Physical search interval

Typical dispersion curves are shown on figure 3.4 with their lower and higher velocity limits.

Figure 3.4: Velocity limits of Love (a) and Rayleigh (b) dispersion curves. Fundamental mode and the two first higher modes are represented with plain, dashed, dotted lines respectively. The horizontal lines are the physical velocity limits.

All real curves have a velocity less than or equal to the maximum S-wave velocity of the model ($V_{s,max}$ ). The minimum possible velocity is not the same for Love and Rayleigh cases. For Love waves, all modes tend to a common velocity at very high frequencies, equal to the minimum S-wave velocity of the model ($V_{s,min}$ ). At high frequency, deep layers are ignored by the surface waves behaviour. For Rayleigh waves, all higher modes tend to $V_{s,min}$ and $V_{s,max}$ at high and low frequencies respectively. For the fundamental mode, the minimum ($V_{r,min}$ ) and maximum ($V_{r,max}$ ) velocities at high and low frequencies are slightly less than $V_{s,min}$ and $V_{s,max}$ , respectively. $V_{r,min}$ and $V_{r,max}$ are equal to the Rayleigh velocity observed for a homogeneous half-space with the velocity of the first layer and the bottom half space, respectively. In this latter case, equations (3.31) and (3.32) simplify to

$$(\frac{1}{V_s^2}-2 \frac{1}{V_r^2})^2=4 \frac{1}{V_r} \sqrt{(1-\frac{1}{V_p^2})(1-\frac{1}{V_s^2})}$$ (3.36)

where $V_s$ is the velocity of S-waves, $V_p$ is the velocity of P-waves, and $V_r$ is the unknown velocity of Rayleigh waves. Equation (3.36) shows that the velocity of Rayleigh waves is constant for all frequencies, and hence no dispersion takes place. $V_{r,max}$ for Rayleigh dispersion curves (fundamental mode) is thus calculated by solving equation (3.36) with $V_s$ and $V_p$ being the slownesses of the layer with the minimum $V_s$ . A few Newton-Raphson (Press et al. (1992)) iterations are necessary to obtain $V_{r,min}$ and $V_{r,max}$ .