Search space design is a fundamental procedure for the inverse problem in Rayleigh wave exploration. In practice, such a crucial task mainly depends on researchers’ experience-based judgment. However, intricate near-surface materials may lead to erroneous search space design, and consequently, accurate inversion results cannot always be ensured. By the forward calculation, it is found that there is a strong relationship between the fundamental dispersion curves of a given earth model and its miniature model; namely, if the parameters (layer thicknesses and $S$ -wave velocities) of the given earth model are shrunk at a certain scale, the phase velocities on the fundamental dispersion curve will decrease at the same scale. Taking advantage of this relation, we propose a broad learning framework for search space design in a data-driven manner. First, a training set of minified models is generated using the forward calculation of dispersion curves. Then, a mapping relationship between dispersion curves and minified earth models is built via a broad learning network. Finally, a search space for the actual earth model is designed using the minified model based on the relation. As the ranges of parameters in the minified model are much smaller, the network can find the model parameters for the corresponding dispersion curve quickly and easily. Numerical simulations and field data applications demonstrate the reliability and effectiveness of the proposed method for search space design. It is concluded that the proposed method can promote accurate estimation of $S$ -wave velocities by Rayleigh wave inversion without experience-based judgment.

Shear-wave (S-wave) velocity is considered an essential parameter for the study of the earth, and Rayleigh wave inversion has been widely accepted and used to determine it. Given high-quality measured dispersion curves, the inversion performance depends on the applied optimization algorithm inside the inversion process. We propose a novel inversion framework to promote efficient and accurate inversion, i.e., a two-stage broad learning inversion framework (TS-BL). The proposed TS-BL not only inherits the powerful mapping capability and simple configured structure of broad learning (BL) network but also makes two significant improvements to better acclimatize itself to Rayleigh wave inversion. First, TS-BL adopts a two-stage inversion strategy to perform optimizing two times. It does not yield the same search space in the two inversion stages. In the first stage, because the inversion aims to find an approximation rather than the accurate value of model parameters, the difficulty in constructing the mapping model is reduced by sacrificing accuracy. Then, an effective BL network can be established using smaller sample sizes. In the second stage, the search space becomes much narrower, commencing with the approximation results obtained in the prior stage. This helps the final BL network to easily and quickly model the actual relationship between measured dispersion curves and unknown model parameters. After that, the forward modeling of measurements rather than the validation data set is exploited for tuning the network’s hyperparameters. The physical model is superior to the validation data set for selecting a suitable network complexity to adapt to the measured dispersion curves because the latter only describes an overall relationship. As a result, accurate S-wave velocities can be efficiently acquired by using the proposed TS-BL with a low cost of training samples. The efficiency and reliability of TS-BL have been demonstrated in numerical and field data examples.

Rayleigh wave inversion is a reliable method for inverting the S-wave velocities to reflect the stiffness status of the soil and rock masses of the subsurface. The optimization potential of neural networks in the inversion task is gaining recognition among researchers. Regarding neural-network-based Rayleigh wave inversion, a closer functional relationship between the training samples and the unknown function to be modeled indicates improved inversion performance. The traditional sampling method involves randomly generating samples within a predefined search space, which can result in some samples deviating from the actual functional relationship, thus reducing the accuracy and stability of the inversion. However, few studies consider the sample selection issue in the inversion process based on neural networks. This study proposes a sample selection method for selecting more appropriate training samples to overcome the neglect of sample selection, enhancing the functional modeling of neural networks for Rayleigh wave inversion. The implementation of the proposed sample selection method involves two procedures. First, the random samples are generated within a predefined search space to create a pool of samples. Afterward, the mean moving correlation coefficients of the samples inside the pool are calculated to select more suitable samples for network training based on the moving correlation calculation. Numerical simulations and field data applications demonstrate the necessity and effectiveness of the proposed sample selection method for neural-network-based Rayleigh wave inversion. It is concluded that the proposed method effectively enhances the performance of S-wave velocity estimation through Rayleigh wave inversion using neural networks.

The electrical resistivity method is widely used in near-surface mineral exploration. At present, the deterministic algorithm is commonly employed in three-dimensional (3-D) electrical resistivity inversion to obtain subsurface electrical structures. However, the accuracy and efficiency of deterministic inversion rely on the initial model. In practice, obtaining an initial model that approximates the true subsurface electrical structures remains challenging. To address this issue, we introduce a broad learning (BL) network to determine the initial model and utilize the limited memory quasi-Newton (L-BFGS) algorithm to conduct the 3-D electrical resistivity inversion task. The powerful mapping capability of the BL network enables one to find the model that elucidates the actual observed data. The single-layer BL network makes it efficient and easy to realize, leading to much faster network training compared to that using the deep learning network. Both the synthetic and field experiments suggest that the BL framework could effectively obtain the initial model based on observed data. Furthermore, in comparison to using a homogeneous medium as the initial model, the L-BFGS inversion with the BL framework-designed initial model improves the inversion accuracy of subsurface electrical structures and expedites the convergence speed of the iteration. This study provides an effective approach for fast initial model design in a data-driven manner when the prior information is unavailable. The proposed method can be useful in high-precision imaging of near-surface mineral electrical structures.

Gravity data inversion is of critical importance in geophysics, encompassing a range of applications, such as the exploration of geological resources, the identification of geological structures, and the detection of groundwater resources. This study proposes a three-dimensional (3D) machine learning approach to enhance the efficiency of the aforementioned exploration tasks by leveraging gravity data. The mapping relationship between gravity data and subsurface density structures is modeled by the broad learning network, distinguished by its high training efficiency and robust modeling capability. Notably, the proposed inversion method obviates the constraints on the number of anomalies prior to the inversion process. This is achieved by setting one anomaly with varied locations for different training samples. Numerical and field data applications demonstrate the efficiency of the proposed 3D machine learning gravity data inversion method, especially in automatically determining the number of anomalies. In particular, the proposed method produced accurate density inversion results in the field application, aiding in the identification of potential oil and gas reservoirs in the target region and offering the potential for broader application in other resource exploration. The proposed inversion method can promote the construction of density structures of subsurfaces based on gravity data.