## 3 Data Testing
### Synthetic Data Testing
Since our algorithm is based on smooth inversion, the density values are lower than the real values of the model, but we can still determine the approximate range of the density distribution of the model relatively. Figure 1: Model I with two prisms. Figure 2: Model I related gravity data. (**a**) Gravity data; (**b**) gravity data added noise. In order to further test the algorithm's inversion effect on the complex model, we established Model II as shown in Figure 4, and its specific parameters are shown in Table 1. The fourth and fifth prisms are combined together to form an L-shaped target. The observation range on both the \(x\) and \(y\) axes is 0 to 2000 m, the observation interval is 25 m and the maximum inversion depth is set as 1000 m. If the underground space is divided into \(80\times 80\times 40\), then the size of each prism unit is \(25\) m \(\times\)\(25\) m \(\times\)\(25\) m. Similarly, 5% Gaussian random noise is added to the synthetic gravity data (Figure 5), and the inversion parameters are set as follows: The density constraint range is controlled within \([-0.1,1.1]\)g/cm\({}^{3}\), the maximum number of iterations is set to \(k\) = 100 and the cutoff error is set to \(\varepsilon=0.002\). After 22 inversion iterations, the inversion result is obtained, and the total inversion time was 359.01 s. In the inversion process, the pre-processing stage took 338.16 s to run, and the iteration process took 20.85 s, resulting in an iteration time of 0.95 s. Four cross sections are shown in Figure 6, and the true boundaries of the prisms are represented by the black boxes. The results indicate that the density distribution is in good agreement with the boundaries represented by the black boxes. Figure 3: Model I inversion results. (**a**) Cross section along \(y\) = 2000 m; (**b**) cross section along \(z\) = 900 m.

Finally, to demonstrate the acceleration advantage of the improved algorithm compared to the traditional algorithm, we use the gravity data obtained from Model I and Model II to perform inversions with mesh numbers of \(40\times 40\times 20,50\times 50\times 20,80\times 80\times 20\),

Figure 4: Model II with multiple prisms. Figure 5: Model II related gravity data. (**a**) Gravity data; (**b**) gravity data added noise. Figure 6: Model II inversion results. (**a**) Cross section along \(y\) = 300 m; (**b**) cross section along \(y\) = 1700 m; (**c**) cross section along \(x\) = 300 m; (**d**) cross section along \(z\) = 500 m.

\(100\times 100\times 20\). Tables 2 and 3, respectively, record the calculation time comparisons for data of different scales in the pre-processing stage and the single calculation time comparisons in the iterative process. As can be seen from the results, due to limitations in computer performance (such as memory), the traditional inversion algorithm was not calculated for the \(100\times 100\times 20\) mesh number, but the new algorithm still achieved a substantial inversion speed under the same computer configuration. It is evident that the improved algorithm has a significant acceleration in the running time of both pre-processing stage and the iteration stage. In the pre-processing stage, the improved acceleration algorithm is typically more than 10 times faster than the traditional algorithm (Table 2), while in the iterative process, the acceleration algorithm has more obvious advantages. As can be seen from Table 3, the fastest acceleration ratio can reach over 1000 times, and the acceleration ratio in the iterative process of the acceleration algorithm typically exceeds 400 times, fully demonstrating the advantages of the improved algorithm in terms of inversion speed. We recorded the maximum memory footprint of 40 \(\times\) 40 \(\times\) 20 grid data in both traditional and accelerated algorithms. The maximum memory usage of the accelerated algorithm is 909.6 MiB. For the same number of grids, the maximum memory usage of the traditional algorithm is 24,295.9 MiB.