Three-layer model

With this geometry, the properties of the first and the last layer (half-space) have the same effect as for the two-layer case. The influence of the intermediate layer characteristics ($V_{s1}$ , $V_{p1}$ , and $\rho _1$ ) is investigated here. First, a large variation range is tested for $V_{s1}$ , between 100 to 2500 m/s (figure 3.18). This variation induces several types of models: a low velocity zone ( $V_{s1} < V_{s0} =$ 200 m/s), a normal increase of the velocity ( $V_{s0} < V_{s1} < V_{s2}$ ), and a High Velocity Zone ( $V_{s1} > V_{s2} =$ 2000 m/s). $V_{s0}$ is fixed to 500 m/s. $V_{s2}$ is set to 2000 m/s. $V_p$ profiles are calculated with a Poisson's ratio of 0.25.

Figure 3.18: Influence of $V_{s1}$ with a constant Poisson's ratio. Rayleigh and Love fundamental modes are represented by plain and dotted lines, respectively. $V_{s1}$ varies from 100 to 2500 m/s. $V_{s0}$ =500 m/s. $V_{s2}$ =2000 m/s. Poisson's ratio is 0.25 and density is 2 t/m$^3$ at all depths. (b) Two-layer model ($V_s$ profile) and the corresponding dispersion curve.

1. Low velocity zone
   At high frequency, Love and Rayleigh waves have approximately the same velocity, which is equal to the minimum $V_s$ of the model ($V_{s1}$ in this case). Love curves are monotonously decreasing. On the contrary, Rayleigh curves present a small minimum. At low frequency, the effects of the low velocity zone disappear.
2. Normal increase of the velocity
   The general shape of the dispersion curves is very comparable with the ones for a two-layer model (figure 3.9(b), dispersion curves for the model with $V_{s0}$ =500 m/s). The only difference is the higher velocity between 6 and 30 Hz which follows the velocity increase of the second layer $V_{s1}$ .
3. High velocity zone
   At high frequency, the Rayleigh curves are similar to the curves that are obtained with a two-layer model with a contrast at 10 m (figure 3.19). At low frequency, the Rayleigh curves tend to $V_{r,max}$ like the other classes of models. For Love, the algorithm ends with an error message. Exceptionally for a fundamental mode, the curve does not exist at all frequencies (available above 10 Hz, or for wavelengths less than 200 m). At low frequency, the first layer (10 m thick) is not "seen" by the propagating waves with a wavelength greater than 200 m. The model is equivalent to a high velocity layer overlying a half space where no real solution exists for the Love surface waves (Aki and Richards (2002), equation 7.6 calculated with complex numbers).

Figure 3.19: The light grey model in figure 3.18 is the same as in this figure between 0 and 50 m. (a) $V_s$ profile. (b) The calculated dispersion for the two-layer model. The dispersion curves are also similar at high frequency.

l0cm

The influence of $V_{p1}$ in the intermediate layer is tested with the same model as in figure 3.10. The sediment layer is split in two in the same way as in figure 3.18 (10 and 40 m). $V_{p1}$ in the intermediate layer of 40 m is changed, keeping other parameters constant. The results are shown in figure 3.20 with two cases for $V_s$ between 0 and 50 m: (a) $V_{s0}=V_{s1}$ =200 m/s, and (b) $V_{s0}=V_{s1}$ =1000 m/s. Figures 3.20 and 3.10 are quite similar, proving that intermediate values $V_{p1}$ also influence moderately the dispersion curve. At high frequency, in figure 3.20(b), all curves tend to same Rayleigh velocity. On the contrary, in figure 3.10(b), for low $V_{p0}$ values, a significant influence is observed above 10 Hz. This difference is entirely due to the velocity values between 0 and 10 m, which control the Rayleigh velocity at very high frequency.

Figure 3.20: Influence of $V_{p1}$ on the Rayleigh dispersion curve for two cases: (a) $V_{s0}=V_{s1}$ =200 m/s, and (b) $V_{s0}=V_{s1}$ =1000 m/s. Poisson's ratio varies from 0 (dark) to 0.45 (light). $V_{s2}$ =2000 m/s. Poisson's ratio is 0.25 in first and bottom half-space. The density is 2 t/m$^3$ at all depths.