Time-Domain Electromagnetic Noise Suppression Using Multivariate Variational Mode Decomposition

Abstract

Noise suppression is essential in time-domain electromagnetic (TDEM) data processing and interpretation. TDEM data are typically in broadband signal, which makes it difficult to separate the signal in the whole frequency band. The conventional methods tend to process data trace by trace, ignoring the lateral continuity between channels. This paper proposes a workflow based on multivariate variational mode decomposition (MVMD) and multivariate detrended fluctuation analysis (MDFA) to deal with the noise in 2-D TDEM data. The proposed method initially employs MVMD to decompose TDEM signals into a series of intrinsic mode functions (IMFs). Subsequently, MDFA is used to calculate the scaling exponent of each IMF, facilitating the selection of signal-dominant IMFs. Finally, the signal IMFs are summed up to reconstruct the TDEM signal. Both simulation and field results demonstrate that, by considering the lateral continuity of data across channels, the proposed method is more effective at noise removal than other single-channel data processing techniques.

1. Introduction

In recent years, the time-domain electromagnetic (TDEM) method has seen a surge in popularity across various fields, including geological research [1,2,3], environmental investigation [4,5,6], and unexploded ordnance detection [7,8,9], due to its non-destructive nature, cost-effectiveness, and ease of use. However, the noise in TDEM data significantly reduces the signal-to-noise ratio (SNR), impacting data quality and complicating processing and interpretation. TDEM signals exhibit a broad frequency band and non-linearity and non-stationarity characteristics, particularly becoming weak and susceptible to noise interference in the late stage.

Currently, two predominant methods exist for noise reduction in TDEM data, with the first being machine learning-based noise reduction. Methods for reducing noise in machine learning include utilizing sparse dictionary distributions [10,11] and various deep learning techniques [12,13,14]. Deep learning methods are particularly advantageous since they can automatically learn features from data, avoiding reliance on subjective human-selected data. They have gradually emerged as the mainstay of modern machine learning methods. Asif et al. developed an expert system based on a deep convolutional auto-encoder to eliminate interference from power lines, fences, and other structures in data [15]. Pan et al. proposed an encoder architecture that combines one-dimensional convolution and vision transformers, retaining both the local perceptual features of convolutional networks and the global perceptual features of transformers. This architecture integrates a multi-task loss function and a densely connected residual structure to optimize noise reduction performance [16]. Chen et al. [17] proposed a deep convolutional neural network CNN-based denoiser to model the noise estimation image for different signal features. Wang et al. [18] suggested employing generative adversarial networks to construct a dataset and train a deep neural network-based denoiser to learn the mapping from noisy TDEM signals to the corresponding noise-free signals. Yu et al. [19] proposed the CG-DAE, a CNN-GRU Dual Auto-Encoder designed for 2-D TDEM data processing. This model processes 2-D TDEM response data as an image input network, improving data processing efficiency. Nevertheless, challenges arise from incorporating deep learning into TDEM data processing. These include high computational costs, the need for parameter adjustments, and performance variations depending on architecture and training data size.

The second method is based on signal decomposition, which offers more convenience and less computational intensity than machine learning-based approaches. Ji et al. initially employed a wavelet threshold to remove background noise from TDEM data, then identified and processed the details after the stationary wavelet transform (WT), and finally reconstructed the denoised TDEM data using an inverse stationary wavelet transform [20]. Wang et al. used empirical mode decomposition (EMD) to decompose TDEM signals under strong noise into multiple intrinsic mode functions (IMFs). They successfully extracted weak signals by employing power detection to extract and reduce the principal components of the noise, thereby enhancing the depth and accuracy of TDEM detection [21]. Liu et al. used ensemble empirical mode decomposition (EEMD) to suppress the motion noise of aviation transient electromagnetic [22]. Feng et al. first utilized the whale optimization algorithm (WOA) to obtain the optimal combination of modal decomposition number K and penalty factor $\alpha$. Subsequently, they applied variational mode decomposition (VMD) for signal decomposition and finally used the Bhattacharyya distance algorithm to distinguish between effective and noise modes, accurately reconstructing the signal [23].

However, these methods have certain limitations. The WT method needs to select the appropriate wavelet basis function according to the characteristics of signal, which lacks self-adaptability and loses signal information in the decomposition process [24,25,26]. Although the EMD method and its variations do not require a priori information, they tend to cause modal aliasing and the decomposed IMFs are easily contaminated by noise. VMD can estimate multiple modes simultaneously in a non-recursive way, which ensures the completeness of the characteristics and improves the computational efficiency [27]. Using VMD and its improved method to decompose TDEM data can filter out noise and reconstruct signal accurately [28]. Nevertheless, a TDEM signal usually is a multi-channel signal, and the existing methods process TDEM data trace by trace, ignoring the continuity of data between channels.

Multivariate variational mode decomposition (MVMD) is a method for decomposing multichannel data [29]. MVMD processes TDEM data across multiple channels in unison, extracting IMFs that share identical center frequencies and bandwidths across various channels [30]. This concurrent processing of multichannel TDEM data enhances IMF continuity and aids in extracting spatial data features, effectively reducing noise [31]. This paper proposes a multichannel workflow for denoising TDEM data using MVMD and multivariate detrended fluctuation analysis (MDFA). Initially, we utilized MVMD to decompose 2-D TDEM signals into a series of IMFs. We then applied MDFA to calculate the scaling exponent of each group of IMFs, using these indices to select signal-dominant IMFs. The final step involved summing the selected signal IMFs to reconstruct the signal. We generated 2-D TDEM data and introduced varying levels of noise. The processing results indicate that the proposed method effectively removes noise, achieving a higher SNR by considering the lateral continuity between channels. The proposed method more accurately restores anomalous shapes compared to methods that process data channel by channel. The processing of field data further confirms that the method effectively eliminates noise in early- and late-stage TDEM data.

2. Methods

In this section, we first briefly introduce the related mathematical theories and expressions. Then, we propose an MVMD-based MDFA workflow to separate and attenuate the random noise contained in the raw TDEM data. Next, we introduce the method for selecting parameters of MVMD. Finally, we provide further elucidation of the workflow, as shown in Algorithm 1.

Algorithm 1 TDEM signal denoising with MVMD-MDFA

Require:  2-D TDEM signal $\mathbf{x}\left(t\right)$ Ensure:  Denoised 2-D TDEM signal ${\mathbf{x}}_{denoise}\left(t\right)$  1: set $K_{min},K_{max}$, $\alpha =2000$, $n=0$, $ϵ={10}^{-7}$  2: for $k=K_{min}; k<K_{max}, k=k+1$ do  3:    applying MVMD and calculating the correlation coefficients ${\mu }_k$  4:    if ${\mu }_k\ge$ threshold then  5:        $K_{best}=k-1$  6:    end if  7: end for  8: while ${\sum }_k{\sum }_c\frac{\parallel u_{k,c}^{n+1}-u_{k,c}^n{\parallel }_2^2}{\parallel u_{k,c}^n{\parallel }_2^2}<ϵ$ do  9:    $n=n+1$ 10:    for $k=1; k<K_{best}, k=k+1$ do 11:      for $c=1; c<C, c=c+1$ do 12:         $u_{k,c}^{n+1}=\underset{u_{k,c}}{\arg \min }\mathcal{L}\left(\left\{u_{i<k,c}^{n+1}\right\},\left\{u_{i\ge k,c}^n\right\},\left\{{\omega }_i^n\right\},{\lambda }_c^n\right)$ 13:      end for 14:    end for 15:    for $k=1; k<K_{best}, k=k+1$ do 16:      ${\omega }_k^{n+1}=\underset{{\omega }_k}{\arg \min }\mathcal{L}\left(\left\{u_{i,c}^{n+1}\right\},\left\{{\omega }_{i<k}^{n+1}\right\}\left\{{\omega }_{i\ge k}^n\right\},{\lambda }_c^n\right)$ 17:    end for 18:    for $c=1; c<C, c=c+1$ do 19:      ${\lambda }_c^{n+1}={\lambda }_c^n+\tau \left(x_c-{\sum }_k{u}_{k,c}^{n+1}\right)$ 20:    end for 21: end while 22: for $k=1; k<K_{best}, k=k+1$ do 23:    calculating the scaling exponent ${\beta }_k$ of $IM{F}_k$ 24: end for 25: calculating the slope of $\beta$, summing the IMFs preceding the point of maximum slope.

2.1. Multichannel Variational Mode Decomposition

The multivariate variational mode decomposition is an extension of the VMD method designed for multichannel data processing [32]. By simultaneously processing multichannel TDEM data, the MVMD algorithm enables the effective extraction of spatial features while enhancing the lateral continuity of data across channels. The principle of MVMD is as follows.

MVMD decomposes the 2-D TME signal $\mathit{d}\left(t\right)=[d_1\left(t\right),d_2\left(t\right),\dots ,d_C\left(t\right)]$ into K IMFs ${\mathit{u}}_k\left(t\right)$.

$$\mathit{d}\left(t\right)=\sum _{k=1}^K{\mathit{u}}_k\left(t\right),$$

(1)

where $\mathit{d}\left(t\right)$ contains C channels, each containing N sampling points.

And the k-th ${\mathit{u}}_k\left(t\right)$ can be represented as

$${\mathit{u}}_k\left(t\right)=\left[\begin{array}{c}{u}_{k,1}\left(t\right)\\ u_{k,2}\left(t\right)\\ ⋮\\ u_{k,C}\left(t\right)\end{array}\right]=\left[\begin{array}{c}{f}_{k,1}\left(t\right)e^{j{\varphi }_{k,1}\left(t\right)}\\ f_{k,2}\left(t\right)e^{j{\varphi }_{k,2}\left(t\right)}\\ ⋮\\ f_{k,C}\left(t\right)e^{j{\varphi }_{k,C}\left(t\right)}\end{array}\right],$$

(2)

where $f_{k,C}\left(t\right)$ and ${\varphi }_{k,C}\left(t\right)$ denote amplitude function and phase function corresponding to the C-th channel of ${\mathit{u}}_k\left(t\right)$.

The aim of MVMD is to extract a collection of multivariate modulated modes ${\mathit{u}}_k\left(t\right){|}_{k=1}^K$ with two key criteria: (i) minimizing total bandwidths of modes extracted from input data, and (ii) ensuring that the sum of derived oscillations accurately reconstructs the original signal ${\mathit{u}}_k\left(t\right)$.

The constrained optimization strategy can be expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underset{u_{k,c},{\omega }_k}{min}& \left\{\sum _{k=1}^K\sum _{c=1}^C{∥{\partial }_t\left[u_{k,c}\left(t\right)+\mathcal{H}{u}_{k,c}\left(t\right)\right]e^{-j{\omega }_{k}t}∥}_2^2\right\},\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \sum _{k=1}^K{u}_{k,c}\left(t\right)=d_c\left(t\right),c=1,2,\dots ,C\hfill \end{array}\end{array}$$

(3)

where $u_{k,c}\left(t\right)$ and $\mathcal{H}{u}_{k,c}\left(t\right)$ denote the k-th IMF of the c-th channel signal $d_c\left(t\right)$ and its Hilbert transform; ${\partial }_t$ represents the partial derivative of time; ${\omega }_k$ denotes the central frequency of $u_{k,c}\left(t\right)$.

Then, an augmented Lagrangian is formulated to solve the optimization problem expressed as (3). This is achieved by adding two penalty terms to the Lagrangian: a quadratic term that enforces accuracy in reconstruction, and a Lagrangian multiplier $\lambda$ term that ensures strict compliance with constraints.

$$\begin{array}{c}\mathcal{L}\left(\left\{u_{k,c}\right\},\right\{{\omega }_k\},\left\{{\lambda }_c\right\})\hfill \\ =\alpha {\displaystyle \sum _{k=1}^K}{\displaystyle \sum _{c=1}^C}{∥{\partial }_t\left[u_{k,c}\left(t\right)+\mathcal{H}{u}_{k,c}\left(t\right)\right]e^{-j{\omega }_{k}t}∥}_2^2\hfill \\ +{\displaystyle \sum _{c=1}^C}\parallel d_c\left(t\right)-{\displaystyle \sum _{k=1}^K}{u}_{k,c}\left(t\right){\displaystyle {\parallel }_2^2}+{\displaystyle \sum _{c=1}^C}〈{\lambda }_c\left(t\right),d_c\left(t\right)-{\displaystyle \sum _{k=1}^K}{u}_{k,c}\left(t\right)〉,\hfill \end{array}$$

(4)

where $〈a,b〉$ represents the inner product of a and b, ${\lambda }_c$ represents the c-th channel of $\lambda$. The optimization problem is solved using the alternating direction method of multipliers (ADMMs), and the formula for updating $u_{k,c}$, ${\omega }_{k,c}$ and ${\lambda }_c$ is given by:

$${\widehat{u}}_{k,c}^{n+1}\left(\omega \right)=\frac{{\widehat{d}}_c\left(\omega \right)-{\sum }_{i\ne k}{\widehat{u}}_{i,c}\left(\omega \right)+\frac{{\widehat{\lambda }}_c\left(\omega \right)}{2}}{1+2\alpha {(\omega -{\omega }_k)}^2},$$

(5)

$${\omega }_k^{n+1}=\frac{{\displaystyle \sum _{c=1}^C}{\int }_0^{\infty }\omega {\left|{\widehat{u}}_{k,c}\left(\omega \right)\right|}^{2}d\omega }{{\displaystyle \sum _{c=1}^C}{\int }_0^{\infty }{\left|{\widehat{u}}_{k,c}\left(\omega \right)\right|}^{2}d\omega },$$

(6)

$${\widehat{\lambda }}_c^{n+1}\left(\omega \right)={\widehat{\lambda }}_c^n\left(\omega \right)+\tau \left({\widehat{d}}_c\left(\omega \right)-\sum _{k=1}^K{\widehat{u}}_{k,c}^{n+1}\left(\omega \right)\right),$$

(7)

where ${\widehat{u}}_{k,c}$, ${\widehat{d}}_c$ and ${\widehat{\lambda }}_c$ are the Fourier transform of $u_{k,c}$, $d_c$ and ${\lambda }_c$. $\tau$ represents the parameter of noise tolerance. Update the above equation until it satisfies the following iteration termination condition:

$$\sum _{k=1}^K\sum _{c=1}^C\frac{\parallel {\widehat{u}}_{k,c}^{n+1}-{\widehat{u}}_{k,c}^n{\parallel }_2^2}{\parallel {\widehat{u}}_{k,c}^n{\parallel }_2^2}<ϵ,$$

(8)

where $ϵ$ is the tolerance parameter of the convergence criterion.

2.2. Multichannel Detrended Fluctuation Analysis

TDEM signals, when decomposed via MVMD, produce a series of IMFs, requiring the introduction of auxiliary discriminant methods to select signal IMFs. Detrended fluctuation analysis (DFA) is extensively employed to identify scaling properties and detect long-range correlations in non-stationary time series, owing to its proficiency in detrending time series data [33]. MDFA simultaneously processes multichannel signals, enhancing the lateral continuity of data across channels [34]. Specifically, MDFA processes multichannel IMFs, effectively removing noise and trend components of various orders attributed to external factors.

For a given C-channel, time series ${\mathit{x}}_i={[x_{i_1},x_{i_2},\dots ,x_{i_C}]}^{\mathrm{T}}$ observation at time i, $y_i$ is computed by

$${\mathit{y}}_i=\frac{1}{N}\sum _{i=1}^N\left({\mathit{x}}_i-\overline{\mathit{x}}\right),$$

(9)

where $i=1,2,\dots ,N$ donates the length of observation time, $\overline{x}=\frac{1}{N}{\sum }_{i=1}^N$.

Next, ${\mathit{y}}_i=y_{i_c}{|}_{c=1}^C$ for all observation time is divided in $2N_s$ disjoint intervals by cutting ${\mathit{y}}_i$ into $N_s=N/s$ segments sequentially and inversely. Then, the local trend ${\tilde{\mathit{y}}}_i=y_{i_c}{|}_{c=1}^C$ is estimated based on the fitting polynomial of each channel

$${\tilde{y}}_{i_c}=a_c·i^2+b_c·i+c_c,i=1,\cdots ,s$$

(10)

where $a_c,b_c,c_c$ are the coefficients of the quadratic term.

Then multivariate fluctuation $F_m^{\Sigma }\left(s\right)$ is formulated using Mahalanobis norm ${∥{\mathit{y}}_i-{\tilde{\mathit{y}}}_i∥}_{\Sigma }$

$$F_m^{\Sigma }\left(s\right)=\sqrt{\frac{1}{2s{N}_s}{\displaystyle \sum _{v=1}^{2N_s}}{\displaystyle \sum _{i=vs+1}^{(v+1)s}}{\left({\mathit{y}}_i-{\tilde{\mathit{y}}}_i\right)}^T{\Sigma }^{-1}\left({\mathit{y}}_i-{\tilde{\mathit{y}}}_i\right)},$$

(11)

where covariance matrix $\Sigma$ delineates the interchannel dependencies within the detrend ${\mathit{y}}_i-{\tilde{\mathit{y}}}_i$.

If the time series is long-range-correlated, it would be reflected in the function $F_m^{\Sigma }\left(s\right)$

$$F_m^{\Sigma }\left(s\right)=s^{\beta },$$

(12)

Finally, scaling exponent $\beta$ is calculated by the slope of $\ln F_m^{\Sigma }\left(s\right)$ and $\ln s$. The signal IMFs and noise IMFs can be screened out by the slope of $\beta$, then summing the signal IMFs to reconstruct the signal.

2.3. Parameter Selection

This section elaborates on the selection of pivotal parameters for MVMD. The number of modes to be decomposed, denoted as K, is crucial for MVMD’s efficacy. To circumvent issues of over or under-decomposition, K is determined based on a dual criterion: a low correlation coefficient and minimal energy loss. Specifically, the correlation coefficient threshold is set at 0.4, and the number of modes for decomposition is chosen within the range from $K_{min}$ to $K_{max}$. MVMD is applied iteratively within this defined range. If the correlation coefficient between the decomposed IMFs exceeds this threshold, it indicates over-decomposition. The optimal decomposition mode, $K_{best}$, is then considered one less than the current K. Furthermore, the bandwidth of the extracted modes is governed by the parameter $\alpha$; a smaller $\alpha$ results in a broader bandwidth. We set $\alpha$ to 2000 to improve the fidelity of the constructed IMFs. Lastly, $ϵ$ signifies the tolerance error between the raw TDEM data and the decomposed IMFs, and we set $ϵ$ at ${10}^{-7}$.

3. Synthetic Results

Synthetic TDEM data were generated to evaluate the effectiveness of the proposed workflow, as depicted in Figure 1a. The survey area is 5 m wide from east to west and 10 m long from north to south. The east–west profile comprises 20 channels, each encompassing 200 measurement points along the north–south direction. To simulate pipes of different diameters and different burial depths, responses from large, medium, and small-sized pipelines were created at distances of 8, 5, and 2 m to the north, respectively. Then, 0 dB Gaussian white noise was added to the synthetic TDEM data. Notably, the responses at 8 m and 2 m can be observed, but the anomaly shapes are not clear, and the response at 5 m is minimally visible, as shown in Figure 1b.

Next, we applied MVMD to the noisy data and extracted a series of IMFs, as illustrated in Figure 2. These IMFs are sequentially organized from low to high frequency. Analysis of the graph reveals that the low-frequency IMFs encompass more energy, signifying the underlying trend of change, while the high-frequency IMFs, although noisier, provide finer details. As evidenced in Figure 2a, at the exact locations of the simulated pipeline responses in the original data, corresponding anomalies are discernible in IMF1, ensuring continuity across adjacent channels. In IMF1, the 160th point across all channels exhibits the maximum response, corresponding to the response of the pipeline that traverses all channels at 8 m in the original data; the responses at points 40 and 100 have similar characteristics. This observation is consistent in IMF2. However, due to increased frequency in IMFs 3 to 9, the anomalous responses indicative of the pipelines become indistinct.

Subsequently, MDFA is conducted on all IMFs to calculate the scaling exponent, $\beta$, which serves as a criterion to differentiate between signal-dominated and noise-dominated IMFs. The slope of $\beta$ signifies the reduction in long-range correlation, with a higher slope indicating a substantial increase in noise content. Consequently, IMFs preceding the point of maximum slope are deemed to predominantly contain relevant signals. In Figure 3, the red line marks the maximum slope point, identifying IMF1 to IMF4 as signal-dominated.

Finally, the signal-dominated IMFs are summed to obtain the TDEM signal after removing the noise. We showcase the signal processing details before and after for channels 2 and 14 and compare these with the results from other methods. As observed in Figure 4, although the responses from the large and medium pipelines are significant, they are still severely interfered with by noise, and the response of the small pipeline is submerged in noise. The proposed method effectively removes the noise, restoring the responses of the large and medium pipelines and accurately extracting the response of the small pipeline.

Subsequently, we visualized the denoised data and compared it with the results obtained from other methods. As shown in Figure 5, anomalies can be identified in all four images, each exhibiting distinct morphologies. The proposed method eliminates noise, resulting in a noticeably cleaner image. More importantly, it accurately extracts weak responses previously obscured by noise while maintaining the continuity of anomalous shapes across channels. In contrast, the results processed by other methods exhibit discontinuities in the anomalous forms, failing to reconstruct the characteristics of the pipelines accurately.

We also processed the data containing 5 dB and 10 dB noise levels using various methods, with the results displayed in

Table 1. SNR of synthetic data processed by different methods.

Methods	ICEEMDAN	EWT	VMD-DFA	Proposed Workflow
0 dB Noise	0.57	1.44	1.02	2.24
5 dB Noise	3.96	6.11	4.55	6.78
10 dB Noise	7.39	10.47	9.26	11.12

. The outcomes obtained with the proposed method surpassed those of other methods across different noise levels, thereby validating its effectiveness. The time costs by different methods are shown in

Table 2. Time costs of synthetic data processed by different methods.

Methods	ICEEMDAN	EWT	VMD-DFA	Proposed Workflow
Time costs (s)	26.7	1.04	7.2	30.4

, where the proposed method takes the longest time. Compared to other single-channel methods, the proposed method considers all channels as a set and the inter-channel correlations, increasing the time required for iteration.

4. Experimental Results

4.1. System Description

The TDEM system, depicted in Figure 6, developed by the Aerospace Information Research Institute at the Chinese Academy of Sciences validated the algorithm. This system consists of a transmitting module, a data acquisition module, and a positioning module. It features a square transmitting coil measuring 1 m on each side and three concentric square receiving coils, each with a side length of 0.5 m. The location of the sensor is precisely determined by a Real-Time Kinematic (RTK) module positioned at the center of the receiving coil, achieving a positioning accuracy of less than 1 cm.

4.2. Test Field Result

Verification experiments were conducted in Liaocheng, Shandong, China. Approximately 1 m beneath the experimental site, a steel gas pipeline with a diameter of about 30 cm was buried. Four parallel measurement lines were planned, spaced approximately 1 m apart, as depicted by the red line in Figure 7. During these tests, the TDEM system moved at an approximate speed of 1 m per second.

Upon completing the measurements, we rendered the original data in graphical form, as depicted in Figure 8. Figure 8a represents the early-stage data, which show that the anomalies have greater amplitude and are more distinct in shape due to a higher SNR in the early signals. However, noise interference results in some roughness at the edges. Figure 8b displays the late-stage data. Over time, the SNR in these stages decreases, and although anomalies remain visible, their precise shapes become indiscernible.

We established the optimal modal decomposition number as 6 and applied MVMD to both early- and late-stage data. This process yielded a series of IMFs, which are depicted in Figure 9 and Figure 10, respectively. From Figure 8, it is evident that due to the differences in signal characteristics between early and late stages, the features of the IMFs obtained from decomposition also differ. In the early-stage signals, IMF1 and IMF2 have higher SNR with distinct anomaly shapes, whereas in the late-stage signals, only the anomaly shape of IMF1 is distinctive.

Subsequently, we employed MDFA to calculate the scaling indices of the IMFs for both early and late stages, with the results displayed in Figure 11. Based on the scaling indices, we reconstructed the early-stage signal by summing IMF1 to IMF3, and the late-stage signal by summing IMF1 and IMF2.

The denoised early- and late-stage signals are presented in Figure 12. It is observable that in the denoised early-stage signals, the pipeline response shape is complete, and the edges are more precise; in the late-stage signals, the SNR has dramatically improved, accurately extracting the pipeline response from the noise, demonstrating the effectiveness of the proposed workflow in eliminating noise and restoring the abnormal shape of the pipeline.

5. Discussion

In TDEM survey tasks, when the scale of subsurface anomalies is substantially large relative to the detection equipment utilized, this scenario typically yields analogous anomalous responses across multiple survey lines, highlighting a prevalent issue in geophysical exploration. The inherent wide dynamic range of TDEM data further complicates the scenario as later-stage data become increasingly prone to a spectrum of interferences, including environmental noise, equipment-induced artifacts, and signal attenuation. Traditional approaches in signal processing predominantly focus on decomposing data trace by trace without addressing the lateral continuity of data between adjacent channels, potentially overlooking significant subsurface features.

This paper pioneers a multichannel TDEM data processing methodology combining MVMD and MDFA. Contrary to VMD and similar single-channel processing methods, which process data trace by trace, our method astutely extracts IMFs with uniform center frequencies across all channels. This critical advancement ensures the preservation and enhancement of lateral data continuity between channels. This approach addresses the challenges above by exploiting the spatial coherence inherent in geophysical datasets.

Simulation studies meticulously crafted identical pipeline responses in adjacent channels, mirroring real-world scenarios of significant subsurface anomalies. To emulate varying degrees of signal fidelity, we introduced three distinct amplitude levels—high, medium, and low—each representing different SNRs. This setup allowed us to rigorously evaluate the efficacy of our proposed method against conventional techniques. After undergoing MVMD processing, the simulated TDEM data yielded a series of IMFs, as shown in Figure 2, where IMF1 consistently exhibits analogous amplitudes and trends across different channels, thereby reinforcing the spatial continuity of the subsurface anomaly signal. Subsequent application of MDFA enables the calculation of scaling indices for the various IMFs. By selecting signal IMFs before the maximum reduction in long-range correlations for summation, we effectively complete the signal denoising process, thereby significantly enhancing the signal clarity and interpretability of TDEM data. The efficacy of our proposed methodology is vividly demonstrated in Figure 5a, where continuous anomalous patterns emerge with remarkable clarity, successfully recovering data previously obscured by noise. The efficacy of this multichannel approach is further corroborated by Table 1, which showcases superior performance metrics across varying SNRs. Then, we used a TDEM system to measure the response of underground gas pipelines, and field data analysis reinforces the method’s robustness, effectively mitigating noise in both early- and late-stage TDEM data, thus accurately depicting the morphology of subsurface targets.

During our evaluation, traditional VMD processing clocked in at 7.2 s, while our novel approach required 30.4 s. The MDFA took 1.6 s, and MVMD accounted for 28.8 s, because of its holistic treatment of all channel data as a unified entity and meticulous consideration of inter-channel correlations. This approach necessitates more iterations, extending the computation time required for the ADMM to converge on the optimization problem. The selection of the parameter $\alpha$, pivotal for the decomposition’s effectiveness, is manually determined based on the specific demands of the dataset, representing a potential area for optimization.

Our future work is poised to delve into more efficient optimization methodologies to reduce computational overhead. Furthermore, we are committed to developing refined criteria for selecting MVMD parameters. Additionally, we will explore other multichannel data processing methods, such as multivariate iterative filtering [35] and multichannel adaptive Fourier decomposition [36], which do not require prior information and exhibit higher robustness to enhance the effectiveness and utility of multichannel TDEM data processing techniques in geophysical exploration.

6. Conclusions

This paper introduces a noise reduction method for multichannel TDEM data. It involves decomposing 2-D TDEM data into a series of IMFs using MVMD, followed by MDFA to distinguish between signal and noise IMFs. The final denoising step is achieved by summing the selected IMFs. Multichannel TDEM data were generated, and various noise levels (0 dB, 5 dB, and 10 dB) were added to compare the effectiveness of different methods. The results indicate that the proposed method enhances lateral continuity between channels and accurately extracts anomalous signal features, effectively reducing noise. Using the TDEM system, we collected responses from underground pipelines. This process validated the proposed method’s ability to effectively handle noise in both early- and late-stage data, significantly improving reliability and interpretability.