4.3.2. Refraction within the Solid Earth

The Rayleigh wave ray paths are expected to bend also when they travel through the solid earth boundaries with significant impedance contrasts. In order to examine possibility of such boundaries and implications for the validity of the triangulation method, the shear velocity (*Vs*) structure of the study area is explored as described below. The Rayleigh wave ray paths are expected to bend also when they travel through the solid earth boundaries with significant impedance contrasts. In order to examine possibility of such boundaries and implications for the validity of the triangulation method, the shear velocity (*Vs*) structure of the study area is explored as described below. The cross-section A-A' presented in Figure 9a shows that under the continental slope there exists

*J. Mar. Sci. Eng.* **2020**, *8*, x FOR PEER REVIEW 16 of 21

The cross-section A-A0 presented in Figure 9a shows that under the continental slope there exists a layer of material having *Vs* less than 2.2 km/s, which is interpreted as sediments in [15]. Our HVSR survey around the eastern foothills of the Appalachian mountain and near the coastal area suggest that the sediment–hard layer interface lies at a depth as depicted by a dotted line in the A-A0 profile in Figure 9a, and that the average *Vs* of the sediment is about 0.9 km/s [5]. According to [51], the shear velocity contrast at the sediment–hard layer interface should be larger than 2.5 to produce a clear a predominant frequency peak on the HVSR spectrum, therefore, the *Vs* of the hard layer should be around 2.2 km/s. Therefore, this interface is inferred to connect to the 2.2 km/s contour line (depicted as the dashed line in Figure 9a). Shear velocity gradients are calculated along A-A0 at two different elevations by [15] shear velocity model, and a transitional zone (TZ) of largest shear velocity gradient (where *Vs* increases from about 1.7 to 3.2 km/s within a horizontal distance of ≈80 km) is identified and outlined as shown in Figure 9a. The TZ at Cape Hatteras (Figure 9c) extend roughly parallel to the continental slope. The absence of notable shear velocity variations at −15.2 km (Figure 9d) suggests that the transitional zone may not extend to this depth. a layer of material having *Vs* less than 2.2 km/s, which is interpreted as sediments in [15]. Our HVSR survey around the eastern foothills of the Appalachian mountain and near the coastal area suggest that the sediment–hard layer interface lies at a depth as depicted by a dotted line in the A-A' profile in Figure 9a, and that the average *Vs* of the sediment is about 0.9 km/s [5]. According to [51], the shear velocity contrast at the sediment–hard layer interface should be larger than 2.5 to produce a clear a predominant frequency peak on the HVSR spectrum, therefore, the *Vs* of the hard layer should be around 2.2 km/s. Therefore, this interface is inferred to connect to the 2.2 km/s contour line (depicted as the dashed line in Figure 9a). Shear velocity gradients are calculated along A-A' at two different elevations by [15] shear velocity model, and a transitional zone (TZ) of largest shear velocity gradient (where *Vs* increases from about 1.7 to 3.2 km/s within a horizontal distance of ≈80 km) is identified and outlined as shown in Figure 9a. The TZ at Cape Hatteras (Figure 9c) extend roughly parallel to the continental slope. The absence of notable shear velocity variations at −15.2 km (Figure 9d) suggests that the transitional zone may not extend to this depth.

As explained above, the Rayleigh waves propagating through the solid earth from the deep ocean near the continental slope (DF1 and DF2 bands) or from the continental shelf (DF3 band) to the inland stations are expected to change propagation direction due to gradual and continuous refraction as they pass through TZ. The total refraction angle ε is defined as shown in Figure 11a only for the great circles passing through TZ which represents the worst scenario of changes. The cumulative probability of ε in the three DF bands is given in Figure 11b. The weighted average (m) of ε in DF1, DF2, and DF3 bands are calculated to be 8.8◦ , 8.2◦ , and 8.7◦ , respectively, and the probabilities of ε values less than these corresponding weighted averages (P < m) are 60.2%, 63.1%, and 59.6%. As explained above, the Rayleigh waves propagating through the solid earth from the deep ocean near the continental slope (DF1 and DF2 bands) or from the continental shelf (DF3 band) to the inland stations are expected to change propagation direction due to gradual and continuous refraction as they pass through TZ. The total refraction angle *ε* is defined as shown in Figure 11a only for the great circles passing through TZ which represents the worst scenario of changes. The cumulative probability of *ε* in the three DF bands is given in Figure 11b. The weighted average (m) of *ε* in DF1, DF2, and DF3 bands are calculated to be 8.8°, 8.2°, and 8.7°, respectively, and the probabilities of *ε* values less than these corresponding weighted averages (P < m) are 60.2%, 63.1%, and 59.6%.

**Figure 11.** (**a**) Defining angle ε to measure total refraction of Rayleigh wave ray passing through the transitional zone (TZ in Figure 9a,c) taking a ray path to station T60A as an example. The black solid and red dotted lines are the great circle projected in this study and "actual" ray path of Rayleigh wave, respectively. (**b**) Cumulative probability of ε for waves in the three DF bands propagating through **Figure 11.** (**a**) Defining angle ε to measure total refraction of Rayleigh wave ray passing through the transitional zone (TZ in Figure 9a,c) taking a ray path to station T60A as an example. The black solid and red dotted lines are the great circle projected in this study and "actual" ray path of Rayleigh wave, respectively. (**b**) Cumulative probability of ε for waves in the three DF bands propagating through TZ.

Another examination of the Rayleigh wave refraction from sediments to bedrock is carried out

Another examination of the Rayleigh wave refraction from sediments to bedrock is carried out by tracking hurricane "Sandy" in Atlantic Ocean on 27 October 2012 using 8 h ambient noise recorded on the bedrock in Tishomingo, Mississippi. The DF peak is found at the central frequency of 0.18 Hz with very high energy. The primary vibration direction at this central frequency is calculated by both *Ra* and Polarization analysis methods and projected as great circles which point to the locations of hurricane "Sandy" successfully (see Figure S4 in the Supplementary Materials to this article). Thus, the great circles can be considered as valid projections to the source areas of the DF energy.
