Subsurface structures in reality commonly have fractal characteristics. However, conventional magnetic exploration methods typically adopt anomalies with uniform physical properties for the forward modeling, which fails to capture the multiscale self-similarity of real geological bodies, and ultimately compromises the accuracy of subsequent inversion interpretations. The Menger sponge is a classic 3-D fractal structure that effectively characterizes the fractal properties of subsurface ore bodies. Therefore, focusing on the characterization of magnetic anomalies in complex fractal structures, this study conducted 3-D forward modeling of magnetic multi-quantities including magnetic anomaly, magnetic three components, and magnetic gradient tensor for Menger sponge magnetic anomaly bodies with different fractal stages. The generalized magnetic potential governing equation was derived based on the fundamental equations of a stable magnetic field, and the discretized format of the governing equation was derived using finite element method(FEM). The method’s validity was verified through comparison with analytical solutions, demonstrating a relative error below 5%. The Menger sponge fractal anomaly bodies with different stages were constructed within 3-D meshs, and the influence of different fractal stages on magnetic anomaly, magnetic three components, and magnetic gradient tensor was analyzed. Furthermore, the effect of magnetic multi-quantities on the recognition of fractal structures was discussed. This study effectively simulates the spatial distribution of magnetic multi-quantities for 3-D fractal structures, which provides a theoretical basis for high-precision inversion of complex geologic bodies, with significant applications in the analysis of ore body structures.

The small-loop time-domain electromagnetic (TDEM) method has been widely used in urban underground space detection in recent years due to the small workspace requirements. Due to the small side length of the transmitting coil of the small-loop electromagnetic method, more difficulties have been introduced in modeling and instrument development. The traditional modeling method cannot include source calculation, and the error becomes significantly large when calculating the initial field of the small-loop, and the cross iteration of electric and magnetic fields also increases the modeling error of the small-loop. Therefore, a high-precision 3-D small-loop source TDEM modeling method need to be proposed to provide the theoretical basis for feature analysis, inversion, and instrument parameter design. In this article, the electromagnetic wave equations are adopted as the controlling equations and discretized based on the Crank–Nicolson finite-difference time-domain (CN-FDTD) method. The entire computational space, including air and ground, is subdivided into sources to support 3-D small-loop TDEM modeling for shallow anomalous body conditions. Furthermore, the iterative formulas of the electromagnetic wave equations in the complex frequency-shifted perfect match layer (CFS-PML) are derived, the reflection errors are largely suppressed, and the modeling accuracy is significantly improved. Finally, the effectiveness of the improved method is verified by homogeneous models, layered models, and complex anomaly models. In addition, analyzing the propagation characteristics of the small-loop can guide the setting of the receiver sampling rate and improve the accuracy of detection. The results show that the improved method can achieve stable, low-memory, and high-precision 3-D small-loop source TDEM modeling, which can provide theoretical support for the application of the TDEM method in urban and shallow detection.

The induced polarization (IP) and superparamagnetic (SPM) multieffect fields are generated due to complex physical characteristics and parameter information of polymetallic particles in grounded-wire source time-domain electromagnetic (TDEM) detection. Recent studies show that the accurate observation of IP characteristics and elimination of SPM interference can improve TDEM interpretation accuracy effectively. However, it is difficult to realize the accurate recognition and effective observation of the multieffect responses in the large range of nonuniform grounded-wire source signals. Therefore, in this study, the 3-D modeling of IP and SPM effects in grounded-wire source TDEM is realized by introducing the fractional Cole-Cole conductivity and susceptibility models and establishing the double-curl electric field convolution matrices. The response characteristics of IP and SPM effects are analyzed, and a three-component observational method of multieffect is proposed. The effectiveness of the proposed method is verified by carrying out observation experiments of IP and SPM equivalent circuits. The results show that observing the magnetic field component parallel to the grounded-wire source can obtain IP and SPM response characteristics earlier, and observing the vertical component can reduce the interference of SPM effect while obtaining more obvious IP responses. Through the comprehensive analysis of three-component responses, the IP and SPM characteristics can be better determined. This study has a guiding role in receiving the information of multieffect responses and improving the accuracy of polymetallic ore detection.

With the improvement in the accuracy of time-domain electromagnetic method (TDEM) observations for geophysical exploration, abnormal diffusion phenomena have become a more salient focus of research. In particular, the induced polarization (IP) effect and magnetic viscosity (MV) effect are often observed in the exploration of polymetallic ore, with IP effect leading to a negative TDEM response, and MV effect generating a power-law decay of −0.6 to 1.4 in the late stage response. Ignoring these effects can lead to incorrect data interpretation. Therefore, to model and analyze both IP effect and MV effect accurately, a 3-D numerical modeling method for IP–MV effect is proposed. The Cole–Cole conductivity and Cole–Cole susceptibility models are approximated in the time domain using the multiple-zero-pole (MZP) method. Then, the diffusion equations for the electric and magnetization intensity fields are derived as control equations, while the 3-D modeling of the IP–MV effect with full waveform is realized based on the improved recursive convolution technique and finite-difference time-domain (FDTD) method. The effectiveness is verified by comparing with the 1-D numerical integration solutions. The response characteristics of IP effect and MV effect are analyzed with full trapezoidal waveform, and a complex model including IP–MV effect is discussed. It is demonstrated that IP–MV effect can be better observed at the ON-time stage. The proposed method can effectively model the diffusion process of IP–MV effect with the full waveform, which can be helpful for improving the inversion accuracy and detection precision for complex geological formations.

Recently, the electromagnetic (EM) anomalous diffusion phenomenon has been observed in time-domain EM (TDEM) surveys. Furthermore, the data interpretation accuracy has been reduced by adopting the traditional EM theory and methods. A number of models, such as random medium and roughness electrical conductivity theory, have been adopted to model the EM anomalous diffusion. However, problems such as modeling difficulty and massive discretization exist regarding characterizing the long-range correlation of EM anomalous diffusion. The space-fractional derivative has been proven to preferably describe the long-range correlation characteristic. Only a handful of studies on TDEM anomalous diffusion with space-fractional derivative have been conducted due to the difficulties in computational engineering problems. Therefore, we performed a series of studies about 2-D TDEM anomalous diffusion with space-fractional derivative. The 2-D TDEM space-fractional diffusion equation was constructed based on the space-fractional Ohm’s law model. Furthermore, the discretization and iteration forms of the control equation were derived based on the finite element method (FEM) by introducing the Riemann–Liouville (R–L)-type Riesz fractional derivatives. The 2-D mountain-shaped function and partial integration method (PIM) were combined to convert the fractional derivative into the primitive function form. Hence, the 2-D modeling of the space-fractional EM diffusion was realized. The effectiveness of our method was verified by the function construction method and wavenumber-domain analytical solution. The spatial and temporal characteristics of the space-fractional EM diffusion were analyzed by different geological models. Furthermore, we discuss the differences with the classical EM diffusion. Our method can effectively model the space-fractional EM diffusion in TDEM surveys and provide theoretical bases for improving the TDEM interpretation accuracy with complex geological conditions.

The ground-source airborne time-domain electromagnetic (GATEM) method is an effective electromagnetic exploration technology. The actual geological medium has rough characteristics; however, the current inversion methods for GATEM data are mostly based on homogeneous medium and extract only resistivity information. In this article, a neural network (NN) is served as extracting the two parameters of resistivity and roughness for rough medium. The structural parameter selection of NN has no fixed formula and is often related to experience. The NN has difficulty converging to the target accuracy if the structural parameters are not selected properly. To realize high-precision inversion of GATEM data, this article introduces Rademacher complexity to limit the generalization error and improve the generalization ability of the NN. Above all, a sample set of the GATEM response, resistivity, and roughness of the rough medium is established. In the next place, a fully connected NN structure is constructed, and a highly generalized NN is obtained by using Rademacher complexity. Then the mapping relationships are established through training, and the NN method is served as inverting the resistivity and roughness. The initial NN and the highly generalized NN are used to invert the GATEM response of rough medium for typical geological models. The results of the highly generalized NN based on Rademacher complexity are closer to the real models. The method is applied to the GATEM field data in Zhuxianzhuang, Anhui Province, China, and the results are consistent with the geological data.