*3.3. Robustness to Other Wave Components*

This numerical experiment was carried out based on elastic wave propagation theory, in which the signals received contain various components, such as P-P, P-S, and S-S waves, surface waves, and direct waves (Figure 19). For these imaging methods, we employed the P-P signal. Other components in the signal were considered to be noise or interference in imaging, and S-waves and surface waves were significant causes of imaging artifacts.

**Figure 19.** Components in a seismic signal.

Figure 20 shows the normalized amplitude curves and Hilbert transforms at *x* = 15 m in model B-2 (Figure 20a,b) and model A-3 (Figure 20c,d). The curves indicate the following: (1) The amplitude in the shallow parts of the Kirchhoff curve and DAS curve is high (greater than the reflection amplitude). This is the result of a surface wave. However, in the MVSS curve, the surface wave amplitude is suppressed. The surface wave is suppressed because the MVSS method amplifies the reflected wave by using the CF matrix. (2) From the Hilbert transform, it can be noted that the MVSS beamforming method can recognize two close interfaces, while the Kirchhoff and DAS methods might combine these interfaces as one. Thus, MVSS beamforming has a better vertical resolution than the other two methods. (3) In the MVSS curve, the reflection amplitudes are greater than those in the Kirchhoff migration and DAS beamforming curves; this is also partly due to the suppression of surface waves. This means that MVSS beamforming offers a better contrast than the other two imaging methods.

**Figure 20.** Imaging amplitude curves at *x* = 15 m and the Hilbert transform of each. (**a**,**b**) model B-2; (**c**,**d**) model A-3.

Figure 21a shows the imaging result of Kirchhoff migration with all signal components. The interfaces between the layers at depths of 5 m and 15 m are clear. However, massive horizontal artifacts can be detected at the top of the image; in addition, artifacts obliquely penetrate the shallower interface, and noise exists beneath the interface at a depth of 15 m. Likewise, Figure 21d shows the imaging result of DAS beamforming with all signal components. The two interfaces are clear, but horizontal artifacts can again be observed at the top of the image, and there are slight artifacts on both sides of the interfaces. Noise also appears beneath the interface at a depth of 15 m. Figure 21g shows the imaging result of MVSS beamforming with all wave components. The two interfaces are clear at depths of 5 m and 15 m. However, horizontal artifacts remain near the top of the image, and noise can be observed beneath the deeper interfaces. Nevertheless, compared with the imaging results of Kirchhoff migration and DAS beamforming, MVSS beamforming can suppress both the horizontal artifacts caused by surface waves and the artifacts on both sides of the image caused by S-waves.

These results confirm that the MVSS beamforming method can significantly suppress artifacts caused by direct waves and surface waves. The suppression of surface waves in reflection seismic exploration has long been a topic of interest [32]. Strong surface waves are significant causes of artifacts in imaging. Hence, to determine how surface waves impact the imaging results, we performed image processing by excluding direct waves and surface waves and then compared the results with the original imaging results. Figure 21b shows the Kirchhoff migration imaging results excluding direct waves and surface waves. Compared with Figure 21a, the surface wave artifact at the top of the image disappears, and the interference across the shallower interface is weaker. However, the artifacts on both sides of the interfaces and beneath the deeper interface still exist. Figure 21e shows the DAS imaging result excluding direct waves and surface waves. Compared with Figure 21d, the surface wave artifact at the top of the image disappears, but the S-wave artifacts across the interfaces still exist. Figure 21h shows the MVSS beamforming result excluding direct

waves and surface waves. Compared with Figure 21g, the artifact at the top of the image disappears, but the interferences beneath the deeper interface still exist. These comparisons suggest that the main consequence of direct waves and surface waves is the presence of artifacts at the top of the image.

**Figure 21.** Imaging results after eliminating either surface waves or S-waves. (**a**) Kirchhoff migration result using the original signals. (**b**) Kirchhoff migration result using signals without surface waves. (**c**) Kirchhoff migration result using signals with only P-waves. (**d**) DAS beamforming result using the original signals. (**e**) DAS beamforming result using signals without surface waves. (**f**) DAS beamforming result using signals with only P-waves. (**g**) MVSS beamforming result using the original signals. (**h**) MVSS beamforming result using signals without surface waves. (**i**) MVSS beamforming result using signals with only P-waves.

S-waves (or converted waves) are another main factor causing artifacts and interference in seismic imaging. S-wave signals and converted waves (reflected S-waves converted from P-waves at interfaces) have similar in-phase axes as P-wave signals. However, the velocities of S-waves and converted waves do not match the estimated velocity (P-wave velocity), so the use of S-waves (or converted waves) can cause errors. Figure 21c shows the Kirchhoff migration imaging result ignoring S-waves. Compared with Figure 21a, artifacts at the top of the image still exist, but the artifacts are thinner and weaker. Moreover, the interferences across the shallower interface and beneath the deeper interface both disappear. Figure 21f shows the DAS beamforming result ignoring S-waves. Compared with Figure 21d, the artifacts at the top of the image still exist but are weaker, and the interferences beneath the deeper interface and on both sides of the shallower interface disappear. Finally, Figure 21i shows the MVSS beamforming result ignoring S-waves. Compared with Figure 21g, the artifacts at the top of the image still exist, albeit with weaker energy, and the interference beneath the deeper interface disappears. In addition, compared with Figure 21f, the artifacts on both sides of the interfaces disappear.

From this discussion, we have found the following:


3. The ability of MVSS beamforming to suppress surface wave artifacts at the top of the image and the S-wave artifacts on both sides of the interfaces is superior. However, the S-wave artifacts beneath the interfaces still affect MVSS beamforming as much as they do the other methods.

### *3.4. Robustness to Random Noise*

The signals in the numerical experiments were clear synthetic signals (Figure 22a). The SNR is defined as follows, where *Psignal* and *Pnoise* are the power levels of the signal and noise, respectively, and *Asignal* and *Anoise* are the amplitudes of the signal and noise, respectively:

*Psignal*

*Asignal*<sup>2</sup>

**Figure 22.** Imaging result using the clear signal (**a**) and noisy signal (**b**). (**c**) Kirchhoff migration imaging result using the clear signal. (**d**) DAS beamforming result using the clear signal. (**e**) MVSS beamforming result using the clear signal. (**f**) Kirchhoff migration imaging result using the noisy signal. (**g**) DAS beamforming result using the noisy signal. (**h**) MVSS beamforming result using the noisy signal.

We added Gaussian noise to the signal (Figure 22b) to make the SNR 0 dB (the power of noise equals the power of the target reflection signals) and then generated images using Kirchhoff migration, DAS beamforming, and MVSS beamforming.

In the Kirchhoff migration result with the noisy signal (Figure 22d), the interference caused by white noise can be observed as dots in the image. In the imaging result of DAS beamforming for the signal with Gaussian noise (Figure 22f), compared with the imaging result containing clear signals (Figure 22e), the white noise in the signal is detectable in the imaging result as well.

Figure 22h shows the MVSS beamforming result for the signal with Gaussian noise. Compared with the Kirchhoff migration and DAS beamforming results, the influence of white noise is much less apparent for the MVSS beamforming results. However, relative to the imaging result with a clear signal (Figure 22h), although the noise is well suppressed, the brightness of the reflection interface decreases. The reason is that the energy of all signals increases when white noise is added, which narrows the energy gap separating the reflection signal and other signals from empty space. As a consequence, the energy of the reflective interface is relatively reduced. This phenomenon corresponds to the principle of keeping the effective signal gain unchanged while minimizing the energy of the whole image in the MVSS beamforming method.

### *3.5. 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, t 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.

**Figure 24.** Signal advance correction to match the peak time to the arrival time.

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.

**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.

This phenomenon also appears in the 30 m × 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

a shallower position. Therefore, this processing scheme will enhance the focusing effect while causing the target position to be slightly shifted.

**Figure 26.** Imaging after applying a time correction in model B-2: (**a**) true velocity model; (**b**) no processing; (**c**) 1/4-cycle advance; (**d**) 1/2-cycle advance; (**e**) 3/4-cycle advance.

### **4. Discussion**

### *4.1. Computational Efficiency*

In terms of the size of our numerical experiment (20 shots, 151 receivers, 1001 time grids per signal), the calculation time needed to implement each imaging method ranged from several seconds to minutes, as summarized in Table 4. DAS beamforming was the most efficient method, with a fast imaging time of 3.9 s, followed by MVSS beamforming at 201.1 s, and Kirchhoff migration took the longest at 520.1 s. If the time delay was calculated according to the background velocity and stored prior to imaging, the calculation times of DAS and MVSS beamforming were further shortened. In the Kirchhoff migration adopted in this paper, the travel times from source to scatter point and from scatter point to receiver were approximated by a Dix equation using the RMS velocity. On the other hand, the travel times were calculated based on the geometry relationship between the source, scatter point, and receiver. To some extent, DAS beamforming is a simplified option of Kirchhoff migration, which results in the computational efficiency difference. Conceivably, if the condition of the medium at each point is known, the Kirchhoff migration method should theoretically have the best imaging results. In this experiment, the geological information was known approximately (background velocity), so the accuracy advantage of Kirchhoff migration was almost negligible, but the efficiency advantage of beamforming persisted. Beamforming methods assume that the wave propagates in a straight line and that the seismic velocity in the area is the same. This assumption is also available for Kirchhoff migration. However, in the adopted algorithm it still took time to estimate the travel time by a Dix equation using the RMS velocity desired from a homogeneous background model. This meant that the travel time was almost same as that in DAS beamforming, in which the travel time was simply calculated according to geometry and background velocity, but the calculation time was quite different. Therefore, the efficiency of DAS beamforming was as high as 130 times that of Kirchhoff migration. For this reason, we would like to apply beamforming imaging to detect underground anomalies in a scene with a single background velocity. The computational efficiency and accuracy of an algorithm have always been contradictory, but in near-surface exploration, the prior information is simple, which is suitable for beamforming methods to improve efficiency.

**Table 4.** Calculation time of each method.

