## 2 Inversion Methodology
Then, the new objective function has the form \[\phi=\|\mathbf{hy}-\mathbf{r}\|_{2}^{2}+\mu^{2}\|\mathbf{W_{m}y}\|_{2}^{2}-2 \lambda\sum\limits_{i=1}^{n}[\ln(y_{i}-y_{\text{min},i})+\ln(y_{\text{max},i}-y_ {i})], \tag{7}\]  
where \(-2\lambda\sum\limits_{i=1}^{n}[\ln(y_{i}-y_{\text{min},i})+\ln(y_{\text{max},i} -y_{i})]\) is the barrier function, and \(\lambda\) is the barrier parameter. The [PERSON] method [13; 14; 15] is used to solve the minimization of (7) iteratively. The regularization parameter \(\mu\) is kept fixed during the iterations, and the barrier parameter \(\lambda\) decreases with iteration [13]. At the \(k\)th iteration, one step of the Newton method is applied for (7) to yield  
\[ \begin{split}\left\{\mathbf{h}^{T}\mathbf{h}+\mu^{2}\mathbf{W_{m }}^{T}\mathbf{W_{m}}+\lambda^{(k)}\bigg{[}\Big{(}\mathbf{X}^{(k)}\Big{)}^{-2} +\Big{(}\mathbf{Y}^{(k)}\Big{)}^{-2}\bigg{]}\right\}\Delta\mathbf{y}^{(k)}=\\ -\mathbf{h}^{T}\Big{(}\mathbf{hy}^{(k-1)}-\mathbf{r}\Big{)}- \mu^{2}\mathbf{W_{m}}^{T}\mathbf{W_{m}}\Big{(}\mathbf{y}^{(k-1)}-\mathbf{y}^{ (0)}\Big{)}+\lambda^{(k)}\bigg{[}\Big{(}\mathbf{X}^{(k)}\Big{)}^{-1}+\Big{(} \mathbf{Y}^{(k)}\Big{)}^{-1}\bigg{]}\mathbf{e}\end{split} , \tag{8}\]  
where \(\mathbf{X}^{(k)}=diag\Big{(}\mathbf{y}^{(k-1)}-\mathbf{y_{min}}\Big{)}\), \(\mathbf{Y}^{(k)}=diag\Big{(}\mathbf{y_{max}}-\mathbf{y}^{(k-1)}\Big{)}\), \(\mathbf{e}\in R^{m\times 1}\) is the vector with all entries one, and \(\mathbf{y}^{(0)}\) is an initial model. The solution \(\Delta\mathbf{y}^{(k)}\) is the search direction at the \(k\)th iteration. The strategy of [PERSON] and [PERSON] [13] is used to obtain the final inversion result, and Algorithm 1 shows the detailed steps. In step 9, the solution \(\Delta\mathbf{y}^{(k)}\) can be obtained by the preconditioned conjugate gradient (PCG) method, and the preconditioner \(\mathbf{P}^{(k)}\) has the form  
\[\mathbf{P}^{(k)}=diag\Bigg{(}\sqrt{diag\Big{(}\mathbf{A}^{(k)}\Big{)}}\Bigg{)}, \tag{9}\]  
where \(\mathbf{A}^{(k)}\) has the form  
\[\mathbf{A}^{(k)}=\mathbf{h}^{T}\mathbf{h}+\mu^{2}\mathbf{W_{m}}^{T}\mathbf{W_ {m}}+\lambda^{(k)}\bigg{[}\Big{(}\mathbf{X}^{(k)}\Big{)}^{-2}+\Big{(}\mathbf{ Y}^{(k)}\Big{)}^{-2}\bigg{]}. \tag{10}\]  
```
Preparation : \(\mathbf{d_{obs}}\), \(\mathbf{G}\), \(\mathbf{W_{d}}\), \(\mathbf{m_{min}}\), \(\mathbf{m_{max}}\) and \(K_{\text{max}}\). 1. Calculate \(\mathbf{W_{m}}\), and \(\mathbf{Z}\). 2. Initialize \(\mathbf{m}^{(0)}=0.001\), and \(k=0\). 3. Calculate \(\mathbf{h}=\mathbf{W_{d}GZ}^{-1}\), \(\mathbf{r}=\mathbf{W_{d}}\mathbf{d_{obs}}\), \(\mathbf{y}^{(0)}=\mathbf{Z}\mathbf{m}^{(0)}\), \(\mathbf{y_{min}}=\mathbf{Z}\mathbf{m_{min}}\), and \(\mathbf{y_{max}}=\mathbf{Z}\mathbf{m_{max}}\). 4. Estimate the regularization parameter \(\mu\). 5. Calculate \(\lambda^{(1)}=\frac{\big{\|}\mathbf{hy}^{(0)}-\big{\|}_{2}^{2}+\mu^{2}\big{\|} \mathbf{W_{m}y}^{(0)}\big{\|}_{2}^{2}}{-2\frac{\phi}{\mu^{2}}\Big{[}\ln\Big{(} \mathbf{y}^{(0)}-\mathbf{y}_{\text{min}}\Big{)}+\ln\Big{(}\mathbf{y}_{\text{ max}}-\mathbf{y}_{i}^{(0)}\Big{)}\Big{]}}\). 6. while \(k<K_{\text{max}}\)
7. \(k=k+1\). 8. Update \(\mathbf{X}^{(k)}=diag\Big{(}\mathbf{y}^{(k-1)}-\mathbf{y_{min}}\Big{)}\) and \(\mathbf{Y}^{(k)}=diag\Big{(}\mathbf{y_{max}}-\mathbf{y}^{(k-1)}\Big{)}\). 9. Solve Equation (8) for \(\Delta\mathbf{y}^{(k)}\). 10. Update \(\mathbf{y}^{(k)}=\mathbf{y}^{(k-1)}+\delta\beta^{(k)}\Delta\mathbf{y}^{(k)}\), where \(\delta=0.925\) and \[\beta^{(k)}=\left\{ \begin{array}{cc}1,&\text{if }\mathbf{y_{min}}<\mathbf{y}^{(k-1)}+\Delta \mathbf{y}^{(k)}<\mathbf{y_{max}}\\ \min\bigg{(}\min\limits_{y_{i}^{(k-1)}+\Delta y_{i}^{(k)}<y_{\text{min}}}\frac{y _{i}^{(k-1)}-y_{\text{min}}}{\big{|}\Delta y_{i}^{(k)}\big{|}},&\min\limits_{y _{i}^{(k-1)}+\Delta y_{i}^{(k)}>y_{\text{max}},i}&\frac{y_{\text{max}}-y_{i}^{( k-1)}}{\big{|}\Delta y_{i}^{(k)}\big{|}}\bigg{)}\,,&\text{otherwise} \end{array} \right.\]
11. Exit the loop if the termination criterion is satisfied. 12. Update \(\lambda^{(k+1)}=\Big{[}1-\min\Big{(}\beta^{(k)},\delta\Big{)}\Big{]}\lambda^{(k)}\). 13. End
14.