## 2 Data and Methods
### L-BFGS Inversion  
Inversion of GDS can generally be treated as seeking the minimum of  
\[\Phi(\mathbf{m},\lambda)\underset{\mathbf{m}}{\rightarrow}\text{min}. \tag{3}\]  
The penalty function \(\Phi(\mathbf{m},\lambda)\) can be defined by  
\[\Phi(\mathbf{m},\lambda)=\Phi_{d}(\mathbf{m})+\lambda\Phi_{m}(\mathbf{m}). \tag{4}\]  
where \(\Phi_{d}(\mathbf{m})\) is the data misfit and \(\Phi_{m}(\mathbf{m})\) is the model roughness; \(\lambda\) is the regularization parameter of the tradeoff \(\Phi_{d}(\mathbf{m})\) and \(\Phi_{m}(\mathbf{m})\) in inversion. \(\mathbf{m}\) is the model vector and is comprised of the conductivity in each prism in three-dimensional (3D) inversions [6]. An objective function can be expressed as the notation of the \(L_{\text{p}}\)-norm measurement, and Equation (4) is expressed as  
\[\Phi(\mathbf{m},\lambda)=\mathbf{W}_{d}(\psi(\mathbf{m})-\mathbf{d})_{p}^{p}+ \lambda\mathbf{W}_{m}(\mathbf{m}-\mathbf{m}_{0})_{p}^{p} \tag{5}\]  
where \(\mathbf{d}\) is the observed data; \(\mathbf{m}_{0}\) is the background model with \(\psi\) as the operator of forward modelling; \(\mathbf{W}_{d}\) is a diagonal matrix comprised of data covariance; and \(\mathbf{W}_{m}\) is a model smoothing matrix designed to smooth the conductivity with adjacent grids in the X, Y, and \(Z\) directions. Smoothness among adjacent cells relates to the value of smoothing coefficients, and the model will be much smoother when coefficients are closer to 1. Differentiating Equation (5) on both sides with respect to the model parameters and performing a Taylor-series expansion, the equation can be expressed as the following if the higher-order terms of the expansion are neglected,  
\[ \begin{split}\left[\mathbf{J}^{\mathrm{T}}\mathbf{W}_{d}^{ \mathrm{T}}\mathbf{R}_{d}\mathbf{W}_{d}\mathbf{J}+\lambda\mathbf{W}_{m}^{ \mathrm{T}}\mathbf{R}_{m}\mathbf{W}_{m}\right]\delta\mathbf{m}\\ =\mathbf{J}^{\mathrm{T}}\mathbf{W}_{d}^{\mathrm{T}}\mathbf{R}_{d} \mathbf{W}_{d}[\psi(\mathbf{m})-\mathbf{d}]+\lambda\mathbf{W}_{m}^{\mathrm{ T}}\mathbf{R}_{m}\mathbf{W}_{m}(\mathbf{m}-\mathbf{m}_{0}).\end{split}   
The nonlinear optimization of Equation (5) is solved using the limited-memory quasi-Newton method (L-BFGS), which has been widely applied in electromagnetic induction detection [30] to seek the solution of penalty functions. The basic iterative progress in L-BFGS is  
\[\mathbf{m}_{k+1}=\mathbf{m}_{k}+\alpha_{k}\mathbf{p}_{k} \tag{8}\]  
where  
\[\mathbf{p}_{k}=-\mathbf{B}_{k}^{-1}\
abla\Phi_{k} \tag{9}\]  
and  
\[\
abla\Phi_{k}=\left.\left(\frac{\partial\Phi}{\partial m_{1}},\frac{\partial \Phi}{\partial m_{2}},\cdots,\frac{\partial\Phi}{\partial m_{N}}\right)^{ \mathsf{T}}\right|_{\mathbf{m}=\mathbf{m}_{k}} \tag{10}\]  
Here, \(k\) is the number of inversion iterations; \(\alpha_{k}\) and \(\mathbf{p}_{k}\) are the searching step and searching direction, respectively; and \(\mathbf{B}_{k}\) is the approximation of the Hessian matrix [31]. Employing the approximation matrix can avoid the direct calculation of the Hessian matrix, thus greatly reducing the amount of computer storage and processing time needed. The calculation of the Jacobian matrix and forward responses are required in the computation of Equation (6). Calculation of the forward responses of a model can be easily carried out. The direct calculation of the Jacobian matrix, however, places high requirements on computing resources, but a more feasible method--adjoint forward technique--is chosen to accomplish the calculation [3; 29; 32]. Using this technique, the products of the Jacobian matrix and data vector can be separated into a few operations of forward modeling, and thus the computational requirements are greatly reducing. Considering the strong fluctuation in recorded data at some stations and that a 3D \(L_{1}\)-norm inversion [3] in which the data misfit is measured by the \(L_{1}\)-norm can suppress the influence of data with substantial noise on the stability of inversion, \(L_{1}\)-norm inversion will be conducted in this paper.