{k})^{T}(\mathbf{d}-\mathbf{F}(\mathbf{m}_{k}))+a_{H}(\mathbf{S}_{x}^{T} \mathbf{W}_{x}\mathbf{S}_{x}\mathbf{m}_{k}+\mathbf{S}_{y}^{T}\mathbf{W}_{y} \mathbf{S}_{y}\mathbf{m}_{k})+\right.\\ a_{V}(\mathbf{S}_{z}^{T}\mathbf{W}_{z}\mathbf{S}_{z}\mathbf{m}_ {k})+a_{p}\mathbf{W}_{p}(\mathbf{m}_{k}-\mathbf{m}_{p})\right]\!, \tag{4}\]  
where \(\mathbf{I}\) is the identity matrix and the elements of the Jacobian matrix \(\mathbf{J}\) are the sensitivities  
\[J_{ij}=\frac{\partial\log|E_{i}|}{\partial log(\rho_{j})}. \tag{5}\]  
Here, the logarithms are employed to reduce the range of values of the objective function since the field amplitudes, as well as the resistivities, can span several orders of magnitude. Reducing these ranges improves the process by allowing a more balanced fitting of the data set. The logarithms also have the advantage of preventing negative values to be attributed to the resistivities. In this application, all resistivities are isotropic; therefore, only one value (one parameter) is ascribed to each layer in a CMP column. To take advantage of the analytical derivatives of the field (\(\partial E_{i}/\partial\rho_{j}\)), which are calculated as presented by [5], the logarithmic sensitivities, as defined in Equation (5), are calculated as  
\[J_{ij}=\rho_{j}Re\left\{\frac{1}{E_{i}}\frac{\partial E_{i}}{\partial\rho_{j} }\right\}. \tag{6}\]  
Starting with a suitable initial model, which can simply be a homogeneous earth (all layers with the same resistivity), the iteration represented by Equation (4) is repeated until the convergence criteria are satisfied. In this application, the stopping criteria are the minimum variation of the model and the minimum variation of data misfit from one iteration to the next.