**1. Introduction**

Most underground geotechnical engineering is carried out near the surface. Unfortunately, the near-surface geological conditions are highly complex due to the abundance of joints, faults, boulders, caves, and so on. Encountering unexplored weak formations and caves can cause a variety of geological disasters, such as uneven settlement or collapse, which can inflict immense economic losses and even death [1,2]. According to the report [3] of the current status of sinkhole collapses in the karst area in China, more than 1500 karst

**Citation:** Peng, M.; Wang, D.; Liu, L.; Liu, C.; Shi, Z.; Ma, F.; Shen, J. Near-Surface Geological Structure Seismic Wave Imaging Using the Minimum Variance Spatial Smoothing Beamforming Method. *Appl. Sci.* **2021**, *11*, 10827. https:// doi.org/10.3390/app112210827

Academic Editors: Guofeng Liu, Zhifu Zhang, Xiaohong Meng and Jong Wan Hu

Received: 29 September 2021 Accepted: 12 November 2021 Published: 16 November 2021

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collapsing events have been recorded and these events formed more than 45,000 sinkholes. More than 75% sinkholes were triggered by human activities. These activities are mainly around the near-surface, including mine drainage, foundation engineering, and tunnel constructions. Subsidence sinkholes result from both subsurface dissolution and the downward gravitational movement of the undermined overlying material. These sinkholes, which are invisible from the surface, are the most important from a hazard and engineering perspective [4]. Hence, to provide guidance for construction and design efforts, geophysical prospecting methods are often used in engineering to detect underground structures as an important supplement to drilling and excavation. However, the near-surface is complex and traditional migration imaging methods may suffer from types of artifacts and interferences. High-precision prospecting in near-surface detection is quite challenging since surface waves and S-waves are powerful near the surface. There is an urgent need to deal with the artifacts and interferences. Furthermore, the detection capability is limited by the construction site and time frame [5]. Therefore, the development of accurate geophysical prospecting methods for near-surface detection is of great significance to geotechnical engineering.

Seismic methods are among the most commonly used geophysical techniques for near-surface detection with the advantages of a high efficiency, a high accuracy, and a low cost; moreover, seismic techniques are nondestructive [6]. Seismic reflection is extensively used in both academic and practical engineering [7]. One of the main goals of seismic data processing is seismic imaging. For instance, seismic migration imaging [8] is used to map underground structures. Although advanced imaging methods have emerged in recent decades, such as reverse time migration [9,10] and Gaussian beam migration [11–13], Kirchhoff migration [14] is still the most popular approach due to its ability to provide an image of sufficient quality and at an affordable computational cost. Zhang et al. [15] processed a model example and seismic field data to demonstrate the validity of prestack Kirchhoff time migration. Yuan et al. [16] applied Kirchhoff prestack time migration to seismic data of coal seam reflections and obtained better images than with poststack time migration. Wang et al. [17] used tomographic travel-time inversion and prestack Kirchhoff depth migration-based migration velocity analysis (MVA) and obtained a precise, highresolution migration velocity model. Liu and Zhang [18] proposed a novel approach that attached a prediction shaping filter to Kirchhoff prestack time migration to mitigate the stretching effect and demonstrated their method with a numerical example and field data. Zhang et al. [7] applied Kirchhoff poststack migration to a case of seismic scattered wave imaging and obtained reliable imaging results in both the synthetic data and field data. Nevertheless, in near-surface seismic exploration, where the depth of the target area is usually less than a hundred meters, Kirchhoff migration still suffers from the following problems: (1) Kirchhoff migration relies on the preprocessing of signals to deal with noise and interference such as white noise, S-wave signals, and multiples; (2) Kirchhoff migration is affected by high-energy surface waves, which appear as massive artifacts in the imaging results. The surface wave can be muted from the observed data, but it requires a manual cost; and (3) Kirchhoff migration is limited by the degradation in the imaging resolution that occurs when the wavelength is not much smaller than the size of the imaging area in near-surface detection.

In the medical field, the widely applied ultrasound beamforming method, which detects human body structures using acoustic properties, gives us another possible way to acquire seismic signals and solve the abovementioned problems. The delay and sum (DAS) beamforming algorithm is the most popular beamforming technique for imaging human body structures because of its real-time imaging capability [19,20]. In recent decades, various adaptive beamforming methods have been proposed to improve the resolution and stability of beamformers in medical ultrasound imaging. The most common approaches are based on the minimum variance (MV) beamformer devised by Capon [21], and numerous MV beamforming algorithms have been developed to continue improving the imaging quality [22–25]. Asl [26] proposed an MV beamforming method combined with

adaptive coherence weighting and achieved an excellent performance in an application to medical ultrasound imaging. Ma et al. [27] introduced the multiple delay and sum with enveloping method to efficiently suppress sidelobes and artifacts. Most recently, Chen et al. [28] proposed the multioperator MV adaptive beamformer to promote real-time imaging. Accordingly, because the transmission and receiver concepts in medical imaging and seismic reflection are similar, the medical beamforming method can be adapted for seismic imaging.

In this study, we propose a minimum variance spatial smoothing (MVSS) beamforming method for near-surface reflection seismic exploration in a homogeneous assumption. Synthetic near-surface geological models are established to carry out a numerical simulation, and the image quality of the proposed MVSS beamforming method is compared with that of Kirchhoff migration and basic DAS beamforming. Moreover, the robustness to interferences and noise, robustness to other wave components and a delay time correction to enhance the focus effect are presented. Finally, the computational efficiency, a comparison between MVSS beamforming and RTM (one of the best imaging methods for seismic data), and the potential application of the CF matrix in time domain prestack migration methods are discussed.
