Adaptive Variational Mode Decomposition and Principal Component Analysis-Based Denoising Scheme for Borehole Radar Data

Abstract

To address the significant impact of noise on the target detection performance of borehole radar (BHR), a key type of ground-penetrating radar (GPR), a denoising scheme based on the whale optimization algorithm (WOA) for adaptive variational mode decomposition (VMD) and multiscale principal component analysis (MSPCA) is proposed. This study initially conducts the modal decomposition of BHR data using an improved adaptive VMD method based on the WOA; it then automatically selects modes meeting specific frequency band standards. The correlation coefficients between these modes and the original signal are computed, discarding weakly correlated modes before signal reconstruction. Finally, MSPCA further suppresses noise, yielding denoised BHR data. Simulations show that the proposed scheme increases the signal-to-noise ratio by 17.964 dB or higher, surpassing the more established denoising techniques of robust principal component analysis (RPCA), MSPCA, and empirical mode decomposition (EMD), and obtains the most favorable results in terms of the RMSE and MSE metrics. The experimental results demonstrate that the proposed scheme more effectively suppresses vertical and random noise signals in BHR data. Both the numerical simulations and experimental results confirm the effectiveness of this scheme in noise reduction for BHR data.

1. Introduction

GPR is widely used for geotechnical surveys, groundwater monitoring, polar glacier studies, lunar exploration, geological surveys, and buried object identification as a key tool in shallow surface geophysics [1,2,3,4,5,6,7]. BHR, identified as a key type of ground-penetrating radar (GPR), is employed as an electromagnetic detection tool for subsurface investigations. BHR is widely applied in mineral resource exploitation, civil engineering, geophysical exploration, and geological surveys [1,8]. Furthermore, BHR is utilized for the detection of underground structures [9,10], mapping of hydrological properties [11], monitoring of oil well perforations [12], and identification of tunnels [13]. Significant noise interference in BHR receiving signals originates from the complexity of the underground environment, thermal noise within electronic components, external electromagnetic interference, and the characteristics of wideband antennas [14,15,16,17]. This noise consists of coherent noise (clutter), which exhibits inter-trace correlation, and incoherent noise (random noise), which is randomly distributed across the data. Both types of noise significantly impact the data fidelity and the signal-to-noise ratio (SNR) [18,19]. Furthermore, the emission of high-frequency electromagnetic waves by the BHR system, in conjunction with the region’s inherent filtering effects, exacerbates the nonlinearity and nonstationarity of the received signals [14,16,17,20,21]. Useful signals are masked by nonlinear and nonstationary noise, resulting in degraded target detection performance and impaired quality in data interpretation, migration imaging, and full-waveform inversion [22,23,24]. Consequently, the suppression of random noise continues to be a critical focus within BHR data processing research [25,26]. Regarding the suppression of random noise, the existing methods have been categorized into five main types, which are complementary when applied to BHR data. In subspace methods, clutter is suppressed through the decomposition of Bscan images into different subspaces. While theoretically straightforward and easy to implement, these methods encounter challenges in accurately estimating the noise subspace when the clutter energy is comparable to that of the target signal, resulting in unstable denoising performance. Low-rank and sparse matrix decomposition (LRSD) facilitates the separation of strong background energy from sparse targets by modeling the data as the combination of a low-rank background and sparse components. However, residual noise within the sparse matrix may be overlooked, and the performance remains highly sensitive to the hyperparameter selection, making the method prone to variations across different scenarios. Multiresolution methods process signals and noise across multiple scales and orientations, allowing for the preservation of local details while attenuating specific types of random noise. However, these methods exhibit limitations when dealing with highly nonstationary noise. Deep learning methods exhibit significant advantages in feature representation and adaptive modeling when sufficient labeled training data are available. However, their effectiveness is constrained by the requirement for large-scale labeled datasets, and their generalization performance remains unstable in new environments with distributional variations. Due to the nonstationary and nonlinear characteristics of BHR signals, modal decomposition enables adaptive signal decomposition into several intrinsic mode functions (IMFs). However, these methods are often constrained by mode mixing, endpoint effects, and the absence of rigorous mathematical frameworks for parameter selection. The necessity of the manual tuning of key parameters further restricts their effectiveness in real-world BHR data processing. Although these approaches complement each other to some extent, challenges remain, including noise-to-target energy similarity, the dependence on large training datasets, and mode mixing in decomposition. These limitations hinder the ability to achieve an optimal balance between denoising accuracy and signal fidelity, particularly in complex environments. To overcome these limitations, a BHR denoising strategy is proposed, integrating adaptive VMD with MSPCA. In particular, the whale optimization algorithm (WOA) is employed to optimize the VMD parameters, enabling the more precise decomposition of nonlinear and nonstationary signals. Residual noise is further suppressed using MSPCA in the multiscale domain. This scheme exploits the strengths of modal decomposition for nonstationary noise suppression while employing multiscale analysis for signal detail preservation. As a result, a superior balance is maintained between the BHR denoising effectiveness and weak target fidelity.

Denoising methods based on subspace decomposition, such as singular value decomposition (SVD), are applied to BHR data to enhance the correspondence with strata resistivity and improve the stratigraphic classification capabilities [10]. Other subspace-based techniques, including independent component analysis (ICA) [27] and principal component analysis (PCA) [28,29], are extensively utilized for radar data denoising. However, these methods rely on accurate noise subspace estimation, assuming significant differences between noise and target responses, with noise primarily captured by dominant components. When the noise intensity is similar to that of the target signals, separation becomes challenging, potentially compromising noise suppression. Morphological component analysis (MCA) is also employed for radar data denoising by decomposing signals into clutter and target components, although its performance heavily depends on the selected dictionary [30].

Low-rank and sparse matrix decomposition (LRSD) methods are widely employed for radar noise suppression. For instance, robust non-negative matrix factorization (RNMF) models radar data as low-rank clutter and sparse target signals to suppress clutter [31], while robust principal component analysis (RPCA) achieves the superior suppression of noise compared to PCA [32,33,34]. Furthermore, other LRSD approaches, such as GO decomposition (GoDec) [35] and robust orthogonal subspace learning (ROSL) [36], represent GPR Bscan data as combinations of low-rank and sparse matrices, corresponding to clutter and target signals. However, their performance is sensitive to the hyperparameter selection, which balances low-rank and sparse components, with the optimal values varying by scenario. Additionally, focusing on dominant components may result in neglecting residual noise in the sparse matrix, reducing the effectiveness when noise is inadequately removed [34].

Multiresolution techniques are extensively applied in radar noise suppression. The wavelet transform (WT), for instance, is frequently utilized for multiscale GPR signal denoising [37,38,39]. The curvelet transform (CT) and nonlocal mean (NLM) filtering have been applied to random noise attenuation in seismic and GPR data [40]. The structural features of radar signal reflection layers are captured by CT across multiple scales and orientations while preserving the primary energy components. Furthermore, CT filtering is robust in noise suppression and ensures signal integrity [40]. Bakshi’s multiscale principal component analysis (MSPCA) combines the strengths of PCA and WT for multivariate process monitoring and noise elimination [41,42,43], and it has been successfully applied to denoise GPR and GNSS signals [14,44]. However, the WT is most effective for stationary signals, limiting its application to highly nonlinear and nonstationary radar data [14,45]. Other multiresolution methods, such as the curvelet and shearlet transforms [39,46], are also applied but remain primarily effective for uniformly distributed random noise [47], struggling with nonstationary noise in BHR data.

Deep learning has been applied in radar noise suppression, including clutter removal neural networks (CR-Nets) [48], convolutional autoencoder (CAE) networks [49], and autoencoder-based methods [50]. Despite their notable advantages, these methods often rely on large training datasets and specific conditions for optimal performance. Limited generalization capabilities, especially in diverse scenarios, remain a major research focus [25]. Furthermore, obtaining clutter-free Bscan images of underground targets for training presents a persistent challenge in practical BHR experiments.

Modal decomposition methods have also been utilized in BHR denoising. Empirical mode decomposition (EMD) adaptively decomposes nonlinear, nonstationary signals into finite subcomponents, thereby highlighting anomalous geological features in BHR data [51]. To address intrinsic mode mixing, variants like ensemble EMD (EEMD) [52], complete ensemble EMD (CEEMD) [53], and improved CEEMD (ICEEMD) [54] have been proposed. However, issues such as mode mixing, endpoint effects, and insufficient mathematical foundations persist [19,47]. Variational mode decomposition (VMD) [55] resolves the challenges associated with empirical mode decomposition (EMD), including mode mixing, endpoint effects, and the absence of a well-defined mathematical framework. This is achieved through a mathematical optimization process, ensuring more precise, stable, and controllable decomposition results. Furthermore, signals are adaptively decomposed into intrinsic mode functions (IMFs) with a finite bandwidth by VMD, thereby effectively enhancing the target signals while suppressing noise. However, its reliance on manual parameter tuning, such as mode number selection, limits its broader applicability [56].

A denoising scheme for BHR data is proposed in this study, integrating adaptive VMD and MSPCA. The core parameters $(K,\alpha )$ of VMD are adaptively optimized through the WOA, followed by signal decomposition using VMD. The frequencies of the decomposed modes are evaluated to ensure alignment with the predefined range of the BHR mean spectrum. Modes with weak correlations are removed based on the Pearson correlation coefficient (PCC). Finally, MSPCA is employed to further suppress residual noise. The performance of the proposed scheme is evaluated against RNMF, EMD, RPCA, MSPCA, and WOA-based Adaptive VMD, demonstrating superior noise suppression capabilities.

The main innovations of this study are as follows. (1) First Application of VMD for BHR Denoising and Reconstruction: Nonlinear and nonstationary BHR signals are decomposed into intrinsic mode functions (IMFs) with finite bandwidths, effectively enhancing target signals masked by noise. Additionally, this method resolves inherent issues in EMD, such as mode mixing and ambiguous physical interpretation. (2) Adaptive VMD: To improve the accuracy and robustness of VMD decomposition, the core parameters $(K,\alpha )$ are optimized using the WOA, which enables rapid convergence and global search capabilities. The maximum sample entropy $H(·)$, recognized for its strong anti-interference capabilities, is employed as the optimization function. Specific frequency band modes are adaptively selected for reconstruction, facilitating the separation of major noise components from the target signals. The correlation between the selected modes and the original signal is evaluated, and highly correlated modes are reconstructed to enhance the denoising performance. These components collectively form the adaptive VMD BHR denoising algorithm. (3) Effective Handling of Nonstationary Signals with Adaptive VMD and MSPCA: Adaptive VMD processes nonstationary signals effectively, separating noise from target components. MSPCA combines the strengths of the WT and PCA, enabling the extraction of primary signal features across multiple scales. While the WT is more effective for stationary signals, it is limited in addressing nonstationary BHR data. Adaptive VMD addresses the nonlinear and nonstationary characteristics of BHR signals, and MSPCA subsequently suppresses residual noise and enhances weak signals. In summary, the proposed scheme achieves effective noise suppression, delivering optimal visualization and metric results.

2. Proposed Scheme

A denoising scheme for BHR data is proposed in this study, integrating adaptive VMD and MSPCA. The algorithmic foundations of this scheme are introduced in this section. Adaptive VMD incorporates WOA-based parameter optimization, frequency-domain filtering, and time-domain correlation-based mode selection. The optimization of the VMD parameters is initially conducted using the WOA, which improves the efficiency of VMD in extracting target signals. In the subsequent step, frequency bands and modes demonstrating a high degree of correlation with the original signal are chosen for reconstruction. MSPCA is then employed to suppress residual noise and to further diminish noise signals that resemble the target within the reconstructed Bscan. The detailed structure of the processing workflow is illustrated in Figure 1. Initially, BHR Bscan data, organized in M rows and N columns, are transformed into Ascan data of length N. This conversion simplifies the data processing required for the WOA optimization of the VMD parameters while preserving key data features. In this data compression method, $A_{i,j}$ denotes the value in the i-th row and j-th column of the BHR data, and $A_{\mathrm{mean}}\left(j\right)$ is the mean value of the j-th column.

$$\begin{array}{c}\hfill A_{\mathrm{mean}}\left(j\right)={\displaystyle \frac{1}{M}}\sum _{i=1}^M{A}_{i,j}(j=1,2,\dots ,N)\end{array}$$

(1)

2.1. Adaptive VMD Algorithm

In this study, VMD is first applied for BHR data denoising, effectively highlighting target signals obscured by noise. In this application, the compressed Ascan data, $A_{\mathrm{mean}}$, are decomposed by VMD into K modal functions, represented as $u_k$.

$$\begin{array}{c}\hfill A_{\mathrm{mean}}=\sum _{k=1}^K{u}_k\end{array}$$

(2)

In VMD, the signal is decomposed by solving a variational problem that ensures minimal bandwidth for each modal function. This approach improves the accuracy and robustness of the decomposition. Through this decomposition process, the modal functions become more physically meaningful, allowing the more accurate representation of the original signal. In this study, ${\omega }_k$ denotes the center frequency of the K-th mode; ${\mathrm{\Delta }}_n$ denotes the difference operator at discrete time n; $\delta \left[n\right]$ denotes the Kronecker Delta function at discrete time; and j denotes the imaginary unit. ${\parallel ·\parallel }_2$ denotes the L2 norm, also known as the Euclidean norm [55].

$$\begin{array}{c}\hfill \underset{u_k,{\omega }_k}{min}\left\{\sum _{k=1}^K{∥{\mathrm{\Delta }}_n\left[\left(\delta \left[n\right]+j{\displaystyle \frac{1}{\pi n}}\right)\ast u_k\left[n\right]\right]e^{-j{\omega }_{k}n}∥}_2^2\right\}\mathrm{subject}\mathrm{to}\sum _{k=1}^K{u}_k=A_{\mathrm{mean}}\end{array}$$

(3)

The optimization problem is formulated through the Lagrangian method by introducing a quadratic penalty term and Lagrange multipliers $\lambda$ into the VMD formula. It is then solved using the Alternating Direction Method of Multipliers (ADMM). In this formulation, the Lagrange multiplier $\lambda$ is used to adjust the constraints during the optimization process. Within the objective function, the penalty parameter $\alpha$ is used to determine the weight of the penalty term, balancing the reconstruction error with modal sparsity in the VMD decomposition. The formula is constructed with three key components. The modal bandwidth measure term minimizes the bandwidth of each decomposed mode, ensuring that the signal components are concentrated in the frequency domain. By reducing the bandwidth, the precision of the signal decomposition process is enhanced. The quadratic penalty term enhances the fidelity of signal reconstruction by penalizing deviations from the original signal. The Lagrange multiplier term ensures that the sum of all modal components strictly equals the original signal, thereby preserving signal completeness and integrity. It enforces strict reconstruction constraints throughout the optimization process. The combination of the quadratic penalty term and the Lagrange multiplier leverages the advantages of both. The quadratic penalty provides robust convergence properties, while the Lagrange multiplier enforces strict constraint satisfaction. Together, these components enable a balanced and stable optimization framework for the decomposition process [56].

$$\begin{array}{cc}\hfill L\left(u_k,{\omega }_k,\lambda \right)=& \alpha \sum _{k=1}^K{∥{\mathrm{\Delta }}_n\left[\left(\delta \left[n\right]+j{\displaystyle \frac{1}{\pi n}}\right)\ast u_k\left[n\right]\right]e^{-j{\omega }_{k}n}∥}_2^2+{∥A_{\mathrm{mean}}\left[n\right]-\sum _{k=1}^K{u}_k\left[n\right]∥}_2^2\hfill \\ \hfill & +\langle \lambda \left[n\right],A_{\mathrm{mean}}\left[n\right]-\sum _{k=1}^K{u}_k\left[n\right]\rangle \hfill \end{array}$$

(4)

The update formula for the mode $u_k$ is presented below:

$$\begin{array}{cc}\hfill u_k^{n+1}\left[n\right]& =arg\underset{u_k}{min}\left\{\alpha {∥{\mathrm{\Delta }}_n\left[\left(\delta \left[n\right]+{\displaystyle \frac{j}{\pi n}}\right)\ast u_k\left(n\right)\right]e^{-j{\omega }_{k}n}∥}_2^2\right.\hfill \\ \hfill & +\left.{∥A_{\mathrm{mean}}\left[n\right]-\sum _{k=1}^K{u}_k\left[n\right]+{\displaystyle \frac{\lambda \left(n\right)}{2}}∥}_2^2\right\}\hfill \end{array}$$

(5)

The formula for the updating of ${\omega }_k$ is presented below:

$$\begin{array}{c}\hfill {\omega }_k^{n+1}=arg\underset{{\omega }_k}{min}\left\{{∥{\mathrm{\Delta }}_n\left[\left(\delta \left[n\right]+j{\displaystyle \frac{1}{\pi n}}\right)\ast u_k\left[n\right]\right]e^{-j{\omega }_{k}n}∥}_2^2\right\}\end{array}$$

(6)

In VMD signal decomposition, the penalty parameter $\alpha$ and the number of mode decompositions K are predetermined and serve as crucial factors. A lower penalty parameter $\alpha$ reduces bandwidth constraints on the modal components, resulting in modal signals with a broader bandwidth. Conversely, an excessively high or low number of mode decompositions K can cause the over-decomposition or under-decomposition of the signal. Optimizing the VMD parameters through the whale optimization algorithm (WOA) enables the automatic selection of parameters suitable for the signal, enhancing the decomposition accuracy and robustness. The whale optimization algorithm (WOA) [57], inspired by the bubble-net hunting behavior of whales, is characterized by its rapid convergence and robust global search capabilities. The WOA dynamically adjusts the convergence factor to balance global exploration and local exploitation. This mechanism is employed to efficiently optimize complex, nonlinear problems and avoid local optima. In this study, the WOA is combined with variational mode decomposition (VMD) to adaptively optimize the penalty parameter $\alpha$ and the number of decompositions K. The global search and local exploitation properties of the WOA ensure the accurate optimization of both integer and continuous parameters, making it particularly suitable for tuning K and $\alpha$. By leveraging the WOA’s stochastic nature and linear convergence factor, the optimization process achieves improved stability and convergence speeds. Collectively, these attributes significantly enhance the decomposition performance of VMD.

The optimization of the variational mode decomposition (VMD) parameters $(K,\alpha )$ is performed using the whale optimization algorithm (WOA) [56,57]. The optimization process involves mapping the positions of the whales to the VMD parameter space, thereby enabling the adaptive adjustment of both the penalty parameter $\alpha$ and the number of modes K. Whale Position Representation: The position of each whale is denoted as $X\left[i\right]=[K_i,{\alpha }_i]$, where $K_i$ refers to the number of decomposition modes and ${\alpha }_i$ corresponds to the penalty parameter. Population Initialization: A population of N whales is initialized randomly, with positions $X_i$ constrained within predefined boundary limits. This ensures that all whale positions remain within the feasible optimization range. Parameter Updates During Iterations: In each iteration of the WOA, the positions $X\left[i\right]$ are updated according to the exploration and exploitation strategies embedded in the algorithm, allowing for the dynamic adjustment of both K and $\alpha$. Following each update, boundary checks are conducted to ensure that the values of K and $\alpha$ remain within their valid ranges.

$$\begin{array}{c}\hfill {\left\{{\overrightarrow{X}}_i\right\}}_{i=1}^N=\{{(K_i,{\alpha }_i)\}}_{i=1}^N\end{array}$$

(7)

Prior to parameter optimization, the fitness function must be defined. The sample entropy, represented as $SampEn(m,r,u_k)$, is used to quantify the complexity and nonlinearity of a time series. The sample entropy is characterized by independence from the data length and strong resistance to interference, enabling stable calculations regardless of the data length. It is also insensitive to noise, allowing accurate complexity measurement despite disturbances. The sample entropy is defined using m as the embedding dimension and r as the tolerance to measure the similarity of subsequences within the time series. Here, $B^m\left(r\right)$ and $B^{m+1}\left(r\right)$ are the total counts of instances where the distances between subsequences are below r in dimensions m and $m+1$, respectively. $u_k$ is the input time series for which the sample entropy is being calculated [58].

$$\begin{array}{c}\hfill SampEn(m,r,u_k)=-ln\left({\displaystyle \frac{B^{m+1}\left(r\right)}{B^m\left(r\right)}}\right)\end{array}$$

(8)

The core parameters of VMD, specifically the number of decomposition modes K and the penalty factor $\alpha$, are optimized through a fitness function derived from the sample entropy (SampEn). This fitness function quantifies the regularity of the intrinsic mode functions (IMFs) and is utilized to assess and enhance the quality of VMD signal decomposition. The primary objective is to minimize the local SampEn of the IMFs. Lower SampEn values signify greater regularity in the IMFs, which in turn improves the quality of signal decomposition. The fitness value calculation process is as follows. (1) Decomposition Step: VMD is applied with varying combinations of the number of decomposition modes K and penalty factor $\alpha$ to decompose the input signal into K IMFs. (2) Entropy Calculation: The SampEn value for each IMF is computed. (3) Fitness Selection: Since the appearance of targets in BHR echo signals is often accompanied by an increase in entropy, the maximum SampEn value among all IMFs is selected as the fitness value. The fitness value derived from the maximum SampEn represents the complexity and irregularity of the IMFs. Minimizing the fitness value ensures that the resulting IMFs are more regular, effectively suppressing noise while retaining the target signal components. This optimization strategy enhances the signal regularity and improves the interpretability of the decomposition results, underscoring its importance in applications such as target detection in BHR signals. The fitness value is calculated using the following formula:

$$\begin{array}{c}\hfill {\mathrm{Fitness}}_i=max\left\{SampEn(m,r,u_k)\right\}{|}_{(K,\alpha )=(K_i,{\alpha }_i)}\end{array}$$

(9)

The adjustment of the $(K,\alpha )$ parameters follows the update rules of the whale optimization algorithm (WOA). In this process, b represents a constant that determines the shape of the logarithmic spiral, while l is a randomly selected value within the range of $[-1,1]$. In the WOA, the parameter p is a random probability value within $[0,1]$. When $p<0.5$, the encircling mechanism is applied; otherwise, if $p\ge 0.5$, spiral updating is employed. The current optimal solution is denoted by ${\overrightarrow{X}}^{\ast }\left(t\right)$, and the vectors $\overrightarrow{A}$ and $\overrightarrow{C}$ control the convergence factors [57].

$$\begin{array}{cc}\hfill {\overrightarrow{X}}_{i+1}& ={{\overrightarrow{X}}^{\ast }}_i-\overrightarrow{A}·|\overrightarrow{C}·{\overrightarrow{X}}_{\mathrm{rand}}-\overrightarrow{X}|,p<0.5\hfill \\ \hfill {\overrightarrow{X}}_{i+1}& =|{{\overrightarrow{X}}^{\ast }}_i-{\overrightarrow{X}}_i|·e^{bl}·cos\left(2\pi l\right)+{{\overrightarrow{X}}^{\ast }}_i,p\ge 0.5\hfill \end{array}$$

(10)

The optimization process is repeated until the predefined maximum number of iterations, set to 16 by default, is reached. Upon the completion of the iterations, the optimal parameters are determined and retained for future application.

$$\begin{array}{c}\hfill (K,\alpha )=arg\underset{{\left\{{\overrightarrow{X}}_i\right\}}_{i=1}^N}{max}\mathrm{Fitnes}{\mathrm{s}}_i\end{array}$$

(11)

Once the optimal parameters $(K,\alpha )$ are obtained, the original BHR Bscan data are decomposed using VMD to generate the modal functions ${\mathrm{IMF}}_k^{i,j}$. The Fourier transforms of the modes are then calculated to identify the peak frequency in the average Fourier spectrum. Subsequently, the average Fast Fourier Transform (FFT) of the original BHR Bscan data is computed. To determine the selected frequency band (SFB), the threshold is defined as the spectrum mean plus or minus one-fourth of the difference between the peak value and the spectrum mean. Modes within the selected frequency band are adaptively chosen for reconstruction, thereby separating the main noise components from the target signal and forming the reconstructed matrix ${\widehat{A}}_{\mathrm{SFB},i,k}$. The correlation coefficient $R_{i,k}$ between the selected mode ${\widehat{A}}_{\mathrm{SFB},i,k}$ and the i-th row of the corresponding original Bscan signal $A_i$ is calculated to assess their relationship. In the formula for the correlation coefficient $R_{i,k}$, $\mathrm{Cov}[A_i,{\widehat{A}}_{\mathrm{SFB},i,k}]$ is the covariance between them, and $\mathrm{Var}\left[A_i\right]$ and $\mathrm{Var}\left[{\widehat{A}}_{\mathrm{SFB},i,k}\right]$ are their respective variances [59].

$$\begin{array}{c}\hfill R_{i,k}={\displaystyle \frac{\mathrm{Cov}[A_i,{\widehat{A}}_{\mathrm{SFB},i,k}]}{\sqrt{\mathrm{Var}\left[A_i\right]\times \mathrm{Var}\left[{\widehat{A}}_{\mathrm{SFB},i,k}\right]}}}\end{array}$$

(12)

Modes with a correlation coefficient $R_{i,k}$ greater than the threshold ${\mathrm{Threshold}}_{\mathrm{COR}}$ are selected for reconstruction. The reconstructed signal is denoted by ${\widehat{A}}_{\mathrm{COR}}$, and ${\widehat{A}}_{\mathrm{SFB},i,k}$ represents the selected mode.

$$\begin{array}{c}\hfill {\widehat{A}}_{\mathrm{COR}}=\sum _{k\in \{k_1,k_2,\dots ,k_m\}}{\widehat{A}}_{\mathrm{SFB},i,k}\mathrm{subject}\mathrm{to}{R}_{i,k}>{\mathrm{Threshold}}_{\mathrm{COR}}\end{array}$$

(13)

2.2. Multiscale Principal Component Analysis (MSPCA)

MSPCA leverages the complementary strengths of the WT and PCA to enable robust multiscale signal processing. The WT excels in decomposing signals across multiple scales, isolating high- and low-frequency components, while PCA identifies the principal features within these scales. Together, these methods facilitate the efficient extraction of meaningful signal components, even under challenging noise conditions. For BHR data, selecting appropriate MSPCA parameters is critical in maintaining signal fidelity and achieving effective noise suppression. The symmetric wavelet basis ‘sym28’ is employed due to its orthogonal and symmetric properties, which preserve the smoothness and essential features of BHR signals [38]. This choice is informed by the complex frequency components and significant noise inherent in BHR data. The high time–frequency resolution of ‘sym28’ enables the effective retention of the target signal characteristics while suppressing noise. To balance the computational complexity and feature extraction accuracy, the decomposition level is empirically set to five. This configuration prevents distortions from over-decomposition and ensures the clear separation of noise from the target signals. BHR signals are inherently nonstationary due to geological variations, subsurface dielectric constant differences, conductivity changes, and environmental interferences. Instrumental noise and subsurface interactions further contribute to their nonlinear and complex nature. Although the WT is highly effective in processing stationary signals, its performance diminishes when applied to nonlinear and nonstationary BHR data, reducing its capability to handle the complex geophysical noise patterns inherent in such signals. To address these limitations, the proposed methodology integrates adaptive VMD with MSPCA. Adaptive VMD is employed as the initial step to decompose the nonlinear and nonstationary BHR signals into intrinsic mode functions (IMFs) with a finite bandwidth. This decomposition mitigates the nonlinearity and nonstationarity of the signals. Subsequently, MSPCA is applied to refine the signal further, isolating critical features while suppressing residual noise. This sequential strategy combines the complementary capabilities of adaptive VMD and MSPCA to enhance the signal clarity and noise suppression. During the reconstruction phase, the WT decomposes the signal ${\widehat{A}}_{\mathrm{COR}}$ into multiscale approximate coefficients ${\mathbf{A}}_j$, which represent overall trends, and detail coefficients ${\mathbf{D}}_j$, which capture finer details. Here, j denotes the decomposition level within the multiscale framework [41,43]. This integrated approach balances signal clarity and noise reduction, ensuring the reliable processing of BHR data.

$$\begin{array}{c}\hfill {\widehat{A}}_{\mathrm{COR}}=\sum _{j=1}^J({\mathbf{A}}_j+{\mathbf{D}}_j)\end{array}$$

(14)

The signal is decomposed using the discrete wavelet transform (DWT) through a high-pass filter $g\left[n\right]$ and a low-pass filter $h\left[n\right]$, which produces two types of coefficients: detail coefficients $D_j\left[n\right]$, capturing high-frequency information, and approximate coefficients $A_j\left[n\right]$, capturing the main trends in the signal at each decomposition level j.

$$\begin{array}{c}\hfill A_j\left[n\right]=\sum _k{\widehat{A}}_{\mathrm{COR}}\left[k\right]h[2n-k]\end{array}$$

(15)

$$\begin{array}{c}\hfill D_j\left[n\right]=\sum _k{\widehat{A}}_{\mathrm{COR}}\left[k\right]g[2n-k]\end{array}$$

(16)

After the approximate and detail coefficients are obtained across different scales, PCA is applied to the wavelet coefficients at each scale. This approach captures essential signal features across various frequency ranges, thereby enhancing noise suppression. At each wavelet decomposition level j, the covariance matrix ${\mathbf{C}}_j$ is calculated using the following formula. Here, ${\mathbf{X}}_j$ denotes the matrix of coefficients at that level, comprising both approximate and detail coefficients.

$$\begin{array}{c}\hfill {\mathbf{C}}_j={\displaystyle \frac{1}{N}}{\mathbf{X}}_j^T{\mathbf{X}}_j\end{array}$$

(17)

The eigenvalue decomposition of the covariance matrix ${\mathbf{C}}_j$ is given by the following formula, where ${\mathbf{P}}_j$ is the eigenvector matrix and ${\mathrm{\Lambda }}_j$ is the diagonal matrix of eigenvalues:

$$\begin{array}{c}\hfill {\mathbf{C}}_j={\mathbf{P}}_j{\mathrm{\Lambda }}_j{\mathbf{P}}_j^T\end{array}$$

(18)

In the PCA stage of MSPCA, high-frequency detail coefficient matrices, primarily containing noise, are discarded, while low-frequency approximate coefficient matrices, which capture primary trends and target features, are retained. The selection of principal components for the approximate coefficient matrices is guided by the cumulative variance criterion (CVC) [29], a method that sorts eigenvalues in descending order and accumulates their variance contributions until the cumulative contribution rate reaches 96%. This threshold ensures the retention of meaningful signal components while effectively reducing noise interference. By eliminating detail coefficient matrices and preserving key information within approximate coefficient matrices, the CVC enhances the fidelity of signal reconstruction, achieving a balance between signal integrity and noise suppression. The retained principal components, denoted as $k_{A,j}$ and $k_{D,j}$, correspond to approximate ${\mathbf{A}}_j^{\mathrm{PCA}}$ and detail coefficient matrices ${\mathbf{D}}_j^{\mathrm{PCA}}$, respectively. These components undergo an inverse PCA transformation, resulting in the reconstructed coefficient matrices ${\mathbf{A}}_j^{\mathrm{recon}}$ and ${\mathbf{D}}_j^{\mathrm{recon}}$. Finally, the denoised signal ${\widehat{A}}_{\mathrm{COR}}^{\mathrm{recon}}$ is obtained through an inverse WT, completing the reconstruction process. This systematic approach demonstrates the effectiveness of MSPCA in processing BHR data, as it ensures robust noise suppression while preserving critical signal characteristics. The methodology leverages the complementary strengths of PCA and the WT to optimize signal reconstruction for nonstationary and noisy data environments.

$$\begin{array}{c}\hfill {\widehat{A}}_{\mathrm{COR}}^{recon}=\sum _{j=1}^J({\mathbf{A}}_j^{recon}+{\mathbf{D}}_j^{recon})\end{array}$$

(19)

3. Examples

3.1. Numerical Simulation Results

3.1.1. Case 1: Underwater Scenario with Multiple Rectangular Targets

The proposed method is applied to denoise BHR data, with the results compared to those of alternative methods. As shown in Figure 2, numerical simulations are conducted using gprMax [60], employing a homogeneous water model containing three iron plate targets, each measuring 20 cm in height and 2 cm in width. Targets 1 and 3 are positioned 60 cm from the BHR, while target 2 is located 50 cm away, with a distance of 60 cm between adjacent targets. The transmitting and receiving antennas, spaced 26 cm apart, are excited with a 230 MHz Ricker wavelet. A 100 ns time window with 3 cm step increments is used for BHR Bscan measurements.

As shown in Figure 3c, the RNMF method highlights the target signal to some extent, although its noise suppression remains limited. The clarity of the target signal is impacted by the significant noise remaining within the red box. RPCA is effective for sparse noise but exhibits limitations in suppressing highly random noise. These limitations result in incomplete noise suppression and hinder the accurate reconstruction of the signals, especially when the noise and target signals overlap. As shown in Figure 3d, significant residual noise is observed within the red box, demonstrating RPCA’s reduced effectiveness in highly random noise environments. Although RPCA performs well in sparse noise environments, its efficiency diminishes significantly under conditions of prevalent random noise. RPCA struggles to fully isolate noise from the target signal, which compromises the clarity and fidelity of the reconstructed signal. As noted in Section 2.2, MSPCA integrates the complementary strengths of PCA and the WT. However, BHR signals are inherently nonstationary and nonlinear in nature. The WT is more effective for stationary signals, which constrains its applicability to BHR data. Additionally, MSPCA exhibits limited capabilities in suppressing random noise. As shown in Figure 3e, MSPCA effectively highlights weak echo signals, especially within the yellow box. However, residual noise remains significant in the red box, highlighting the limitations of its noise suppression capabilities.

EMD effectively decomposes nonlinear, nonstationary signals into intrinsic mode functions (IMFs), facilitating the enhancement of weak echo signals and achieving effective noise attenuation. However, the effectiveness of the method is constrained by mode mixing, which introduces high-frequency spike noise. This limitation diminishes its practical reliability and computational applicability, especially in scenarios involving complex signal interference. As depicted in Figure 3f, the residual noise patterns caused by mode mixing are evident, highlighting the challenges that EMD faces in maintaining signal clarity and fidelity. As demonstrated in Figure 3g, the adaptive VMD method excels in both noise suppression and highlighting weak echo signals, with the noise within the red box significantly reduced and the target signal within the yellow box appearing clearer. Noise and signals are effectively separated by adaptive VMD, making it suitable for the processing of nonstationary signals. As shown in Figure 3h, the proposed scheme (adaptive VMD + MSPCA) enhances both noise suppression and signal clarity. The noise within the red box is more effectively suppressed, and the target signal within the yellow box appears clearer and more distinct. The multiscale feature extraction advantages of adaptive VMD and MSPCA are combined in this scheme, resulting in more efficient noise suppression and signal enhancement. As described in Section 2.2, the sequential application of adaptive VMD and MSPCA is shown to achieve superior noise suppression and enhance weak signals. The effectiveness of the proposed scheme (adaptive VMD + MSPCA) in both noise suppression and target signal enhancement is validated through a comparative analysis of the denoising effects, demonstrating its suitability for the processing of noisy BHR signals.

A comprehensive comparison of the denoising performance, using metrics such as the signal-to-noise ratio (SNR), root mean square error (RMSE), and mean square error (MSE), is presented in Figure 4. These metrics are offered as critical insights into the effectiveness of the various denoising methods. The SNR quantifies the ratio of the signal amplitude to noise. Higher SNR values indicate improved signal clarity and reduced noise interference, which is essential for the accurate detection and interpretation of BHR data. For instance, an SNR increase of 19.088 dB, as achieved by the proposed method, results in significantly enhanced target signal interpretability in real-world applications, such as subsurface imaging and anomaly detection. The RMSE and MSE are quantified to indicate the fidelity of signal reconstruction. Lower RMSE and MSE values indicate effective noise suppression while preserving key signal features. For BHR applications, this ensures that the geological characteristics of the subsurface are accurately captured without distortion, enabling more reliable data interpretation.

The initial SNR of the noisy Bscan data is −5.826 dB. After applying the various denoising methods, the following results are observed. RNMF: The SNR increases marginally to −5.816 dB, representing a slight improvement of 0.01 dB. This minimal improvement indicates RNMF’s limited capabilities in distinguishing between noise and signals. RPCA: As described in Section 3.1.1, RPCA achieves an SNR improvement to −5.184 dB (approximately 0.642 dB increase). However, RPCA is less effective in suppressing high levels of random noise, leading to suboptimal results for BHR signal processing. MSPCA: Through the integration of PCA and the WT, MSPCA achieves a notable SNR improvement to 0.180 dB (approximately 6.006 dB increase). However, as highlighted in Section 2.2, MSPCA is more effective for stationary signals, restricting its applicability to the inherently nonstationary and nonlinear BHR signals. EMD: EMD achieves an SNR of 3.290 dB, an improvement of approximately 9.116 dB. While this indicates reasonable noise suppression, mode mixing issues, as described in Section 3.1.1, introduce high-frequency spike noise, which limits its effectiveness. Adaptive VMD: Adaptive VMD tunes the frequency bandwidth of modal components, resulting in a substantial improvement in the SNR to 8.997 dB (an increase of approximately 14.823 dB). This demonstrates its effectiveness in suppressing noise and enhancing the signal clarity. Proposed Method (Adaptive VMD + MSPCA): The proposed method combines the strengths of adaptive VMD and MSPCA, resulting in an SNR improvement to 13.262 dB, corresponding to a substantial increase of approximately 19.088 dB. This significant enhancement not only suppresses noise but also enhances the visibility of weak signals, making it the most effective method for BHR signal processing. Additionally, its superior performance in terms of the RMSE and MSE metrics further validates the method’s robustness.

From the perspective of the simulation data, the increase in the SNR correlates with more distinct target features and reduced noise interference. For instance, as shown in Figure 3h and Figure 5h, at an SNR improvement of 19.088 dB, previously obscured signals emerge more clearly, significantly enhancing the visibility and interpretability of geological features such as targets or stratigraphic boundaries. The reduction in the RMSE and MSE ensures higher accuracy in signal reconstruction, minimizing the likelihood of false positives or missed detections. In real-world scenarios, these improvements result in tangible benefits in BHR data processing. These include enhanced target detectability, as subsurface features become more distinguishable, aiding geophysical surveys and infrastructure inspections; increased data reliability, as lower noise levels and higher reconstruction fidelity reduce the risk of misinterpretation, supporting better decision-making in resource exploration and environmental monitoring; and greater applicability in complex environments, as the proposed method’s ability to handle nonstationary signals ensures robust performance across diverse geological conditions.

Figure 5 presents the denoising results of the 160th simulated Ascan signal. The region subjected to significant random noise is delineated by the red box in Figure 5. This simulated noise is introduced to test the effectiveness of the proposed denoising method under high-interference conditions. In Figure 5, the yellow box encompasses the target signal and the tail edges of the target reflections. This area demonstrates how the proposed denoising method effectively enhances the target signal and its tail edges, even in the presence of substantial noise. The target signal is enhanced to some extent by RNMF and RPCA, but the noise is not effectively suppressed. As noted in Section 2.2, MSPCA integrates the complementary strengths of PCA and the WT and is suitable for multiscale signal processing. As seen from the target signal in the yellow box, weak echo signals are effectively highlighted by MSPCA. However, its noise suppression capabilities are weak in highly random noise environments.

As described in Section 3.1.1, while EMD highlights weak echo signals and achieves good noise suppression to some extent, its application is constrained by mode mixing, which introduces high-frequency spike noise. This limitation is evident as the noise within the red box remains inadequately suppressed. The high-frequency noise within the red box persists, and the signal quality is affected. The noise suppression and signal enhancement are significantly improved by adaptive VMD, resulting in a higher SNR. The noise within the red box is significantly reduced, and the target signal within the yellow box is clearer. Good separation of the noise and signal is demonstrated. The noise suppression and signal enhancement are further improved by our proposed method, which combines adaptive VMD and MSPCA. The noise within the red box is further suppressed, and the target signal within the yellow box becomes clearer and more distinct. The multiscale feature extraction advantages of adaptive VMD and MSPCA are combined in this scheme, demonstrating the best performance in terms of noise suppression and signal enhancement.

The effectiveness of the proposed method is assessed using synthetic BHR data at various noise levels. The SNR of the input data with noise varies from −25 dB to 0 dB. In Figure 6, the performance of various algorithms is shown in terms of the output SNR, MSE, and RMSE across the considered input SNR levels. The algorithms under evaluation include RNMF, RPCA, MSPCA, EMD, adaptive VMD, and the proposed adaptive VMD + MSPCA scheme. As depicted in Figure 6a, the output SNR improves for all algorithms as the input SNR increases. Specifically, the adaptive VMD method demonstrates a significant improvement in the SNR, with increases spanning from 14.4 dB to 14.8 dB. When integrated with the MSPCA method, the proposed adaptive VMD + MSPCA scheme attains even greater SNR improvements, ranging from 18.4 dB to 24.5 dB. This improvement is particularly pronounced under low input SNR conditions (e.g., −25 dB to −15 dB), where noise suppression poses significant challenges. The increase in the SNR results in more distinguishable target features in the BHR data. With a higher SNR, the signal becomes significantly clearer, reducing the interference from background noise. This enables the better detection and identification of subsurface features that may have been obscured in low-SNR conditions. Through the improvement in the SNR, the signal previously concealed by the noise is made significantly more visible, thereby greatly enhancing the target’s detectability. For instance, in geological survey applications, an enhanced SNR helps in distinguishing between different geological layers or anomalies, improving the reliability of the results. The adaptive VMD method optimizes the number of modal components and their frequency bandwidths, thereby effectively separating noise from the signal. This optimization is critical in processing nonstationary signals, as frequently encountered with BHR data. Nonstationary signals, which vary over time, necessitate adaptive techniques to track changes in the signal’s characteristics. The adaptive VMD method’s ability to adaptively adjust the frequency bandwidths of modal components ensures that the signal is better separated from the noise, leading to an overall improvement in the signal quality. When combined with the multiscale feature extraction capabilities of MSPCA, the adaptive VMD + MSPCA scheme provides the best output SNR across all input SNR conditions. The multiscale nature of MSPCA allows for enhanced feature extraction at various resolutions, further improving the accuracy of signal detection and noise suppression. This combination proves highly effective in processing nonstationary BHR signals.

In Figure 6b,c, the adaptive VMD + MSPCA scheme is shown to exhibit the best performance in terms of both the output MSE and RMSE. The MSE and RMSE are essential metrics in assessing the accuracy of the reconstructed signal. A reduction in these values indicates the more accurate recovery of the signal, with minimal distortion or noise artifacts. The proposed method achieves the lowest MSE and RMSE values across all algorithms tested, indicating that it effectively removes noise while preserving the key features of the original BHR signal. The reduction in the MSE and RMSE, alongside the improvement in the SNR, ensures that the processed signal is both clearer and more accurate. Lower MSE and RMSE values mean that the denoised signal more closely matches the true subsurface conditions, reducing the risk of false positives or missed detections in subsequent analysis. A comparison of the denoising effects across different methods and input SNR levels reveals that the best performance in terms of noise suppression and signal enhancement is achieved by the adaptive VMD and the adaptive VMD + MSPCA schemes. These results validate the effectiveness of the proposed methods in denoising BHR signals. Specifically, the proposed adaptive VMD + MSPCA scheme not only maximizes the SNR but also minimizes the error between the original and denoised signals, making it the most effective method for practical BHR data processing.

3.1.2. Case 2: Underwater Scenario with Multiple Circular Targets

As shown in Figure 2, numerical simulations were conducted using gprMax [60], a specialized tool designed for the modeling of ground-penetrating radar scenarios. The simulation details align with those outlined for Case 1 in Section 3.1.1, with the primary difference being that the targets in Case 2 comprise five circular objects instead of three rectangular ones. The circular targets are positioned such that targets 1, 3, and 5 are 60 cm from the BHR, while targets 2 and 4 are 50 cm away. The center-to-center distance between adjacent targets is 80 cm. This modification creates a more intricate subsurface structure, establishing a robust benchmark for the assessment of the denoising performance of the proposed methods under diverse conditions. Figure 7 depicts the simulated Bscan data for BHR processed using various denoising techniques. The Bscan data for Case 2 were processed using several denoising techniques, as in Section 3.1.1 for Case 1, and the results are illustrated in Figure 7. Figure 7c shows that the RNMF method partially enhances the target signal (highlighted in the yellow box); however, significant residual noise remains, particularly in high-interference regions (highlighted in the red box), as observed in Section 3.1.1 for Case 1. Figure 7d shows that the RPCA method is effective in handling sparse noise but struggles with random noise, resulting in residual interference that compromises the clarity of the target signal. Figure 7e shows that, while the MSPCA method enhances weak echo signals, its performance in noise suppression is limited in regions with high levels of random noise. Figure 7f shows that the EMD method achieves noise suppression but is affected by mode mixing, which introduces high-frequency spike noise, impairing its performance with nonstationary signals. Figure 7g shows that the adaptive VMD method significantly mitigates noise, leading to a clear enhancement in the target signal. Figure 7h shows that the combination of adaptive VMD and MSPCA achieves the best denoising performance, effectively suppressing the noise and significantly improving the clarity of the target signal, consistent with the results observed in Section 3.1.1 for Case 1.

Figure 8 summarizes the SNR, RMSE, and MSE results for the various methods applied to Case 2, similarly to those in Section 3.1.1 for Case 1. The initial SNR of the noisy Bscan data is −7.63 dB. After the application of the various denoising methods, the results were as follows. RNMF: A marginal improvement in the SNR, reaching −7.617 dB, with minimal reductions in the RMSE and MSE. RPCA: Improves the SNR to −7.084 dB but fails to effectively reduce the RMSE and MSE in the presence of high levels of random noise. MSPCA: Achieves a moderate SNR improvement to −1.611 dB, with corresponding reductions in the RMSE and MSE (1.216 and 1.478, respectively). EMD: Improves the SNR to 1.209 dB, with the RMSE and MSE further reduced to 0.878 and 0.772, respectively. Adaptive VMD: Outperforms all previous methods, with an SNR of 8.045 dB, RMSE of 0.400, and MSE of 0.160. Adaptive VMD + MSPCA: The proposed method achieves the highest SNR of 10.334 dB, the lowest RMSE (0.307), and the lowest MSE (0.094), thereby demonstrating its superior denoising capabilities. The adaptive VMD + MSPCA method provides superior denoising performance across multiple metrics, achieving the highest SNR and the lowest RMSE and MSE. These results reinforce the method’s effectiveness, particularly in complex geophysical scenarios involving nonstationary and nonlinear signals, as demonstrated in Section 3.1.1 for Case 1. This method combines the adaptive frequency separation of VMD with the multiscale feature extraction of MSPCA, leading to improved noise suppression and clearer signal enhancement across various conditions.

Figure 9 presents the denoising results for the 160th simulated Ascan signal, similarly to those in Section 3.1.1 for Case 1. The region affected by significant random noise is delineated by the red box, which was introduced to evaluate the effectiveness of the proposed denoising methods under high-interference conditions. The yellow box highlights the target signal and the trailing edges of its reflections, demonstrating how the proposed denoising method enhances both the target signal and its trailing edges, even in the presence of significant noise. The RNMF method (Figure 9c) and RPCA method (Figure 9d) marginally enhance the target signal but fail to suppress the noise effectively. The MSPCA method (Figure 9e) integrates the complementary strengths of PCA and the WT, making it suitable for multiscale signal processing. As seen in the yellow box, weak echo signals are effectively highlighted by MSPCA, but its noise suppression capabilities are limited, particularly in environments with high levels of random noise, as noted in Section 3.1.1 for Case 1. As discussed in Section 3.1.1 for Case 1, the EMD method (Figure 9f) provides good noise suppression and enhances weak echo signals but is constrained by mode mixing, which introduces high-frequency spike noise. As a result, the noise within the red box remains inadequately suppressed, and the signal quality is compromised. The noise suppression and signal enhancement are significantly improved by the adaptive VMD method (Figure 9g), resulting in a higher SNR. The noise within the red box is significantly reduced, and the target signal in the yellow box becomes clearer, demonstrating the effective separation of the noise and signal. The proposed method, adaptive VMD + MSPCA (Figure 9h), further improves the noise suppression and enhances the signal quality. The noise in the red box is more effectively suppressed, and the target signal within the yellow box becomes clearer and more distinct, further enhancing the signal clarity. By combining the multiscale feature extraction capabilities of adaptive VMD and MSPCA, this method demonstrates superior performance in terms of both noise suppression and signal enhancement, as confirmed in Section 3.1.1 for Case 1.

Figure 10 illustrates the performance of various denoising algorithms under different input SNRs, using three metrics, the output SNR, MSE, and RMSE, as demonstrated in Section 3.1.1 for Case 1. The input SNR is varied from −25 dB to 0 dB to simulate varying levels of noise interference, providing a comprehensive evaluation of the algorithms’ effectiveness. The methods evaluated include RNMF, RPCA, MSPCA, EMD, the proposed adaptive VMD method, and the combined adaptive VMD + MSPCA scheme. Figure 10a shows the output SNR as a function of the input SNR. As the input SNR increases, all algorithms exhibit improved performance. However, the adaptive VMD and adaptive VMD + MSPCA methods outperform the other methods across all SNR levels, especially under low-SNR conditions (−25 dB to −15 dB). The adaptive VMD method achieves significant SNR improvements by optimizing the modal components and suppressing the noise effectively. Furthermore, the adaptive VMD + MSPCA scheme enhances these gains, leveraging multiscale feature extraction and adaptive decomposition, with output SNR improvements exceeding 17.307 dB at 0 dB and up to 25.068 dB at lower SNR levels. Figure 10b presents the MSE results for the different input SNRs. Lower MSE values indicate better noise suppression and signal reconstruction. Among all methods, the adaptive VMD + MSPCA scheme consistently achieves the lowest MSE, significantly outperforming other methods, particularly at lower input SNRs. This superior performance is attributed to its integrated approach, combining adaptive frequency optimization and PCA to comprehensively reduce residual noise. Figure 10c shows the RMSE results, highlighting the robustness of the proposed methods in maintaining the signal’s integrity while suppressing noise. Again, the adaptive VMD + MSPCA scheme achieves the lowest RMSE, demonstrating its ability to retain critical signal features while minimizing distortion. In contrast, RNMF and RPCA show limited noise suppression capabilities, particularly under high-noise conditions, resulting in higher RMSE values. Overall, Figure 10 confirms the effectiveness of the adaptive VMD and adaptive VMD + MSPCA schemes in denoising BHR signals. As demonstrated in Section 3.1.1 for Case 1, adaptive VMD is particularly effective in handling nonstationary signals, while the combination of adaptive VMD and MSPCA provides the best overall performance. These results validate the superiority of the proposed methods in suppressing noise and enhancing the signal quality, confirming their suitability for geophysical applications involving noisy BHR data.

A comprehensive analysis of Cases 1 and 2 confirms the consistency and superiority of the adaptive VMD + MSPCA denoising scheme. The two cases are represented by different target layouts and noise environments; however, the adaptive VMD + MSPCA scheme consistently shows high denoising efficiency and effective signal enhancement across all conditions. Denoising Performance Consistency: In both the simple underwater scenario with targets (Case 1) and the more complex multitarget setup (Case 2), the adaptive VMD + MSPCA scheme significantly improves the SNR and reduces noise interference, thereby enhancing the detectability and interpretability of the target signal. Performance Consistency: In both cases, the scheme exhibits superior denoising performance and signal enhancement, particularly under low-SNR conditions. The adaptive VMD + MSPCA combination outperforms other traditional methods, such as RPCA, MSPCA, and EMD, in terms of its signal processing capabilities. Signal Processing Accuracy: The proposed adaptive VMD + MSPCA scheme significantly improves the SNR and optimizes the performance in terms of the RMSE and MSE metrics, indicating its ability to retain key signal features while reducing data distortion. Algorithm Adaptability: The introduction of adaptive VMD enhances the algorithm’s adaptability in processing nonstationary signals. This ensures that the algorithm performs efficiently in diverse and complex environments. This comprehensive validation of both Cases 1 and 2 confirms the superior performance of the adaptive VMD + MSPCA denoising scheme. This method shows consistent advantages across various scenarios and noise environments, confirming its broad application potential in complex underground target detection and BHR data processing. This scheme improves the quality of BHR signals and enhances the clarity of target signals, providing reliable technical support for practical applications such as geological exploration and resource surveying.

3.2. Experimental Results

Figure 11 presents a complete diagram of the BHR equipment. The total length of the BHR equipment is approximately 1481.4 cm, with separation of 330 cm between the transmitting and receiving antennas. The equipment consists of the receiving circuit, receiving antenna, transmitting antenna, and transmitting circuit, with the transmitting circuit installed and connected sequentially at the bottom end. The electric cable connects the ground control unit to the underwater BHR. The ground control unit, wireless data transmission system, and mechanical structure, along with the electric cable, control the operating state of the underwater BHR and receive radar data. The BHR underwater experiment, conducted in Qiandao Lake, Hangzhou, China, is designed to validate the performance of the proposed scheme. A pulse BHR with a 230 MHz antenna is used to detect underwater iron plate targets. The target, a metal plate measuring 200 cm in length, 100 cm in width, and 5 cm in thickness, is positioned 100 cm from the instrument. Ascans are collected at 3 cm intervals, each consisting of 633 sampling points at a frequency of 2.5 GHz. The overall delay of the Ascan is set to 45 ns according to the ground control unit settings, and each Ascan is repeated and stacked 60 times at each position to form the final Ascan. This results in a Bscan composed of 371 Ascans. Basic processing steps, including background removal, are initially applied. The experimental setup is shown in Figure 2.

Figure 12 is presented, illustrating the denoising performance of various algorithms under different noise conditions, with the target signal marked in yellow and the noise highlighted within the red boxes in Figure 12a. In the two leftmost vertical red noise boxes, the RNMF (Figure 12b) and RPCA (Figure 12c) methods fail to effectively suppress the noise, leaving significant residual noise signals. Although MSPCA (Figure 12d) achieves some noise suppression, residual noise persists. The EMD (Figure 12e) method reduces the noise but introduces high-frequency spike noise due to mode mixing, as explained in Section 3.1.1. In contrast, the adaptive VMD method (Figure 12f) significantly reduces the noise, leading to a marked improvement in signal clarity. The adaptive VMD + MSPCA scheme (Figure 12g) nearly eliminates the residual noise, providing the most effective suppression. In the middle vertical red noise boxes, the RNMF and RPCA methods again leave substantial residuals. While MSPCA (Figure 12d) provides moderate noise suppression, some residual noise remains. The EMD method (Figure 12e) reduces the noise effectively, and adaptive VMD (Figure 12f) achieves substantial noise suppression. Finally, the adaptive VMD + MSPCA scheme (Figure 12g) outperforms the other methods, nearly eliminating the noise and significantly enhancing the clarity and resolution of the target signal.

In Figure 12a, the bottom horizontal red noise box highlights the regions where poor noise suppression is exhibited by RNMF (Figure 12b), leaving significant residuals. Both RPCA (Figure 12c) and MSPCA (Figure 12d) show some noise reduction; however, their effectiveness is limited, particularly in this high-interference area. Effective noise reduction is achieved by EMD (Figure 12e), although some residual noise remains. In contrast, the adaptive VMD method (Figure 12f) shows significant noise reduction, leading to a marked improvement in signal clarity. Furthermore, the adaptive VMD + MSPCA scheme (Figure 12g) provides the most effective suppression, leaving almost no residual noise. When considering the upper and lower high-frequency red noise boxes on the right side of Figure 12a, RNMF (Figure 12b), RPCA (Figure 12c), and MSPCA (Figure 12d) exhibit limited suppression of the high-frequency noise. Noise reduction is achieved by EMD (Figure 12e), but this comes at the cost of introducing new high-frequency noise. The adaptive VMD method (Figure 12f) performs significantly better in reducing high-frequency noise, resulting in improved signal quality. The adaptive VMD + MSPCA scheme (Figure 12g) is the most effective in suppressing high-frequency noise, providing the best results in this context.

In the top yellow target signal area of Figure 12a, the RNMF (Figure 12b) and RPCA (Figure 12c) methods show limited effectiveness in enhancing the target signal, with substantial noise interference still present. MSPCA (Figure 12d) improves the target signal’s clarity, but its noise suppression capabilities remain limited. The EMD method (Figure 12e) achieves effective noise reduction and some improvement in terms of target signal enhancement, but high-frequency noise remains a concern. In contrast, the adaptive VMD method (Figure 12f) significantly enhances the target signal’s clarity while minimizing noise interference. The best results in terms of target signal enhancement with minimal noise interference are provided by the adaptive VMD + MSPCA scheme (Figure 12g), which is particularly well suited for the underwater uniform medium scenario.

The denoising effects of each algorithm indicate that the target signal’s features are enhanced to some extent by RNMF (Figure 12b); however, its poor performance in terms of noise suppression complicates the distinction between the noise and signal. As discussed in Section 3.1.1, RPCA demonstrates effectiveness in managing sparse noise environments but is limited in suppressing high levels of random noise, which hinders effective noise suppression and signal separation (Figure 12c). MSPCA, as mentioned in Section 2.2, integrates the complementary strengths of PCA and the WT, enabling efficient multiscale signal processing. While weak echo signals are effectively highlighted by MSPCA, its noise suppression capabilities are limited in high-noise environments, making it more suitable for the processing of stationary signals (Figure 12d). EMD, as described in Section 3.1.1, successfully decomposes signals and suppresses noise (Figure 12e), but its practical application is limited by mode mixing, which can introduce high-frequency spike noise, thereby reducing its reliability in scenarios with complex signal interference. Adaptive VMD optimizes the number of mode components and their frequency bandwidths (Figure 12f), effectively separating noise from signals and providing excellent noise suppression and signal enhancement for nonstationary signals. The combination of adaptive VMD and MSPCA achieves the most effective noise suppression and signal enhancement (Figure 12g), with the denoised results closely resembling those of the underwater homogeneous medium scenario, thus validating the robustness of the proposed method. The sequential application of adaptive VMD and MSPCA, as detailed in Section 2.2, takes advantage of the frequency optimization provided by adaptive VMD and the multiscale feature extraction capabilities of MSPCA. Adaptive VMD is initially employed to process nonstationary signals, with MSPCA applied afterward for further optimization. This method effectively suppresses noise, enhances weak signals, and preserves critical features at various scales.

4. Conclusions

This study presents a denoising scheme for BHR signals, based on VMD and MSPCA. The core innovation of the scheme involves the application of the WOA to adaptively optimize the VMD parameters K and $\alpha$, thereby achieving the automatic and efficient selection of decomposition modes and enhancing the adaptability of VMD in complex noise environments. Specifically, VMD is initially employed to decompose the nonlinear and nonstationary BHR signals into intrinsic mode functions (IMFs), effectively overcoming common issues such as mode mixing and the lack of clear physical interpretation associated with EMD. The WOA, in conjunction with the maximum sample entropy, is utilized to optimize the VMD parameters, facilitating precise mode selection and enhancing the separation between the noise and signal. By calculating the correlation between each mode and the original signal in both the time and frequency domains, modes exhibiting weak correlations are identified and discarded, effectively reducing the noise while preserving valuable signal information. Finally, MSPCA is applied to perform the multiscale processing of the signal, further extracting weak signals and suppressing residual noise to ensure the integrity of the original signal structure.

The proposed scheme is rigorously validated through numerical simulations and underwater BHR experiments. The simulations show that the proposed scheme increases the signal-to-noise ratio by 17.964 dB or higher, significantly outperforming traditional denoising methods. Additionally, the underwater BHR experiments confirm the scheme’s practical applicability, particularly its effectiveness in suppressing vertical and random noise, commonly encountered in geophysical exploration, as well as its robustness in handling nonstationary noise environments. This method is especially suitable for geophysical exploration scenarios characterized by complex noise environments and nonstationary signals. By enhancing the SNR and reducing noise, this method facilitates the detection of geological features, subsurface structures, and resource distributions, thereby significantly improving the data quality and interpretability.

Despite its notable effectiveness, this scheme presents the following limitations. High computational intensity: The multistage processing involving VMD, MSPCA, and the WOA demands greater computational resources compared to simpler methods such as mean filtering or RNMF. Risk of over-denoising: The overlapping of noise characteristics with the signal may lead to the inadvertent removal of valuable signal components. To address these limitations and broaden its applicability, future research will concentrate on reducing the computational overhead by developing more efficient algorithms for parameter optimization and mode decomposition to decrease the processing time; enhancing noise–signal differentiation by improving the scheme’s ability to distinguish between noise and signals, thereby minimizing the risk of over-denoising; and expanding the application range by adapting the scheme for use in heterogeneous geological environments and on various types of radar and geophysical data.