# Three-Dimensional Broad Learning Gravity Data Inversion Using Single-Anomaly Training Samples
## 3 Results and Evaluation of Performance
### Numerical Simulations  
This section introduces three numerical density models to verify the proposed inversion method. The first model consists of multiple anomalies with rectangular shapes, designed to assess the proposed method's ability to identify the number of anomalies accurately. The second one emphasizes evaluating the capability to recover the irregular shapes of the anomalies. The third model comprises anomalies with different shapes, densities, and positions, aimed at evaluating the robustness of our approach. As shown in Figure 4, the simulated gravity data (Figure 4b) for a density model with multiple regularly shaped anomalies (Figure 4a) are first applied to examine the proposed inversion method. The length, width, and height of each anomaly are 1000 m,  
Figure 3: Objective function curves under varying network complexities. 1000 m, and 600 m, respectively, with a density of 1.0 g/cm\({}^{3}\). These surface gravity data, generated through the forward modeling of the numerical model, were used as the input to the trained network constructed using single-anomaly training samples. The network subsequently produced the corresponding inverted density model. Figure 5a provides 3D views of the inverted density model. The results clearly demonstrate that the trained BL network successfully identified four density anomalies without any prior assumption regarding the number of anomalies. It also shows a good recovery of the shape of the identified anomalies in the inverted results compared with the actual model. Additionally, the objective function value between the actual and inverted models is 0.0249, showcasing satisfactory performance in terms of inverting gravity anomaly values. To provide a more comprehensive evaluation of the inversion results, we calculated the coefficient of determination (R\({}^{2}\)) between the observed gravity anomaly (Figure 4b) and the gravity anomalies derived from the inversion results (Figure 5b), yielding a value of 0.999. The residuals between the inverted and observed gravity anomaly data were also computed, as shown in Figure 5c. The inversion residuals are predominantly concentrated within the range of \(-1.0\) to 1.0 mGal, which are relatively small compared with the gravity anomalies in Figure 4b. Undoubtedly, the functional mapping modeled by the trained BL network using single-anomaly training samples effectively captures the relationship between a multiple-anomaly density model and its measured gravity data. It is reasonable to analyze that the success of this approach lies in the variation in anomaly size and location across the training samples and, more importantly, in the BL network's ability to generalize to different numbers of anomalies. This numerical example demonstrates the effectiveness of the proposed inversion method when applied to a subsurface density model with multiple regularly shaped anomalies. The effectiveness of the proposed inversion method is further assessed using a numerical earth model with irregularly shaped anomalies. Figure 6 shows the actual density model  
Figure 4: Numerical density model with four rectangular-shaped anomalies and corresponding surface gravity anomaly observations. (**a**) Three-dimensional representation of the actual density model. (**b**) Surface gravity anomaly observations. Figure 5: Inversion results of the numerical density model with four rectangular-shaped anomalies. (**a**) Three-dimensional representation of the inverted density model. (**b**) Gravity anomaly data forwarded from the inversion result. (**c**) Residuals between inverted and observed surface gravity anomaly data. and corresponding theoretical surface gravity anomaly data. Unlike the first numerical model, this model contains two irregularly shaped anomalies with a density of 1.0 g/cm\({}^{3}\).