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 seismoelectric effect, characterized by the coupling of seismic and electromagnetic (EM) waves in fluid-saturated porous media, offers a promising avenue for subsurface exploration and earthquake seismology. However, the complexity of the Earth’s subsurface, particularly the presence of anisotropic velocity and attenuation, poses challenges for accurate modeling and interpretation. Here, we introduce a novel approach to model seismoelectric wave propagation in viscoelastic anisotropic porous media by incorporating the nearly constant Q model into the seismoelectric wave equations. Using a series of 2-D and 3-D models, we analyze the influence of both velocity and attenuation anisotropy on the propagation of seismoelectric waves. Our results show that both velocity and attenuation anisotropies contribute to the attenuation of the seismoelectric wavefield. Spatial changes in the elastic and electrical parameters of the subsurface media generate interface response of EM waves, although these are weaker compared to coseismic electric fields. Our extended seismoelectric models provide a robust description of wave propagation in poro-viscoelastic anisotropic media, which can allow for enhancing the current understanding of the Earth’s interior.

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.

A high-resolution global magnetic gradient tensor (MGT) model is an essential tool for aided navigation. However, the high-order expansion using the traditional spherical harmonic (SH) method will lead to the non-convergence in calculating locations near the Earth’s polar regions, and poles are treated as singularities, making it challenging to meet navigation’s high-resolution requirements. This paper proposes a calculation method for the non-singular MGT model based on oblate spheroidal harmonics (OSH). Firstly, we introduce a semi-normalization factor to the second-kind associated Legendre function to ensure equal contribution from each degree function. To avoid calculating complex numbers in the above procedures, a renormalization process is used alongside a recursive expansion of the operation, utilizing a suitable hypergeometric transformation. The first-order and second-order derivative explicit equations are derived. The effectiveness of higher-degree forms (up to 2000) is evaluated to verify the model’s accuracy, and we establish the conversion relation between OSH and SH under semi-normalized coefficients. For the pole singularity, the differential constant equation ofPn(x)is used to eliminate the sinusoidal co-latitude contained in the denominator of MGT. Finally, the high-resolution modeling of MGT is realized. We discuss the distribution of the MGT at the heights of 0.2 km and 300 km, respectively, and draw their spectra. The results of the numerical experiments verify the validity of the model and the accuracy of the calculations at the geographic poles, which are difficult for conventional methods.

The ground-source airborne time-domain electromagnetic (GATEM) system is an efficient geophysical survey system for geological surveys and mineral surveys. The geological resistivity structure is obtained by inversion methods, and however, the deterministic inversion methods can only provide an optimal resistivity model. The Bayesian inversion method can provide the posterior probability distribution; however, it requires a large amount of calculation. In this article, to improve the efficiency, a surrogate modeling based on neural network (NN) is applied to replace forward simulation calculation in the Bayesian inversion method. The accuracy of the NN-based surrogate modeling is related to the training sample set. To obtain high-precision inversion results, the surrogate modeling based on NN will be update adaptively online. Above all, an initial NN-based surrogate modeling is trained on a sample set of prior information. The high-fidelity surrogate modeling based on NN is obtained through new sample sets of GATEM data that are generated to update the surrogate modeling, if the NN-based surrogate modeling is inaccurate when the Bayesian inversion method is running. An optimal solution model and the posterior probability distribution of model parameters for GATEM inversion results are calculated through the Bayesian method. The effectiveness of the Bayesian inversion method with a surrogate modeling based on NN is verified by the GATEM responses for typical geological models.

When using magnetic resonance tomography (MRT) for imaging 2-D or 3-D water-bearing structures in a subsurface, the transmitting frequency must be the same as the Larmor frequency. Due to the inhomogeneity and noise interference in a geomagnetic field, it is difficult to determine the precise Larmor frequency using a magnetometer, resulting in unknown frequency offsets and inaccurate estimations of water content and relaxation time ($T_{2}^{*}$). To solve the 2-D MRT imaging problem in the case of an unknown frequency offset, a frequency cycling method is proposed in this article. This method takes the estimated Larmor frequency as the center, uses two frequencies with the same offset for transmitting, then combines the acquired MRT signals to obtain frequency-cycled data, and finally uses the off-resonance kernel function for inversion. Based on MRT forward modeling and QT inversion, we conduct synthetic data experiments on a complex model with three water-bearing structures and test the 2-D imaging results of the frequency-cycled data. The results show that the water content and$T_{2}^{*}$distribution obtained by the inversion of the frequency-cycled data can accurately reflect the water-bearing structure, which is better than the results of the assumed on-resonance case. In addition, the phase correction method presented in this article significantly improves the accuracy of 2-D MRT estimated aquifer properties under low resistivity conditions. Finally, the validity and accuracy of the frequency cycling method are verified by comparing the inversion results of the data with known drilling data collected in field measurements.

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.

Considering the viscoelastic anisotropy and electrical depression characteristics of the complex geological media, we introduce the generalized standard linear solid (GSLS) model to describe the relaxation effect of the solid skeleton and the Cole-Cole model to describe the frequency dependence of electric conductivity. The seismoelectric model of the poro-viscoelastic anisotropic medium was constructed, and the corresponding wave and diffusion equations in the time domain were derived. We then analyze the characteristics of seismoelectric wavefields in viscoelastic transverse isotropic (TI) media with a homogenous model, a two-layer model and a layered-model with depression. Results show that the TI anisotropy, viscosity of fluid, tilt angle all have significant effects on the propagation of seismoelectric waves in the homogenous model. The strong attenuation of seismoelectric waves in the two-layer model shows the validity of the relaxed skeleton and frequency dependent conductivity used in our approach, which could also effectively capture the reflection and transmission phenomena in the seismic and associated EM fields in the layered model with depression.

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.

Electromagnetic (EM) sounding methods in the time and frequency domains are widely used in polymetallic ore exploration and can measure both EM induction and induced polarization (IP) responses simultaneously. Depth inversion is a crucial factor affecting the accuracy of data interpretation. The effect of IP on EM wave propagation has been ignored in traditional studies of diffusion depth and skin depth. This article is based on the generalized effective-medium theory of the IP effect (GEMTIP) model to represent the IP effect. The frequency domain solution based on GEMTIP is obtained. To address the problem whereby the time-domain solution cannot be solved by inverse Laplace transform, rational approximation and linear programming methods are used to simulate the variation of pulse field source with time and distance. We analyze the influences of geometrical parameters and physical properties of rocks and ores with the IP effect on the depth. Finally, we consider the influence of intrinsic chargeability and improve the classic diffusion depth formula. The formulae for estimating the time-domain diffusion depth of the IP effect under certain conditions are given and verified. These methods are of great significance to better understand the relationship between the subsurface dispersive media and the propagation of EM waves.

The effect of magnetic viscosity (MV) has continued to be observed with the development and application of the time-domain electromagnetic method (TDEM). Field and laboratory data show that the MV effect is characterized by a -1±0.4 scope power-law delay in the late stage of electromagnetic response. Research on the MV effect can improve the detection accuracy of TDEM and assist in prospecting for ferromagnetic minerals. Most of the studies are based on the Chikazumi susceptibility model and the one-dimensional modeling method. However, the late-stage electromagnetic response shows a -1 power-law delay, which is inconsistent with the measured data. The log-uniform distribution of relaxation time τ in the Chikazumi model is not always appropriate. This study considers the Cole-Cole susceptibility model with a log-normal distribution of relaxation time. The three-dimensional (3D) modeling method of the MV effect is raised based on the rational function approximation algorithm and recursive convolution technique; the control equations and iterative process were adjusted based on finite-different time-domain (FDTD) method. The effectiveness was verified via half-space and layered models; the effects of susceptibility parameters on the response were clarified; moreover, the MV effect of the 3D anomalous model was analyzed. Our method can model the fractional propagation process of the MV effect more efficiently and help to improve the prospecting accuracy of the TDEM method under complex magnetic geological conditions.