# Electrical Conductivity Evidence for the Existence of a Mantle Plume Benath Tarim Basin
## 2 Materials and Methods
### Inversion Algorithm  
#### 2.2.1. SA Inversion  
One-dimensional inversion of GDS can be achieved by minimizing the following objective function:  
\[\hat{E}(\mathbf{\sigma},\omega,\lambda)=\sum_{N_{\omega}}\left[\lg\left|C^{\text{ obs}}(\omega)\right|-\lg\left|C^{\text{mod}}(\omega)\right|\right]^{2}+\lambda f_{ \text{m}}(\sigma), \tag{3}\]  
where \(\lambda\) is the regularization factor; \(N_{\omega}\) is the number of frequencies; \(C^{\text{obs}}(\omega)\) is the C-response computed from the observed data; and \(C^{\text{mod}}(\omega)\) is the model theoretical C-response. \(f_{\text{m}}\) is the model constraint describing the smoothness of the model. \(\sigma\) is the conductivity model vector and is the parameter that needs to be inverted. In this case, the thicknesses of the thin spheres that make up the Earth are fixed and do not participate in the inversion. The process of minimizing the objective functional is similar to the process of cooling a high-energy solid material to reach a steady state of minimal energy [34]. Thus, the simulated annealing nonlinear optimization algorithm can be adapted for the one-dimensional GDS inversion. In this case, the objective function defined by Equation (3) is taken as the energy function in the annealing process. By reducing the temperature in an appropriate manner, the state corresponding to each molecule when the value of the objective function reaches the minimuml can be accepted as the solution of an inversion problem. The improved simulated annealing (ISA) algorithm proposed in this paper is based on traditional simulated annealing combined with statistical analysis to improve the robustness of inversion. The main steps in the flowchart include:  
1. Initialization: \(T_{0}\), \(M\), \(\sigma^{\text{max}}\), \(\sigma^{\text{min}}\), \(\sigma_{0}\), \(L\), \(RMS^{\text{max}}\), \(i=0\). 2. Set \(l=1\) and perform a Markov chain loop \(L\) times. 1. Generate a model \(\sigma_{i}^{\text{ }l}\) randomly; calculate its corresponding objective function value and RMS; 2.2 When RMS is less than \(RMS^{\text{max}}\), save the model as a member of the solution set for statistical analysis in step 5; 2.3 When the objective function value \(\hat{E}\) corresponding to the model \(\sigma_{1}^{\text{ }1}\) meets the Metropolis criterion, the model is used as the current optimal solution. The loop continues if the value \(\hat{E}\) corresponding to the model does not meet the Metropolis criterion; 2.4 Update \(l\): \(l=l+1\); 3. Decrease temperature T; \(i=i+1\). 4. When \(i>M\), the annealing cycle ends; otherwise, return to step 2. 5. Statistical calculation of the optimal solution: Perform statistical analysis on all models with RMS less than \(RMS^{\text{max}}\) to determine the final solution. In the above iteration process, \(T_{0}\) is the initial temperature; \(i\) is the current number of iterations; \(M\) is the upper limit of the number of iterations; \(l\) is the current number of cycles in the chain, which is controlled by the chain length \(L\) of the Markov chain. When the value of \(L\) increases, the number of global searches at the same temperature increases, and as a result, there is a greater probability of obtaining the global optimal solution [35]. Considering the time cost, after our experimental verifications, \(L=10\) can achieve the search purpose. \(\sigma^{\text{max}}\) and \(\sigma^{\text{min}}\) are the conductivity value boundaries, and they can be estimated based on known petrophysical models or previous experience [36]. \(\sigma_{0}\) is the initial value of the model. SA inversion does not require a fixed initial model, so the initial value is a random number within the valid range. \(RMS^{\text{max}}\) represents the threshold of RMS, while RMS here is defined by  
\[RMS=\sqrt{\frac{1}{N_{\omega}}\sum_{i=1}^{N_{\omega}}\left(\frac{C^{\text{ mod}}(\omega)-C^{\text{obs}}(\omega)}{C^{\text{obs}}(\omega)}\right)^{2}}. \tag{4}\]An appropriate choice of \(RMS^{\max}\) is essential for inversion [25].