## 3 Data Testing  
### Synthetic Data Testing  
To demonstrate the computational efficiency of the proposed method, we designed the model based on the published article by [PERSON] et al. (2014). Regarding the accelerated algorithm we presented, we aim to compare the acceleration advantages of the proposed method with traditional algorithms and the GPU parallel algorithm proposed by [PERSON] et al. (2014). To better explain the impact of our method, we built two models to test the effect of the improved algorithm. The parameters of the models are shown in Table 1. In this paper, the steps (1) to (6) in Algorithms 1 and 2 are referred as the preprocessing stage, and the steps (7) to (11) are referred to as the iteration stage. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Model I** & **Model II** \\ \hline The number of prisms & 2 & 5 \\ Density (g/cm\({}^{3}\)) & 0.5 & 1.0 \\ Size (m\({}^{3}\)), Top depth (m) & & \(1000\times 500\times 250\), 350 \\ Size (m\({}^{3}\)), Top depth (m) & \(1000\times 1500\times 600\), 600 & \(400\times 300\times 300\), 200 \\ Size (m\({}^{3}\)), Top depth (m) & \(1000\times 1500\times 600\), 600 & \(400\times 400\times 300\), 200 \\ Size (m\({}^{3}\)), Top depth (m) & & \(400\times 400\times 300\), 200 \\ Size (m\({}^{3}\)), Top depth (m) & & \(250\times 750\times 300\), 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related parameters of Model I and Model II. ```
(1) Input: \(S,d\) Set: \(k=0,m_{0}=0\) (2) Calculate the weighted parameter matrix: \(W_{d}^{M\times n}=diag(diag(SS^{T})^{\frac{1}{2}})\), \(W_{n}^{N\times N}=diag(diag(S^{T}S)^{\frac{1}{2}})\) (3) Calculate the weighted parameter: \(S_{W}=W_{4}SW_{n}^{-1}\), \(d_{W}=W_{4}d\), \(m_{W0}=0\) (4) Calculate intermediate variables: \(Q=S_{W}^{N}S_{W}+a_{0}W_{n}^{-2}\), \(q=S_{W}^{T}d_{W}\) (5) Calculate the initial search direction: \(p_{0}=r_{0}=q-Qm_{W0}\), and calculate the initial search step size: \(t_{0}=\frac{r_{0}^{T}r_{0}}{r_{0}^{T}Qm_{0}}\) (6) Update the initial solution and calculate the error: \(m_{out}=m_{u0}+t_{0}\times p_{0},\ m_{0}=W_{n}^{-1}m_{W0}\) \(m_{0}(m_{0}>m_{max})=m_{max},\ m_{0}(m_{0}<m_{min})=m_{min}\) \(m_{u0}=W_{n}m_{u0},\ loss=\frac{1}{M}\|Sm_{0}-d\|^{2}\) (7) Set the algorithm iteration condition: \(while\ \ loss>\varepsilon\ \sigma\ k<N_{max}\): (8) Upset parameter \(Q\), and then update negative gradient \(r_{2}\): \(k=k+1\) \(Q_{k}=S_{W}^{T}S_{W}+a_{k}W_{n}^{-2}\) \(r_{k}=q-Q_{k}m_{u0,k-1}\) (9) Calculate the intermediate parameter \(\beta\)



of the prisms is 600 m, and the horizontal distance of two prisms is 1000 m as shown in Figure 1. The observation plane is located on \(z=0\), and the measurement area covers a range of 4000 m \(\times\) 4000 m with a sampling interval of 100 m. The maximum inversion depth is set to 2000 m. To ensure the stability of the algorithm, 5% Gaussian random noise is added to the synthetic gravity data. Figure 2 shows the gravity anomaly data with and without noise. The proposed algorithm in this paper is applied to invert the gravity data anomaly with noise, and the inversion parameters were set as follows: the density constraint range was controlled at \([-0.1,0.6]\) g/cm\({}^{3}\), the maximum number of iterations \(k=100\), and the cutoff error was set at \(\varepsilon=0.002\). After 17 inversion iterations, the data fitting difference reached 0.001, thus ending the iteration. In the inversion process, the pre-processing stage took 6.27 s, and the iteration process took 1.36 s, resulting in a single iteration time of 0.08 s. As shown in Figure 3, we select a cross section along \(y=2000\) m and \(z=900\) m to display the inversion results, in which the black boxes in the section represents the true boundary of the established model. It can be seen that the inversion results are in good agreement with the real model position, indicating that the improved algorithm has anti-noise performance.