Reverse Time Migration Method for Joint Imaging of Multiples and Primaries in Vertical Seismic Profiling

Abstract

Vertical seismic profiling (VSP) has garnered widespread attention because of its ability to provide high-quality seismic data. Owing to the unique characteristics of VSP observation systems, conventional multiple-wave imaging methods designed for surface seismic data are often not suitable for VSP data. Additionally, the existing research lacks methods capable of simultaneously imaging surface and interbed multiples in asymmetric observation systems. Here, the reverse time migration (RTM) imaging problem of multiple VSP waves is addressed, and a joint RTM imaging method is proposed for primary and multiple waves. Without the need to predict or separate multiple waves, this method utilizes seismic signals containing primary and multiple waves to replace wavelet excitation at the source location. Primary waves, surface multiples, and interbed multiples are simultaneously imaged, addressing the limitations of traditional VSP imaging methods that can only image areas near observation wells. Numerical tests demonstrate that the proposed method provides a broader effective imaging range and higher image quality than traditional VSP RTM methods.

1. Introduction

Seismic imaging is an effective method for exploring subsurface structures with the aim of obtaining accurate images of the true reflectivity of underground formations. It is widely regarded as one of the most effective techniques for mapping subsurface structures using seismic data [1]. Among seismic imaging methods, reverse time migration (RTM) is particularly notable because of the use of the two-way wave equation, which is not constrained by the dip angle of subsurface structures. Consequently, this method can be used to image various complex geological formations with high accuracy [2,3,4].

Considerable research has been conducted to enhance the quality of seismic imaging results. Liu et al. reduced low-frequency noise in imaging results by separating up-going and down-going wavefields [5]. Yoon and Marfurt improved imaging quality by incorporating the Poynting vector into the imaging condition [6]. Ha et al. achieved longitudinal and transverse wave separation in elastic wave reverse-time migration [7]. Moreover, the quality of seismic data significantly influences the results of seismic imaging methods. Low-quality seismic data can result in numerous artifacts in the imaging results, compromising both the reliability and accuracy of seismic imaging outcomes. Therefore, in conventional RTM methods, primary waves are typically used as effective data to ensure seismic data quality, whereas multiple waves are treated as noise and are suppressed [8,9,10]. However, multiple waves can also provide useful information. Compared with primary reflections, multiple waves exhibit smaller reflection angles and longer propagation paths, offering a broader illumination coverage and richer subsurface structural information [11,12].

Therefore, with the advancement in seismic exploration technology, an increasing number of researchers have attempted to utilize multiple waves as effective signals [10]. Guitton applied the one-way wave equation, using seismic data to replace the seismic wavelet and propagated multiple waves instead of traditional seismic data to obtain multiple-wave imaging results [13]. Berkhout and Verschuur converted multiple waves into primary waves and employed conventional seismic imaging methods [14]. Liu et al. integrated multiple-wave imaging with RTM based on the two-way wave equation and proposed an RTM method for multiple waves [15]. However, incorrect cross-correlations between multiple waves can substantially degrade imaging quality. To address this issue, Wong et al. combined least-squares migration with multiple-wave imaging to reduce artifacts in imaging results [16]. Zhang et al. proposed a least-squares migration for water-bottom-related multiples, which further eliminated crosstalk in multiple-wave imaging and improved the spatial resolution of the imaging results [12].

Vertical seismic profiling (VSP) refers to a seismic observation method in which the source is located at the surface, while seismic receivers are deployed in observation wells that are vertical to the ground [17]. Compared with surface seismic methods, VSP provides stronger seismic signals because the receivers are closer to the target structures and can better mitigate the effects of low-velocity layers on seismic data. Consequently, VSP offers higher-quality imaging in areas near observation wells [18]. Addressing the improvement of VSP RTM, Zhang et al. combined wavefield decomposition imaging conditions with the Poynting vector, thereby enhancing imaging quality [19]. Guo et al. significantly reduced the impact of migration noise on imaging quality by constructing dip angle gathers and estimating coverage redundancy in the dip domain [20]. Zheng et al. employed Hilbert transform to separate up-going and down-going waves, reducing the influence of low-frequency noise and artifacts on the imaging results [21]. Owing to the characteristics of the VSP observation system, seismic data received via the receivers contain a significant number of down-going waves, which are primarily composed of multiples. Therefore, mirror imaging is an effective multiple-wave imaging method for VSP. Jamali et al. converted first-order surface multiples into primary reflections above the water surface, thereby enabling VSP multiple-wave imaging [22]. Zhong et al. proposed a novel joint RTM method. By placing mirror receivers above the water surface, they converted down-going multiples into up-going waves, enabling the simultaneous imaging of both multiples and primary waves [23].

Interferometric imaging is another effective multiple-wave imaging method. Schuster et al. proposed that common propagation paths of seismic signals from the source to different receivers can be canceled under a high-frequency approximation [24]. Snieder et al. transformed passive-source data into reflected seismic data using a deconvolution algorithm [25]. Jiang et al. achieved the migration of arbitrary-order multiples by using semi-natural Green functions and interferometric imaging [26].

The utilization and imaging of multiples have been challenging. For VSP observation systems, the vertical placement of receivers renders the traditional surface seismic multiple-wave imaging methods unsuitable. Furthermore, the existing research lacks methods capable of simultaneously imaging surface and interbed multiples in asymmetric observation systems.

To achieve the joint imaging of primary waves, surface multiples, and interbed multiples in VSP observation systems, a common-receiver-based multiple-wave reverse time migration (CR-MRTM) method is proposed in this study. This approach replaces the commonly used Ricker wavelet with original seismic data containing multiple waves, which are excited at the source location and then reverse-propagated, balancing the complexity of source and receiver wavefields in traditional RTM and enhancing the quality of VSP multiple-wave imaging results. Compared to existing studies, the method proposed in this paper enables the simultaneous imaging of primary reflections and multiple waves without the need for additional signal separation, thereby enhancing the utilization of VSP seismic data. Furthermore, traditional VSP multiple-wave imaging methods typically handle only surface-related multiples, whereas the method presented here performs well for both surface-related and interlayer multiples. Theoretical models and numerical experiments demonstrate that the proposed method can simultaneously image primary and multiple waves, thereby improving the illumination range of VSP imaging. This approach partially addresses the limitation of traditional VSP observation systems, which are restricted to imaging only the vicinity of the observation well, and significantly enhances the overall quality of VSP imaging.

2. Theory and Methodology

2.1. Illumination Analysis and Artifact Formation in Conventional RTM

RTM imaging is generally considered to consist of three main steps: First, the selected seismic wavelet is used as the source for forward extrapolation to generate the complete source wavefield. Due to the presence of noise, which significantly impacts the imaging quality and consequently the accuracy of seismic wavelet extraction from the seismic data, RTM imaging typically employs artificial sources such as the Ricker wavelet [27]. Next, seismic data recorded at the receivers are used as the boundary and final conditions for back-propagation along the time direction (t = 0) to obtain the receiver wavefield. Finally, imaging results are obtained by applying specific imaging conditions to the wavefields.

Figure 1 illustrates RTM imaging in a horizontal single-layer model using a VSP observation system. The vertical observation well is represented by a green rectangle, with three geophones indicated by triangles. The solid red line represents the seismic wavelet that reaches the receiver after a single reflection at the reflection point, illustrating the wave propagation path. In contrast, the solid blue line depicts the propagation path of multiple waves, which reach the receiver after undergoing multiple reflections within the subsurface medium. Figure 1 shows that the primary reflections in a VSP observation system provide a narrow illumination range limited to the vicinity of the observation well. In contrast, multiples with longer propagation paths can carry information on a wider range of subsurface structures. Therefore, effectively utilizing multiples for imaging can partially compensate for the narrow illumination range inherent to VSP imaging.

However, due to the complexity of multiple-wave propagation paths, directly applying conventional RTM methods to process multiples often results in a large number of artifacts, significantly affecting the quality of the final imaging result. Thus, multiples are typically treated as noise and removed in RTM. Figure 2 illustrates the mechanism of artifact formation during the migration of multiples in conventional RTM. The model consists of two horizontal reflectors, with the source location marked by a red dot and the observation well represented by a green rectangle containing one receiver indicated by a triangle. The seismic wavelet is excited at the source location and propagates along the paths shown by solid lines in the figure, where blue and orange represent down and up-going waves, respectively. The wavefield at M1 is back-propagated and cross-correlated with the source wavefield at R1, forming a correct image. Simultaneously, the wavefield at M1 reflects off the horizontal layer and can be considered a secondary source. Its wavefield cross-correlates with the back-propagated wavefield at M2 to form a correct image at R2. However, if the source wavefield continues to propagate downward after reaching R1, it may cross-correlate with the back-propagated higher-order multiples at M2, forming artifacts F1 and F2, as indicated by the dashed lines in Figure 2.

2.2. Theory of Common-Receiver Reverse Time Migration

In contrast to conventional RTM methods, the common-receiver reverse time migration (CRRTM) method utilizes primary reflections instead of seismic wavelets to propagate at source locations. To ensure the validity of the results, forward propagation was replaced with back-propagation. Simultaneously, seismic wavelets are forward-propagated from the receiver location to generate a receiver wavefield. Finally, the source and receiver wavefields were calculated based on the imaging conditions to obtain the imaging results [28].

Figure 3 presents a simple model to demonstrate the effectiveness of CRRTM. The model consists of a horizontal reflector, with the source location represented by a red dot and the observation well and receiver indicated by a green rectangle and a triangle, respectively. Without a loss of generality, the source-normalized cross-correlation imaging condition is applied in this study to process the wavefields, as expressed in Equation (1) [29].

$$I\left(x,z\right)={\displaystyle \frac{\int S\left(x,z,t\right)·R\left(x,z,T_{max}-t\right)dt}{\int S\left(x,z,t\right)·S\left(x,z,t\right)dt}}.$$

(1)

According to Fermat’s principle, the actual path of elastic wave propagation in a medium is the one that minimizes the travel time from the starting point to the endpoint. This implies that the propagation path from the source to the receiver remains unchanged for the wavefield components that contribute to accurate imaging when the excitation positions of the seismic wavelet and seismic data are interchanged. That is, in both conventional RTM and CRRTM, the seismic wavelet and seismic data travel the same propagation path and take the same amount of time to travel from the source location to the reflection point R1 and from R1 to the receiver. This ensures that, in the CRRTM method, the source and receiver wavefields at the reflection point also satisfy the imaging condition described in Equation (1).

In contrast, the seismic wavelet was excited at the source location and propagated downward to the reflection point R1. This process can be expressed as follows:

$$D(z_1,z_0)=X(z_1,z_0)S\left(z_0\right)$$

(2)

Here, $D(z_1,z_0)$ represents the seismic data at location R1, $X(z_1,z_0)$ denotes the response matrix of the subsurface medium for seismic wave propagation from the source to the reflection point, and $S\left(z_0\right)$ represents the source matrix. Subsequently, the seismic data at R1 are reflected and propagated upward, where they are received at the receiver location:

$$D(z_2,z_1)=D(z_1,z_0)R\left(z_1\right)X(z_2,z_1)$$

(3)

Here, $D(z_2,z_1)$ represents the seismic data received at the receiver location, $X(z_2,z_1)$ denotes the response matrix of the subsurface medium for seismic wave propagation from the reflection point to the receiver, and $R\left(z_1\right)$ is the reflection coefficient of the reflector. The effect of the geophone on the seismic data is ignored in this context. Inserting Equation (3) into Equation (2) yields the following:

$$D(z_2,z_0)=X(z_1,z_0)S\left(z_0\right)R\left(z_1\right)X(z_2,z_1)$$

(4)

During the reverse extrapolation of the seismic data, the seismic data received at the receiver location were moved to the source location for reverse propagation. Theoretically, the response matrix of the subsurface medium is the inverse of the forward propagation matrix during reverse propagation:

$$D^{\prime }(z_1,z_0)=X^{-1}(z_1,z_0)X(z_1,z_0)S\left(z_0\right)R\left(z_1\right)X(z_2,z_1)=S\left(z_0\right)R\left(z_1\right)X(z_2,z_1)$$

(5)

Meanwhile, the seismic wavelet propagates from the receiver location, producing a wavefield, as shown below:

$$D^{\prime }(z_1,z_2)=S\left(z_0\right)X(z_2,z_1)$$

(6)

Substituting the source and receiver wavefields of CRRTM into the cross-correlation imaging condition in Equation (1) yields $R\left(z_1\right)$ at the reflection point, which represents the true reflectivity of the reflector.

At the correct reflector, the source and receiver wavefields generated via CRRTM still satisfy the imaging condition, allowing for the accurate imaging of the subsurface structure. However, for artifacts caused by incorrect cross-correlation, changes in the propagation path alter the artifact positions generated via the erroneous cross-correlation of wavefields, which are subsequently suppressed during the stacking process.

2.3. Common-Receiver-Based Multiple-Wave Reverse Time Migration

From the illumination analysis shown in Figure 1, it can be observed that using only primary reflections as effective seismic data results in a limited imaging range because of the unique characteristics of the VSP observation system. Therefore, it is necessary to utilize multiples in the VSP seismic data to enhance the illumination range of VSP imaging. In conventional RTM methods, imaging multiples directly often introduces numerous artifacts into the imaging results, thereby degrading the imaging quality. However, CRRTM can effectively suppress artifacts in imaging results. Based on CRRTM, we propose a common-receiver-based multiple-wave reverse time migration (CR-MRTM) imaging method. By exciting both primary reflections and multiples at the source location and simulating the forward propagation of the seismic wavelet at the original receiver location, this method achieves joint imaging of primary reflections and multiples. We demonstrate the effectiveness of CR-MRTM under the cross-correlation imaging condition.

The cross-correlation imaging condition was first proposed by Claerbout in 1971 [30]. The imaging results were obtained through the correlation of the source and receiver wavefields at each time step, followed by a summation of the results. Equation (1) has already presented the source-normalized cross-correlation imaging condition. For the convenience of the subsequent theoretical derivation, the cross-correlation imaging condition without source normalization can be expressed as shown in Equation (7):

$$I_{(x,z)}=\sum _{t=0}^{T_{\max }}S(x,z,t)·R(x,z,T_{\max }-t)$$

(7)

The receiver wavefield $R(x,z,T_{\max }-t)$ was obtained by back-propagating seismic data through the subsurface medium. As the original seismic data include both primary and multiple reflections, Equation (7) can be expressed as follows:

$$I_{(x,z)}=\sum _{t=0}^{T_{\max }}S(x,z,t)·\left[M_0(x,z,T_{\max }-t)+M_1(x,z,T_{\max }-t)+M_2(x,z,T_{\max }-t)+\cdots \right]$$

(8)

Here, $I_{(x,z)}$ represents the imaging value at the point $(x,z)$, $S(x,z,t)$ denotes the source wavefield generated via the seismic wavelet initiated at the source location, propagating forward through the subsurface medium, $M_0(x,z,T_{\max }-t)$ represents the wavefield formed through the back-propagation of the primary reflection from the seismic data, and $M_1(x,z,T_{\max }-t)$ and $M_2(x,z,T_{\max }-t)$ represent the wavefields formed via the back-propagation of the first- and second-order multiples, respectively. Furthermore, because seismic data undergo transmission and reflection during their propagation through the subsurface medium, Equation (8) can be further simplified as follows:

$$\begin{array}{cc}\hfill I_{(x,z)}& =\sum _{t=0}^{T_{\max }}[S(x,z,t)·\left(M_{00}(x,z,T_{\max }-t)+M_{11}(x,z,T_{\max }-t)+M_{22}(x,z,T_{\max }-t)+\cdots \right)\hfill \\ \hfill & +S(x,z,t)·\left(M_{01}(x,z,T_{\max }-t)+M_{02}(x,z,T_{\max }-t)+M_{03}(x,z,T_{\max }-t)+\cdots \right)\hfill \\ \hfill & +S(x,z,t)·\left(M_{10}(x,z,T_{\max }-t)+M_{12}(x,z,T_{\max }-t)+M_{13}(x,z,T_{\max }-t)+\cdots \right)\hfill \\ \hfill & +\cdots ]\hfill \end{array}$$

(9)

Here, $T_{\max }$ represents the maximum recording time of the seismic data, while $M_{00}$ denotes the wavefield obtained through the reverse propagation of the primary wave along its original path. Similarly, $M_{11}$ and $M_{22}$ correspond to the wavefields generated via the reverse propagation of the first-order and second-order multiple waves along their respective propagation paths. In contrast, $M_{01}$ and $M_{02}$ represent the wavefields resulting from the reflection and transmission of the primary waves along the propagation paths of the first-order and second-order multiples within the subsurface medium, which are erroneously back-propagated. The first term on the right side of Equation (9) includes the result of the cross-correlation between the seismic data propagated along its original path to the reflection point and the source wavefield. This cross-correlation contributes to the correct imaging of the subsurface reflection interfaces. However, the other terms arise from incorrect cross-correlation between the receiver and the source wavefields, leading to the formation of artifacts that significantly affect the imaging quality of RTM. Furthermore, Equation (9) also indicates that the original seismic data, which included both primary and multiple waves, exhibited a highly complex propagation path during back-propagation. Wavefield components formed by different propagation paths may interfere with each other, which, to some extent, also affects the imaging quality.

In the CR-MRTM method, seismic data, including both primary and multiple waves, are used to stimulate the source wavefield at the source location, replacing the seismic wavelets. Based on this approach, the potential for secondary reflections in the back-propagation of the seismic data can be avoided. Under these conditions, the cross-correlation imaging condition can be expressed as below, where $S_0$ to $S_3$ represent the source wavefields generated via the seismic wavelets propagating along different paths, and $M_0$ to $M_3$ denote the wavefields obtained via the reverse propagation of primary and multiple waves through the subsurface medium. In addition to the previously listed components, the true source and receiver wavefields also contain higher-order multiple wave components, which are indicated here by ellipses.

$$\begin{array}{cc}\hfill I_{(x,z)}& =\sum _{t=0}^{T_{\max }}[\left(S_0(x,z,t)+S_1(x,z,t)+S_2(x,z,t)+S_3(x,z,t)+\cdots \right)\hfill \\ \hfill & ·\left(M_0(x,z,T_{\max }-t)+M_1(x,z,T_{\max }-t)+M_2(x,z,T_{\max }-t)+\cdots \right)]\hfill \end{array}$$

(10)

In CR-MRTM, originally complex propagation paths are split into source and receiver wavefields, significantly reducing the potential for interference between the wavefield components. In addition, because seismic data containing subsurface structural information no longer undergo additional reflections during back-propagation, this information is preserved to the greatest extent.

Seismic data propagated via CR-MRTM include both primary and multiple waves, eliminating the need for multiple-wave prediction or separation. Furthermore, because the CR-MRTM does not interfere with the propagation paths of seismic waves through the subsurface medium, it can simultaneously handle both surface and interbed multiples. In the next section, we provide a more detailed explanation of the effectiveness of CR-MRTM in multiple-wave imaging, specifically in terms of noise suppression and illumination coverage enhancement, based on simulation experiments.

3. Numerical Simulation

In this section, we demonstrate the effectiveness of CR-MRTM using a simple three-layer model and the Marmousi model, and we discuss its advantages over conventional RTM methods.

In the numerical experiments presented in this section, we used the staggered-grid finite difference method to solve the wave equation. The specific first-order velocity–stress equation for elastic waves is given via the following formula [31,32]:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial v_x}{\partial t}}={\displaystyle \frac{1}{\rho }}\left({\displaystyle \frac{\partial {\tau }_{xx}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{xz}}{\partial z}}\right)\hfill \\ {\displaystyle \frac{\partial v_z}{\partial t}}={\displaystyle \frac{1}{\rho }}\left({\displaystyle \frac{\partial {\tau }_{xz}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{zz}}{\partial z}}\right)\hfill \\ {\displaystyle \frac{\partial {\tau }_{xx}}{\partial t}}=(\lambda +2\mu ){\displaystyle \frac{\partial v_x}{\partial x}}+\lambda {\displaystyle \frac{\partial v_z}{\partial z}}\hfill \\ {\displaystyle \frac{\partial {\tau }_{zz}}{\partial t}}=(\lambda +2\mu ){\displaystyle \frac{\partial v_z}{\partial z}}+\lambda {\displaystyle \frac{\partial v_x}{\partial x}}\hfill \\ {\displaystyle \frac{\partial {\tau }_{xz}}{\partial t}}=\mu \left({\displaystyle \frac{\partial v_x}{\partial z}}+{\displaystyle \frac{\partial v_z}{\partial x}}\right)\hfill \end{array}\right.$$

(11)

Here, $v_x$ and $v_z$ represent the horizontal and vertical particle velocities, respectively; ${\tau }_{xx}$, ${\tau }_{xz}$, and ${\tau }_{zz}$ are the components of the stress tensor, $\rho$ is the medium density, and $\lambda$ and $\mu$ are the Lamé parameters, which are determined by the medium’s shear wave velocity, compressional wave velocity, and density.

In addition, for cases in which boundary reflections must be eliminated, perfectly matched layer (PML) techniques were utilized for the numerical experiments in this study. After the imaging results were obtained, a Laplace filter was uniformly applied to process them and mitigate the effects of low-frequency noise.

3.1. Simple Three-Layer Model

Figure 4 shows the simple three-layer model used to test the performance of CR-MRTM. The model consisted of 2000 grid points in the horizontal direction and 600 grid points in the vertical direction, with both horizontal and vertical grid sizes of 5 m. In addition, a PML with a grid size of 60 was added around the model to eliminate boundary reflections during seismic wave propagation. The region outside the yellow dashed box in Figure 4 represents the PML.

The setup of the VSP observation system is as follows. The system included 960 sources with a fixed distance of 10 m between adjacent ones. And all sources were positioned at a depth of 200 m. The positions of the left- and rightmost source points were 200 m and 9800 m, respectively. The vertical observation well was located at a horizontal position of 1500 m, with 300 receiver points spaced 5 m apart. The upper- and lowermost receiver points were located at 500 m and 2000 m, respectively. The positions of the source and receiver points are marked with red and blue solid lines, respectively, in Figure 4. In numerical experiments, a Ricker wavelet with a central frequency of 20 Hz was used as the excitation source, with a maximum recording time of 5 s and a time sampling interval of 0.5 ms. These parameters were selected to ensure that the recorded seismic data accurately reflect the subsurface medium while minimizing computational resource consumption. Furthermore, the parameters must avoid numerical dispersion and ensure stability when simulating seismic wave propagation using the finite difference method. Based on the numerical experiments, we conclude that the chosen parameters meet the required criteria. For typical scenarios and the model selected, the grid size, time sampling interval, and central frequency of the seismic wavelet must satisfy the following equations [32]:

$$\mathrm{\Delta }tmax\left(v\right)\sqrt{{\displaystyle \frac{1}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{1}{\mathrm{\Delta }{z}^2}}}\le C$$

(12)

$$\mathrm{\Delta }t={\displaystyle \frac{min\left(v\right)}{max\left(f\right)·\mathrm{\Delta }x}}$$

(13)

Here, $\mathrm{\Delta }t$ represents the time sampling interval, $\mathrm{\Delta }x$ is the grid size, f is the seismic wave frequency, v is the wave velocity, and the value of C can be obtained from the tables. In the numerical experiments presented in this study, C is 0.7774.

Figure 5a shows the seismic data corresponding to the source located at $x=500$ m. To highlight the experimental results, original seismic data were separated in order to ensure that only the compressional (P-wave) component was included as input. The portion of the seismic data within the red dashed box is enlarged in Figure 5b. In addition to the direct wave and the first reflection wave, significant multiple reflections can be observed in the data. As the model is surrounded by PML layers, these multiples only include interbed multiples, excluding surface multiples. In conventional RTM imaging methods, these multiples manifest as artifacts in the imaging results, significantly affecting the quality of the images.

Figure 6 shows the image obtained using the conventional RTM method. Given the setup of the velocity model and the position of the observation system in our numerical experiments (the velocity model contains a large-scale anticline, and the observation well covers a limited depth range), it can be inferred that most of the subsurface structural information cannot reach the receiver positions through primary reflections alone. As indicated by the dashed box in Figure 6, the imaging results from the conventional RTM method show significant interference from artifacts for our velocity model. Additionally, the magnified section within the blue solid line box, displayed at the top of Figure 6, highlights the limited illumination area provided via the conventional RTM method.

In contrast to conventional RTM imaging methods, CR-MRTM requires the extraction of seismic data from different sources for the same receiver point and their reorganization into a common-receiver gather. Seismic data are then excited at the source positions, while the seismic wavelet is propagated forward at the receiver positions to obtain the imaging results. Figure 7 shows the imaging results obtained using CR-MRTM. It is evident that, compared to the conventional RTM method shown in Figure 6, our proposed approach significantly improves the effective imaging range, as illustrated by the section within the blue solid line box in Figure 7. The magnified result for this area is displayed at the top of Figure 7. Additionally, our method effectively suppresses the artifacts present in conventional RTM imaging. Furthermore, because the numerical experiment includes PML layers around the velocity model, input seismic data do not contain surface multiple reflections. Therefore, it can be demonstrated that the proposed method effectively handles interbed multiples in seismic data.

To further illustrate the impact of CR-MRTM on imaging quality, we extracted the imaging results of both CR-MRTM and conventional RTM methods at a depth of 1 km, as shown in Figure 8. The reference reflectivity, indicated by the solid green line, is calculated based on the velocity model and other relevant information. It is clearly evident that, compared to the conventional RTM method, the imaging results obtained using CR-MRTM are much more consistent with the reference reflectivity. Furthermore, we designed an evaluation criterion to assess the discrepancy between the imaging results and the reference reflectivity. Specifically, we referenced and adjusted the calculation method for the Peak Signal-to-Noise Ratio (PSNR) to ensure that the resulting values are not influenced by errors introduced during the wavefield propagation process. The signal-to-noise ratios at a 1 km depth for both CR-MRTM and conventional RTM methods are presented in

Table 1. Imaging quality of different RTM methods, evaluated based on the signal-to-noise ratio.

Method	SNR(db)
conventional RTM	18.6
CR-MRTM	22.5

.

3.2. Marmousi Model

We further demonstrate the effectiveness of CR-MRTM in complex scenarios using the Marmousi model, as well as the differences in the imaging quality between CR-MRTM and conventional RTM methods. The velocity model is illustrated in Figure 9. The model consisted of 2720 grids in the x direction and 560 grids in the z direction, with a grid spacing of 5 m in both directions. A 500-meter-wide PML layer was added around the model to absorb boundary reflections. The region outside the dashed yellow box in the figure represents the PML.

The VSP observation system was set up as follows. The observation well is located at $x=3000$ m, with 500 receiver points uniformly spaced every 5 m. The depth of the uppermost receiver point was 300 m. The source depth was uniformly set to 250 m, with 1335 sources uniformly distributed every 10 m. The positions of the sources and receivers are marked by the red and blue solid lines, respectively, in the figure. We used a Ricker wavelet with a central frequency of 40 Hz to generate the VSP seismic data. The maximum recording time was 6 s, and the sampling interval was 0.5 ms. Only P-waves were retained in the seismic data generated.

We first applied the conventional RTM method to image the seismic data corresponding to the 500 sources on the left side. The results are shown in Figure 10. The solid red line indicates the distribution of the sources involved in the imaging. When the offset distance is small, the energy in the seismic data is primarily composed of the first reflection wave. In this case, the reflective layers within the illuminated region can be effectively imaged using the conventional RTM method.

As the number of sources involved in the imaging continues to increase, the imaging results using the conventional RTM method are shown in Figure 11. Figure 11a presents the imaging result obtained by stacking the data from the 900 sources on the left side, while Figure 11b shows the result from stacking all sources. As the offset distance increases, seismic data from sources farther from the observation well exhibit a significant increase in multiples. The conventional RTM method is not effective at handling these multiples. Therefore, compared with the results in Figure 10, the imaging range in Figure 10 does not show significant improvement, and the imaging results contain more artifacts, resulting in a lower imaging quality.

The results obtained using CR-MRTM are shown in Figure 12. Compared with the conventional RTM method, our approach provides a larger illumination range and higher imaging energy. In the region indicated by the blue solid line box, our method clearly highlights the steep structural features present in the original velocity model, with the magnified results displayed on the right side of Figure 12. Additionally, in the area near the observation well, our method also provides a higher imaging quality.

Furthermore, to demonstrate that the CR-MRTM is also effective for surface multiples, we removed the PML layer from the upper part of the velocity model and regenerated the VSP seismic data for imaging. Regenerated VSP seismic data contain surface multiples, with an energy significantly greater than that of the interbed multiples. The imaging results using the regenerated seismic data are shown in Figure 13. Compared with the previous experimental results, high-quality imaging results can also be obtained using surface multiples. Furthermore, as shown in the blue solid line box area, the illumination range of the imaging results is further enhanced compared with using only interbed multiples. The magnified results for this region are shown at the bottom of Figure 13. The experiment showed that our method can effectively and simultaneously handle first-arrival waves, surface multiples, and interbed multiples.

4. Discussion

Based on the use of the CR-MRTM method proposed in this study, surface and interbed multiples can be effectively imaged in a VSP observation system without the need for the signal separation of seismic data. Our method improves the effective illumination range of VSP imaging, thereby aiding subsequent tasks such as oil and gas reservoir exploration.

Similar to other RTM methods, the approach proposed in this study also requires solving the two-way wave equation to compute the forward- and back-propagating wavefields. Therefore, to ensure the final imaging quality, the proposed method requires an accurate velocity model. Future research could focus on the construction of accurate velocity models, such as integrating physics-informed neural networks with full-waveform inversion, which would improve the accuracy of full-waveform inversion results [33].

Moreover, current RTM studies predominantly employ artificial sources, such as Ricker wavelets, as signal sources, which help mitigate the contamination of seismic data using noise. However, utilizing wavelets with greater randomness as seismic signals can further enhance the simulation of seismic wave propagation in the subsurface medium. Genovese and Palmeri proposed a method that uses circular harmonic wavelet transforms to randomly generate artificial accelerations, which more accurately reflect the non-stationarity of seismic motions [34]. Therefore, the use of random seismic wavelets in reverse-time migration imaging methods represents a promising area for future research.

In addition, the quality of seismic data also significantly affects the final imaging quality, particularly when the elastic wave equation is used to simulate the wavefield propagation process. This is also one of the key factors influencing the RTM imaging quality in practical applications. Although the proposed method can effectively image multiples in seismic data, research on signal separation and noise removal remains necessary. Furthermore, in the case of elastic waves, the wavefield consists of a significant number of converted shear waves that also contain a large amount of information regarding the subsurface structure. In existing research, converted waves are typically treated as noise and removed. Therefore, exploring how to effectively utilize converted waves for imaging is a promising research direction.

5. Conclusions

Compared to surface seismic surveys, VSP offers advantages such as lower acquisition costs, a higher signal-to-noise ratio, and better resolution. However, it also faces challenges, including a limited imaging range and difficulty in removing artifacts. In response to the difficulty in efficiently utilizing multiples in VSP seismic data in existing research, the challenges faced by conventional RTM methods when processing multiples were analytically examined in this study. One key issue was identified, that is, the presence of significant artifacts during the imaging process. To address this issue, a CR-MRTM method was proposed that replaces seismic wavelets with seismic signals at source locations to achieve the joint imaging of primary reflections, surface multiples, and interbed multiples in complex VSP data. Compared to previous studies, the method we propose effectively utilizes both primary reflections and multiple waves from VSP seismic data, significantly improving imaging quality and the effective imaging range without a substantial increase in computational cost. This approach overcomes the limitation of traditional VSP reverse-time migration, which is restricted to imaging only the vicinity of the observation well. Furthermore, our method eliminates the need for additional seismic signal separation, enabling the joint imaging of primary reflections and multiple waves and thereby enhancing the stability and efficiency of the imaging process. The proposed method demonstrates excellent performance for both interlayer multiples and surface-related multiples, addressing the shortcoming of conventional VSP multiple-wave imaging methods, which typically only image surface-related multiples.