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.
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
and then from the reflection point to the receiver at time
. This process is illustrated in
Figure 2 by red and blue lines, respectively. If the maximum recording time at the receiver is
, the part of the seismic data recording the reflection point information is located at
. During reverse-propagation, these data are excited by the receiver at time
. After propagation for time
, it reaches the reflection point at time
. 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 is required to reach the reflection point. Thus, the seismic data arrive at the reflection point at time . Simultaneously, the seismic wavelet propagating from the receiver also requires time 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:
where
and
represent the source and receiver wavefields at location
and time
t, respectively,
t represents time, and
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:
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:
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
where
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:
Here, T represents the period of the seismic wavelet used, and 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.
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.