## 3 Numerical Experiments
### Focus Enhancing by a Signal Advance Correction  
In real-world applications, the velocity assigned to the background is inaccurate. Although the phenomenon we discuss in this section was not apparent in our numerical experiments, it does exist in practice. Therefore, another model was designed at a smaller scale with a higher velocity to observe more obvious phenomena and to verify how MVSS beamforming works when an inaccurate background velocity is provided. Figure 23 displays the imaging results when different background velocities were provided while the true background velocity was 3000 m/s. When the background velocity reaches 2250 m/s, MVSS beamforming can barely differentiate three different targets (dots) in the region and loses focus. Interestingly, MVSS beamforming does not appear to work optimally under an accurate velocity (3000 m/s), whereas it performs successfully under a velocity of 2750 m/s. The reason is that unlike in oil and gas exploration, the shot wavelet length in near-surface exploration cannot be ignored because the scale of the model is small. Figure 23: Imaging results using different background velocities while the true velocity is 3000 m/s. (**a**) True velocity model; (**b**) 2250 m/s; (**c**) 2500 m/s; (**d**) 2750 m/s; (**e**) 3000 m/s; (**f**) 3250 m/s. In the signals received in a near-surface situation, the time from zero to the first arrival is t. However, it is not the most appropriate value to use because the time employed in the algorithm is the peak-to-peak time or the time from first arrival to first arrival (Figure 24). Thus, we fixed all signals by half a cycle to correct this phenomenon, called 'defocusing', to match the wave peak times, and the signal was advanced by a quarter of a cycle, half a cycle, and three-quarters of a cycle of the source wavelet, producing a set of images using MVSS beamforming. The results show that the imaging algorithm works best when the signal is advanced by half a cycle. However, the location of the objective becomes shallower than the real location as a result. To better observe the signal correction described above, we implemented the imaging algorithm in a model with smaller point anomalies and a high background velocity and excluded both surface waves (direct waves) and S-waves. The results with no processing, a 1/4-cycle advance, a 1/2-cycle advance, and a 3/4-cycle advance are shown in Figure 25. Upward-concave artifacts can be detected near the point target in the image without processing; this is the phenomenon known as defocusing. The reason for this defocusing is the mismatch between the delay information and the reflection signals. With an increase in the signal correction amount, the best focusing effect was achieved when the correction value was 1/2 of a cycle. When the correction value exceeds this value, the imaging defocuses in the other direction (concave-downward); the reason for this is again the mismatch between the delay information and the reflection signals. This phenomenon also appears in the 30 m \(\times\) 20 m model, although it is not particularly obvious due to the lower frequency and slower wave velocity. The red line in Figure 26 is a reference line at a depth of 5 m (Figure 26a), and the red box marks the comparison of different time advance corrections. After the signal advance correction was applied, the focusing effect improved, and the focusing effect was best when the correction value was 1/2 of a cycle (Figure 26d). When the correction value increased to 3/4 of a cycle (Figure 26e), the image began to defocus to the other side (shallower side). It is worth noting that due to the signal advance correction, all the imaging targets moved to  
Figure 24: Signal advance correction to match the peak time to the arrival time. Figure 25: Imaging after time advance corrections: (**a**) no processing; (**b**) 1/4-cycle advance; (**c**) 1/2-cycle advance; (**d**) 3/4-cycle advance. a shallower position.