Our discussion will concentrate on whether VR36 or VR40 should be used in the predictive equation to determine the fast proxy of Vs30 for various synthetic and real data velocity profiles. Based on the dynamic stiffness of 53 velocity profiles, the fundamental-mode of Rayleigh wave dispersion curves were calculated. The value of Vs30 calculated by the predictive equation is compared to the actual value of Vs30 determined by the formula recommended by the international building code, as shown in (Eq. 2.3). With the exception of a few exceptional cases, the difference between the actual and predictive values of Vs30 is usually within +/- 10% error bonds. The following result will summarize the selection of a more suitable predictive equation, as well as the evaluation of Vs30 proxy.

3.1) GROUP A

As shown in Figure 1 (a, b, c, d), there are twenty-five 2-layer synthetic earth profiles with varied velocity contrast. The fundamental-mode model dispersion curve was calculated using the dynamic stiffness matrix method on the first layer with varied depth and constant shear wave velocity (120 m/s), with varying half space shear velocity to a depth of 30 m. The choices of VR36 or VR40 in the predictive equation for determining Vs30 will be summarized in the following section.

3.1.1) Weak soil layer overlaying stiff soil

Figure (1a) shows the two-layer constant velocity profile with varying depth of the 1st layer. Due to the low velocity contrast, overall the actual Vs30 have a good correlation with the Vs30 determined by the both the predictive equation and lies within the +/-10% error bond, but the Vs30 calculated by using VR40 in the predictive equation is closely related to the actual Vs30 as shown in (Fig 2). By using VR36 and VR40, at a height of (H15 Top layer height) 15 m and (H20) 20 m both results are underestimated.

3.1.2) Weak Soil Layer Overlying Soft Rock

Fig. 1b illustrates the weak soil overlaying on the soft rock in a similar way to Fig. 1a. The results of the Vs30 and Rayleigh wave phase velocity (VR36, VR40) correlations point to the same conclusion. Vs30 was determined by applying VR36 and VR40 in the prediction equations at heights of 5 m and 25 m, and the results of Vs30 are in +/-10 percent error bonds. The Vs30 computed by VR36 is underestimated for heights of 10 m, 15 m, and 20 m. Overall, VR40 is in good agreement with genuine Vs30, as shown in the graph (Fig.3).

3.1.3) Weak Layer Overlaying on Rock

Fig. 1c represent the case of weak layer over rock. Result shows that, due to high velocity contrast, Vs30 calculated by the predictive equations are underestimated and the correlation between Vs30 and VR at a wavelength of 36 m and 40 m are not in a reasonable range. The results are basically dominated by using VR50 in the predictive equation as shown in (Fig. 4).

3.1.4) Weak Layer Overlaying on Hard Rock

In the way similar of the other two-layer velocity profile, velocity of the 1st layer was keep constant with variable height as shown in (Fig. 1d). Due to high velocity contrast between the two layers, it is clear from the results that the Vs30 calculated by the predictive equation by using Rayleigh wave velocity at VR36, VR40 is not promising and not even a single data point lies within +/-10% acceptable error bond. Vs30 estimated by using VR50, two velocity profile, for instance, H20, and H25 is in +/-10% bound as shown in (Fig 5).

3.2) Synthetic Data Representation

Synthetic data representation as illustrated in Fig.6 is the most sophisticated correlation, to obtained a quick idea about the selection of the Rayleigh wave velocity at a specific wavelength for instance, VR36, VR40, and VR50, to use in the predictive equations, for the calculation of Vs30 within the acceptable +/-10% error bound zone. The Rayleigh wave velocity at a wavelength of 50 m i.e. VR50 covered more area for instance, high velocity contrast than VR40 and VR36 respectively. When the velocity contrast is not significantly higher between the 1st layer and half space, as shown in Fig. 1(a) with 1st layer thickness 22 m < H < 13 m, Vs30 determined by the predictive equation, by using the Rayleigh wave velocity at wavelength of VR36 lies within the acceptable zone. In this case, by using VR40, the Vs30 lies in the acceptable zone as shown by dark black circle on the VR40 line. In case of weak layer over soft rock, the Vs30 calculated by the predictive equation by using VR36 and VR40 will lies in the acceptable error bound zone only, if the thickness of the 1st layer lies between 22 m < H < 10 m and 20 m < H < 10m respectively, but on the other hand, VR50 will be in the acceptable zone. As the velocity contrast between the top layer and the haft space becomes more significant, as shown in Fig. 1(c,), by using VR40, and VR50, Vs30 will be in acceptable zone, if the height of the 1st layer H > 22 m and 5 m > H > 10 m respectively. For weak soil over hard rock, due to a very high velocity contrast, Vs30 calculated by VR50 lies in the acceptable zone as shown in the figure below.

3.3) Group B Synthetic Gradient Models

At many sites, under typical geological conditions, soil stiffness increases gradually with depth due to its geological age, cementation, compaction, overburden pressures etc., and the effect of the non-fundamental mode Rayleigh wave energy on the dispersion curve is minimal. In this situation the common expectation includes engineering fill over stiff sediments, asphalt, concrete, and soft lose material on the compacted base materials or soft soil over shallow soft rock. To represent these condition in field, six gradient synthetic models were selected as illustrated in Fig.1(e), to calculate its theoretical dispersion curves. Vs30 was determine by the predictive equations using VR36, VR40 and VR50. From the results as shown in (Fig. 7) it is concluded that Vs30 has a very good correlation with the phase velocity of Rayleigh wave at a wavelength of 36 m i.e. VR36. All the data points are in the acceptable zones. The Vs30 calculated by using VR40 and VR50 in the predictive equation are overestimated.

3.4) Real Data Models

As indicated in Fig. 1, twenty-seven genuine data models were acquired from the survey report of the University of Puerto Rico in collaboration with the US Geological Survey (1f). They investigated near-surface shear wave velocity and compressional wave velocity using noninvasive seismic refraction-reflection profiling techniques. The average shear waves velocity at the top 30 meters was computed using Eq. 2.3, and the findings were compared to those obtained from the prediction equation using VR36 and VR40. The majority of the results are acceptable, with a few exceptions as shown in Fig. (8) and Fig. (9). (9). The two-layer synthetic technique is used in real-world data models. The results are not always acceptable due to high velocity contrast.