Enhanced Offshore Wind Farm Geophysical Surveys: Shearlet-Sparse Regularization in Multi-Channel Predictive Deconvolution

Abstract

This study introduces a novel multi-channel predictive deconvolution method enhanced by Shearlet-based sparse regularization, aimed at improving the accuracy and stability of subsurface seismic imaging, particularly in offshore wind farm site assessments. Traditional multi-channel predictive deconvolution techniques often struggle with noise interference, limiting their effectiveness. By integrating Shearlet transform into the multi-channel predictive framework, our approach leverages its directional and multiscale properties to enhance sparsity and directionality in seismic data representation. Tests on both synthetic and field data demonstrate that our method not only provides more accurate seismic images but also shows significant resilience to noise, compared to conventional methods. These findings suggest that the proposed technique can substantially improve geological feature identification and has great potential for enhancing the efficiency of seabed surveys in marine renewable energy development.

1. Introduction

Global warming has been a hot issue of widespread concern, which in turn has led to an increasing demand for clean energy. As a mode of new energy development, offshore wind power harnesses clean, renewable wind resources, causing no pollution to the environment [1,2]. In the preliminary geological survey for offshore wind farms, the main concern is the shallow subsurface geological conditions, such as the exposure of bedrock and the presence of buried ancient river channels. These features typically occur near the seabed, so the focus is on understanding the shallow geological information. Buried ancient river channels can cause uneven bearing capacity due to the inconsistency in material filling compared to the surrounding sediments. This can lead to risks such as building displacement or tilting during construction activities, such as piling. The exposure of bedrock or buried bedrock increases the bearing capacity of the strata, which complicates piling operations [3]. Currently, the methods for investigating seabed strata are relatively limited, primarily relying on shallow and deep drilling to obtain seabed stratigraphic data. This approach is costly, time-consuming, and labor-intensive. Additionally, drilling cannot be too dense, as it may result in discontinuous seabed stratigraphic data.

The single-channel seismic [4,5] exploration method, as a towed geophysical surveying technique, can achieve continuous detection of the seafloor interface [6]. Its advantages include strong source energy, great detection depth, high resolution, lightweight equipment, and reduced cost and time requirements. These characteristics enable the effective continuous detection of submarine geological interfaces, as shown in Figure 1. Consequently, it aids in understanding the spatial morphology and relationships among seabed strata, determining geological features, identifying phenomena like collapses and troughs, and calibrating stratigraphy. It also assesses the undulating morphology and distribution of bedrock surfaces, addressing drilling limitations effectively [7]. To date, the single-channel seismic method has been applied in preliminary geological surveys for offshore drilling and fault monitoring in the hydrocarbon industry [8,9,10].However, in practical applications, the development of free-surface multiple reflections is complicated by the presence of strong reflections from the sea surface. Multiple overlapping reflections from various areas and primaries from geological interfaces pose challenges to strata identification, thereby increasing the risk of misinterpretation [11]. Multiple reflection-attenuation techniques are divided into two main categories. The first category includes filtering methods, which suppress multiples by exploiting the differences in characteristics and properties between multiple reflections and primaries, such as predictive deconvolution and F-K transform [4]. The second category involves wave equation-based prediction subtraction methods, whereby a multiple reflections model is first predicted from the seismic data and then adaptively subtracted from the data, such as SRME [12]. By analyzing the original data profiles, it was found that the seabed strata are relatively flat and exhibit good continuity. In offshore wind farm geophysical surveys, the focus is on shallow short-period multiple reflections, which are highly predictable. Under these conditions, predictive deconvolution effectively attenuates these multiples. Additionally, this method is cost-effective, straightforward to implement, and allows for the rapid and efficient processing of seismic data, meeting the needs of engineering applications.

Based on the above, predictive deconvolution can be used as our technical approach, but traditional single-channel predictive deconvolution [13,14,15] only considers time-domain variations. Multi-channel predictive deconvolution has been introduced to exploit the spatial consistency between adjacent seismic traces for predicting multiple reflections [16,17,18,19]. By computing time and space prediction filters to accommodate lateral variations in subsurface structures [20], this method better preserves primaries and attenuates multiple reflections. However, noise is inevitably present in practical acquisition processes, which can affect the continuity of seismic signals and thus limit the performance of multi-channel predictive deconvolution.

When solving for multi-channel prediction deconvolution filters, many underdetermined problems arise, leading to non-unique solutions. Therefore, regularization terms are needed as constraints to seek a stable solution [21,22]. To overcome this challenge, sparse regularization optimization inversion has received widespread attention. This method combines sparse representation theory with optimization algorithms to represent seismic signals sparsely, removing redundant information. The sparse signal is then inverted, benefiting from its sparsity, which makes it less likely to get trapped in local minima during inversion [23,24].

Among various sparse transformation methods, Shearlet transform has advantages such as sparsity, directionality, and multiscale properties [25,26]. Its fine scale can excellently represent different frequency band characteristics, providing a unified transformation in both discrete and continuous domains, and can optimally sparsely represent anisotropic images. Its ability to separate noise from the effective signal in the Shearlet domain allows for the effective suppression of multiple reflections and the preservation of primaries even in the presence of noise interference. The use of Shearlet transform for removing random noise has become well-established in seismic exploration [27,28]. Therefore, we introduce sparse regularization into the multi-channel predictive deconvolution framework, presenting a Shearlet-based sparse regularization-enhanced multi-channel predictive deconvolution technique [29,30]. This innovative approach not only more effectively suppresses multiple reflections under noisy conditions, but also significantly reduces redundant data compared to traditional methods.

In the following, we first introduce the basic principles of multi-channel predictive deconvolution and sparse representation. Subsequently, we integrate sparse regularization into the multi-channel predictive deconvolution framework, present the principles of our proposed method, and solve for the 2D predictive filters. Following this, tests on synthetic and field data are utilized to demonstrate the effectiveness of the proposed method. Lastly, in the Section 4, the practicality and potential of parameter settings and the precise suppression of multiple reflections are explored. The Section 5 provides a comprehensive summary of the article.

2. Methods

2.1. Background

In this section, we will introduce the basic background of the method. On one hand, we show the fundamental principle of multi-channel prediction deconvolution, emphasizing the limitation of single-channel predictive deconvolution operating only in the time domain. On the other hand, we introduce the sparse representation of seismic signals, providing a theoretical basis for the subsequent Shearlet sparse transformation of the data.

2.1.1. Multi-Channel Predictive Deconvolution

Single-channel predictive deconvolution typically uses 1D time filters to filter seismic data, effectively suppressing multiple reflections in periodic time intervals. The basic equation for single-channel predictive deconvolution can be expressed as [4,31]

$$p_{s+i}=d_{s+i}-{\displaystyle \sum _{k=-K}^K{f}_k{d}_{k+i},i=1,2,\dots ,T_0,}$$

(1)

where $s$ represents the time prediction step in a single-channel data sample, $2K+1$ denotes the length of the 1D time filter in terms of time samples, and $T_0$ stands for the total time length of the single-channel data sample. The equation can be expressed in matrix form as:

$$p=d-fD,$$

(2)

In this equation, $\mathbf{p}={[p_{s+1},p_{s+2},\dots ,p_{s+T_0}]}^T$ represents the estimated primaries after filtering, $\mathbf{d}={[d_{s+1},d_{s+2},\dots ,d_{s+T_0}]}^T$ denotes the original seismic data, and $\mathbf{f}={[f_{-K},f_{-K+1},\dots ,f_K]}^T$ is the 1D predictive filter. $\left[\begin{array}{cccc}{d}_{1-K}& d_{2-K}& \cdots & d_{1+K}\\ d_{2-K}& d_{3-K}& \cdots & d_{1+K}\\ ⋮& ⋮& \ddots & ⋮\\ d_{T_0-K}& d_{T_0-K+1}& \cdots & d_{T_0+K}\end{array}\right]$ represents the convolution matrix of the data, $\mathbf{d}={[d_{s+1},d_{s+2},\dots ,d_{s+T_0}]}^T$.

However, achieving effective suppression in the time domain with single-channel predictive deconvolution relies on the assumption [20] that multiple reflections of different orders on the seafloor exhibit strict periodicity [32,33]. Even with a horizontal seafloor, strict periodicity in multiple reflections is only observed at zero offset. The use of single-channel predictive deconvolution has clear limitations. When periodicity assumptions are not met, it can result in increased interference from multiple reflections. Furthermore, in order to attenuate multiple reflections of different orders in each time interval, single-channel predictive deconvolution is prone to overfitting in regions where the original signal overlaps with multiple reflections. This results in the distortion of the primaries data [5].

To overcome the limitations of single-channel prediction deconvolution, a multi-channel prediction deconvolution using 2D prediction filters has been proposed [13,34,35,36,37]. This method not only considers the temporal variations in seismic data, but also takes into account the spatial continuity of the data. This enables the 2D predictive filters to better adapt to lateral changes in subsurface structures, thereby reducing the impact of noise on signal continuity. The basic equation for multi-channel predictive deconvolution can be expressed as [11]:

$$y=m-M\mathbf{L},$$

(3)

where

$$y=\left[\begin{array}{c}{p}_1\\ p_2\\ ⋮\\ p_{X_0}\end{array}\right],$$

(4)

$$m=\left[\begin{array}{c}{d}_1\\ d_2\\ ⋮\\ d_{X_0}\end{array}\right],$$

(5)

$$\mathbf{L}=\left[\begin{array}{c}{f}_1\\ f_2\\ ⋮\\ f_{X_0}\end{array}\right],$$

(6)

$$M=\left[\begin{array}{cccc}{D}_{-q}& D_{-q+1}& \cdots & D_q\\ D_{-q+1}& D_{-q+2}& \cdots & D_{q+1}\\ ⋮& ⋮& \ddots & ⋮\\ D_{-q+X_0-1}& D_{-q+X_0}& \cdots & D_{q+X_0-1}\end{array}\right].$$

(7)

In the above equations, $X_0$ represents the total spatial length of the data sample, $2q+1$ represents the spatial length of the 2D prediction filter window. Vectors $y$,$m$ and $\mathbf{L}$ represent the primaries, original data, and 2D prediction filter. $M$ represents a convolution matrix, and the following equation obtains $T_0{X}_0\times (2K+1)(2q+1)$, where $T_0$ represents the time length of the 2D data window, and $2K+1$ represents the temporal length of the 2D prediction filter.

2.1.2. The Sparse Representation Theory for Seismic Signals

In this section, we introduce the fundamental theory of sparse representation for seismic signals.

Based on the latest research advancements, seismic signals $d\in R^n$ can be sparsely represented using a sparse sampling matrix $\Psi \in R^{n\times m}$. If the amplitude decrease gradients of the sparse domain coefficients after sampling are significant, the seismic signal can be directly represented as the product of the sparse domain inverse transform and the sparse domain coefficients, and then incorporated into the optimization inversion problem. The specific process can be described by the following linear equation [38,39]:

$$c=\Psi d\mathrm{s}.\mathrm{t}.\min {‖c‖}_0,$$

(8)

where the sampling matrix $\Psi$ can be represented as

$$\Psi =\left\{\Phi \in {\mathrm{R}}^{\mathrm{n}}|\gamma \in \Gamma ,‖\Phi ‖=1\right\},$$

(9)

where, $c\in {\mathrm{R}}^{\mathrm{m}}$ represents the coefficients in the transformed domain and satisfies ${‖c‖}_0\ll \mathrm{m}$. ${‖c‖}_0$ denotes the ${\mathrm{L}}_0$ norm of the coefficients in the transformed domain, indicating the number of non-zero elements, which reflects the sparsity of the seismic signal. The process of coefficient representation is showed in Figure 2. $\Phi$ represents the vector form of the sampling matrix $\Psi$. Due to the difficulty of optimally solving the ${\mathrm{L}}_0$ norm, the ${\mathrm{L}}_1$ norm is often used as a substitute constraint to simplify the optimization problem [40]:

$$c=\Psi d\mathrm{s}.\mathrm{t}.\min {‖c‖}_1,$$

(10)

In the actual process of seismic data acquisition, noise $n$ is unavoidable. The above equation can be further expressed as:

$$d={\Psi }^{\mathrm{T}}c+n,$$

(11)

where ${\Psi }^{\mathrm{T}}$ is the transpose matrix of $\Psi$ (usually chosen as an orthogonal transformation or a tight frame). Equation (11) represents an underdetermined system of equations, implying the existence of infinitely many solutions. To mitigate the impact of noise, Equation (11) can be transformed into the following optimization problem:

$$P_1=\left\{\begin{array}{c}\underset{c}{\min }{‖c‖}_0 \mathrm{subject} \mathrm{to}{‖d-{\Psi }^{\mathrm{T}}c‖}_2\le \epsilon \\ \widehat{c}=\Psi \widehat{d}\end{array}\right.,$$

(12)

similarly, due to the difficulty in optimally solving the ${\mathrm{L}}_0$ norm, the ${\mathrm{L}}_1$ norm is used as a substitute to simplify the optimization problem as its constraint:

$$P_2=\left\{\begin{array}{c}\underset{c}{\min }{‖c‖}_1 \mathrm{subject} \mathrm{to}{‖d-{\Psi }^{\mathrm{T}}c‖}_2\le \epsilon \\ \widehat{c}=\Psi \widehat{d}\end{array}\right.,$$

(13)

where $\widehat{d}$ and $\widehat{c}$, respectively, represent the optimal results derived in the data domain and sparse domain, while $\epsilon$ denotes the maximum of noise level.

2.2. Sparse Representation Method Based on Shearlet Transform

Based on the preceding discussion, the premise for sparse optimization inversion is to find a suitable transformation method to transform the data into a sparse domain, aiming to achieve the best sparse representation of seismic signals. Commonly used sparse transforms in seismic signal processing include Fourier transform and short-time Fourier transform [41,42], continuous wavelet transform [43], Curvelet transform [44] and Shearlet transform [45,46], among others.

Due to its directional and tight support structure, the Shearlet transform excels in expressing different frequency band characteristics with fine scales [25,26], allowing for the effective suppression of multiple reflections and the preservation of primaries even in the presence of noise interference. It provides a unified transformation for both discrete and continuous variations, offering optimal sparse representation for anisotropic images. Therefore, we choose the Shearlet transform to sparsely represent seismic data.

To demonstrate the varying effects of different sparse representations on seismic signals, we applied wavelet, curvelet, and Shearlet transforms to synthetic data corrupted with noise. Based on the sparse representation of seismic signals, the key is to use the coefficients in the sparse domain to map the data [47]. Effective signals and noise can be separated by setting a threshold, making the sparsity of the signal an important parameter. To assess sparsity, we selected the top 3% of coefficients with the largest amplitudes in the sparse domain for reconstruction. Figure 3 shows the comparative results, with arrows indicating noise interference on the effective signal. Additionally, we conducted Root Mean Square Error (RMSE) and Signal-to-Noise ratio (SNR) tests after reconstruction, with detailed information provided in the Supplementary Materials. The SNR value of the original data is 3.011 dB. The SNR values for reconstructions using wavelet transform, curvelet transform, and Shearlet transform were 30.0269 dB, 15.3765 dB, and 51.8630 dB, respectively. It is evident that Shearlet transform outperforms wavelet and curvelet transforms, providing the optimal sparse representation of the signal.

2.3. Sparse Regularization Optimization in Multi-Channel Prediction Deconvolution

To introduce sparse regularization into the multi-channel predictive deconvolution, further transformations of Equation (3) are necessary. However, the convolution operation in Equation (3) involves significant computational overhead. We discovered that introducing the identity matrix for convolution allows the filters to directly operate on the original data. The error between these two methods is only 0.072, demonstrating that this approach reduces unnecessary computations. Therefore, we optimized Equation (3) as follows:

$$y=m-Qm=(I-Q)m=Fm,$$

(14)

where $F$ represents the 2D prediction filter. We want to represent the original data using primaries as $m={\mathbf{F}}^{-1}y$. However, in geophysics, due to poor stability and computational complexity, inverse operations are generally avoided [48]. So, we define $Q$ as a multiple reflections extraction filter in our study with a fixed prediction stride determined by the autocorrelation of the original data. When it is applied to the original data and the original data contain only primaries after prediction filtering, the predicted multiple reflections can be obtained. Thus, given $Qy=Q\mathbf{m}=g$, where $g$ represents the predicted multiple reflections, we have $y+Qy=m$. By denoting, $I+Q=A$ we can derive:

$$m=y+Qy=(I+Q)y=Ay,$$

(15)

In field data acquisition, noise is unavoidable. The above equation can be further expressed as:

$$m=Ay+n.$$

(16)

When using traditional inversion methods to solve for 2D filters, noise can complicate the propagation relationships between wavefields, leading to instability in the inversion process and a tendency to get trapped in local minima, thereby affecting the accuracy of the final results [49].

To overcome this challenge, we introduce sparse regularization into the multi-channel prediction deconvolution framework. In contrast to the limitations of traditional inversion methods, sparse domain optimization inversion is a novel inversion approach based on sparse representation theory and optimization algorithms [50].

Therefore, we present the use of the Shearlet transform to sparsely represent the 2D prediction filter. Equation (16) can be rewritten as:

$$m=S^{\mathrm{T}}\mathbf{k}\mathbf{y}+n,$$

(17)

where $S^{\mathrm{T}}$ represents the inverse Shearlet transform operator. In the sparse domain, a specific sparsity ratio is established to retain the main features while removing redundant information, thereby minimizing the impact of noise on the filter’s predictive performance. To determine the sparse regularization results for the 2D filter, we applied the FISTA algorithm [11] to solve the 2D filter, and formulated an objective function [51,52]. The objective function can be expressed by Equation (13):

$$f(\mathbf{k})={‖m-S^{\mathrm{T}}\mathbf{k}\mathbf{y}‖}_2^2+\lambda \mathrm{{\rm Z}}(\mathbf{k}),$$

(18)

where $\mathbf{k}$ represents the inversion result (prediction filter in the Shearlet domain), $\lambda \mathrm{{\rm Z}}(\mathbf{k})$ is the regularization term, and $\lambda$ is the parameter controlling the strength of the regularization. Incorporating regularization terms can further stabilize the solution process and prevent overfitting. By introducing additional constraints, the method effectively mitigates noise amplification issues during the deconvolution process, resulting in smoother seismic signals.

We use the ${\mathrm{L}}_1$ norm as the regularization term in the objective function to promote sparsity in the solution. This approach effectively suppresses noise and enhances the stability and accuracy of the inversion results. The objective function can then be further rewritten as:

$$f(\mathbf{k})={‖m-S^{\mathrm{T}}\mathbf{k}\mathbf{y}‖}_2^2+\lambda {‖\mathbf{k}‖}_1,$$

(19)

where ${‖\mathbf{k}‖}_1$ represents the ${\mathrm{L}}_1$ norm of the 2D prediction filter. Finally, the obtained filter is transformed back to the time-space domain using the Shearlet inverse transform, and then applied to the original data to obtain the final primaries results.

Figure 4 provides a concise overview of the sparse optimization inversion process in the Shearlet domain. In the following example, we demonstrate the effectiveness of multi-channel prediction deconvolution based on Shearlet sparse promotion.

3. Results

In this section, the proposed method is applied on both synthetic data and field data. For synthetic data, we chose synthetic data without noise for single-channel predictive deconvolution, and compared them with data from traditional multi-channel predictive deconvolution and our proposed method. The aim was to show the limitations of single-channel predictive deconvolution while showing that our method suppresses multiple reflections more effectively than traditional methods. Before experimenting with field data, we preprocessed it by removing direct waves and swell-correcting for reverberations. We then compared these three methods with synthetic data, highlighting that our approach achieved the better suppression of multiple reflections and the removal of redundant information, especially under conditions of noise interference.

3.1. Synthetic Data

The synthetic data are generated by ray-tracing with 2D acquisition geometry, which has 41 sources and receivers. The shot and receiver locations are identical. The shot and receiver are 10 m apart. The time sample number is 41 with a sample interval of 2.25 ms in every trace. The synthetic model consists of a horizontal surface, an undulating surface, and an inclined surface.

Figure 5a shows the primaries obtained using ray tracing, showing the subsurface layer positions in the synthetic data. Based on the primaries event in Figure 5a, we selected a 200 ms time prediction lag. This decision led to the generation of synthetic data, as depicted in Figure 5b, which highlights the overlap between primaries generated by the inclined interface and multiple reflections from other interfaces. Figure 5c shows the effect of single-channel prediction deconvolution on the synthetic data, with the red arrows indicating overfitting caused by the single-channel method’s disregard for the relationships between adjacent traces, and the orange arrows indicating distortions of the primaries due to overlapping with multiple reflections.

Figure 6a,b respectively demonstrate the effects of using traditional multi-channel prediction deconvolution and our proposed method on synthetic data. Figure 7a,b correspond to the enlarged views of the red boxes in Figure 6a,b. The multi-channel prediction deconvolution takes into account the relationships between seismic traces, allowing for better adaptation to lateral variations in subsurface structures. This approach circumvents the overfitting and distortion of the primaries caused by the overlap of primaries and multiple reflections observed in the single-channel prediction deconvolution. Moreover, in the region where multiple underground reflections overlap with each other, highlighted by the orange arrows in magnified view in Figure 7a, the traditional multi-channel prediction deconvolution leaves multiple residual reflections, whereas our proposed method in Figure 7b effectively resolves this issue.

Through the comparison of the three methods, it is evident that multi-channel prediction deconvolution, which considers the spatial coherence between adjacent traces, provides more accurate results than single-channel prediction deconvolution. Furthermore, our proposed method based on Shearlet sparse promotion for multi-channel prediction deconvolution offers a more precise solution when dealing with complex wavefield components. As further demonstrated by the synthetic data example, Figure 6b shows the result obtained using our proposed method, and Figure 8 presents the error curves between the results of the three approaches and the dataset containing only primaries, in which the error curves were produced using the same mean square error equation as in Equation (20). Light blue and dark blue, respectively, represent single-channel prediction deconvolution and traditional multi-channel prediction deconvolution, while red represents our proposed method. It is noteworthy that our method exhibits the smallest error, confirming the effectiveness of our method in handling data with multiple reflections.

$$MSE={\displaystyle \frac{1}{\mathrm{n}}}{{\displaystyle \sum _{i=1}^{\mathrm{n}}({\mathrm{Y}}_i-{\widehat{\mathrm{Y}}}_i)}}^2.$$

(20)

3.2. Field Data

The field data were obtained from a certain site in the Beibu Gulf of Guangxi. Specific seismic line locations are provided in the Supplementary Materials. Data collection was conducted using a survey vessel, and using single-channel seismic profiling with a trace spacing of 5 m, covering a total survey line length of 15 km. The seismic source used was an electric spark source with a triggering interval of 1000 ms and a recording duration of 200 ms with the time interval of 0.5 ms. The receiving cable consisted of AAE 20-element hydrophones, and data acquisition was performed using the CODA DA2000 system. During the survey, the vessel towed the seismic source and a series of receivers, continuously collecting seismic data along the direction of our survey lines. The data from this region exhibited the development of significant multiple reflections, as shown in the rectangular box in Figure 9, which affected subsequent geological interpretation and posed challenges for our multiple reflection suppression efforts.

In the actual geological survey, drilling was also conducted, with one of the drilling results shown in

Table 1. Drilling data from a certain site in the Beibu Gulf.

Number	Layer Bottom Elevation (m)	Layer Bottom Depth (m)
1	−23.76	3.8
2	−26.16	6.2
3	−33.06	13.1
4	−53.96	34.0
5	−70.06	50.1

reaching a depth of approximately 50 meters below the seabed. This indicates that our focus is on the shallow seabed geological conditions. Using the drilling data, we constructed a constant velocity model with a water velocity of 1400 m/s and a stratum velocity of 2000 m/s. The reference depth scale obtained by time-depth conversion is shown on the right side of Figure 9.

Figure 9 shows the original data of seismic line a4. Upon closer inspection in the zoomed-in portion of Figure 9, distinct multiple reflections are observed in the original collected data. A time prediction lag of 30 ms is selected based on the autocorrelation analysis of a4, shown in Figure 10. The processed results of single-channel prediction deconvolution, traditional multi-channel prediction deconvolution, and our proposed method applied to the original data are obtained. For a more visual comparison of the three methods used in estimating the primaries, Figure 11 shows a comparison of the zoomed-in section from 55 ms to 90 ms, located from 0.8 km to 1.3 km.

In Figure 11a, the original data showing the development of multiple reflections are presented. Figure 11b shows the result of single-channel prediction deconvolution, where residual multiple reflections are still visible at the location indicated by the yellow arrow, along with some data distortion indicated by the blue arrow. In contrast, Figure 11c demonstrates that using traditional multi-channel prediction deconvolution effectively reduces data distortion, as noise interference leaves some residual multiple reflections. Figure 11d shows the amplified results derived using our proposed method. As indicated by the arrows in the figure, our method demonstrates the superior suppression of multiple reflections and the removal of redundant information under noisy conditions compared to traditional methods. This provides a robust theoretical foundation for geological interpretation.

Figure 12 presents a comparison of single-channel results between the traditional multi-channel predictive deconvolution method and our proposed method (time from 0 ms to 160 ms and location from 0 m to 85 m), with arrows indicating the different processing results for single multiple reflections. It can be seen more intuitively that the performance of the traditional multi-channel predictive deconvolution method is limited by noise interference. In contrast, our proposed method still exhibits high performance under noise interference.

4. Discussion

The synthetic data were generated using ray-tracing with a 2D acquisition geometry, which includes 41 sources and receivers. The parameters for this setup are consistent with those used in the previously described synthetic data. To test the robustness of our method against noise, we added independently distributed Gaussian white noise to the synthetic data. The presence of noise complicates the propagation relationship between primaries and multiple reflections. We set the signal-to-noise ratio to 3.011 dB, at which level the noise disrupts the continuity and accuracy of the signal, thereby providing a better representation of real-world conditions. Figure 13a shows primaries data and Figure 13b shows synthetic data containing multiple reflections with noise.

Single-channel predictive deconvolution uses 1D predictive filters that consider only temporal variations in each trace, which limits its effectiveness in handling stratigraphic undulations and overlapping wavefields. In contrast, multi-channel predictive deconvolution employs 2D predictive filters, accounting for the relationships between seismic traces, making it better suited for adapting to lateral variations in subsurface structures without causing distortion in primaries. However, both methods are susceptible to noise. Our approach integrates sparse regularization into the multi-channel predictive deconvolution framework, allowing for the inversion of 2D predictive filters and ensuring that the final filter’s performance is not adversely affected by noise. Comparing the three methods discussed earlier, Figure 13c shows the result of single-channel prediction deconvolution, Figure 14a presents the outcome of traditional multi-channel prediction deconvolution, and Figure 14b showcases our proposed method’s final result. It is evident that single-channel and traditional multi-channel prediction deconvolution methods are susceptible to noise interference, resulting in poorer prediction outcomes. Conversely, our method maintains superior performance even under noise interference conditions. Additionally, we performed SNR tests for the three methods. The SNR value of the original data is 3.011 dB. The SNR values for the results obtained using single-channel prediction deconvolution, traditional multi-channel prediction deconvolution, and our proposed method were 17.1834 dB, 25.2530 dB, and 40.9028 dB, respectively.

Typically, in multi-channel prediction deconvolution, the parameters of the 2D data window $W$, filter length $F$, and sliding window step size $step$ are crucial factors, and it is worth conducting in-depth research to select the most suitable parameter combinations. We typically set the filter length based on the length of the seismic wavelet from the actual data, usually choosing a length equivalent to one wavelet to ensure effective capture of the signal’s local features. This choice enhances signal processing accuracy and improves the suppression of multiple reflections. The size of the data window is adjusted according to the continuity of the data: for data with good continuity, we use a larger data window to capture more signal information, whereas for data with poorer continuity, we select a smaller window to improve data continuity. The sliding window step size is generally set to half or slightly more than the data window length to balance processing accuracy and computational efficiency. We conducted comparative experiments to explore how three parameters affect prediction results. Considering the correlation between 2D data window lengths and sliding window step sizes from previous studies, we designed the first set of experiments. This set involved keeping the 2D data window and sliding window step size constant while varying the filter length to observe resulting changes. The second set involved keeping the filter length constant while varying the lengths of the 2D data window and sliding window step size. The specific parameter settings were as follows: Figure 15a with $W=40,step=25,F=40$; Figure 15b with $W=40,step=25,F=30$; Figure 15c with $W=30,step=20,F=30$. By comparing the yellow arrows in the figures, it can be observed that the parameter setting in Figure 15b was more effective, suppressing more irrelevant information.

Building upon the accurate suppression of multiple reflections achieved using our proposed method, we further demonstrate the ability to visualize the obtained profiles in 3D, enabling the spatial identification of submarine basement surfaces, as shown in Figure 16.

In practical exploration, multi-channel predictive deconvolution considers lateral spatial coherence to suppress multiple reflections. However, its performance may be compromised in areas with significant seabed variations, such as shallow waters where strong refraction and coherent noise are prevalent due to small critical angles. Despite these limitations, multi-channel predictive deconvolution can offer a more robust suppression of multiple reflections through algorithm optimization or the addition of appropriate regularization terms. Furthermore, due to the computational efficiency and storage advantages of our proposed method, it is capable of overcoming the challenges of large computations and high memory consumption in 3D data processing. These improvements are critical for minimizing data processing losses and enhancing the feasibility of deploying multi-channel predictive deconvolution in complex geological settings.

In practical applications, as our focus is on shallow information, we have primarily removed short-period multiple reflections, while long-period multiple reflections remain. These long-period multiple reflections, originating from deeper geological structures, present challenges due to their low-frequency and long-period characteristics, which complicate the accurate of prediction. Future research should focus on developing longer and more stable filters, improving the handling of low-frequency noise and signals, and incorporating deep learning to enhance predictive capabilities. These advancements will improve the accuracy and applicability of our method under complex geological conditions.

5. Conclusions

This study highlights the limitations of single-channel prediction deconvolution, which relies solely on 1D prediction filters and often results in the significant distortion of primary data, as demonstrated through both synthetic and field data analyses. In response, we developed a multi-channel prediction deconvolution technique enhanced by Shearlet-based sparse regularization. This innovative approach not only more effectively suppresses multiple reflections under noisy conditions, but also reduces redundant data significantly more than traditional methods. This technology holds great potential for advancing marine geological surveys, particularly in the exploration of renewable energy resources, offering a more cost-effective and accurate tool for the industry.