Upgoing and Downgoing Wavefield Separation in Vertical Seismic Profiling Guided by Signal Knowledge Representation

Abstract

Effective vertical seismic profiling (VSP) of upgoing and downgoing wave separation is essential for high-quality imaging. However, VSP wavefield separation is particularly challenging under complex geological conditions. Existing solutions encompass one derived from the mathematical characteristics of upgoing and downgoing waves, employing signal decomposition methodologies, and another that utilizes data-driven machine learning techniques, achieving wavefield separation by training sample data to identify the distinct characteristics of upgoing and downgoing waves. This study introduces a VSP wave-separation method using signal knowledge representation, primarily by constructing knowledge representations of upgoing and downgoing waves. Physics-informed recurrent neural network FWI and Poynting vector physical knowledge representation yielded accurate velocity models. Axial gradient information was utilized to construct morphological knowledge representations of upgoing and downgoing waves. Directional differentiation knowledge representations were established based on kinematic characteristic disparities between upgoing and downgoing waves in the time-depth domain. These wave knowledge representations (KRs) built a dual convolutional autoencoder. Its distinct branches extracted up/down wave information, while the KRs, transformed into loss functions, enabled knowledge-driven unsupervised VSP wave separation. The proposed methodology was validated using a homogeneous layer and Marmousi models, demonstrating the effective separation of upgoing and downgoing waves from the VSP seismic records.

1. Introduction

Seismic waves are fundamental to geophysical exploration, serving as the primary carriers of information about the Earth’s subsurface. These elastic waves, predominantly categorized into compressional (P-waves) and shear (S-waves), propagate through geological media with velocities and amplitudes that are sensitive to variations in rock properties, fluid content, and structural features. Seismic exploration methods, therefore, aim to generate and record these waves to image subsurface structures and characterize reservoir properties. Vertical Seismic Profiling (VSP) is a crucial borehole seismic technique in seismic data acquisition. It typically involves deploying an array of receivers at various depths within a wellbore to record the seismic signals generated by a source positioned at or near the surface. This acquisition geometry allows VSP to directly record both the downgoing wavefield and the upgoing wavefield. The direct measurement of the downgoing wavefield provides invaluable information for determining the seismic source wavelet, accurate interval velocities, and formation attenuation characteristics, while the upgoing wavefield carries detailed information about reflectors and structural features around the wellbore. Consequently, compared to surface seismic exploration, VSP generally offers superior signal-to-noise ratios and higher vertical resolutions. The VSP thus enables more precise measurements of formation velocity parameters, P- and S-wave properties, anisotropy, amplitude information, and lithological characteristics, thereby significantly enhancing seismic data-interpretation capabilities in structural, stratigraphic, and lithological analysis [1]. In VSP-based data processing, the separation of upward and downward wavefields is a prerequisite for seismic migration imaging [2]. Upgoing waves typically contain valuable information from subsurface reflectors. A clean upgoing wavefield forms the basis for performing Amplitude Versus Angle (AVA) or Off-set (AVO) analysis to identify fluid properties, calculating formation absorption attenuation (Q-value) parameters to evaluate reservoir characteristics and achieving accurate well-seismic time–depth calibration. Downgoing waves, on the other hand, mainly consist of direct waves and multiples [3]. Among these, clear direct waves provide precise seismic wave travel time information, serving as crucial input for calculating accurate interval velocities and establishing reliable velocity models. This is vital for subsequent migration imaging and depth–domain interpretation [4,5]. Furthermore, analyzing the downgoing waves and the multiples they generate aids in understanding the mechanisms of multiple generation within seismic data, subsequently guiding the suppression of multiples in surface seismic data processing. Downgoing waves can also be utilized for extracting the seismic wavelet, performing deconvolution to enhance resolution, and similarly, for estimating the attenuation characteristics of the formation media. In summary, the accuracy of VSP wavefield separation directly dictates the quality of the input data for subsequent processing steps and ultimately determines the reliability and geological significance of the final interpretation results. However, complex geological conditions frequently result in overlapping wavefields contaminated with noise, thereby complicating conventional separation methods [6].

Traditional wavefield-separation approaches can be classified into two categories. The first comprises time–depth separation methods, including Singular Value Decomposition (SVD) [7] and median filtering [8]. SVD leverages differences in seismic wave lateral coherency; however, due to its non-orthogonal nature, SVD performs poorly with VSP wavefields. Median filtering utilizes distinct dip angles in the time-depth domain but requires precise first-break picking and may introduce frequency distortion in high-frequency signals. The second category encompasses transform-domain filtering methods, primarily f-k filtering [9] and Radon transformation [10,11]. These methods separate wavefields by converting time-depth domain data into specific parameter domains. However, both approaches suffer from truncation effects and spatial aliasing, which potentially compromise separation accuracy through manual window selection. These challenges stem partly from the interference of wavefields within VSP records and the potentially time-varying spectral characteristics of the signal itself. Accurately characterizing and simulating such complex signal properties is itself a research focus; for instance, Genovese and Palmeri utilized wavelet transforms to generate random seismic ground motion records with non-stationary characteristics, aiming to more realistically reflect the complex nature of seismic signals [12].

Recently, there has been increasing interest in deep learning for complex non-linear optimization problems [13,14,15]. Deep learning has successfully addressed the challenges of seismic inversion [16] and denoising [17] in seismic data processing. Compared to the limitations of traditional methods, neural networks possess powerful non-linear fitting capabilities. They are able to learn from data and model the complex and highly non-linear mapping relationship between the mixed wavefield and the separated up/downgoing wavefields. This gives them a theoretical advantage in handling complex wavefield superposition problems that are difficult for traditional linear methods to solve. Data-driven Convolutional Neural Networks (CNNs) [5] and Generative Adversarial Networks (GANs) [18], including variants like CGANs with specialized convolution blocks, effectively capture weak wavefield characteristics and mitigate the effects of amplitude disparity. However, the performance of these approaches depends heavily on the quality of the training data and their generalization capabilities. Although Tao [6] and Margrave et al. [19] addressed this limitation using synthetic dataset generation, challenges persist in practical applications due to limited real-world samples. Dalai [20] proposed an unsupervised deep learning method for denoising receiver function data. This approach is based on an encoder–decoder framework, with its core idea being that the network learns the inherent structure of the signal by reconstructing the noisy input itself rather than a clean target during training, thereby achieving denoising and effectively eliminating the need for large-scale labeled data. Lu [21] employed a dual-convolutional autoencoder architecture to realize unsupervised wavefield separation in VSP data. The pursuit of robust unsupervised and physics-guided methods continues to be an active area of research, with investigations into hybrid model-data-driven strategies [22] and the application of physics-informed principles to further constrain network learning.

This study proposes a knowledge-guided unsupervised deep learning method (KGCAE, Knowledge-Guided Convolutional Autoencoder) whose core innovation lies in constructing and integrating multi-dimensional knowledge representations to guide the training process. These include (1) physics-derived representations based on physical principles, (2) morphological representations reflecting structural priors, and (3) directional discrepancy representations characterizing wave propagation patterns. By translating these knowledge representations into loss functions to guide the wavefield separation process, the method aims to combine the representational power of deep learning with physical constraints for up/down-going wave separation, overcoming limitations of traditional approaches and purely data-driven supervised methods.

The proposed KGCAE leverages its dual-autoencoder architecture and unsupervised, knowledge-driven learning paradigm to eliminate dependence on large, labeled datasets, which is typically required by supervised deep learning methods. Through the integration of diverse knowledge representations, this approach guides the network to generate physically plausible and structurally coherent separation results, particularly under challenging geological conditions with severe wavefield aliasing and low signal-to-noise ratios. This knowledge-guided mechanism effectively constrains the solution space, avoiding non-physical artifacts or stability issues common in purely data-driven or morphology-based unsupervised methods, thereby achieving more robust and stable separation performance. However, the method’s performance depends on the accuracy of the input knowledge representations. The physical knowledge relies on the quality of the velocity model obtained through full-waveform inversion (FWI), while the morphological knowledge currently requires manual dip picking.

This study aims to achieve VSP upgoing and downgoing wavefield separation by establishing a comprehensive multi-dimensional knowledge representation system for signal analysis and constructing a knowledge-driven separation framework based on a dual convolutional autoencoder, transforming the knowledge representation system into constraint loss functions applied to the autoencoder. An unsupervised separation of the upgoing and downgoing waves was achieved. Systematic validation experiments conducted on both the homogeneous layer and Marmousi models demonstrated that the proposed methodology effectively processes the VSP upgoing and downgoing wavefield separations under complex geological conditions, effectively avoiding the reliance on large-scale labeled samples inherent in traditional data-driven methods.

2. Knowledge Representation for VSP Wavefield Separation

This section will detail the construction of knowledge representations for the target signals, used to guide the subsequent VSP upgoing and downgoing wave separations. This study extracts and utilizes characteristic information of the wavefield from multiple perspectives, constructing multiple forms of knowledge representations. These include physical knowledge representations derived from the energy flux density vector (Poynting vector), morphological knowledge representations based on shape priors, and directional difference knowledge representations based on wavefield propagation characteristics. These knowledge representations collectively form a comprehensive system for wavefield description.

2.1. Physical Knowledge Representation Based on Energy Flow

The core principle for physically distinguishing upgoing and downgoing waves lies in their direction of energy propagation. This study leverages the energy flux density vector, commonly known as the Poynting vector e, to capture this directional information [23,24,25]. The extraction process involves three key steps: first, obtaining an approximate subsurface velocity model through Full Waveform Inversion (FWI); second, performing elastic wave forward modeling using this model; and third, calculating the Poynting vector from the simulated wavefield to derive the knowledge representations.

To obtain accurate geological model parameters, we adopted a Physics-Informed Recurrent Neural Network (PIRNN)-based full-waveform-inversion method [26]. The geological model parameters mainly comprise the P- and S-wave velocities ($c_p,c_s$) and density $\mathsf{\rho }$, which are key parameters describing the elastic properties of underground media. We can establish the mapping relationship between the geological model parameters and the resulting VSP wavefield. Specifically, we assume that the P- and S-wave velocities in the geological model are represented as $c_p\in R^{N\times M}$ and $c_s\in R^{N\times M}$ respectively, where N and M represent the number of vertical and horizontal sampling points. The forward modeling process can be expressed abstractly as

$$\tilde{X}=F\left(c_p,c_s\right),$$

(1)

where $\tilde{X}\in R^{T\times N}$ represents the VSP wavefield obtained through forward modeling, $T$ is the number of time sampling points, and $F\left(\cdot \right)$ represents the forward process, which is specifically implemented using the PIRNN method. To invert the geological model parameters, $c_p$ and $c_s$, using the observed VSP wavefield, $X$, as the label, the inversion problem of the geological model parameters can be converted into a constrained optimization problem:

$${\begin{array}{l}{\min }_{c_p,c_s}||X-\tilde{X}||\\ \mathrm{subject} \mathrm{to} \tilde{X}=F(c_p,c_s)\end{array}}_{,}$$

(2)

where $\tilde{X}\in R^{T\times N}$ is the observed VSP wavefield. By solving the above optimization problem, we can obtain the optimal solution of the geological model parameters, $c_p^{\mathrm{*}} and c_s^{\mathrm{*}}$, and obtain the approximate solution of the seismic wavefield, ${\tilde{X}}^{\mathrm{*}}$, through forward modeling. To achieve VSP wavefield simulation, the PIRNN incorporates the physical constraints of the wave equation into the neural network using the RNN’s time-stepping mechanism for forward modeling. Through the backpropagation algorithm, the PIRNN can invert the geological model parameters from the observed VSP wavefield, thereby providing a foundation for the subsequent upgoing- and downgoing wave knowledge representation.

After obtaining $c_p^{\mathrm{*}}$ and $c_s^{\mathrm{*}}$, the approximate solution of the seismic wavefield, ${\tilde{X}}^{\mathrm{*}}$, is obtained through a forward modeling process. During this process, the energy flux density vector [27,28] is introduced to describe the propagation direction and distribution of energy within the wavefield. We simulated the wavefield propagation in the two-dimensional depth-offset space. The behavior of the displacement field $u=\left(u_x,u_z\right)$ can be effectively described by decomposing it using the dilatation scalar $\mathsf{\theta }=\nabla \cdot \mathrm{u}$ and the rotation vector $\mathsf{\omega }=\nabla \times u$. The elastic wave equation, expressed in terms of these P-wave and S-wave related potentials, is given by [28,29]

$${\displaystyle \frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{t}}^2}}={c_p^{*}}^2\nabla \mathsf{\theta }-{c_s^{*}}^2\nabla \times \mathsf{\omega },$$

(3)

where the estimated velocities $c_p^{*}$ and $c_s^{*}$ are used. Solving Equation (3) numerically yields an approximate full wavefield solution $X^{*}$. From this solution, we can derive the necessary quantities for energy analysis, including the particle velocity $\mathrm{v}={\displaystyle \frac{\partial u}{\partial t}}$ the dilatation scalar $\mathsf{\theta }$, and the rotation vector $\mathsf{\omega }$.

Lastly, based on the simulated wavefield $X^{*}$, we calculate the energy flux density vector, $e$, to understand the direction and distribution of energy propagation [27,28]. The vector e is composed of P-wave and S-wave components, $e_p$ and $e_s$, respectively:

$$\left\{\begin{array}{l}{e}_p=-\theta v_p\\ e_{\mathrm{s}}=v_s\times \omega \end{array}\right.,$$

(4)

where $e_p=\left(e_{px},e_{pz}\right)$ is the energy flux density vector of the P waves and $e_s=\left(e_{sx},e_{sz}\right)$ is the energy flux density vector of the S waves. $v_p$ is the P-wave particle vibration velocity and $v_s$ is the S-wave particle vibration velocity. Equation (4) clarifies the interrelationship between the P- and S-waves during time propagation and elucidates the mechanism whereby they couple with the dilatation scalar and rotation vectors. The total energy flux density vector is expressed as follows:

$$e=e_p+e_s.$$

(5)

The energy flux density vector, $e$, not only describes the direction and intensity of energy propagation but also provides a basis for distinguishing between upgoing and downgoing waves. From Equation (5), the P-wave component, $e_p$, is related to the dilatation scalar and the P-wave particle vibration velocity field, $v_p$, whereas the S-wave component, $e_s$, is related to the rotation vector, $\mathsf{\omega }$. We utilize the sign of the vertical component of the P-wave energy flux vector, $e_{pz}$, to determine the primary wave propagation direction. A positive $e_{pz}$ generally corresponds to upward energy flow, and a negative value to downward flow. This directional information allows us to mask the seismic wavefield approximate solution, $X^{*}$, to extract the knowledge representation of the upward and downward wavefields:

$$\begin{array}{l}{U}_{rp}(t,z_r)=\left\{\begin{array}{ll}{\tilde{X}}^{*}(t,z_r),\hfill & \mathrm{if} e_{pz}\ge 0\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.\\ D_{rp}(t,z_r)=\left\{{\begin{array}{ll}{\tilde{X}}^{*}(t,z_r),\hfill & \mathrm{if} e_{pz}<0\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}}_{,}\right.\end{array}$$

(6)

where $U_{rp}$ represents the knowledge representation of the upward P-waves, $D_{rp}$ represents the knowledge representation of the downward P-waves, and $e_{pz}$ is the vertical component of the P-wave energy flux density vector, $e_p$. Figure 1 shows the extraction process of the physical knowledge representation, where (a) and (e) represent the parts of the P-wave energy flux density vector, with $e_{pz}>0$ and $e_{pz}<0$ obtained through the above forward process, showing the distribution of the upward and downward waves. By applying this equation to the seismic wavefield approximate solution, $X^{*}$, in (c), we can obtain the physical knowledge representation of the upgoing and downgoing waves in (d) and (e), respectively. This knowledge representation, which is derived directly from physical models, has a clear physical meaning and can reflect the direction of wavefield propagation to provide important information on physical constraints for subsequent upgoing and downgoing wave separation.

2.2. Morphological Knowledge Representation

In near-VSP data, odd-order-reflected waves (downward waves) typically propagate downward, whereas even-order-reflected waves (upward waves) typically propagate upward. This difference in the direction of propagation confers distinct apparent velocity and slope characteristics for the upward and downward waves in the depth-time domain. Specifically, the upgoing wavefield forms a negative-slope inverse pattern in the depth-time domain, whereas the downgoing wavefield forms a positive-slope direct pattern. As shown in Figure 2, the near-offset VSP record of a simple layered model reveals that upgoing and downgoing waves have a set of axial dip angles with the horizontal interface, which is denoted as $\mathsf{\alpha }=\left\{{\mathsf{\alpha }}_i|1\le i\le p,{\mathsf{\alpha }}_i\in R\right\}$ and $\mathsf{\beta }=\left\{{\mathsf{\beta }}_i|1\le i\le q,{\mathsf{\beta }}_i\in R\right\}$. To utilize these morphological features to constrain the upgoing and downgoing wavefields, we introduced a knowledge representation based on morphological priors. This representation aims to emphasize the axial continuity and slope differences between the upward and downward wavefields, thereby improving the accuracy of wavefield separation.

To extract this knowledge representation, we used a modified Sobel operator to calculate the gradients in the axial direction. The Sobel operator is a commonly used discrete differential operator for calculating the approximate gradients of images and signals. It detects edges and directional changes in images using convolution operations. Figure 3a,b shows the basic Sobel operators, which represent the horizontal and vertical convolution kernels, respectively. To accommodate the slope characteristics of the upward and downward wavefields, the Sobel operator must be modified to capture the gradient information in different directions. Specifically, we rotated the Sobel operator as follows:

$${\nabla }_{{\alpha }_1}=\cos {\alpha }_1\ast A+\sin {\alpha }_1\ast B,$$

(7)

where ${\mathsf{\alpha }}_1$ represents the rotation angle, $A$ and $B$ represent the horizontal and vertical Sobel operators shown in Figure 3a,b, and ${\nabla }_{{\mathsf{\alpha }}_1}$ represents the rotated gradient operator. Taking ${\mathsf{\alpha }}_1=45°$ as an example, the rotated Sobel operator is shown in Figure 3c, which can effectively capture the axial gradient information of the downgoing waves that form a 45° angle with the horizontal plane.

Lu et al. [21] proposed a gradient knowledge representation based on morphological priors. Let $\tilde{U}$ be the network’s prediction for upgoing waves and $\tilde{D}$ be the prediction for downgoing waves. The gradient along the axial direction is calculated as follows:

$$\begin{gathered} G\left(\alpha ,\tilde{U}\right)=min\left\{{\nabla }_{{\alpha }_i}\ast \tilde{U}|1\le i\le p,p\ge 1\right\} \\ G\left(\beta ,\tilde{D}\right)=min\left\{{\nabla }_{{\beta }_i}\ast \tilde{D}|1\le i\le q,q\ge 1\right\}, \end{gathered}$$

(8)

where $\ast$ represents convolution operation; min represents element-wise minimum operation; $\mathsf{\alpha }=\left\{{\mathsf{\alpha }}_i|1\le i\le p,{\mathsf{\alpha }}_i\in R\right\}, \mathsf{\beta }=\left\{{\mathsf{\beta }}_i|1\le i\le q,{\mathsf{\beta }}_i\in R\right\}$ represents the axial dip angles of upgoing and downgoing wavefields in the depth-time domain; ${\nabla }_{{\mathsf{\alpha }}_i}$ and ${\nabla }_{{\mathsf{\beta }}_i}$ represent the modified Sobel operators that are constructed in accordance with the axial-dip angle sets of upgoing and downgoing waves; and $G\left(\mathsf{\alpha },\tilde{U}\right)$ and $G\left(\mathsf{\beta },\tilde{D}\right)$ are the calculated gradient information matrices. As upgoing and downgoing waves have characteristics of axial slope differences and axial continuity, the wavefields should be smooth along their axial directions (which imply small variations in their axial directions). This knowledge representation can be used to construct loss functions based on $G\left(\mathsf{\alpha },\tilde{U}\right)$ and $G\left(\mathsf{\beta },\tilde{D}\right)$, guiding the network to optimize axial continuity during training, thus achieving morphological constraints on upgoing and downgoing waves. Figure 3 shows the basic horizontal and vertical Sobel operators (Figure 3a,b) and an example of a Sobel operator rotated by 45° (Figure 3c), which form the basis for constructing directionally sensitive gradient detectors. Figure 4 illustrates the process of extracting the morphological prior-knowledge representation. First, the axial dip angle sets, $\mathsf{\alpha }$ and $\mathsf{\beta }$, of the upgoing and downgoing waves are manually picked from the original VSP data. Then, modified Sobel operators are constructed based on the dip angle sets. Finally, the axial gradient information is extracted from the predicted upgoing and downgoing wavefields, $\tilde{U}$ and $\tilde{D}$, using their corresponding modified Sobel operators.

2.3. Directional Difference Knowledge Representation

In the VSP records, the upgoing and downgoing waves exhibited significant directional differences. The zero-mean characteristic of the VSP wavefield is shown in the VSP wavefield diagram of the layered model in Figure 5. There is a notable difference between the upward and downward waves, which can be represented by the angle between them; the wavefield mean approaches zero. Based on this observation, we introduced cosine similarity to characterize this knowledge representation. Cosine similarity is defined as the cosine of the angle between two vectors, with values ranging from [–1, 1].

When two vectors are orthogonal, the cosine similarity is 0, and when two vectors are identical or opposite, the value is 1 or –1. However, because the VSP wave-separation problem scenario includes negative signals, we retained the cosine similarity in the interval [0, 1] through an absolute value operation. Let $\tilde{U}$ be the predicted upgoing wavefield and $\tilde{D}$ be the predicted downgoing wavefield. Then, the cosine similarity is defined as

$$\cos (\tilde{U},\tilde{D})={\displaystyle \frac{\tilde{U}\cdot \tilde{D}}{‖\tilde{U}‖‖\tilde{D}‖}},$$

(9)

where $\tilde{U}\cdot \tilde{D}$ represents the dot product of the two wavefields, and $‖\tilde{U}‖$ and $‖\tilde{D}‖$ represent the L2 norms of the two wavefields, respectively. In the knowledge representation of wavefield difference, we used the absolute value of the cosine similarity as a measure of the directional difference:

$$L_{direction}=\left|\cos (\tilde{U},\tilde{D})\right|.$$

(10)

3. Methodology

3.1. Modeling of Upgoing and Downgoing Wave Separation Problem

The goal of signal separation is to recover the unknown upgoing wavefield, $U\in R^{T\times N}$, and downgoing wavefield, $D\in R^{T\times N}$, from the observed VSP record, $X\in R^{T\times N}$. This study employed a U-Net architecture using two independent networks to predict the upward and downward wavefields. When deriving the solution, physical knowledge representations ($U_{rp}$ and $D_{rp}$), morphological knowledge representations, and wavefield directional difference knowledge representations were explicitly introduced as constraints to enhance the physical consistency and accuracy of the wavefield separation. Specifically, our modeling of the upgoing and downgoing wave-separation problems is as follows:

$$\begin{gathered} \begin{array}{c}argmin\\ {\theta }_u,{\theta }_d\end{array}\mathrm{ }\mathrm{ }\mathrm{ }\underset{\mathrm{Reconstruction}}{\underbrace{{\left|\right|\mathrm{ }X\mathrm{ }-\mathrm{ }\tilde{X}\mathrm{ }\left|\right|}_h}}\mathrm{ }+\mathrm{ }{\lambda }_1\mathrm{ }\underset{\mathrm{Physical} \mathrm{rules}}{\underbrace{{\left|\right|\mathrm{ }{U}_{rp}\mathrm{ }-\mathrm{ }\tilde{U}\mathrm{ }\left|\right|}_h}}\mathrm{ }+\mathrm{ }{\lambda }_2\mathrm{ }\underset{\mathrm{Physical} \mathrm{rules}}{\underbrace{{\left|\right|\mathrm{ }{D}_{rp}\mathrm{ }-\mathrm{ }\tilde{D}\mathrm{ }\left|\right|}_h}}\mathrm{ }+\mathrm{ }{\lambda }_3\mathrm{ }\underset{\mathrm{Morphological}}{\underbrace{{\left|\right|\mathrm{ }G\left(\alpha ,\mathrm{ }\tilde{U}\right)\mathrm{ }\odot \mathrm{ }G\left(\beta ,\mathrm{ }\tilde{D}\right)\mathrm{ }\left|\right|}_F}}\mathrm{ }+\mathrm{ }{\lambda }_4\mathrm{ }\underset{\mathrm{Direction} \mathrm{Difference}}{\underbrace{\left|\mathrm{ }\cos \left(\tilde{U},\mathrm{ }\tilde{D}\right)\mathrm{ }\right|}} \\ \mathrm{s}.\mathrm{t}.\mathrm{ }\mathrm{ }\mathrm{ }\tilde{X}=\mathrm{ }\tilde{U}+\tilde{D} \\ \mathrm{ }\tilde{U}={\mathcal{G}}_{\mathcal{u},{\theta }_{\mathcal{u}}}\left(X\right)\in R^{T\times N} \\ \tilde{D}={\mathcal{G}}_{\mathcal{d},{\mathsf{\theta }}_{\mathcal{d}}}\left(X\right)\in R^{T\times N} , \end{gathered}$$

(11)

where $X$ represents the original VSP record, $\tilde{X}$ represents the predicted reconstructed wavefield, and the remaining terms are the constraints introduced from the knowledge representations, including the constraints in the physical, morphological, and propagation-direction difference. Furthermore, $\left|\right|\cdot \left|\right|$：$R^{T\times N}\to R^{+}$ is an operator for measuring error, $\odot$ represents element-wise multiplication, ${\mathcal{G}}_{u，}{\mathcal{G}}_d$：$R^{T\times N}\to R^{T\times N}$ represent the two-branch network-mapping functions for predicting upgoing and downgoing wavefields, which recover $\tilde{U}$ and $\tilde{D}$ from the original wavefield $X$, and ${\mathsf{\theta }}_u$,${\mathsf{\theta }}_d$ represent the parameters of $\mathcal{G}$.

This modeling process not only needs to measure the reconstruction error between the observed record,$X$, and the reconstructed wavefield, $\tilde{U}+\tilde{D}$, but also the error between the predicted wavefields, $\tilde{U}$ and $\tilde{D}$, and their knowledge representations. This constrains the continuity and directional consistency of the wavefield to ensure the physical reasonableness of wavefield separation.

3.2. Network Structure

This study adopts a dual convolutional autoencoder network architecture guided by signal knowledge representation, called the Knowledge-Guided Convolution Autoencoder (KGCAE), to achieve upgoing and downgoing wavefield separation [21]. Figure 6 shows the constructed autoencoder network architecture, where Net-U and Net-D are responsible for predicting the upgoing and downgoing wavefields, respectively. The input was the original VSP record and the output was the predicted upgoing and downgoing wavefields of the same shape as the original record.

Using Net-U as an example, the specific structures and parameters are listed in

Table 1. Structure and Parameter Configuration of Each Layer in the Net-U (net1) Convolutional Autoencoder.

Layer			Output Channels	Kernel Size	Stride	Activation	Output Shape
Input			1	-	-	-	1600 × 100 × 1
Encoder Layer	Conv2D		32	3 × 3	1 × 1	tanh	1600 × 100 × 32
	MaxPool		-	2 × 2	2 × 2	-	800 × 50 × 32
	Conv2D		64	3 × 3	1 × 1	tanh	800 × 50 × 64
	Conv2D		32	3 × 3	1 × 1	tanh	800 × 50 × 32
	MaxPool		-	2 × 2	2 × 2	-	400 × 25 × 32
Decoder Layer		Conv2D	32	3 × 3	1 × 1	tanh	400 × 25 × 32
		UnSampling2D	-	2 × 2	-	-	800 × 50 × 32
		Conv2D	64	3 × 3	1 × 1	tanh	800 × 50 × 64
		Conv2D	32	3 × 3	1 × 1	tanh	800 × 50 × 32
		UnSampling2D	-	2 × 2	-	-	1600 × 100 × 32
		Conv2D	1	3 × 3	1 × 1	tanh	1600 × 100 × 1

. Here, we assumed that the original input VSP record had the shape (1600, 100, 1), representing 1600 time points, 100 detectors, and 1 channel, and Input data are normalized into the [−1, 1] interval for better convergence. The overall structure consisted of encoding and decoding layers. The encoding layer is employed for feature extraction and compression of VSP data, as illustrated in Figure 7. It specifically comprises three convolutional layers and two pooling layers. The first convolutional layer is set to 32 channels to capture basic, low-level features; an excessive number of channels would increase computational redundancy. The second layer expands to 64 channels to encode more complex, high-level features, thereby enhancing the feature representation capability. The third layer compresses the channel count back to 32, aiming to avoid excessive feature redundancy while retaining critical information for the decoding layer. To balance the receptive field and simultaneously reduce the loss of boundary information, the convolutional kernel size is set to 3 × 3. The pooling layers utilize 2 × 2 max pooling to enhance the feature extraction capability. The decoding layer comprised 4 convolutional and 2 upsampling layers for data feature reconstruction and recovery. The channel counts for the convolutional layers were set to 32, 64, 32, and 1, respectively; the final layer outputs single-channel reconstructed data with dimensions consistent with the original input VSP data. The convolutional kernel size was 3 × 3. The upsampling layers employed 2 × 2 nearest-neighbor upsampling. We use the tanh activation function instead of ReLU because the value interval contains both positive and negative values, The use of the ReLU function would lead to value overflow. Figure 7 shows the upgoing and downgoing wave-separation processes.

3.3. Loss Function

To utilize knowledge representations for guiding wave separation, this study modeled them as a loss function, $L$, which includes the reconstruction, representation, gradient, and cosine losses. The specific loss function is defined as follows:

$$L=L_{rec}+{\mathsf{\lambda }}_1{L}_{rep}+{\mathsf{\lambda }}_2{L}_{grad}+{\mathsf{\lambda }}_3{L}_{cos},$$

(12)

where $L_{rec}$ represents the reconstruction loss that measures the difference between the original and reconstructed wavefields, $L_{rep}$ represents the representation loss, which measures the difference between the predicted and physics-knowledge-represented wavefields, $L_{grad}$ represents the gradient loss obtained from the morphological knowledge representation, emphasizing the wavefield continuity, and $L_{cos}$ represents the cosine loss obtained from the knowledge representation of the difference in the propagation direction, used to reduce the aliasing phenomena in wavefield separation. Next, we describe the principles of each loss function and their roles in wavefield separation.

3.3.1. Reconstruction Loss

The reconstruction loss, $L_{rec}$, was used to measure the difference between the sum of the upgoing wavefield, $\tilde{U}$, and the downgoing wavefield, $\tilde{D}$, predicted by the dual convolutional autoencoder and the original VSP record, $X$, specifically defined as

$${\mathrm{L}}_{\mathrm{r}\mathrm{e}\mathrm{c}}={\left|\left|X-\tilde{U}-\tilde{D}\right|\right|}_h,$$

(13)

where $\left|\right|\cdot |{|}_h$ represents the error measure based on the Huber function, which is specifically defined as:

$$f_h\left(s\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}\ast s^2, if \left|s\right|\le d\\ d\ast \left|s\right|-{\displaystyle \frac{1}{2}}\ast d^2, if \left|s\right|>d\end{array}\right..$$

(14)

The Huber function uses squared loss for small errors and linear loss for larger errors, thus combining the advantages of both the mean squared error and absolute error. This effectively reduces the sensitivity to outliers. The introduction of the reconstruction loss ensures that when the model predicts the upgoing and downgoing wavefields, the overall wavefield can accurately reconstruct the original observed data. This loss term encourages the model to maintain the overall consistency and physical reasonability of the wavefield during the separation of the upward and downward waves.

3.3.2. Morphological Knowledge Representation Loss

The gradient loss, $L_{grad}$, is derived from morphological knowledge and is used to measure the axial continuity of the upward and downward wavefields predicted by the network, specifically defined as

$$L_{grad}={\left|\left|G\left(\mathsf{\alpha },U\right)\odot G\left(\mathsf{\beta },D\right)\right|\right|}_F^2,,$$

(15)

where $\odot$ represents element-wise multiplication, $\left|\right|\cdot |{|}_F$ represents the L2 norm, and $\alpha =\left\{{\alpha }_i|1\le i\le p,{\alpha }_i\in R\right\},\beta =\left\{{\beta }_i|1\le i\le q,{\beta }_i\in R\right\}$ represent the slope sets of upgoing and downgoing wavefields, respectively.

Lu [21] built a dual convolutional autoencoder for wavefield separation based on the gradient loss function, called dualCAE, and achieved certain results. However, the gradient loss is based only on morphological priors; the dip-angle sets of the upward and downward wavefields must be manually selected. For wavefield separation with a complex morphology, the stability is relatively poor and requires complex parameter adjustments. In the next section, we introduce the representation loss and improve the separation results by incorporating physical rules.

3.3.3. Physical Knowledge Representation Loss

The representation loss, $L_{rep}$, is derived from physical knowledge and aims to enhance the accuracy and stability of the upgoing and downgoing wavefield separation by introducing physical rules. The representation loss measures the difference between the model-predicted upward wavefield, $\tilde{U}$, and downward wavefield, $\tilde{D}$. The knowledge representation wavefields, $U_{rp}$ and $D_{rp}$, are extracted based on the energy flux density vectors, specifically defined as

$$L_{rep}={\left|\left|U_{rp}-\tilde{U}\right|\right|}_h+{\left|\left|D_{rp}-\tilde{D}\right|\right|}_h,$$

(16)

where $\left|\right|\cdot |{|}_h$ represents the error measurement using the Huber function. By introducing representation loss, the model not only focuses on the overall wavefield reconstruction but also emphasizes the consistency between the predicted wavefields and the knowledge representation of the upward and downward wavefields extracted through physical methods. This loss ensures that, during the separation of the upgoing and downgoing waves, the model can accurately capture the physical characteristics and propagation directions of the wavefield, thereby improving the separation accuracy. Furthermore, in complex geological structures, the representation loss provides additional constraints that prevent unreasonable wavefield reconstructions during separation, thereby enhancing the stability of the separation process.

3.3.4. Directional Difference Knowledge Representation Loss

The Cosine Loss, $L_{cos}$, is derived from the knowledge representation of the differences in propagation direction and is primarily used to further optimize the directional differences between the upgoing and downgoing wavefields. The cosine loss is calculated as follows:

$$L_{cos}={\displaystyle \frac{1}{N}}{\sum }_{i=0}^{N-1}{\left|\left|cos\left(u_i,d_i\right)\right|\right|}_2,$$

(17)

where $N$ is the number of receivers, $u_i$ and $d_i$ represent the values of the predicted upgoing wavefield, $\tilde{U}$, and downgoing wavefield, $\tilde{D}$, at the i-th receiver from the autoencoder, respectively, and $\left|\right|\cdot |{|}_2$ denotes the L2 norm of a vector. The introduction of cosine loss aims to create distinct differences between the upgoing and downgoing wavefields, reduces aliasing phenomena in wavefield separation, and ensures that the separated wavefields conform better to actual physical laws.

4. Numerical Examples

This section evaluates the effectiveness of the proposed up/down-going wave separation method, which is based on target signal knowledge representation, through numerical experiments. Two velocity models were selected for the experiments: a simple horizontally layered geological model and the two-dimensional (2D) elastic Marmousi model. We conducted ablation studies and comparative experiments on the horizontally layered velocity model and also plotted comparison curves showing single-trace separation results from different methods. On the Marmousi model, comparative tests were performed against the traditional f-k method to validate the effectiveness of knowledge representation in the process of separating up- and down-going waves. All numerical simulations and model training were completed under a unified hardware environment, utilizing an NVIDIA A6000 GPU computing platform and implemented using the TensorFlow deep learning framework.

4.1. Validation on Homogeneous Layered Velocity Model

The horizontally layered velocity model consisted of four distinct layers with P-wave, S-wave, and density parameters, as illustrated in Figure 8b. Velocity and density inversion interfaces formed within a four-layer geological structure are crucial for studying phenomena such as the generation of reflected waves and inter-layer multiples. The synthetic model used in this research aims to provide a clear and controllable environment to validate and demonstrate the core ideas and methods of our study. Although the density values are on the lower side relative to their corresponding P-wave velocities according to Gardner’s empirical formula, this allows for the inclusion of atypical conditions under specific geological settings, such as high porosity or particular fluid fills, thereby effectively supporting the core argument of this method. The observation system is illustrated in Figure 8b, which features a source point positioned 50 m below the ground with a 12 m offset. Receivers were distributed at depths ranging from 250 to 4250 m, with 10 m intervals between each receiver. A Ricker wavelet with a dominant frequency of 50 Hz was used as the source signal. The VSP data were generated through elastic wave forward modeling; the modeling parameters are listed in

Table 2. Homogeneous model forward parameters.

Parameters	Forward Modeling
Central Frequency of Source	50 Hz
Sampling Time	4 ms
Location of launch point	50 m below the ground
Offset	12 m
Distance between adjacent receivers	10 m
Receivers’ distribution range	Range: from 250 to 4250 m underground
Depth of model	4500 m
Width of model	50 m

. The resulting raw VSP records are presented in Figure 8a, with the corresponding f-k spectra shown in Figure 8c, where the energy concentration is apparent at approximately 50 Hz.

4.1.1. Ablation Experiments on the Homogeneous Layered Model

In accordance with Equation (11), the loss function weights for the experiments in this section were configured as listed in

Table 3. Loss function parameter settings for three cases.

Parameters	Case 1	Case 2	Case 3
${\lambda }_1$	$0.2$	$0.2$	$0.2$
${\lambda }_2$	$0$	$1\times {10}^{-4}$	$1\times {10}^{-4}$
${\lambda }_3$	$0$	$0$	$1\times {10}^{-3}$
$\alpha$	$0$	$(129°,139°)$	$(129°,139°)$
$\beta$	$0$	$(41°,52°)$	$(41°,52°)$
Epoch	10,000	$\mathrm{10,000}$	$\mathrm{10,000}$

. Each case runs for 10,000 iterations, with each epoch taking 100 ms.

To validate the effectiveness of physical knowledge representation and gradient loss in constraining the upgoing and downgoing wavefields, three scenarios were examined:

• Application of reconstruction loss and physical knowledge representation loss;
• Application of reconstruction loss, Physical Knowledge Representation Loss, and Morphological Knowledge Representation Loss;
• Application of reconstruction loss, Physical Knowledge Representation Loss, Morphological Knowledge Representation Loss, and Directional Difference Knowledge Representation Loss.

Figure 9a–h shows the experimental results for the three cases. Specifically, (a) and (e) show the knowledge representation of the layered model; (b) and (f) illustrate the separation results from the first case; (c) and (g) present the outcomes from the second case; and (d) and (h) display the results from the third case. The first and second rows represent the separated upgoing and downgoing wavefields, respectively. Based on (a) and (e), the knowledge representation of the upgoing and downgoing wavefields was not entirely accurate. As the energy flux density vectors can only reflect the total energy information of the wavefield, the wave knowledge representation derived from their directions exhibits notable separation artifacts. Consequently, significant axial discontinuities appear at the convergence of upgoing and downgoing waves. Figure 9b,f demonstrates the network output when applying reconstruction and representation losses. Although the output should align with its knowledge representation when only representation loss is applied, it notably improved the axial discontinuity issue. This enhancement is attributed to parameter sharing in the CNNs and the multiscale invariance of the wave axial characteristics. However, the separation artifacts remained visible in the output. Figure 9c,g shows the network output under the second configuration. With the addition of gradient regularization, the separation artifacts in the wavefields were effectively reduced, although not completely eliminated, whereas the axial continuity of the wavefields was enhanced. However, due to the strong residual energy in the first arrival portion of the upgoing wave representation, some downgoing wave residuals persisted in this region. Figure 9d,h presents the network output under the third configuration. The incorporation of cosine loss encourages the model to learn distinctly different directional features for the upgoing and downgoing waves, thereby improving the separation accuracy. The results demonstrated significantly enhanced wavefield separation with cleaner separation outcomes.

4.1.2. Comparative Experiments on the Homogeneous Layered Model

To further investigate the effectiveness of the proposed KGCAE methodology, we conducted wavefield separation on the aforementioned simple-layered model and VSP data, comparing it with the conventional f-k filtering method and the morphological feature-guided (dualCAE) approach [21].

Table 4. Simulation parameters of KGCAE and dualCAE.

Parameters	KGCAE	dualCAE
${\lambda }_1$	$0.2$	$0$
${\lambda }_2$	$1\times {10}^{-3}$	$1\times {10}^{-3}$
${\lambda }_3$	$1\times {10}^{-4}$	$1\times {10}^{-4}$
$\alpha$	$(129°,139°)$	$(129°,139°)$
$\beta$	$(41°,52°)$	$(41°,52°)$
$\mathrm{E}\mathrm{p}\mathrm{o}\mathrm{c}\mathrm{h}$	$\mathrm{10,000}$	$\mathrm{10,000}$

lists the hyperparameter configurations for KGCAE and dualCAE methods. The proposed KGCAE method runs for 10,000 iterations, with each epoch consuming 100 ms. For the f-k filtering, a rectangular window was applied in the f-k domain. The separation was performed based on the sign of the apparent velocity associated with the wave energy: energy corresponding to negative apparent velocities was attributed to the upgoing wavefield, while energy with positive apparent velocities was assigned to the downgoing wavefield.

Figure 10 presents a comparison of the separation results from the three methods. The f-k filtering results are shown in (a) and (d), (b) and (e) display the separation results using the dualCAE method, (c) and (f) show the results obtained using the proposed KGCAE methodology. The corresponding frequency-beam diagrams are illustrated in (g–l). From a comprehensive perspective, three methodologies successfully achieved distinct wavefield separation, mataining axial continuity and zero-mean characteristics in both the upward and downward waves. However, as observed in Figure 8c, spatial aliasing exists in the raw VSP data. When the energy distributions of upgoing and downgoing waves overlap near zero apparent velocity, energy leakage occurs in the separation results (Figure 10a,d). Additionally, ringing artifacts may emerge during the conversion from the frequency-wavenumber (f-k) domain to the time-depth domain, as highlighted within the dashed box in Figure 10a. In contrast, the proposed KGCAE method demonstrated superior performance in terms of energy intensity and separation clarity. A notable advantage of our methodology lies in its ability to process raw VSP data directly without domain transformation. It automatically allocates frequency-domain aliasing to its respective upgoing and downgoing components without requiring explicit frequency-domain filtering, thereby effectively mitigating spectral aliasing challenges. The energy is correctly partitioned into upgoing and downgoing wavefields, reducing leakage and yielding clearer f-k spectra in the separated wavefields. Compared to conventional f-k filtering and dualCAE methods, this approach achieves higher fidelity.

Comparing the convergence processes of KGCAE and dualCAE algorithms, we focused on the reconstruction loss evolution rather than the total loss, because the loss function compositions and weights differ. As illustrated in Figure 11, the blue curve (dualCAE) exhibits significant fluctuations during convergence, whereas the red curve (KGCAE) demonstrates notably smaller oscillations. The reconstruction loss convergence curves indicate that the proposed method surpasses dualCAE in terms of convergence speed and stability. The f-k method, being a direct filtering process based on the described criterion, does not have an analogous iterative convergence behavior.

4.2. Validation on Marmousi Model

To evaluate the performance and stability of our proposed KGCAE method against the dualCAE method [21] and F-K filtering method for complex structures, this study employed a widely recognized Marmousi model.

The Marmousi model used in this experiment is shown in Figure 12. Specifically, Figure 12a–c represents the P-wave velocity, S-wave velocity, and density distributions of the Marmousi model, respectively. The observation system configuration, as indicated in Figure 12a, consists of a source positioned at the surface with a 40 m offset while receivers are distributed underground at intervals of 5 m, spanning depths from 0 to 500 m. A Ricker wavelet with a dominant frequency of 30 Hz was employed as the source signal. The VSP data were generated through elastic wave forward modeling, with the specific simulation parameters detailed in

Table 5. Marmousi forward parameters.

Parameters	Forward Modeling
Central Frequency of Source	$30 \mathrm{H}\mathrm{z}$
Sampling Time	$5 \mathrm{m}\mathrm{s}$
Offset	$40 \mathrm{m}$
Distance between adjacent receivers	$5 \mathrm{m}$
Receivers’ distribution range	0 to 500 m underground
Depth of model	$500 \mathrm{m}$
Width of model	$500 \mathrm{m}$

. The resulting VSP forward modeling record is presented in Figure 12d, and the corresponding f-k spectra are shown in Figure 12e. The f-k spectrum clearly demonstrates the presence of distinct upgoing and downgoing wave energies in the original VSP record, with energy concentrated around the 30 Hz frequency band.

Comparative Experiments on the Marmousi Model

Comparative experiments were conducted using the dualCAE method and F-K filtering method. The hyperparameters for the KGCAE method and the dualCAE method were configured as listed in

Table 6. Simulation parameters of KGCAE and dualCAE.

Parameters	KGCAE	dualCAE
${\lambda }_1$	$0.2$	$0$
${\lambda }_2$	$1\times {10}^{-3}$	$1\times {10}^{-3}$
${\lambda }_3$	$1\times {10}^{-4}$	$1\times {10}^{-4}$
$\alpha$	$(96°,102°)$	$(96°,102°)$
$\beta$	$(78°,81°)$	$(78°,81°)$
$Epoch$	$\mathrm{10,000}$	$\mathrm{10,000}$

, in accordance with the loss function defined in Equation (12). KGCAE is trained for 10,000 iterations at 90 ms per epoch.

Figure 13a,b illustrates the physical knowledge representation of the upgoing and downgoing waves, respectively, demonstrating the wavefield distribution extracted based on the energy flux density vectors. Figure 13c,d shows the frequency–beam spectra of (a,b). Figure 14 shows the separation results and their frequency-wavenumber spectra for three methods applied to the Marmousi model.

Figure 14a,d,g,j displays the separation results and their f-k spectra using the F-K filtering method. As evident from the f-k spectrum of the original VSP data in Figure 12c, severe spatial aliasing exists between upgoing and downgoing waves. Since F-K filtering separates wavefields by distinguishing opposite apparent velocity directions, noticeable aliasing artifacts remain in the separation results, as indicated by the black arrows in Figure 14d where upgoing wave components contaminate the downgoing wavefield. Moreover, similar to the comparative experiments on the homogeneous layered model, the F-K filtering results still exhibit leakage and ringing effects. Figure 14c,f,i,l shows the separated upgoing and downgoing waves and their frequency–wavenumber spectra obtained by the KGCAE while Figure 14b,e,h,k shows the corresponding results from the dual CAE method, demonstrating that, compared to the dual CAE, the KGCAE more effectively utilizes the axial and numerical distribution information from the knowledge representation of the wavefield. This leads to clearer axial characteristics in the separated wavefields, better preservation of the zero-mean properties, and more effective elimination of high-frequency noise, thereby maintaining the frequency characteristics of the original signal. This indicates that the KGCAE exhibits superior performance in maintaining physical consistency and authentic signal properties when processing complex wavefields.

Although the dualCAE method achieves relatively clean separation results through morphological prior constraints with clear and continuous axial characteristics in the target wavefield, it has certain limitations. Specifically, the first arrival portion of the upgoing wavefield appears chaotic; DC noise contamination is evident at the wavefield boundaries due to edge effects (as shown in Figure 14b,e). The frequency–wavenumber spectra further confirm these observations, revealing the limitations of dualCAE in managing boundary effects, particularly in complex geological structures, where the separation performance is less stable and precise than that of KGCAE. Furthermore, Figure 15 presents a comparative analysis of the reconstruction loss, $L_{rec}$, for both methods, with the red curve representing KGCAE and the blue curve representing dualCAE. Analysis revealed that the proposed method demonstrated faster convergence and greater stability during training. This indicates that constraining the wavefield separation through physics-based knowledge representation leads to a more efficient optimization, thereby more rapidly achieving stable separation results.

5. Conclusions

In this article, we constructed a VSP upgoing and downgoing wavefield-separation method utilizing signal knowledge representation. First, our methodology involved establishing a signal knowledge representation system and its acquisition methodology, followed by the construction of a dual convolutional autoencoder knowledge-driven separation framework. By transforming the knowledge representation into constraint loss functions, an effective unsupervised separation of the upward and downward waves was achieved. While this study utilized a dual convolutional autoencoder framework, the proposed signal knowledge representation system and its corresponding constraint losses could potentially be adapted to guide other neural network architectures, such as U-Net, for VSP wavefield separation. Systematic validation experiments on horizontally layered and Marmousi models demonstrate that this method effectively addresses the VSP wavefield separation challenges under complex geological conditions.

However, this method has several limitations. The accuracy of physical knowledge representation acquisition depends on the precision of the full waveform inversion, whereas the axial dip angles in morphological knowledge representation require manual picking. Furthermore, although the knowledge representation of the propagation directional difference reflects the orthogonality of the upgoing and downgoing waves, their propagation directions in the actual VSP data are not completely orthogonal. This potentially affects the generalization capability of the model. To address these limitations, future studies should employ more robust inversion algorithms to obtain a more accurate representation of physical knowledge and investigate intelligent feature extraction algorithms for automated morphological feature recognition.