The distortion of magnetotelluric (MT) data due to various noise has emerged as a universal yet challenging issue in geophysical exploration. When MT surveys conducted in mining areas and urban areas, there are always full of spatially and temporally random artificial electromagnetic sources, violating the conventional plane-wave assumption, resulting in near-field effects. While, most MT data processing methods have challenges of difficulty in suppressing near-field effect, even with losing useful signals. Thus, we propose a novel method based on multiscale decomposition combined with coherence clustering. First filtering out small-scale signals from the time series, then classifying MT data into useful signals, incoherent noisy data, and coherent noisy data. This approach generates a mixed MSD-Coclustering structure with enhanced robustness in restoring weak natural signals while attenuating highly coherent anthropogenic interference, particularly in strong interference environments. The performance of MSD-Coclustering was tested using synthetic data and MT measured data, the signal-to-noise ratio (SNR) of noisy data is improved by nearly ten times, with no limitation to noise properties and data quality. Notably, the proposed method effectively suppresses the near-field effect without loss of useful signals and economic costs generated by setting up remote reference stations. This advancement significantly improves the interpretation accuracy and reliability of MT data in complex environments, with MT application fields extending.

The natural electromagnetic field observed in magnetotelluric (MT) sounding is non-stationary, making it challenging to obtain reliable frequency spectrum information using Fourier transform. In practical measurements, long-duration observations of the electromagnetic field signal are often required to obtain accurate low-frequency impedance, resulting in significant technical difficulties and high economic costs. We provide a method for estimating MT impedance using the instantaneous spectrum obtained with variation mode decomposition (VMD). By replacing the Fourier spectrum with the instantaneous spectrum, this approach mitigates the requirement for extended signal observation time when estimating low-frequency impedance. Compared to previous studies on impedance estimation, we analyze the influence of signal period number on spectrum reliability, emphasizing the effectiveness and reliability of instantaneous spectrum in dealing with non-stationary signal. The feasibility of obtaining low-frequency MT impedance from short-duration observations is also discussed. The proposed method employs VMD to extract the instantaneous spectrum and utilizes hat matrix (A projection matrix) and signal noise separation (SNS) techniques to suppress noise interference in the spectrum, thereby enhancing the reliability of the instantaneous spectrum and the resulting impedance estimates. The method is applied to synthetic data with added noise and real data collected in the Qilian region of China, and the results are compared with those obtained by using different methods. The experiments demonstrate that VMD instantaneous spectrum reliably reflects the spectral characteristics of the signal, exhibiting minimal changes as the signal duration decreases. Therefore, this method can obtain robust low-frequency impedance even with short MT time series.

With the increasing development of geothermal energy resources, obtaining precise subsurface thermal conductivity structures has become crucial. However, current geophysical inversion methods lack a detailed technique for directly estimating subsurface thermal conductivity, especially when utilizing both borehole temperature field data and surface heat flow data as constraints. To address challenges posed by sparse borehole data and the limited resolution of borehole temperature field data, this study introduces a novel approach. It first utilizes boundary detection techniques to refine the extent of anomalous regions using heat flow data. Subsequently, by incorporating inversion results from borehole temperature data, a reference model is established, enabling a joint inversion technique that leverages both borehole temperature field data and surface heat flow data. Model experiments demonstrate the feasibility and effectiveness of this joint inversion method, significantly improving subsurface thermal conductivity imaging. Finally, the analysis of field data further validates the practicality and efficiency of this approach.

Due to the intricate and uncertain nature of petrophysical properties, the practical application of petrophysical joint inversion poses significant challenges, and the realization of multigeophysical joint inversion methods is particularly demanding. In this study, we present an effective and versatile joint inversion method based on a coreference model and wide-range petrophysical constraints, building upon the foundation of traditional petrophysical constrained joint inversion techniques. This method is applied to the 3-D joint inversion of magnetotelluric (MT), magnetic, and gravity data. Our proposed joint inversion framework, utilizing the coreference model, decomposes the multigeophysical joint inversion into a combination of pairwise joint inversions, thereby reducing the number of weighting coefficients required. The wide-range constraints employ model transformation to introduce a priori information, while the coupling terms and range constraints facilitate the coupling of different petrophysical parameters within specified ranges, enhancing the fault tolerance of the petrophysical constraints. Model texts demonstrate the method’s ability to identify anomalies in complex interface models that traditional approaches may overlook, thereby effectively improving the inversion performance of gravity and magnetic methods. Furthermore, the validity and practicality of the method are confirmed through the processing of real measurement data obtained from the Songliao Basin, China.

Magnetotelluric (MT) impedance estimation requires a high signal-to-noise ratio (SNR). When low-SNR data are processed, it is difficult to obtain a robust MT response. In this article, based on the spectral characteristics of noise sequences, the influence of the scale and waveform of noise sequences on impedance estimates is studied, and a multiscale denoising method for MT signals is proposed. This method applies the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) to decompose the multiscale noise into different components, and then, the influence of noise on the spectrum is evaluated through the spectrum obtained by the short-time Fourier transform (STFT) of each component. This ICEEMDAN- and STFT-based MT (ICMT) denoising method can, thus, filter out the small-scale abrupt noise that has a great impact on the MT response and retain the large-scale smooth noise that has a small impact to suppress noise and reduce the loss of the effective signal at the same time. Various noises are added to a pure MT signal to test the performance of ICMT. It is demonstrated that ICMT has a small loss of effective signals and can obtain MT response results with small recovery errors, even when the SNR of the signal is as low as −20 dB. Finally, ICMT is applied to process the heavy noisy MT data in an ore concentration area. The results suggest that ICMT can effectively suppress noise, and robust impedance estimates can be obtained.

Traditional magnetotelluric (MT) impedance estimations are based on Fourier theory and carried out in the frequency domain, which has a strict stationarity requirement for the analyzed signal. However, the stationarity assumption cannot be satisfied when the data possess a low signal-to-noise ratio (SNR) and/or a short observation period. These shortcomings can cause significant errors in the MT impedance estimations, especially in the low-to-medium frequency bands. The alternating direction method of multipliers (ADMMs) and polarization analysis of the electromagnetic signal can be attached to the time-domain MT impedance estimation to address these shortcomings. This time-domain technique calculates the impedance in the time domain without Fourier transformation, and the ADMM and polarization analysis are applied to further improve the stability and convergence of the impedance estimation. Here, we present an ADMM-based method for MT impedance estimations (ADMM-MT). Various noise are added to a noise-free MT signal to test the performance of ADMM-MT. The results show that ADMM-MT yields impedance estimates with relative recovery errors below 0.2, even when the SNR of the data is 0 dB and the observation period is 60 min. We then apply ADMM-MT to field data in Inner Mongolia, China. The results indicate that the apparent resistivity curves obtained by ADMM-MT using a short time series are smooth over the 0.001–1-Hz band, which is consistent with the remote reference (RR) processing results obtained using a long time series. In contrast, the curves obtained by traditional robust estimation method are strongly biased.

The cross-gradient joint inversion of gravity and magnetic data is a commonly used way, which requires dividing the subsurface into closely arranged cells with structured or unstructured grids to solve the physical properties of the discrete subsurface. Unstructured grids can more effectively fit undulating terrain and irregular geological bodies than structured grids. However, the existing cross-gradient joint inversion method with structured grids cannot be used to complete the inversion with unstructured grids. We proved that the second-order Taylor formula is more suitable for calculating the cross-gradients of unstructured grids; therefore, we first proposed and implemented the second-order Taylor formula cross-gradient (2-TCG) method for the cross-gradient joint inversion of gravity and magnetic data with unstructured grids, which can calculate the cross-gradient item more easily than other ways because the first few coefficients of Taylor formula are gradient items. The model tests showed that the 2-TCG joint inversion method can accurately achieve the 3-D joint inversion of gravity and magnetic data with unstructured grids, and this method can better describe the detailed information of irregular geological bodies in complex terrain. To explore the minerals and analyze the metallogenic model in the Huanggang–Ganzhuermiao metallogenic belt of the southern Greater Khingan Range, Inner Mongolia, we measured gravity and magnetic data on a scale of 1:10000. We estimated the mineral distributions with 150–280 m depth by 2-TCG joint inversion. We analyzed the relationship between ore bodies and faults, and we established a mixed metallogenic model of the skarns and hydrothermal veins in the survey area.

Magnetotelluric (MT) sounding data are easily contaminated by various noise sources, especially noise with a long duration (even full-time noise), which makes it difficult to obtain accurate values when calculating the weight factors of the response function, resulting in distortion of the response results. Based on blind source separation theory, fast independent component analysis (FastICA) can separate this kind of noise. However, this method is challenged by the number of separated field sources and the unequal signal amplitude before and after processing. We have developed a novel signal noise separation method, improving the traditional FastICA, where a correlation coefficient is used to compute the number of field sources, and the fast iterative shrinkage threshold algorithm (FISTA) is used to adjust the signal amplitude problem before and after FastICA decomposition. Compared with common field source division methods and the traditional FastICA, the experimental results indicate that our method can separate and remove noise with a long duration, increase the signal-to-noise ratio of the data, and improve the MT response curves. Meanwhile, case studies of measured data illustrate that our method obtains a more robust magnetotelluric response than the conventional robust method and FastICA.

When magnetotelluric (MT) sounding data are measured in mining areas and urban areas, the useful signals are buried under the surrounding interference sources in the whole period, which completely covers up the useful signals, resulting in jump points and distortion of the response curves. Sparse representation uses atoms in a dictionary to process noisy signals. Based on the arbitrariness of the length of these dictionary atoms, they can effectively suppress noise even if the noise fills the entire observation period. We propose an improved sparse representation based on an adaptive dictionary, which can construct a dictionary according to the characteristics of the data itself (extracting the noise and the useful signals separately) and then automatically filter the noise atoms (which are regarded as noise) in the dictionary to suppress long-duration noise (noise lasts for a long time). For the synthetic data and the measured data, in a comparison with the common signal noise separation methods, the results indicate that the proposed method can more completely extract the profiles of the long-duration noise and greatly increase the signal-to-noise ratio. Moreover, for various noise sources, the proposed method indicates better improved performance, obtaining smoother and more reliable response results.

Besides Karatsuba’s algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design$GF(2^n)$subquadratic multipliers. However, when$GF(2^n)$elements are represented using a polynomial basis or its generalization—shifted polynomial basis—this approach is currently appliable only to fields$GF(2^n)$generated by an irreducible trinomial or a special type of irreducible pentanomials, but not to a general irreducible pentanomial. The reason is that no transformation matrix, which transforms the Mastrovito matrix into a Toeplitz matrix, has been found. In this article, we propose such a transformation matrix and its inverse matrix for an arbitrary irreducible pentanomial.