Efficient Method for Enhancing Reverse-Time Migration Images Using Vertical Seismic Profiling Data

Abstract

Featured Application

The work reported in this paper contributes to the further improvement of the reverse-time migration imaging accuracy of VSP data, which has broad application prospects for oil and gas exploration of unconventional reservoirs.

Abstract

Vertical seismic profiling has garnered widespread attention in the industry as a supplement to seismic exploration due to its higher data quality compared to surface seismic data. However, its unique observation system in which geophones are only distributed within observation wells results in uneven coverage of subsurface structures. This can lead to significant noise when directly applying conventional reverse-time migration techniques used in surface seismic imaging. This study addresses the issue of noise suppression in reverse-time migration imaging associated with walk-away vertical seismic profiling and presents two main innovations. First, a common-receiver reverse-time migration imaging method is proposed, which uses the observation signals as excitation signals for the corresponding shots after reverse-time processing. Second, an excitation-time-constrained cross-correlation imaging condition is introduced to eliminate non-contributing portions of the wavefield, thereby modifying the traditional cross-correlation imaging condition to include an excitation time constraint. The combination of these methods enhances imaging quality by effectively suppressing noise, as demonstrated through theoretical analysis and numerical simulations with synthetic models.

1. Introduction

Seismic data processing methods play crucial roles in the exploration of oil and gas resources. The results of seismic data processing provide a foundation for subsequent tasks, such as hydrocarbon reservoir identification and seismic inversion [1,2]. Among these techniques, seismic imaging is particularly significant because it produces high-resolution images of the true reflectivity of subsurface structures. As such, it is regarded as one of the most effective methods for mapping underground structures and properties using seismic data [3,4].

Traditional seismic migration methods, such as ray-based migration and one-way wave equation migration, typically depend on approximate solutions to the wave equation. These methods often perform inadequately when dealing with complex geological structures and may even exhibit instability [5]. By contrast, reverse-time migration (RTM) is a migration method based on the two-way wave equation. The advantages of the two-way wave equation in accuracy and reliability allow waves to propagate in all directions, making them less constrained by the velocity model and propagation direction [6]. Compared to traditional migration methods, RTM is capable of accurately handling complex geological structures, including steeply dipping layers, and is theoretically the most precise among all seismic migration techniques [7,8,9].

However, the quality of an RTM image is heavily influenced by both the quality of the seismic data and the imaging method used. A good standard for evaluating seismic imaging methods is fidelity, which is the ability to map the target features to their correct locations [10]. Despite significant advancements in seismic data acquisition technology in recent years, the quality of seismic data remains susceptible to various types of noise. This challenge is particularly pronounced in academic research, where resources are more limited compared to industrial applications [11,12]. Therefore, it is essential to preprocess raw seismic data to meet the requirements of subsequent migration imaging. Li et al. proposed using singular value decomposition (SVD) filtering to attenuate direct and refracted waves in seismic records, thereby highlighting the shallow reflection wavefield [13]. Canales [14] utilized f-x domain predictive filtering to suppress random noise in seismic data. Another effective noise suppression technique is median filtering, which can efficiently remove salt-and-pepper noise present in seismic data [15]. Besides improving seismic data quality, there is also substantial research focused on optimizing imaging methods to enhance the quality of seismic images. For instance, separating the upgoing and downgoing waves in a wavefield can reduce the low-frequency noise in the imaging results [16,17]. Ha [18] separated P-waves and S-waves in elastic wavefields. Lee [19] proposed a reconstructed wavefield RTM algorithm using a backward-reproduced wavefield instead of back-propagating the observed data to achieve higher-quality imaging results. Lee and Pyun [6] effectively reduced artifacts in RTM by combining de-primary RTM algorithms and impedance matching, resulting in clearer imaging outcomes. Zhong et al. introduced a joint RTM (JRTM) method that improves the quality of the imaging results by simultaneously using both upgoing and downgoing waves [20].

In the RTM process, after obtaining the wavefield, different imaging conditions (ICs) are applied to the image space to generate a migrated image of the subsurface structures. The choice of the IC significantly influences the final quality and fidelity of the RTM image. The initial ICs were divided into excitation time and cross-correlation imaging conditions. The excitation time imaging condition does not require the storage of the entire wavefield but requires computation of the excitation time [21]. This excitation time is typically the first-arrival time or the time of maximum amplitude and can also be calculated using ray-tracing methods [22,23].

The cross-correlation imaging condition, first proposed by Claerbout, provides stable structural images [24,25]. To address the lack of a clear physical relationship between the cross-correlation imaging results and reflection coefficients, the normalization of the imaging results using the source or receiver wavefields can be employed to obtain results that correlate with the true reflection coefficients [26].

Building on these ICs, Arntsen et al. proposed the deconvolution imaging condition [27], Neto et al. introduced a time–space shift imaging condition [28], and Liang et al. developed an improved backscattering imaging condition to separate the effects of impedance and velocity perturbations on reflectivity [29].

The primary objective of vertical seismic profiling (VSP) is to image small-scale structures effectively and provide high-quality imaging results in the vicinity of an observation well [30]. In VSP observation systems, the seismic source is typically positioned horizontally on the surface, while geophones are placed vertically within the observation well. This configuration allows the geophones to be closer to the target layers, thereby reducing the influence of low-velocity layers on seismic data. Consequently, VSP seismic data generally exhibit a higher quality compared to conventional surface seismic data. However, the unique characteristics of the VSP system lead to good coverage only in the vicinity of the observation well, with significantly uneven coverage across the complete subsurface structure. This uneven coverage can introduce substantial noise when traditional surface seismic imaging methods are directly applied.

Noise suppression in migration imaging has been a longstanding challenge. Moreover, existing research has not provided effective methods for suppressing noise generated during the migration imaging process under asymmetric observation systems. To address this issue and achieve effective noise suppression in migration imaging within VSP systems, this study proposes a common-receiver RTM (CRRTM) imaging method. By shifting the observation signals to their source locations, this method suppresses the noise caused by the uneven coverage characteristics of the VSP observation system. In addition, we propose an excitation time-constrained cross-correlation imaging condition (ETC-IC). By controlling the effective time range within the cross-correlation imaging condition, this method suppresses the imaging noise caused by multiple reflections. The analysis of theoretical models and simulation experiments demonstrates that the method proposed in this paper can effectively suppress noise generated during VSP migration imaging. Comparative experiments show that the common-receiver point imaging method offers an approximately 12% improvement in image quality compared to traditional methods.

The subsequent chapters of this paper are organized as follows:

Section 2 analyzes the theoretical causes of migration noise in the RTM imaging process and demonstrates the rationale and effectiveness of the CRRTM and the ETC-IC through theoretical proofs.

Section 3 starts with a simple model and employs simulation experiments to illustrate how noise affects imaging quality in traditional VSP migration imaging methods and how the proposed method suppresses noise.

Section 4 presents specific experimental results to compare the imaging outcomes of the traditional VSP RTM method with those of the CRRTM method.

Section 5 contains the discussion.

Section 6 provides the conclusion.

2. Theory and Methodology

In this section, we provide a detailed theoretical analysis to explain the principles of CRRTM. Additionally, we demonstrate how this method can accurately image subsurface structures.

2.1. Causes of Noise in RTM

Traditional RTM imaging methods generally consist of three steps. First, an appropriate forward-modeling method is chosen to propagate the seismic wavelet forward and obtain the complete source wavefield. Second, the seismic records obtained from the receivers are used as boundary and final conditions. Then, back-propagation is performed to reconstruct the complete receiver wavefield. Finally, based on certain imaging conditions, the imaging results for a single source-receiver pair are obtained. The final imaging result is achieved by stacking the imaging results of all source-receiver pairs.

In RTM, in addition to reflections from the interfaces, certain types of noise are generated, primarily low-frequency artifacts and arc noise. This is caused by the incorrect cross-correlation of the source and receiver wavefields in regions beyond the reflection interfaces. In surface seismic imaging, this noise can be attenuated by stacking, resulting in high-quality imaging results. However, for VSP imaging, the unique characteristics of the observation system result in the receivers being distributed only within the preferred region of the observation well. Consequently, single imaging results corresponding to different sources exhibit a strong correlation. This indicates that noise near the observation well cannot be effectively attenuated during stacking. In addition, because this noise is located around the observation well and has the same frequency as the correct reflection events, it severely affects the quality of the imaging results.

To address this issue, this study proposes a common-receiver algorithm. By altering the excitation location of the wavefield, the generated noise is shifted to the sides of the effective imaging area, thereby reducing its impact on the imaging quality within the illumination range in VSP RTM imaging. This improvement enhances the overall quality of the imaging results.

2.2. Theory of CRRTM

In contrast to traditional RTM imaging, CRRTM requires a different approach, as illustrated in Figure 1. In this method, the seismic data obtained at the receivers are shifted to the corresponding source locations before back-propagation is performed. Conversely, the seismic wavelet must be forward-propagated from the receiver locations. Finally, the wavefields obtained from these two propagations are used to compute the imaging results under specific imaging conditions.

To demonstrate the rationality of the CRRTM, we use the model shown in Figure 2 and the zero-lag cross-correlation imaging condition as an example. The model includes a horizontal reflective layer, with the observation well represented by a green rectangle containing a single receiver R1 depicted by a triangle. Given that the propagation speeds of P-waves (primary waves) and S-waves (secondary waves) differ significantly in most cases, we simplify our discussion by considering only the propagation of P-waves. The case of S-waves is addressed in subsequent sections.

In traditional RTM imaging, the seismic wavelet propagates from the source to the reflection point at time $t_1$ and then from the reflection point to the receiver at time $t_2$. This process is illustrated in Figure 2 by red and blue lines, respectively. If the maximum recording time at the receiver is $T_{max}$, the part of the seismic data recording the reflection point information is located at $t_1+t_2$. During reverse-propagation, these data are excited by the receiver at time $T_{max}-\left(t_1+t_2\right)$. After propagation for time $t_2$, it reaches the reflection point at time $T_{max}-t_1$. Therefore, at the reflection point, the source and receiver wavefields satisfy traditional imaging conditions.

In contrast, for CRRTM imaging, the seismic data recorded at the receiver are moved to the source location for back-propagation. The time required for the reverse-propagated seismic data to return to the reflection point also differs. According to Fermat’s principle, a seismic signal reverse-propagated from the source travels along the same path as the initial seismic wavelet from the source to the reflection point. Time $t_1$ is required to reach the reflection point. Thus, the seismic data arrive at the reflection point at time $T_{max}-t_2$. Simultaneously, the seismic wavelet propagating from the receiver also requires time $t_2$ to reach the reflection point according to Fermat’s principle. By ensuring that the wavefields met the imaging conditions at the reflection point, CRRTM was validated for the accurate imaging of subsurface structures.

Therefore, at the reflection point, the wavefields satisfied the imaging conditions for both methods. This demonstrates that the common-receiver point imaging method can accurately image subsurface structures.

2.3. Theory of ETC-IC

The choice of imaging conditions also significantly influences the final imaging results. Currently, the imaging conditions used in pre-stack RTM imaging are primarily of two types: excitation time and cross-correlation.

The cross-correlation imaging condition was first proposed by Claerbout (1971) [25]. This involves performing zero-lag cross-correlation between the source and receiver wavefields at each time step to obtain the imaging result for each point. The specific imaging condition is as follows:

(1)$I_{(x,z)}=\int S\left(x,z,t\right)·R\left(x,z,T_{max}-t\right)dt,$

where $S\left(x,z,t\right)$ and $R\left(x,z,t\right)$ represent the source and receiver wavefields at location $(x,z)$ and time t, respectively, t represents time, and $T_{max}$ represents the maximum recording time at the receiver. From a numerical perspective, the result of the cross-correlation imaging condition is the product of the energies of the source and receiver wavefields at the imaging point. However, this product does not have any direct physical significance. To address this issue, a source-normalized cross-correlation imaging condition can be used as follows:

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

By compensating for the attenuation of seismic waves during propagation, the imaging results of the source-normalized cross-correlation imaging condition become more interpretable and accurately reflect subsurface features, thereby improving the reliability of seismic imaging.

Another widely used imaging condition is the excitation-time imaging condition. This method does not require storing the entire source wavefield but does require obtaining the first-arrival travel time from the source wavefield to each imaging point. During back-propagation of the receiver wavefield, the amplitude at the first-arrival travel time is taken as the imaging result for that point. The specific process is defined as follows:

(3)$I_{(x,z)}=R\left(x,z,T_{max}-t\right),$

where t represents the travel time of the first arrival from the source wavefield to the imaging point.

The excitation-time imaging condition can be considered a special case of the cross-correlation imaging condition. Because the source wavefield is not stored, its value in the cross-correlation imaging condition is replaced with a unit amplitude. In this case, the excitation-time imaging condition can be expressed as

(4)$I_{(x,z)}={\int }_t^{t+ϵ}R\left(x,z,T_{max}-\tau \right)·1d\tau ,$

where $ϵ$ represents an arbitrarily small number and t represents the first-arrival travel time from the source wavefield to the imaging point.

In the RTM process, only a few sources generate effective seismic records that are reflected back to the receiver position for each imaging point. Most sources do not contribute to the imaging. In surface seismic methods, this does not result in the loss of imaging quality because the sources and receivers are randomly distributed. However, because of the specificity of the VSP observation system, multiple imaging processes can lead to noise accumulation. Therefore, it is necessary to introduce additional methods to eliminate the noise effects.

Based on the above analysis, it is essential to propose optimized imaging conditions suitable for WVSP RTM. Building on the two imaging conditions mentioned earlier, this study proposes an excitation-time-limited cross-correlation imaging condition expressed as follows:

(5)$I_{ETC-IC}\left(x,z\right)={\displaystyle \frac{{\int }_{t-\Delta }^{t-\Delta +T}S\left(x,z,t\right)·R\left(x,z,T_{max}-t\right)dt}{{\int }_{t-\Delta }^{t-\Delta +T}S\left(x,z,t\right)·S\left(x,z,t\right)dt}}$

Here, T represents the period of the seismic wavelet used, and $Delta$ is a customizable window to balance the potential errors in the first-arrival travel times.

The ETC-IC retains only portions of the source and receiver wavefields that are related to the first-arrival travel times of the seismic wavelet to each imaging point, typically within one period. This approach effectively excludes parts of the wavefield that do not contribute to imaging, such as noise from multiples, while significantly reducing the amount of data that must be stored, thereby alleviating computational resource demands.

In the next section, we analyze noise generation in traditional RTM and CRRTM methods using a simple model. This will demonstrate the noise-suppression effectiveness of CRRTM.

3. Numerical Analysis

3.1. Noise Analysis of W-VSP RTM

This section uses a simple model as an example to demonstrate the sources of noise in traditional VSP imaging and how noise affects the final imaging quality. The seismic data used in this study were calculated using the finite-difference method, retaining only the upgoing P-wave components and setting a perfectly matched layer (PML) to eliminate boundary reflections.

Here is the actual velocity model used, along with the VSP observation system. The model consists of $400\times 600$ grids with vertical and horizontal intervals of 5 m. Additionally, a 100-grid PML was added around the model to absorb boundary reflections. The velocity within the absorption layer was consistent with the velocity at the initial model boundary. The model contained a flat layer at $z=1500$ m, and the S-wave and P-wave velocities were constant between the different layers. The density at all the positions in the model was 2000 kg/m³. Figure 3 shows the velocity model with the PML absorption layer added, where the area outside the yellow dashed box represents the PML.

The VSP observation system used was as follows. There were 300 sources spaced at fixed intervals of 10 m, located at the surface. A vertical observation well was positioned at $x=2000$ m, containing 120 numerical geophones evenly distributed from $z=800$ m to $z=1400$ m at intervals of 5 m. A Ricker wavelet with a central frequency of 20 Hz was used as the excitation source. The maximum recording time was 3 s with a sampling interval of 0.5 ms. The arrangement of sources and receivers ensured stable seismic data acquisition. The spatial and temporal intervals, along with the central frequency of the seismic wavelet, were optimized for computational efficiency and result quality. Too small an interval drastically reduces computational efficiency, while too large an interval leads to numerical dispersion and other issues. These parameters must also ensure the stability of the finite difference method.

Figure 4a shows a shot gather of the seismic data received by the geophones at $z=800$ m. After the signal separation, only the upgoing P-waves were retained for subsequent imaging, as shown in Figure 4b.

In traditional W-VSP RTM methods, the seismic data recorded by the receiver are back-propagated, whereas the seismic wavelet is forward-propagated from the source position to generate the source wavefield. Figure 5 illustrates the cross-correlation between the source and receiver wavefields at various time intervals.

During the cross-correlation process between the source and receiver wavefields, the direct wave of the source wavefield and reflected wave of the receiver wavefield exhibit a strong correlation along the entire propagation path. This is a major source of severe low-frequency noise in the imaging results. Simultaneously, the source wavefield is incorrectly correlated with the direct wave of the receiver wavefield, leading to arc-like artifacts near the observation well in the imaging results. Unlike low-frequency noise, these artifacts have the same frequency as the effective part of the signal, which makes them difficult to remove using methods such as Laplace filtering. Figure 6 shows the single-shot imaging results of traditional W-VSP RTM.

3.2. Analysis of Noise Suppression by CRRTM

Unlike traditional RTM, CRRTM relocates seismic records to the source location for excitation. Consequently, for each receiver point, it was necessary to extract and save the seismic data corresponding to each source. Figure 7a shows the seismic data received by the receiver at $z=800$ m in the observation well. Figure 7b shows the upgoing P-waves retained after signal separation.

After organizing the seismic data into a common-receiver gather, as shown in Figure 7b, the seismic records were excited at the source location. Simultaneously, the seismic wavelet was forward-propagated at the receiver location. Subsequently, the resulting wavefields were cross-correlated with obtain imaging results. The specific cross-correlation between the source and receiver wavefields is shown in Figure 8.

It is evident that CRRTM effectively relocates migration artifacts, which are otherwise challenging to remove using conventional methods, to the edges of the image. These regions fall outside of the scope of theoretical imaging and, thus, do not affect the areas of interest. Figure 9 illustrates the single-image results obtained via CRRTM.

Unlike acoustic waves, the S-wave components in elastic waves often provide higher-resolution imaging results owing to their lower propagation speeds. CRRTM can also be applied to S-wave imaging. This principle is similar to that of compressional waves, which require the seismic wavelet propagated at the receiver location to be set as an S-wave source. Figure 10 shows the cross-correlation between the source and receiver wavefields when S-waves are used for imaging.

When shear waves are used for imaging, the significant difference in propagation speed between S- and P-waves leads to a weaker correlation between the source and receiver wavefields. This results in a reduction in low-frequency noise in the imaging results. However, owing to the same propagation speed disparity, the direct wave from the source wavefield and the reflected wave from the receiver wavefield correlate incorrectly, creating additional arc noise. These artifacts appear around the observation well and accumulate over multiple imaging iterations. Figure 11 shows a single imaging result obtained using S-waves.

3.3. Analysis of Noise Suppression by ETC-IC

ETC-IC improves the quality of imaging results by modifying the integration limits of the traditional cross-correlation imaging conditions. This method reduces the influence of parts of the wavefield that do not contribute to the imaging process. Specifically, this involved truncating the source and receiver wavefields based on the first-arrival travel times at each point in the model. Figure 12 shows the results of applying the first-arrival travel time limits to the receiver wavefield in the CRRTM at different depths.

Apart from the correct reflection axis, most of the other parts of the wavefield were excluded by the excitation time, thereby not affecting the final imaging results.

4. Results

4.1. Simple Model

This section presents the experimental results using the model from the previous section and compares CRRTM with the traditional W-VSP RTM method. All parameters in the model and observation system were consistent with those described in the previous section. All data presented in this section underwent the same processing to eliminate the effects of low-frequency noise and areas outside the theoretical imaging range.

Figure 13 presents the final imaging results obtained using the conventional W-VSP RTM method. As shown in the figure, the traditional method exhibits significant migration arches around the observation well. Additionally, severe noise is observed at the locations of the observation well.

To further suppress the noise in the traditional W-VSP RTM method, we modified the imaging conditions for ETC-IC. This approach requires obtaining the first-arrival travel times from the receiver points to each grid point in the model. There are two primary methods for selecting the first-arrival travel times: the maximum amplitude method and ray-tracing method. The first-arrival travel times obtained by these two methods are not identical; the maximum amplitude method generally yields slightly shorter travel times than the ray-tracing method. In this study, we primarily used the ray-tracing method to obtain the first-arrival travel time; the final imaging results obtained using the ETC-IC are shown in Figure 14.

When using the ETC-IC, the noise around the observation well was significantly reduced. This noise primarily originates from seismic data because its generation time differs from the first-arrival travel times of the seismic waves. Hence, the ETC-IC effectively eliminates this noise. However, modifying the imaging conditions did not significantly reduce the arc noise around the observation well.

Furthermore, when employing CRRTM, the migration artifacts were relocated to the sides of the effective imaging range. Consequently, in areas of greater interest, the quality of the imaging results was notably enhanced. The imaging results obtained using CRRTM are shown in Figure 15.

We evaluated the imaging quality by comparing the amplitude of the energy within the effective reflection axis with the total energy amplitude in the final imaging results obtained using the three methods. The results are presented in Table 1. These findings indicate that CRRTM effectively suppresses noise, thereby enhancing the imaging quality.

Furthermore, CRRTM can also be utilized for S-wave imaging, and the final imaging results are depicted in Figure 16. It is observed that, when S-waves are employed for imaging, the resulting images exhibit higher precision.

4.2. Modified Marmousi Model

This section employs the Marmousi model to demonstrate the imaging performance of CRRTM in complex scenarios, along with its distinctions from the traditional W-VSP RTM.

Figure 17 illustrates the model utilized, which was composed of $1240\times 2040$ grid points with vertical and horizontal intervals of 2.5 m between grids. The model was surrounded by PML layers comprising 60 grid points on each side. In addition, to mitigate unnecessary strong reflections, minor smoothing of the model was necessary. In this study, a Gaussian filter with sigma = 0.5 was used to smooth the model.

The VSP observation system utilized is described as follows: 480 seismic sources were employed, evenly spaced at 10 m intervals on the surface. The observation well was positioned at x = 2500 m, with 280 numerical receivers distributed along it, spaced at 10 m intervals. VSP seismic data were generated using a Ricker wavelet with a frequency of 20 Hz, with a maximum recording time of 3 s, and a time step interval of 0.25 ms between each time step.

Figure 18 shows the common-receiver point-gathers used in the experiment.

Figure 19 illustrates the final imaging results obtained using CRRTM. This serves as evidence of the applicability of this method to complex models.

5. Discussion

The CRRTM method proposed in this paper effectively images seismic data in VSP observation systems while maintaining a low noise level in the final imaging results. This improvement is advantageous for subsequent applications, such as hydrocarbon reservoir exploration.

Similar to traditional W-VSP RTM imaging methods, CRRTM is sensitive to the accuracy of the velocity model. Future research could focus on developing more accurate velocity models, such as through full-waveform inversion methods. Furthermore, with the continuous advancements in machine learning technologies, there is potential to integrate velocity inversion with machine learning to achieve even more precise velocity models (Lu and Zhang 2023) [31].

The quality of seismic data and wavefields plays a significant role in determining imaging quality, particularly when using elastic waves as the excitation source. This is also a critical factor influencing the quality of RTM imaging in practical research. Unlike acoustic waves, elastic waves comprise both P-waves and S-waves, which makes signal separation more complex. Therefore, it is essential to remove noise from the raw seismic data, such as direct waves and multiples. Additionally, separating S-waves and P-waves in the raw seismic data is necessary to enhance imaging accuracy. Currently, most signal separation methods damage the shallow reflection part of seismic data for direct wave removal. The methods used for wave separation also lead to a loss in seismic data quality. Therefore, implementing high-quality seismic signal separation methods can significantly enhance imaging quality.

Furthermore, the CRRTM proposed in this paper and the ETC-IC can be combined with least-squares reverse-time migration (LSRTM). This combination can be further extended to LS-CRRTM, reducing the negative impact of the forward operator transpose and improving the imaging quality.

6. Conclusions

Compared with surface seismic surveys, VSP provides higher precision and less noise. However, it also introduces issues that are not present in surface seismic surveys, primarily owing to the uneven coverage of subsurface structures by the VSP observation system. To address the issue of significant noise in imaging results caused by uneven coverage, this study proposes a new common-receiver imaging method that relocates seismic data to the source location and uses this as the boundary condition for reverse-propagation, effectively enhancing the illumination range of single reverse-time migration imaging.Furthermore, in traditional cross-correlation imaging conditions, both the source and receiver wavefields contain components that do not contribute to imaging, resulting in significant noise in the final image. This noise is exacerbated by the unique characteristics of the VSP system. Therefore, constraining the wavefield using the excitation time helps improve the quality of VSP migration imaging results.

Common-receiver RTM has a stable workflow and is easy to operate. The numerical experimental results demonstrate that, compared to traditional W-VSP reverse-time migration imaging, common-receiver RTM can effectively eliminate noise artifacts in the imaging results and significantly improve imaging quality.

Author Contributions

Conceptualization, C.L.; methodology, C.L. and Y.L.; software, Y.L.; validation, C.L. and Y.L.; formal analysis, C.L. and Y.L.; investigation, C.L. and Y.L.; resources, C.L.; data curation, C.L. and Y.L.; writing—original draft preparation, Y.L.; writing—review and editing, C.L. and Y.L.; visualization, C.L. and Y.L.; supervision, C.L.; project administration, C.L.; funding acquisition, C.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

We are grateful to the editor in chief, the associate editor, and the reviewers for their valuable comments and suggestions that helped to improve the manuscript. We would like to thank Xiaoyan Nie for their valuable supervision and administrative support during the project. Their guidance and assistance were greatly appreciated.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

RTM	Reverse-time migration
W-VSP	Walk-away vertical seismic profiling
CRRTM	Common-receiver reverse-time migration
ETC-IC	Excitation-time-constrained cross-correlation imaging condition

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1. Workflow of traditonal RTM and CRRTM.

Figure 2. Schematic of the propagation path for seismic waves in VSP survey.

Figure 3. Simple P-wave velocity model, where the yellow dotted box represents the boundary of the PML.

Figure 4. Shot gather in the simple model: (a) original data; (b) separated upgoing data.

Figure 5. Cross-correlation between source and receiver wavefields in traditional VSP RTM at various times.

Figure 6. Single-shot imaging result in the simple model using the traditional W-VSP RTM method.

Figure 7. Reorganized receiver gather from different shot gathers in the simple model: (a) original data; (b) separated upgoing data.

Figure 8. Cross-correlation between source and receiver wavefields in CRRTM at various times.

Figure 9. Single imaging result in the simple model with CRRTM.

Figure 10. Cross-correlation between source and receiver wavefields using S-wave in CRRTM at various times.

Figure 11. Single imaging result in the simple model using CRRTM S-wave imaging.

Figure 12. Limitation of first-arrival time to the receiver wavefield at different depths.

Figure 13. Final migration result of the traditional W-VSP RTM method.

Figure 14. Final migration results of the traditional W-VSP RTM method and ETC-IC.

Figure 15. Final migration results of CRRTM and ETC-IC.

Figure 16. Final migration results of CRRTM when using S-waves.

Figure 17. True P-wave velocity of the modified Marmousi model.

Figure 18. Three receiver gathers of various receivers for the Marmousi model, with receivers located at depths of (a) 150 m, (b) 800 m, and (c) 1500 m. Only the separated upgoing wave has been retained.

Figure 19. Final migration results for the Marmousi model using CRRTM.

Table 1

Imaging quality of three different RTM methods, evaluated based on the energy ratio.

Methods	Energy Ratio (% Percent)
W-VSP RTM	86.02
W-VSP RTM with ETC-IC	85.15
CRRTM with ETC-IC	98.21

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