Introduction

Diffraction is a very common wave type that can be observed in almost every seismic data set. Small-scale geological bodies (such as faults and fractures in carbonate rocks) are the main cause. As the simple hydrocarbon reservoirs have been fully explored and exploited, the unconventional hydrocarbon reservoirs controlled by these small geological bodies are now drawing attentions of exploration geophysicists. The well-preserved diffraction can provide rich information about these reservoirs (Decker et al., 2017; Gelius, 1995; Landa and Keydar, 1998). However, in the traditional seismic exploration workflow, diffractions are treated as a kind of noise and removed from the original data. In addition, the weakness of diffraction amplitudes leads to final imaging challenges. Thus, developing effective and accurate methods for diffraction imaging would be a beneficial goal.

Generally, the key of diffraction imaging is to accurately separate the reflection and diffraction. Based on this idea, many separating methods have been proposed , which can be divided into two main categories according to their mathematical algorithms, path-summation-based methods and wave-equation-based methods. Additionally, some other approaches such as machine learning (Tschannen et al., 2020) are also effective tools for imaging the diffraction (Protasov et al., 2019).

For the path-summation-based methods, the traveltime (event shape) is of great importance to the separation of reflection and diffraction (Kanasewich and Phadke, 1988; Santos et al., 2012). Berkovitch et al. (2009) use a multifocusing algorithm to separate diffractions from reflections by designing a novel time-correction formula to approximate the diffraction events. This method can make the energy of diffraction more focused while scattering that of reflection in the stacking result. Dell and Gajewski (2011) separate diffraction and reflection energy in the time domain using a common-reflection-surface algorithm. Different synthetic examples showed the validity of this method. Waheed et al. (2013) propose a new method for fitting the diffraction traveltimes in the TI media. This approach can accelerate the traditional process for solving equation and lower the computational complexity. Additionally, it adaptively selects the best parameters for the traveltime equation without complex modeling processes. Asgedom et al. (2013) combine two novel algorithms, i.e., the modified common‐reflection‐surface method and the replacement‐media technique, to enhance the weak diffraction in the strong reflection background. Coimbra et al. (2018) propose the finite-offset double-square-root diffraction traveltime equationthat can accurately separate diffraction from the background reflection. A simplified version was also proposed to accelerate the calculation of the equation parameters while keeping the quality of result unchanged. The synthetic and field data results show the outstanding performance of this improvement. Bakhtiari Rad et al. (2018) compare the influence of pre-stack and post-stack diffraction separation to imaging performance, and showed that the former method based on the wavefront attributes can help improve the illumination compared with the latter method. Li and Zhang (2019) design a vertical traveltime difference gather and its plus version. Compared with the traditional two-dimensional dip gather, this gather has the advantage of occupying less storage space. After Kirchhoff time migration, the diffraction is flattened, and the reflection events still have upward dip. By this difference, the reflection can be cut off, which will further strengthen recovered diffraction amplitudes. Merzlikin et al. (2019) introduce the separation of diffraction into an inversion framework, and designed a new decomposition algorithm by combining Kirchhoff modeling, plane-wave deconstruction and an integral operator. This method simultaneously separates weak diffraction and strong reflection accurately, while sufficiently supressing noise. Schwarz (2019) design an adaptive filter to separate diffraction in the background of strong reflection. This method is mainly based on the summation of reflection data rather than diffraction data, and uses a variety of wavefront filters in the stacking domain to separate diffraction and reflection. Using seismic and GPR data examples, they demonstrate that this method could recover weak diffraction signals effectively.

Wave-equation-based methods take all the wavefield information into consideration (Yuan et al., 2019; Sava et al., 2005). Klokov and Fomel (2012) derive a novel analytical equation to accurately separate the diffraction and reflection. After the migration process, the Radon transform can distinguish the diffraction effectively by its shape feature. This method is very stable on both synthetic and field data examples. Zhang and Zhang (2014) image the weak diffraction in the shot and opening-angle gathers by muting the strong reflection and enhancing the weak diffraction. This method generates satisfactory results even if the diffraction interferes with the reflection. The velocities for migration can be gradually updated based on the migrated results to achieve the best result. The real and synthetic data all showed the validity of this method. Liu et al. (2016) propose a fast migration method for the imaging of diffraction. They first use linear Radon transform to obtain the local plane waves, and then implement the zero-lag correlation between these plane waves and incident waves. The final diffraction image is generated after the energy of reflected wave was attenuated by a median filter. Zhang et al. (2019) divide the wavefield into two parts, left and right, by using reverse time migration. Based on the fact that the reflection is always related to a specific dip angle, and can only exist in either the left or right wave field. However, due to the nature of diffraction, associated energy will simultaneously appear in both components at point-by-point multiplication between two components can suppress reflection while the imaging result of diffraction.

Fomel et al. (2007) proposes a two-step diffraction separation and imaging framework. The first step separates diffraction and reflection wavefields based on the spatial coherence using the plane-wave destruction (PWD) method. The PWD method assumes that the reflection waves have better spatial coherence than the diffraction waves, and thus can be predicted by neighbor traces following the smooth local slope field. In the second step, the separated diffraction waves are migrated using different velocities. Their corresponding focusing performance on the images are measured to determine the migration velocity that optimally focuses the image. In this way, one can achieve simultaneous velocity estimation and diffraction imaging. Based on this framework, the diffraction separation becomes a pure signal processing task, where no prior subsurface velocity models are required. Thus, this method is easy to apply and sometimes can obtain very good performance in real data applications. In a similar framework, Zhou et al. (2017) and Zhou and Sun (2018) develop a method to extract diffractions from high-resolution coal seismic data using localized moving-average-error filter (MAEF) to flattened reflection seismic data and or along the reflection dips based on the estimated dip by gradient calculation. However, the success of this method depends on the accuracy of the estimated slope. Because of the smoothness of the local slope, there is leakage of reflection energy into the diffraction estimate which contaminates the imaging results.

In this paper, we propose a new diffraction separation method based on a localized rank-reduction method (LRR). The LRR method also uses a local plane-wave assumption, but bypasses the step of slope estimation. The compromise between the removal of diffractions and preservation of reflections is controlled by the rank in each local window. We propose an adaptive rank-selection method for each localized window to select the optimal cut-off rank that best distinguishes between reflections and diffractions. After the diffraction separation step, an migration operation is applied to obtain the migration diffraction image, e.g., by Kirchhoff migration (Fomel, 2002a), velocity continuation (Fomel, 2003).

The paper is organized as follows: we first introduce the principles of the LRR method. Then, we introduce how we adaptively select the ranks in each local window. The hybrid LRR and adaptive rank selection method is referred to as the localized rank-reduction method with adaptively chosen ranks (LRRA). Next, we use several representative examples to demonstrate the performance of the proposed diffraction separation method, and more importantly how the new method can improve the resolution and fidelity of the final image.