**2. Methods**

The MVSS method was proposed in this paper to improve the signal-to-noise ratio (SNR), to enhance the focusing effect, and to improve the resolution. The core concept of MVSS beamforming is to assign a weight to each receiver. When higher weights are assigned to receivers with higher SNRs, the imaging quality is improved. The receiver array is divided into several subarrays to improve the robustness when obtaining the weights. In each subarray, to improve the resolution of the image and increase the SNR, we designed a weight for each receiver according to the covariance matrix of signals. After the imaging results of each subarray are superimposed, the obtained image is multiplied by the signal coherence matrix to reduce the influences of sidelobe interference and focusing errors. The proposed method is based on a homogeneous subsurface assumption, and it is a time domain raytracing method, which means the propagation path of waves is static. Notably, MVSS beamforming reduces to basic DAS beamforming without spatial smoothing, neglecting to calculate the weight of each receiver, and excluding the coherence factor (CF) matrix. Though DAS beamforming and Kirchhoff migration are coded in different ideas, they share similar principles, such as they are both prestack time migration and raytracing methods. Background velocity plays an important role in Kirchhoff migration and other migration methods, but it does not in DAS beamforming. DAS beamforming can be understood as an extremely simplified Kirchhoff migration, in which we assume that the wave propagates along a straight line and the subsurface is homogeneous.

## *2.1. MVSS Beamforming Imaging*

Figure 1 shows the workflow of MVSS beamforming, which is divided into three parts: (1) signal delay; (2) superposition with calculated weights from minimum variance matrix in subarrays; and (3) imaging and processing, including time-depth conversion and shot stacking. The detailed mathematics can be found in Appendix A.

First, the signal delay times are calculated according to the distance from the target point to each receiver and the background velocity. At the top of Figure 1 is a target point in the detection area. The waves reflected from the target point are received by the array, which is arranged in a straight line. The signals recorded by the receivers have different arrival times because the receivers are situated at different distances from the target point. Thus, delay processing (*τ*<sup>1</sup> ... *τ*5) is applied to these signals so that the fluctuations from this target point are aligned on the time axis.

Second, the signals from different receivers are superimposed with an MVSS beamformer. Calculating the superposition (∑) with weights (*ω*<sup>1</sup> ... *ω*5) of the delayed signals amplifies the signals from the target point while suppressing the reflections from other

scattering points in the imaging area. The weights are calculated from a minimum variance matrix of subarrays with diagonal loading to enhance robustness. The weights are designed to minimize the output interference-plus-noise power while maintaining a distortionless response to the target signal. Then, the CF is used as a weight matrix to enhance the image, which can obviously amplify the reflection signals.

**Figure 1.** Workflow of MVSS beamforming imaging.

Finally, after performing time-depth conversion and stacking the images of all shots, we obtained the beamforming result. Underground structures can be determined by the amplitudes in the image, where a higher amplitude corresponds to a greater possibility of a reflection interface or a stronger reflection.
