# Three-Dimensional Broad Learning Gravity Data Inversion Using Single-Anomaly Training Samples
## 4 Discussion  
This study proposes a novel method for 3D gravity data inversion. The single-layered architecture of the BL network facilitates a time-efficient training process. The introduction of the BL network enables the proposed method to perform real-time inversion, providing superior time efficiency compared with traditional inversion algorithms. Additionally, by using single-anomaly training samples, the proposed method eliminates the need to consider the number of anomalies during the training process. This section explores these three aspects in detail. The only unknown in the BL network is the network weighting matrix, which involves far fewer unknown parameters compared with a deep learning network. As a result, the training process of the BL network-based inversion is more efficient than that of the deep learning network-based inversion. In fact, this has been validated in previous surface wave inversion studies. For example, ref. [42] provides a comparison between the BL network and the Levenberg-Marquardt algorithm-based deep learning network for inverting subsurface shear wave velocities, showing that the BL network-based inversion requires much less training time for modeling. In real-world applications, cross-zone gravity exploration is often unavoidable, necessitating network retraining due to changes in the functional mapping relationship. A gravity data inversion method with a time-efficient training process can significantly improve the efficiency of field exploration, and the method proposed in this study is precisely such an approach. The functional mapping learned from the training samples can approximate the relationship between the measured surface gravity data and the density model. As a result, the machine learning-based inversion method can achieve real-time inversion capabilities. To better evaluate the performance of the proposed method, the inversion of the first numerical density model in Section 3.1 was also conducted using the Gauss-Newton algorithm for comparison purposes. After an inversion time of 4709 s, the resulting inverted density model was determined, as shown in Figure 12a. The objective function value between the actual and inverted models is 0.0316, which is close to the value obtained by the proposed method (0.0249). The residuals between the inverted and observed surface gravity anomaly data are distributed within the range of \(-\)0.1 to 0.1 mGal (Figure 12c), which is slightly smaller than those produced by the proposed method. The inversion results  
Figure 11: Inversion results of the field data over the Vinton salt dome. (**a**) Three-dimensional representation of the inverted density model. (**b**) Horizontal slice at a depth of \(Z\) = 300 m. (**c**) Vertical slices at X = 2000 m and Y = 1600 m. (**d**) Gravity anomaly data forwarded from the inversion result. (**e**) Residuals between inverted and observed surface gravity anomaly data. shown in Figure 12 suggest that the Gauss-Newton algorithm demonstrates a comparable ability to search for the optimal value relative to the proposed method. Additionally, the boundary detection capabilities for the anomalies were compared between the traditional and proposed methods for further evaluation. Figure 13 presents the vertical slices at Y = 1500 m (Figure 13a for the traditional method and Figure 13b for the proposed method) and the density value curves (Figure 13c) along a straight line passing through the centers of two density anomalies on the vertical slices. The density value curves from the actual model, the traditional method, and the proposed method in Figure 13c clearly show that the curve from the proposed method more closely matches the actual values than that of the traditional method. The results demonstrate that the proposed method outperforms the traditional method in terms of boundary detection capabilities. Particularly, the proposed method can predict the subsurface density model based on the measured gravity data in real time, without requiring more than an hour to compute the inversion results.