Surface-Related Multiple Suppression Based on Field-Parameter-Guided Semi-Supervised Learning for Marine Data

Abstract

Surface-related multiple suppression is a critical step in seismic data processing, while traditional adaptive matching subtraction methods often distort primaries, resulting in either the leakage of primaries or the residue of surface-related multiples. To address these challenges, we propose a field-parameter-guided semi-supervised learning (FPSSL) method to more effectively eliminate surface-related multiples. Field parameters refer to the time–space coordinate information derived from the seismic acquisition system, including offsets, trace spaces, and sampling intervals. These parameters reveal the relative positional relationships of seismic data in the time–space domain. The FPSSL framework comprises a supervised network module (SNM) and an unsupervised network module (USNM). The input and output data of the SNM are a small sample of full wavefield data and the weights of a polynomial function, respectively. A linear weighted sum method is employed to represent the SNM outputs (weights), the full wavefield data, and field parameters as a polynomial function of the primaries, which is matched with adaptive subtraction label data. The trained SNM generates preliminary estimates of the primaries and multiples with improved lateral continuity from full wavefield data, both of which are used as inputs to the USNM. The USNM is essentially an optimization operator that refines the underlying nonlinear mapping relationship between primaries and full wavefield data using the local wavefield feature loss function, thereby obtaining more accurate prediction results with respect to primaries. Examples from synthetic data and real marine data demonstrate that the FPSSL method surpasses the traditional L1-norm adaptive subtraction method in suppressing multiples, significantly reducing the leakage of primaries and the residuals of surface-related multiples in the estimated demultiple results. The effectiveness and efficiency of our proposed method are verified through two sets of synthetic data and one marine data example.

1. Introduction

In seismic data processing, effectively eliminating surface-related multiples is a fundamental requirement for obtaining accurate and reliable subsurface imaging [1]. Marine seismic data often contain abundant multiples, among which, surface-related multiples can degrade the signal-to-noise ratio, distort the reflection waveforms of target layers, and significantly interfere with the processing and interpretation of marine data. Surface-related multiples not only affect the accuracy and reliability of seismic imaging but also hinder the correct interpretation of seafloor structures [2]. The surface-related multiple elimination (SRME) method, which is based on wave equations, typically involves two steps: predicting surface-related multiples and performing adaptive matching subtraction. The choice of adaptive matching subtraction methods is crucial, as it directly influences the effectiveness of the SRME method in eliminating surface-related multiples [3]. The traditional approach to adaptive matching subtraction involves constructing a least-squares objective function in the time domain, thus transforming the matching problem into an optimization problem [4,5,6]. However, this least-squares adaptive subtraction method can distort primary reflections and leave residual multiples [7]. In theory, an ideal adaptive subtraction method is highly nonlinear and non-stationary, aligning closely with the principles of nonlinear theory in deep learning. Consequently, many researchers have integrated deep learning (DL) techniques into the adaptive matching subtraction process [8].

The supervised deep-learning-based adaptive matching subtraction (SDLAMS) method is conceptually straightforward; the input consists of the original full wavefield, and the output comprises the estimated primaries, which are matched to the primary label datasets. Although this method has proven to be effective, two key concerns arise. (1) The first is that the real challenge lies in the training datasets. Supervised deep neural networks are highly dependent on the quality of label datasets. However, it is theoretically impossible to provide true and sufficiently comprehensive label datasets for primaries in field or marine data [9]. As a result, the most advanced demultiple approaches are often used to obtain the best label datasets of primaries. Consequently, the performance of the DL-based adaptive subtraction method cannot surpass that of current demultiple techniques [10,11,12]. (2) The simple provision of SDLAMS with full wavefield data and the label datasets of primary waves lacks physical constraints, rendering it difficult to guide deep neural networks in accurately identifying physical relationships between primary waves and multiples in both the time and space domains [9,13,14]. To address these challenges, some researchers have introduced data augmentation methods during the training phase of SDLAMS. For instance, Wang [15] rotated the datasets by a fixed angle interval (h times starting from 0°) and appended the rotated datasets relative to the original data. This approach enhanced the training datasets of the supervised deep neural network and effectively prevented overfitting during the network training process. Even with a small sample size, improved training results can be achieved, thereby increasing the robustness and performance of the supervised deep neural network method. Additionally, Durall [16] created pairs of primary data and full wavefield data based on a large synthetic dataset (including common-offset gather data, angle gather data, and NMO correction data) for input into the SDNN for multiple removals. This method can suppress multiples in various seismic domains (i.e., offset or angle domain, time or depth domain) regardless of the domain or nature of the seismic gathers. The trained deep neural network can successfully suppress multiples while preserving the high-frequency components characteristic of the data, and it can carry out generalization relative to different datasets without the need for retraining. However, data augmentation inevitably increases training costs. In addition to the aforementioned data augmentation methods, the reliable recovery of reflection data can also be employed. One reason is that the success of surface-related multiple estimation (SRME)-related algorithms is sensitive to the quality of the near-offset reconstruction. When it comes to a larger missing gap and a shallower water bottom, the state-of-the-art near-offset gap construction method—the parabolic Radon transform—fails to realize the reliable recovery of shallow reflections due to limited information provided by the data and the highly curved events at the near offsets with strong lateral amplitude variations. One effective approach is to first deploy a DL-based reconstruction of shallow reflections and then use reconstructed data as inputs for subsequent surface-related multiple suppression methods [17]. Besides expanding the training datasets for deep neural networks, different seismic datasets can also be used as inputs. Liu [18] used the initial global estimates of surface-related multiples (predicted in the first step of the SRME method [3]) as inputs to deep learning, which helps reduce computational costs and maintain a balance between multiple attenuation and primary wave preservation. Additionally, good prediction results can be obtained by inputting both full wavefield data and the initial global estimates of surface-related multiples as dual-channel inputs to the network, matching them with primary label data [19].

The improved supervised deep learning adaptive matching subtraction (SDLAMS) method, whether based on data augmentation or multi-channel network inputs, has resulted in reliable primary wave prediction results. However, it still faces limitations in addressing problems (1) and (2). To overcome these challenges, many researchers have turned to semi-supervised and unsupervised methods. These methods have resulted in several novel approaches in seismic exploration, such as acoustic impedance inversion based on semi-supervised learning [20,21,22], 3D salt body interpretation [23], and seismic AVO (amplitude variation with offset) inversion [24]. For example, Wang [25] developed a self-supervised deep neural network method based on a local wavefield characteristic loss function (SDNN-LWCLF) for suppressing surface-related multiples. This approach does not require labeled data but necessitates retraining the SDNN for each new shot or 2D cord. Unsupervised learning can also be regarded as a nonlinear operator. Typically, surface-related multiples are first predicted using traditional methods. However, these predicted multiples often do not match the true surface-related multiples in terms of amplitude, phase, and arrival time. The difference between the predicted surface multiples and the original data is then constrained via unsupervised learning to obtain an optimal solution, thereby completing the suppression of surface-related multiples [26]. The ability to train complex deep neural network models with little or no labeled data using unsupervised and semi-supervised methods has greatly expanded the potential of these networks by exploring the distribution characteristics of unlabeled data and yielding more powerful representations [27].

In this study, we propose a semi-supervised deep learning method guided by field parameters (FPSSL) for attenuating surface-related multiples. The parameter utilized comprises time–space coordinate information derived from the seismic acquisition system, which reveals the relative positional relationships of seismic data in the time–space domain. This enables the deep neural network to explore the deeper nonlinear mapping relationships between primaries and multiples, which are guided by the provided field parameters. The FPSSL framework consists of two modules: a supervised network module (SNM) and an unsupervised network module (USNM). The input and output data of the SNM are, respectively, a small sample of full wavefield data and the weights of a polynomial function. A linear weighted sum method is employed to represent the SNM outputs (weights), full wavefield data, and field parameters as a polynomial function of the primaries, which is then matched to adaptive subtraction label data. The reconstructed polynomial function expression of the primaries acts as a constraint, leveraging the physical characteristics of the primaries and multiples in the time–space domain to guide the training process of the SNM. This approach results in primary and multiple estimates that, while not perfectly accurate, exhibit better lateral continuity. The USNM functions as an optimization operator, utilizing the local wavefield characteristic loss function of the primaries and multiples to refine the accuracy of the primary wave predictions. The FPSSL method proposed in this study does not rely on large amounts of labeled data and realizes superior prediction results compared to labeled primaries using only a small amount of labeled data. The synthetic data and real marine data studies demonstrate the effectiveness and robustness of the proposed method.

2. Methods

2.1. Surface-Related Multiple Elimination Method

The seismic wave generated by source S after undergoing the action of the underground impulse response X₀ returns to the surface and is detected by a geophone, obtaining the primary wave P₀, which can be described as follows:

$${\mathbf{P}}_0={\mathbf{X}}_0\mathbf{S}$$

(1)

Formula (1) represents the multi-dimensional convolution operation of source S and the underground impulse response M₀ in the time–space domain. In marine seismic data acquisition, due to the different wave impedances between seawater and air, the upgoing wavefield P will reflect downward on the surface of the seawater, producing a secondary source ${\mathbf{R}}^{-}\mathbf{P}$, which propagates underground again. This wavefield is received by the detector after being acted upon by X₀, producing free surface-related M₀ multiples.

$${\mathbf{M}}_0={\mathbf{X}}_0{\mathbf{R}}^{-}\mathbf{P}$$

(2)

Here, ${\mathbf{R}}^{-}$ represents the reflection coefficient matrix of the seawater’s surface, which is approximately ${\mathbf{R}}^{-}=-\mathrm{I}$. As shown in Figure 1, the downgoing wavefield includes source S and secondary source ${\mathbf{R}}^{-}\mathbf{P}$. If the upgoing wavefield P is regarded as the result of the downgoing wavefield passing through the underground impulse response, the observed seismic data can be expressed as follows:

$$\mathbf{P}={\mathbf{X}}_0\mathbf{S}+{\mathbf{X}}_0{\mathbf{R}}^{-}\mathbf{P}$$

(3)

Here, ${\mathbf{X}}_0\mathbf{S}$ and ${\mathbf{X}}_0{\mathbf{R}}^{-}\mathbf{P}$ represent the primary and free surface-related multiple, respectively. Formula (3) is expanded into an infinite series form:

$$\mathbf{P}={\mathbf{X}}_0\mathbf{S}+{\displaystyle \sum _{n=1}^{\infty }{\left({\mathbf{X}}_0{\mathbf{R}}^{-}\right)}^n}{\mathbf{X}}_0\mathbf{S}$$

(4)

${\left({\mathbf{X}}_0{\mathbf{R}}^{-}\right)}^n{\mathbf{X}}_0\mathbf{S}$ denotes n-order free surface-related multiples. Figure 2 describes the physical process of marine seismic data feedback iteration. Verschuur (2002) [2] introduced surface operator $\mathbf{A}={\mathbf{S}}^{-1}{\mathbf{R}}^{-}$ in order to rewrite Formula (4) into a data-driven form:

$${\mathbf{M}}_0={\mathbf{P}}_0\mathbf{AP}$$

(5)

Surface-related multiples can be predicted via the spatiotemporal convolution of primaries with full wavefield data. Surface operator $\mathbf{A}={\mathbf{S}}^{-1}{\mathbf{R}}^{-}$ contains information on the source wavelet and the surface reflection coefficient. If the directionality of the source is ignored (assuming a dipole source), the surface operator $\mathbf{A}={\mathbf{S}}^{-1}{\mathbf{R}}^{-}$ can be expressed as a diagonal matrix:

$$\mathbf{A}=A\left(\omega \right)\mathbf{I}$$

(6)

where $A\left(\omega \right)$ is a single-frequency scalar. Using the above-simplified surface operator, Formula (5) can be further expressed as follows:

$${\mathbf{M}}_0=A\left(\omega \right){\mathbf{P}}_0\mathbf{P}$$

(7)

Figure 3 depicts the physical prediction process of free surface-related multiples using the SRME method. The abovementioned free surface-related multiple prediction process requires the use of unknown primary and surface operators; thus, it cannot be directly applied to real seismic data. However, using theoretical prediction Formula (7) for free surface-related multiples, seismic data can be represented using a fully data-driven feedback model:

$$\mathbf{P}={\mathbf{P}}_0+A\left(\omega \right){\mathbf{P}}_0\mathbf{P}$$

(8)

Formula (8) reveals the implicit relationship between seismic data and the primary, and P₀ can also be expressed by P and surface operator A:

$${\mathbf{P}}_0=\mathbf{P}{\left[\mathbf{I}+A\left(\omega \right)\mathbf{P}\right]}^{-1}$$

(9)

The Taylor series expansion of Formula (9) can be written as

$${\mathbf{P}}_0=\mathbf{P}-A\left(\omega \right){\mathbf{P}}^2+A^2\left(\omega \right){\mathbf{P}}^3-A^3\left(\omega \right){\mathbf{P}}^4+\cdots$$

(10)

Formula (10) shows that the primary can be regarded as the result of the weighted superposition of a single frequency component of seismic data and its series of matrix products, and the corresponding weight coefficients are $1,-A\left(\omega \right),A^2\left(\omega \right),-A^3\left(\omega \right)$, etc. Under the assumption that the primary amplitude is at the minimum, surface operator $A\left(\omega \right)$ can be solved using a nonlinear optimization algorithm. In order to avoid solving the nonlinear problem of $A\left(\omega \right)$, Guitton and Verschuur (2004) [3] proposed an iterative SRME method starting from the implicit expression (8) of the feedback model. The iterative algorithm is shown in Formula (11):

$$\left\{\begin{array}{c}{\mathbf{P}}_{0,k+1}=\mathbf{P}-A_{i+1}\left(\omega \right){\mathbf{P}}_{0,j}\mathbf{P}\\ {\mathbf{P}}_{0,k=0}=\mathbf{P}\end{array}\right.$$

(11)

Here, the initial iteration value ${\mathbf{P}}_{0,j=0}$ of the primary can be estimated via other multiple suppression methods (such as filtering methods) or directly set to the original seismic data P; k represents the number of iterations. The predicted surface-related multiples can be expressed as follows:

$${\mathbf{M}}_{0,j+1}=A_{i+1}\left(\omega \right){\mathbf{P}}_{0,j}\mathbf{P}$$

(12)

The objective function for suppressing surface-related multiples can be obtained by minimizing the following L₁ norm:

$$e_{L2}=\mathit{arg}\underset{\mathbf{A}}{\mathit{min}}{‖\mathbf{P}-\mathbf{AM}‖}_1$$

(13)

2.2. Polynomial Function Representation of the Primary

Field parameters refer to the time–space coordinate information derived from the seismic acquisition system, including offset, trace space, and sampling interval values. This type of information reveals the relative positional relationships of seismic data in the time–space domain. According to the Nyquist sampling theorem, when the seismic acquisition system is designed, the sampling interval of the survey line grid (including the line spacing and point spacing of the geophone survey line and the line spacing and point spacing of the shot survey line) and the time interval of seismic records must be evenly and densely distributed in order to ensure that the collected seismic data are not distorted.

The common-shot gather $\mathbf{P}$ from Formula (3) is a three-dimensional data volume related to the time–space domain, in which the first dimension represents shots, the second dimension represents traces, and the third dimension represents time; here, $P\left(i,j,n\right)$ represents the seismic data value at the j-th shot point, i-th trace point, and n-th time point. The trace spacing interval I_trace, shot spacing interval J_shot, and time spacing interval dt are the field parameters derived from the seismic acquisition system. Therefore, the parameter matrix of the three dimensions of $P\left(i,j,n\right)$ is shown in Formula (14):

$$\left\{\begin{array}{c}{L}_p\left(i\right)=\left(i-1\right)\times I_{trace}\\ L_p\left(j\right)=\left(j-1\right)\times J_{shot}\\ L_p\left(n\right)=\left(n-1\right)\times dt\end{array}\right.$$

(14)

$L_p\left(i\right)$ represents the parameter value of the i-th trace point, which is obtained by multiplying the i − 1 by the trace sampling interval. $L_p\left(j\right)$ represents the parameter value of the j-th shot point, which is obtained by multiplying j − 1 by the shot sampling interval. $L_p\left(n\right)$ represents the parameter value of the n-th time point, which is obtained by multiplying n − 1 by the time sampling interval. For example, assuming that the trace spacing is 10 m and each trace has 4 sampling points, the trace–parameter matrix of the i = 1, 2, 3, 4, 5 traces of the j = 1 shot can be described using Formula (15):

$$\begin{array}{l}{\mathbf{L}}_P\left(i=1,2,3,4,5,6\right)=\left[\begin{array}{cccccc}\left(1-1\right)\times I_{trace}& \left(2-1\right)\times I_{trace}& \left(3-1\right)\times I_{trace}& \left(4-1\right)\times I_{trace}& \left(5-1\right)\times I_{trace}& \left(6-1\right)\times I_{trace}\\ \left(1-1\right)\times I_{trace1}& \left(2-1\right)\times I_{trace}& \left(3-1\right)\times I_{trace}& \left(4-1\right)\times I_{trace}& \left(5-1\right)\times I_{trace}& \left(6-1\right)\times I_{trace}\\ \left(1-1\right)\times I_{trace}& \left(2-1\right)\times I_{trace}& \left(3-1\right)\times I_{trace}& \left(4-1\right)\times I_{trace}& \left(5-1\right)\times I_{trace}& \left(6-1\right)\times I_{trace}\\ \left(1-1\right)\times I_{trace}& \left(2-1\right)\times I_{trace}& \left(3-1\right)\times I_{trace}& \left(4-1\right)\times I_{trace}& \left(5-1\right)\times I_{trace}& \left(6-1\right)\times I_{trace}\end{array}\right]\\ =\left[\begin{array}{cccccc}0& 10& 20& 30& 40& 50\\ 0& 10& 20& 30& 40& 50\\ 0& 10& 20& 30& 40& 50\\ 0& 10& 20& 30& 40& 50\end{array}\right]\end{array}$$

(15)

Moreover, the shot–parameter matrix of the j = 1 shot can be described using Formula (16):

$$\begin{array}{l}{\mathbf{L}}_P\left(j=1\right)=\left[\begin{array}{cccccc}\left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}\\ \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}\\ \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}\\ \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}& \left(1-1\right)\times J_{shot}\end{array}\right]\\ =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\end{array}$$

(16)

Assuming that the time sampling interval is 0.05 s, each trace has 4 sampling points, and a shot record contains 6 traces. In this case, the time–knowledge matrix for each shot is the same, which can be described using Formula (17):

$$\begin{array}{l}{\mathbf{L}}_P\left(n=1,2,3,4,5,6\right)=\left[\begin{array}{cccccc}\left(1-1\right)\times dt& \left(1-1\right)\times dt& \left(1-1\right)\times dt& \left(1-1\right)\times dt& \left(1-1\right)\times dt& \left(1-1\right)\times dt\\ \left(2-1\right)\times dt& \left(2-1\right)\times dt& \left(2-1\right)\times dt& \left(2-1\right)\times dt& \left(2-1\right)\times dt& \left(2-1\right)\times dt\\ \left(3-1\right)\times dt& \left(3-1\right)\times dt& \left(3-1\right)\times dt& \left(3-1\right)\times dt& \left(3-1\right)\times dt& \left(3-1\right)\times dt\\ \left(4-1\right)\times dt& \left(4-1\right)\times dt& \left(4-1\right)\times dt& \left(4-1\right)\times dt& \left(4-1\right)\times dt& \left(4-1\right)\times dt\end{array}\right]\\ =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0.05& 0.05& 0.05& 0.05& 0.05& 0.05\\ 0.1& 0.1& 0.1& 0.1& 0.1& 0.1\\ 0.15& 0.15& 0.15& 0.15& 0.15& 0.15\end{array}\right]\end{array}$$

(17)

We use ${\mathbf{L}}_p\left({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}}\right)$ to represent the field parameters matrix, including the time–parameter matrix ${\mathbf{L}}_p\left({\mathbf{t}}_{\mathbf{n}}\right)$, the trace–parameter matrix ${\mathbf{L}}_p\left({\mathbf{r}}_{\mathbf{i}}\right)$, and the shot–parameter matrix ${\mathbf{L}}_p\left({\mathbf{s}}_{\mathbf{j}}\right)$, where ${\mathbf{r}}_{\mathbf{i}}=0\times I_{trace},1\times I_{trace},2\times I_{trace},\cdots ,\left(i-1\right)\times I_{trace}$ ${\mathbf{s}}_{\mathbf{j}}=0\times J_{shot}1\times J_{shot},2\times J_{shot},\cdots ,\left(j-1\right)\times J_{shot}$ ${\mathbf{t}}_{\mathbf{n}}=0\times \mathit{dt},1\times \mathit{dt},2\times \mathit{dt},\cdots ,\left(n-1\right)\times \mathit{dt}$. In this study, the linear weighted sum method is employed to express the SNM outputs (weights), full wavefield data, and field parameters as a polynomial function of the primaries. This polynomial function is described using Formula (18):

$$\begin{array}{l}{\mathbf{P}}^{\prime }=\mathbf{P}\left({\mathbf{x}}_{ri},{\mathbf{x}}_{sj},{\mathbf{T}}_{tn}\right)+{\mathit{\lambda }}_{\mathsf{\Theta }}({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}})\ast {\mathbf{L}}_p\left({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}}\right)\\ =\mathbf{P}\left({\mathbf{x}}_{ri},{\mathbf{x}}_{sj},{\mathbf{T}}_{tn}\right)+{\mathit{\lambda }}_{\mathsf{\Theta }}({\mathbf{r}}_{\mathbf{i}i})\ast {\mathbf{L}}_p\left({\mathbf{r}}_{\mathbf{i}}\right)+{\mathit{\lambda }}_{\mathsf{\Theta }}({\mathbf{s}}_{\mathbf{j}})\ast {\mathbf{L}}_p\left({\mathbf{s}}_{\mathbf{j}}\right)+{\mathit{\lambda }}_{\mathsf{\Theta }}({\mathbf{t}}_{\mathbf{n}})\ast {\mathbf{L}}_p\left({\mathbf{t}}_{\mathbf{n}}\right)\end{array}$$

(18)

${\mathbf{P}}^{\prime }$ represents the reconstructed primary polynomial function. ${\mathit{\lambda }}_{\mathsf{\Theta }}({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}})$ represents the weight matrix. $\ast$ denotes the pointwise multiples. Moreover, ${\mathbf{M}}^{\prime }={\mathit{\lambda }}_{\mathsf{\Theta }}({\hat{\mathbf{r}}}_{\mathbf{i}},{\hat{\mathbf{s}}}_{\mathbf{j}},{\hat{\mathbf{t}}}_{\mathbf{n}})\ast {\mathbf{L}}_p\left({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}}\right)$ can be regarded as a polynomial function of the surface-related multiple.

2.3. Field-Parameter-Guided Semi-Supervised Learning

Semi-supervised learning refers to the use of labeled sample data and unlabeled sample data to jointly train neural networks. In the field of medical segmentation, labeled sample data usually account for 10–50% of the total sample data [28], while in some tasks such as image classification and clustering, it accounts for 1–20% [29]. The amount of labeled sample data in this study is 10% (see Section 3 for details). First, we introduce the process of predicting primaries using the supervised deep-learning-based adaptive matching subtraction method (SDLAMS). ${\mathit{Net}}_{SDLAMS}$ represents the DNN model in the SDLAMS, and the input data for the DNN model are represented by ${\mathbf{x}}_1$. During the training process of the SDLAMS, when the DNN parameter $\mathsf{\Theta }$ is learned, the nonlinear mapping of $\mathbf{N}e{t}_{SDLAMS}$ can be expressed as follows:

$${\mathbf{y}}_1={\mathit{Net}}_{SDLAMS}\left({\mathbf{x}}_1;\mathsf{\Theta }\right)$$

(19)

During the testing phase, assuming that, under the ideal case of the optimal DNN parameter $\mathsf{\Theta }$, we input all full wavefield data ${\mathbf{R}}_{in}$ into the DNN, according to Formula (19), the primary predicted by DNN can be expressed as follows:

$${\mathbf{P}}_{out}={\mathit{Net}}_{SDLAMS}\left({\mathbf{R}}_{in};\mathsf{\Theta }\right)$$

(20)

The FPSSL structure sketch designed in this research study is shown in Figure 4, which includes two modules: a supervised network module (SNM) and an unsupervised network module (USNM). The SNM module consists of one input, a 3D U-net, and two outputs. The USNM module consists of two inputs, a 3D U-net, and an output. The core nonlinear optimization of FPSSL comprises U-net [30,31] with a convolutional neural network structure. Its unique bisymmetric paths and cross-layer skip connections help the model build detailed information during the prediction process. The U-net structure sketch is shown in Figure 5. The contracting path on the left (encoder) consists of 4 parts, and each part includes 2 convolution blocks and max-pooling. Each convolution block includes a convolution layer, a BN layer, and a LeakyReLU activation function. The right expansive path (decoder) is also composed of 3 parts in addition to 2 convolution blocks. Each part includes an up-conv, a cross-layer skip connection, and 2 convolution blocks. Each convolution block includes a convolution layer, a BN layer, and a LeakyReLU activation function, and all convolution kernel sizes are 3 × 3 × 3.

The FPSSL method proposed in this study trains SNM first. The trained SNM generates preliminary estimates of the primaries and multiples with improved lateral continuity from the full wavefield data, both of which are used as inputs to the USNM. Then, the USNM is trained to finally obtain more accurate results with respect to primaries. Before training and testing SNM, full wavefield data $\stackrel{⌢}{\mathbf{P}}$ (10% of $\mathbf{P}$) and labeled primary data ${\stackrel{⌢}{\mathbf{P}}}^{\prime }$ need to be normalized and segmented. In this study, amplitude normalization operator k is used to normalize the amplitudes of full wavefield data $\stackrel{⌢}{\mathbf{P}}$ and labeled primary data ${\stackrel{⌢}{\mathbf{P}}}^{\prime }$ between −1 and 1:

$$\left\{\begin{array}{c}\tilde{\mathbf{P}}=k\stackrel{⌢}{\mathbf{P}}={\displaystyle \frac{\stackrel{⌢}{\mathbf{P}}}{e_{\max }}}\\ {\tilde{\mathbf{P}}}^{\prime }=k{\stackrel{⌢}{\mathbf{P}}}^{\prime }={\displaystyle \frac{{\stackrel{⌢}{\mathbf{P}}}^{\prime }}{e_{\max }}}\end{array}\right.$$

(21)

Here, $\tilde{\mathbf{P}}$ represents the full wavefield data after amplitude normalization. ${\tilde{\mathbf{P}}}^{\prime }$ represents the labeled primary data after amplitude normalization, and it represents the value with the largest absolute value of elements in data $\tilde{\mathbf{P}}$ and ${\tilde{\mathbf{P}}}^{\prime }$. Then, we use division operator $S_a$ to segment seismic data $\tilde{\mathbf{P}}$ and ${\tilde{\mathbf{P}}}^{\prime }$ with the original size into data patches with sizes of $a\times a\times a$; we then obtain the input data ${\mathbf{x}}^{\prime }$ of SDNN and labeled data ${\mathbf{y}}^{\prime }$.

$$\left\{\begin{array}{c}{\mathbf{x}}^{\prime }={\mathbf{S}}_{a}k\tilde{\mathbf{P}}\\ {\mathbf{y}}^{\prime }={\mathbf{S}}_{a}k{\tilde{\mathbf{P}}}^{\prime }\end{array}\right.$$

(22)

During the SNM training phase, we input pre-processed (data normalization and segmentation) data ${\mathbf{x}}^{\prime }$ into the SNM, and we obtain the optimal SNM parameter $\hat{\mathsf{\Theta }}$ and the output ${{\mathbf{P}}_0}^{″}$ of the SNM by minimizing the loss function (see Section 2.3 for details). In the SNM testing stage, we input pre-processed (data normalization and segmentation) full wavefield data $\mathbf{P}$ into the SNM, and we obtain the output patches of the SNM through the optimal SNM parameters $\hat{\mathsf{\Theta }}$ and Formula (20); then, these are inputted into reconstructed primary polynomial Formula (18) to obtain the predicted primary ${\mathbf{P}}^{″}$. We use the inverse of division operator $S_a^{-1}$ to splice ${\mathbf{P}}^{″}$ to its original size, and we restore it to its original amplitude through the inverse of amplitude normalization operator $k^{-1}$, thereby obtaining the primary ${\mathbf{P}}^{‴}$ with the correct amplitude and size:

$$\begin{array}{l}\mathbf{x}=S_{a}k\mathbf{P};{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}})={\mathit{Net}}_{SNM}\left(\mathbf{x};\hat{\mathsf{\Theta }}\right)\\ {\mathbf{P}}^{″}=\mathbf{P}\left({\mathbf{x}}_{ri},{\mathbf{x}}_{sj},{\mathbf{T}}_{tn}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}})\ast {\mathbf{L}}_p\left({\hat{\mathbf{r}}}_{\mathbf{i}},{\hat{\mathbf{s}}}_{\mathbf{j}},{\hat{\mathbf{t}}}_{\mathbf{n}}\right)\\ =\mathbf{P}\left({\mathbf{x}}_{ri},{\mathbf{x}}_{sj},{\mathbf{T}}_{tn}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}({\mathbf{r}}_{\mathbf{i}i})\ast {\mathbf{L}}_p\left({\hat{\mathbf{r}}}_{\mathbf{i}}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}({\mathbf{s}}_{\mathbf{j}})\ast {\mathbf{L}}_p\left({\hat{\mathbf{s}}}_{\mathbf{j}}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}({\mathbf{t}}_{\mathbf{n}})\ast {\mathbf{L}}_p\left({\hat{\mathbf{t}}}_{\mathbf{n}}\right)\\ {\mathbf{P}}^{‴}=k^{-1}{S}_a^{-1}{\mathbf{P}}^{″}\end{array}$$

(23)

Here, ${\mathbf{M}}^{″}={\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}({\mathbf{r}}_{\mathbf{i}},{\mathbf{s}}_{\mathbf{j}},{\mathbf{t}}_{\mathbf{n}})\ast {\mathbf{L}}_p\left({\hat{\mathbf{r}}}_{\mathbf{i}},{\hat{\mathbf{s}}}_{\mathbf{j}},{\hat{\mathbf{t}}}_{\mathbf{n}}\right)$ can be regarded as a polynomial function of the surface-related multiple. The process of inputting ${\mathbf{M}}^{″}$ and ${\mathbf{P}}^{″}$ into USNM to obtain the predicted primary is represented using Formula (24):

$${\mathbf{P}}_0=k^{-1}{S}_a^{-1}{\mathit{Net}}_{USNM}\left({\mathbf{P}}^{″};{\mathrm{M}}^{″};\stackrel{⌢}{\mathsf{\Theta }}\right)$$

(24)

2.4. Method for Evaluating the Model

2.4.1. Loss Function for the FPSSL

SNM uses the mean square error loss function (MSE) to constrain the nonlinear optimization training of U-net. This process can be described via Formula (25):

$$\begin{array}{l}{L}_{MSE}(\hat{\mathsf{\Theta }})={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^m{({\mathbf{P}}_0^{″}-{\mathbf{y}}^{\prime })}^2}{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}^{\prime }({\hat{\mathbf{r}}}_{\mathbf{i}},{\hat{\mathbf{s}}}_{\mathbf{j}},{\hat{\mathbf{t}}}_{\mathbf{n}})={\mathit{Net}}_{SNM}\left({\mathbf{x}}^{\prime };\hat{\mathsf{\Theta }}\right)\\ {\mathbf{P}}_0^{\prime }=\stackrel{⌢}{\mathbf{P}}\left({\mathbf{x}}_{ri},{\mathbf{x}}_{sj},{\mathbf{T}}_{tn}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}^{\prime }({\hat{\mathbf{r}}}_{\mathbf{i}},{\hat{\mathbf{s}}}_{\mathbf{j}},{\hat{\mathbf{t}}}_{\mathbf{n}})\ast {{\mathbf{L}}^{\prime }}_p\left({\hat{\mathbf{r}}}_{\mathbf{i}},{\hat{\mathbf{s}}}_{\mathbf{j}},{\hat{\mathbf{t}}}_{\mathbf{n}}\right)\\ =\stackrel{⌢}{\mathbf{P}}\left({\mathbf{x}}_{ri},{\mathbf{x}}_{sj},{\mathbf{T}}_{tn}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}^{\prime }({\hat{\mathbf{r}}}_{\mathbf{i}})\ast {{\mathbf{L}}^{\prime }}_p\left({\hat{\mathbf{r}}}_{\mathbf{i}}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}^{\prime }({\hat{\mathbf{s}}}_{\mathbf{j}})\ast {{\mathbf{L}}^{\prime }}_p\left({\hat{\mathbf{s}}}_{\mathbf{j}}\right)+{\mathit{\lambda }}_{\hat{\mathsf{\Theta }}}^{\prime }({\hat{\mathbf{t}}}_{\mathbf{n}})\ast {{\mathbf{L}}^{\prime }}_p\left({\hat{\mathbf{t}}}_{\mathbf{n}}\right)\end{array}$$

(25)

The symbol m represents the number of data ${{\mathbf{P}}_0}^{″}$ or ${\mathbf{y}}^{\prime }$. If we divide the original size of the data into v data blocks of size $a\times a\times a$ and then calculate the average of the MAE loss function of the v data blocks, we finally obtain the loss function of SNM:

$$L_{SNM}(\hat{\mathsf{\Theta }})={\displaystyle \frac{1}{v}}{\displaystyle \sum _{ii=1}^v\left[\frac{1}{m}{\displaystyle \sum _{i=1}^m{({{\mathbf{P}}_0}^{\prime }-{\mathbf{y}}^{\prime })}^2}\right]}$$

(26)

USNM uses the local wavefield characteristic loss function to constrain the nonlinear optimization training of U-net. The USNM is essentially an optimization operator that can use the local wavefield characteristic loss function of the primaries and multiples to obtain more accurate prediction results with respect to primaries. The local wavefield characteristic loss function monitors whether the USNM is over-fitted or under-fitted by calculating the local similarity and orthogonality between data ${\mathbf{P}}_0$ and $\mathbf{M}$ [32]; this method avoids the leakage of primaries or the residual of surface-related multiples. The local wavefield characteristic loss function $L_{LWC}(\mathsf{\Theta })$ is defined as follows:

$$\begin{array}{l}{L}_{LWC}(\stackrel{⌢}{\mathsf{\Theta }})={\left({\displaystyle \frac{Cov\left({\mathbf{P}}_0,\mathbf{M}\right)}{\sqrt{Var({\mathbf{P}}_0)Var(\mathbf{M})}}}\right)}^2,\mathbf{M}=\mathbf{P}-{\mathbf{P}}_0;\\ ={\left[{\displaystyle \frac{{\displaystyle \sum _{i=1}^{m^{\prime }}\left({\mathbf{P}}_0-{\overline{\mathbf{P}}}_0\right){\displaystyle \sum _{i=1}^{m^{\prime }}\left(\mathbf{M}-\overline{\mathbf{M}}\right)}}}{\sqrt{{\displaystyle \sum _{i=1}^{m^{\prime }}{\left({\mathbf{P}}_0-{\overline{\mathbf{P}}}_0\right)}^2{\displaystyle \sum _{i=1}^{m^{\prime }}{\left(\mathbf{M}-\overline{\mathbf{M}}\right)}^2}}}}}\right]}^2\end{array}$$

(27)

The symbol $Cov$ represents the covariance operator, $Var$ represents the variance operator, and ${\overline{\mathbf{P}}}_0$ and $\overline{\mathbf{M}}$ are the mean values of data ${\mathbf{P}}_0$ and $\mathbf{M}$, respectively. The symbol $m^{\prime }$ represents the number of data ${\mathbf{P}}_0$ or $\mathbf{M}$. If we divide the original size data into u data blocks of size a × a × a and calculate the average of the loss function $L_{LWC}(\mathsf{\Theta })$ of the u data blocks, we finally obtain the loss function $L_{LWC}(\stackrel{⌢}{\mathsf{\Theta }})$ of USNM:

$$\begin{array}{l}{L}_{USNM}(\stackrel{⌢}{\mathsf{\Theta }})={\displaystyle \sum _{ii=1}^u{\left(\frac{Cov\left({\mathbf{P}}_0,\mathbf{M}\right)}{\sqrt{Var({\mathbf{P}}_0)Var(\mathbf{M})}}\right)}^2}+{\displaystyle \frac{1}{u}}{\displaystyle \sum _{ii=1}^u\left[\frac{1}{m}{\displaystyle \sum _{i=1}^m{(\mathbf{P}-\mathbf{M})}^2}\right]}\\ ={\displaystyle \sum _{ii=1}^u{\left[\frac{{\displaystyle \sum _{i=1}^{m^{\prime }}\left({\mathbf{P}}_0-{\overline{\mathbf{P}}}_0\right){\displaystyle \sum _{i=1}^{m^{\prime }}\left(\mathbf{M}-\overline{\mathbf{M}}\right)}}}{\sqrt{{\displaystyle \sum _{i=1}^{m^{\prime }}{\left({\mathbf{P}}_0-{\overline{\mathbf{P}}}_0\right)}^2{\displaystyle \sum _{i=1}^{m^{\prime }}{\left(\mathbf{M}-\overline{\mathbf{M}}\right)}^2}}}}\right]}^2}+{\displaystyle \frac{1}{u}}{\displaystyle \sum _{ii=1}^u\left[\frac{1}{m}{\displaystyle \sum _{i=1}^m{(\mathbf{P}-\mathbf{M})}^2}\right]}\end{array}$$

(28)

Combining Formulas (26) and (28), the loss function of FPSSL can be comprehensively expressed via Formula (29):

$$\begin{array}{l}{L}_{PKSSL}(\mathsf{\Theta })=L_{SNM}(\hat{\mathsf{\Theta }})+{\lambda }_1{L}_{USNM}(\stackrel{⌢}{\mathsf{\Theta }})\\ ={\displaystyle \frac{1}{v}}{\displaystyle \sum _{ii=1}^v\left[\frac{1}{m}{\displaystyle \sum _{i=1}^m{({{\mathbf{P}}_0}^{\prime }-{\mathbf{y}}^{\prime })}^2}\right]}+{\lambda }_1{\displaystyle \sum _{ii=1}^u{\left(\frac{Cov\left({\mathbf{P}}_0,\mathbf{M}\right)}{\sqrt{Var({\mathbf{P}}_0)Var(\mathbf{M})}}\right)}^2}\\ +{\displaystyle \frac{1}{u}}{\displaystyle \sum _{ii=1}^u\left[\frac{1}{m}{\displaystyle \sum _{i=1}^m{(\mathbf{P}-\mathbf{M})}^2}\right]}+{\lambda }_2{‖\stackrel{⌢}{\mathsf{\Theta }}‖}_1+{\lambda }_3{‖\hat{\mathsf{\Theta }}‖}_1\\ ={\displaystyle \frac{1}{v}}{\displaystyle \sum _{ii=1}^v\left[\frac{1}{m}{\displaystyle \sum _{i=1}^m{({{\mathbf{P}}_0}^{\prime }-{\mathbf{y}}^{\prime })}^2}\right]}+{\lambda }_1{\displaystyle \sum _{ii=1}^u{\left[\frac{{\displaystyle \sum _{i=1}^{m^{\prime }}\left({\mathbf{P}}_0-{\overline{\mathbf{P}}}_0\right){\displaystyle \sum _{i=1}^{m^{\prime }}\left(\mathbf{M}-\overline{\mathbf{M}}\right)}}}{\sqrt{{\displaystyle \sum _{i=1}^{m^{\prime }}{\left({\mathbf{P}}_0-{\overline{\mathbf{P}}}_0\right)}^2{\displaystyle \sum _{i=1}^{m^{\prime }}{\left(\mathbf{M}-\overline{\mathbf{M}}\right)}^2}}}}\right]}^2}\\ +{\displaystyle \frac{1}{u}}{\displaystyle \sum _{ii=1}^u\left[\frac{1}{m}{\displaystyle \sum _{i=1}^m{(\mathbf{P}-\mathbf{M})}^2}\right]}+{\lambda }_2{‖\stackrel{⌢}{\mathsf{\Theta }}‖}_1+{\lambda }_3{‖\hat{\mathsf{\Theta }}‖}_1\end{array}$$

(29)

where ${\lambda }_1$ represents the regularization operator, determined by the ratio of the energy difference between outputs ${\mathbf{P}}_0$ and $\mathbf{M}$ of USNM [33,34]. ${\lambda }_2$ and ${\lambda }_3$ represent the sparsity operator, which can provide sparsity for the USNM parameter $\hat{\mathsf{\Theta }}$ and $\stackrel{⌢}{\mathsf{\Theta }}$. The value of ${\lambda }_2$ and ${\lambda }_3$ in this paper is 1 × 10⁻⁷, which other scholars have also verified as the optimal value [35,36].

2.4.2. Primary Reconstruction Percentage

In order to quantitatively measure the improvement effect of the proposed method FPSSL on the problem of the leakage of primaries, we introduce the primary reconstruction percentage (PRP) [37] for calculation. The expression of PRP is as follows:

$$\mathit{PRP}=1-{\displaystyle \frac{\left|{\mathbf{P}}^{\mathit{pre}}-{\mathbf{P}}^{true}\right|}{\left|{\mathbf{P}}^{\mathit{true}}\right|}}\times 100\%$$

(30)

Here, ${\mathbf{P}}^{true}$ represents the real primary, and ${\mathbf{P}}^{\mathit{pre}}$ represents the primary result predicted using other methods. The closer the PRP result is to the real primary, the higher the PRP, which has an upper limit of 100%. Therefore, the effectiveness of the method FPSSL proposed in this study can be well judged according to Formula (30).

3. Results

3.1. Pluto Data Result

The Pluto dataset is a 2D elastic dataset designed to emulate the deep water subsalt prospects found in the Gulf of Mexico. It contains realistic free surface and internal multiples over a structure that is relatively easy to image. The Pluto dataset is very large. For efficiency reasons, we only selected a part of the Pluto dataset for testing our method. The selected part of the Pluto dataset comprises a total of 400 sources (shots), with 390 receivers (traces) per shot. The spacing between adjacent shots is 22.86 m, as are the adjacent traces. Each trace contains 1120 time samples with an interval of 0.008 s. In total, we selected 10% (40 shots) of the Pluto dataset and used it as the training-labeled dataset for training SNM, and the other 90% was used as a testing dataset to test the suppression results of surface-related multiples. We selected different percentages of the labeled dataset from the Pluto dataset to test the effectiveness of our proposed method in suppressing multiples. Generally speaking, when there were more labeled data, the U-net prediction results were closer to the true result. However, in real field or marine data, we often cannot obtain sufficiently high-quality labeled datasets. Therefore, different percentages of labeled data are selected in order to mimic the real field or marine data situation and to test the effect of our proposed field parameter guiding U-net in suppressing multiples under a small amount of label data. We selected 5% (20 shots), 10% (40 shots), 20% (80 shots), and 30% (120 shots) of the Pluto dataset as training-labeled datasets for training SNM, and the remaining 95% (380 shots), 90% (360 shots), 80% (320 shots), and 70% (380 shots) were selected as test datasets to testing SNM.

Figure 6 shows the common-offset profile. The red arrows in Figure 6a denote the complex surface-related multiples, which are a result of the multisalt dome (green arrows) structures of the Pluto model. Figure 6b and Figure 6c show real surface-related multiples and primaries, respectively. It can be observed that surface-related multiples mainly appear from 4.5 s to 8.96 s. A flowchart for SNM for suppressing surface-related multiples and obtaining primary and surface-related multiples is shown in Figure 4.

According to the flowchart shown in Figure 7, the SNM is trained using Formulas (18), (21), (22), (23), and (25). There are 500 epochs in the training process. Figure 8 shows the loss error graph of the field-parameter-guided U-net (PKU-net) and traditional U-net methods during the training phase, with labeled data accounting for 10%.

Figure 9, Figure 10, Figure 11 and Figure 12 show the results of FPU-net and traditional U-net methods with labeled data, accounting for 30%, 20%, 10%, and 5%, respectively. Figure 9a, Figure 10a, Figure 11a and Figure 12a show the primary results estimated using the FPU-net method. Figure 9b, Figure 10b, Figure 11b and Figure 12b show the surface-related results of the multiples obtained by subtracting Figure 9a, Figure 10a, Figure 11a and Figure 12a from Figure 6a, respectively. Figure 9c, Figure 10c, Figure 11c and Figure 12c show the primary results estimated using the traditional U-net method. Figure 9d, Figure 10d, Figure 11d and Figure 12d show the surface-related results of multiples obtained by subtracting Figure 9c, Figure 10c, Figure 11c and Figure 12c from Figure 6a, respectively. In Figure 9c, Figure 10c, Figure 11c and Figure 12c, the red arrow shows some obvious residuals of the multiples, while the FPU-net results in Figure 9a, Figure 10a, Figure 11a and Figure 12a exhibit no obvious multiple residues. In Figure 9d, Figure 10d, Figure 11d and Figure 12d, the blue arrow shows some leakages of the primaries, while the FPU-net results in Figure 9b, Figure 10b, Figure 11b and Figure 12b have no obvious primary leakages.

According to Formula (30), the higher the PRP value, the closer the estimated primary is to the real primary, and the higher the accuracy of the estimated primary. In order to further verify the accuracy of the FPU-net method in estimating the primary, we calculated the PRP values of 400 shots in Figure 9a, Figure 10a, Figure 11a and Figure 12a and Figure 9c, Figure 10c, Figure 11c and Figure 12c, which are summarized in

Table 1. PRP values of FPU-net and the traditional U-net methods relative to the different percentages of the labeled datasets.

	Different Percentage	30%	20%	10%	5%
PRP Value
FPU-net		95.2%	91.8%	84.3%	78.04%
Traditional U-net		80.8%	77.2%	68.1%	63.3%
Difference in PRP value in FPU-net and TU-net		14.4%	14.6%	16.2%	14.73%
Average PRP difference value		15%

. Figure 13 is an example of Table 1 (labeled data account for 10% of the total data). The PRP values of FPU-net for all 400 shots data are higher than those of the traditional U-net, with an average PRP value of 84.3%. Table 1 shows the PRP values of FPU-net and the traditional U-net methods relative to the different percentages of the labeled datasets. The results show that the PRP values of FPU-net are higher than those of traditional U-net methods. It was proven that the accuracy of primary estimations guided by the field parameter is higher than that of traditional U-net, and the accuracy is improved by at least 15%. We used complex synthetic Pluto data to demonstrate the different results of field-parameter-guided U-net and traditional U-net (no field-parameter-guided U-net) for the suppression of surface-related multiples. The results show that the U-net method based on field parameters exhibits higher accuracies in suppressing multiples. This demonstrates that field parameters can guide U-net in more accurately identifying multiples and suppressing them from the original data while effectively reducing the leakage of primaries. Next, we will introduce field parameters into a semi-supervised U-net model.

3.2. Sigsbee Data Result

These synthetic Sigsbee2b data model the geologic settings found on the Sigsbee2b escarpment in the Gulf of Mexico. This model contains a complex salt structure and a sedimentary sequence broken up by a number of normal and thrust faults. For efficiency reasons, we only selected a portion of the Sigsbee2b dataset to test our proposed FPSSL method. The selected part of the Sigsbee2b dataset has a total of 200 sources (shots) with 240 receivers (traces) per shot. The spacing between adjacent shots is 143 m, and the spacing between adjacent traces is 76.2 m. Each trace contains 900 time points with an interval of 0.008 s. We selected 10% (20 shots) of Sigsbee2b data as a labeled dataset to train the SNM, and we selected the remaining 90% (180 shots) to train the USNM. The estimated primaries of the SRME are described as labeled data results.

Figure 14 shows the common-offset profile of the Sigsbee2b dataset. Additionally, there is a complex salt structure in the model, as shown by the green arrows in the profile; this resulted in surface-related multiples, as shown by the red arrows in the profile. As shown in Figure 14, surface-related multiples mainly exist between 4 s and 9.6 s. We used three methods to suppress surface-related multiples in Sigsbee2b data. The SRME method is a multi-dimensional inversion algorithm that eliminates surface-related multiples based on the feedback model shown in Figure 2 (see Section 2.1 for details). A flowchart of the FPSSL method is shown in Figure 15. A flowchart for the SSL method (without field parameter) is shown in Figure 16.

According to the flowchart shown in Figure 15, the U-net network is trained using Formulas (18), (21), (22), (23) and (25). We trained 300 epochs for SNM and 150 epochs for USNM. Figure 17a,b show the loss error graph of the SNM and the USNM during the training phase. According to the flowchart shown in Figure 16, the U-net network is trained using Formulas (18), (21), and (25). We trained 300 epochs for SNM and 150 epochs for USNM. Figure 18a,b show the loss error graph of the SNM and the USNM during the training phase. Figure 19 and Figure 20 comprise three-dimensional displays of the Sigsbee2b data. The common-offset profile is represented by the time and shot number, and the common-shot gather profile is represented relative to the time and trace number. Figure 19a shows the full wavefield data, and Figure 19b is the primary result of L1-norm adaptive subtraction, which comprises labeled data during the SNM training process. Figure 19c shows the primary result of the FPSSL method proposed in this study, and Figure 19d shows the primary result of the SSL method. The L1-norm method cannot completely suppress the surface-related multiples shown using red arrows in Figure 19b. It is clear from Figure 19c that the surface-related multiples are almost completely suppressed. In Figure 19d, the purple arrow shows the damaged primaries, which is also pointed out in Figure 20c. Figure 20a–c show the results of surface-related multiples obtained by subtracting Figure 19b–d from Figure 19a. The purple arrows in Figure 20 show the leakage of primaries. And the events of the surface-related multiple are blurred, as shown by the orange arrows in the Figure 20c. In contrast, there is almost no primary leakage in Figure 20c, and the events of the multiple’s profile are clearer.

We used complex Sigsbee2b data to test the impact of the three methods on the results of surface-dependent multiple suppression. The results show that the FPSSL method proposed in this study is superior to SRME-L1 and SSL methods in suppressing surface-related multiples, which shows that the FPSSL method can effectively suppress multiples, significantly reduce the residue of multiples, and effectively protect primaries.

3.3. Real Marine Data Result

We used real marine data to test the effectiveness of the FPSSL method in this study with respect to improving residual primary wave and residual multiple wave imaging problems. This dataset comprises single-sided ocean data, from which 430 sources (shots) are selected; there are 120 receivers (trace) per shot, and the spacing between adjacent shots is 22.86 m, as are the adjacent traces. Each trace contains 1300 time samples with an interval of 0.004 s. Figure 21 is the common-offset profile of 2D marine data after automatic gain control, where the yellow arrow indicates the seafloor’s interface, the green arrow indicates the top of the salt dome, and the blue arrow indicates the two reflection interfaces. The surface-related multiples are mainly related to these four interfaces and are relatively complex due to the rugged and undulating seabed structure. The propagation speed of seismic waves in salt domes is faster than that of the surrounding bedrock, which results in the generation of strong-amplitude, surface-related multiple waves when seismic waves propagate between salt domes and seafloor reflection interfaces. After seismic wavelets propagate to the top of the salt, they are reflected. The upgoing wavefield generated by this reflection will propagate to the seabed and be reflected. Then, the downgoing wavefield generated will propagate to the top of the salt again and be reflected again. Finally, the generated upgoing wavefield will eventually be received by the detector, forming the surface-related multiple, as shown by red arrow 1. Similarly, red arrows 2 and 3 represent surface-related multiples generated by two reflection interfaces (as shown by the blue arrows), and the target reflection layer is the top of the salt dome. Red arrow 4 represents first-order water-bottom multiple waves. Red arrows 5 and 6 represent surface-related multiple waves related to the salt dome and the two reflection interfaces. Red arrow 7 represents the salt multiple. Starting from the red arrow, more surface-related multiple waves will appear in the seismic data recorded later; thus, our main multiples target area is located in the red rectangular box.

We used three methods to suppress surface-related multiples in real marine data. The SRME method is a multi-dimensional inversion algorithm that eliminates surface-related multiples based on the feedback model shown in Figure 2 (see Section 2.1 for details). A flowchart of the FPSSL method is shown in Figure 22. A flowchart for the SSL method (without field parameter) is shown in Figure 23.

According to the flowchart shown in Figure 22, the U-net network is trained using Formulas (18), (21), (22), (23), and (25). We trained 300 epochs for SNM and 150 epochs for USNM. Figure 24a,b show the loss error graph of the SNM and the USNM during the training phase. According to the flowchart shown in Figure 23, the U-net network is trained using Formulas (18), (21), and (25). We trained 300 epochs for SNM and 150 epochs for USNM. Figure 25a,b show the loss error graph of the SNM and the USNM during the training phase. Figure 26 and Figure 27 are three-dimensional displays of Sigsbee2b data. The common-offset profile is represented by the time and shot number, and the common-shot gather profile is represented by the time and trace number. Figure 11a shows full wavefield data. Figure 26b shows the primary data results of L1-norm adaptive subtraction, which also comprises the labeled data used during the SNM training process. Figure 26c shows the primary data results predicted via the FPSSL method proposed in this study. Figure 26d is the primary data results predicted using the SSL method. As shown by red arrows 2, 3, and 4 in Figure 26b, the free surface-related and reflection-interface-related multiples are significantly suppressed, and almost no multiples remain. However, the surface-related multiples shown using arrows 1, 5, 6, and 7 have strong amplitudes, and this is due to the fact that the reflective interface of salt domes is a high-velocity body, resulting in the residuals of multiples. The L1-norm method cannot completely suppress the surface-related multiple shown by the red arrow in Figure 26d. Finally, these surface-related multiples have almost been completely eliminated in Figure 26c. Figure 27a–c show the results of the surface-related multiples obtained by subtracting Figure 26b–d from Figure 26a, respectively. The primary leakage situation can be clearly observed via the purple arrow in Figure 27a, and the primary amplitude leakage in the salt dome is more serious. However, the FPSSL method proposed in this study (Figure 27b), as represented by the purple rectangular box in the figure, obviously improves the primary leakage problem and also has good amplitude preservation for the primary. The purple arrows in Figure 27a,c shows some leakages of primaries and the linear noise. The events of the surface-related multiples are blurred, as shown by the orange arrows in the Figure 27c. The events of the surface-related multiple profile in Figure 27b are significantly clearer. We use real marine data to test the impact of the FPSSL method on the suppression results of surface-dependent multiple waves. The proposed FPSSL method can identify primaries and multiples and improve the leakage of primaries and the problematic residue of multiples by training field-parameter-guided semi-supervised neural networks based on their nonlinear mapping relationship and continuous coordinate information in the spatiotemporal domain.

4. Conclusions

Field parameters refer to the time–space coordinate information derived from the seismic acquisition system, including offsets, trace spaces, and sampling interval values. The deeper nonlinear mapping relationship between primaries and multiples can be explored using deep neural networks relative to the guides of field parameters. The idea of semi-supervised learning is implemented in FPSSL, which consists of a supervised network module (SNM) and an unsupervised network module (USNM). It can make full use of the unknown features of a small sample of labeled data and a large sample of unlabeled data to train neural networks. The reconstructed polynomial function expression of the primaries can be regarded as a constraint method, which uses the physical characteristics of primaries and multiples in the time–space domain to constrain the training process of the SNM. The USNM is essentially an optimization operator that can use the local wavefield characteristic loss function of the primaries and multiples to obtain more accurate primaries prediction results. This study proves that a U-net based on field parameters has high multiple suppression accuracies. Field parameters can guide U-net to more accurately identify multiples and suppress them using full wavefield data, while effectively reducing the leakage of primaries. Additionally, the FPSSL method can improve the leakage of primaries or the problematic residue of multiples caused by damage inflicted on the primaries during the adaptive subtraction process of the SRME method. Finally, the combined architecture of SNM and USNM allows the results of FPSSL to exceed the results of the labeled data, providing a new direction for exploring the combination of semi-supervised learning and traditional methods.