Bracketing the root candidates

The roots are searched starting from the highest to the lowest frequency within the range defined by the user. The method is illustrated in figure 3.5. The grey curves correspond to the same dispersion curves as in figure 3.3. In this section, the Rayleigh case is discussed, but the same method applies to Love's case as well, by replacing $V_{r,min}$ by $V_{s,min}$ , and $R_{1212}(z_0)$ by $l_{21}(z_0)$ . $f_1$ is the highest frequency of the user range and $f_2$ is the second highest frequency. The fundamental and the first two higher modes are represented. The plus and minus signs represent the polarity of function $R_{1212}(z_0)$ . The polarity below the fundamental curve (initial polarity) is computed for $\frac{V_{r,max}}{1.05}$ and at low frequency ( $\frac{1}{2\pi}$  Hz). The minimum limit is divided by 1.05 to be sure that the fundamental mode is not missed. The polarity alternates when crossing a modal curve.

Figure 3.5: Method for bracketing roots.

The first root with the minimum velocity, that corresponds to the fundamental mode at the highest frequency ($f_1$ ), is bracketed by increasing the velocity from $\frac{V_{r,max}}{1.05}$ with an adaptive step until finding a sign change. It always exists as the fundamental mode is present for all frequencies (grey dots and black dots when a root is found). The search step is calculated by multiplying the lower limit of the current interval by a constant step ratio. Either for Love and Rayleigh, half the difference between $V_{s,min}$ and $V_{r,min}$ is taken as a reference to adjust the initial velocity step. Hence, the step ratio is defined by

$$\frac{V_{s,min}-V_{r,min}}{2\:V_{s,min}}$$ (3.37)

This method is particularly justified in this case because the ratio of the minimum and maximum velocities of the admissible range is usually around 4 or 5. The step ratio is eventually reduced and the precision is increased, if mode jumping is detected (section 3.1.6). Once a root has been bracketed, its upper and lower bounds are refined down to the current precision using the algorithm described in the next paragraph. The higher bound of the refined interval is kept as the calculated curve. The modal velocity is then computed for the next frequency sample $f_2$ .

The starting velocity for the new search is the velocity calculated for the preceding frequency sample, $f1$ in this case (the higher bound is taken). The search direction depends upon the polarity observed at $f_2$ and at the starting velocity. If it is the same as the initial polarity, the true dispersion is located at a higher velocity (as in figure 3.5) and the root is refined after the same type of search as in $f_1$ . In the other case, the dispersion has a non-monotonous shape, characteristic of models with low velocity zones (section 3.1.6). No search is made because a polarity change has already been found; the root is directly refined. The same process is applied for all frequency samples until the lowest. A final test described in section 3.1.6 is performed on the obtained modal curve. Afterwards, the curve is definitively accepted.

For higher modes, the minimum of velocity ranges are reduced to the values of the refined higher bounds of the preceding mode. The initial polarity is inverted. The modal curve may not be defined for all frequency samples. If so, the velocity search stops at $V_{s,max}$ . If the polarity at $V_{s,max}$ is the same as the initial polarity, no root exists and the computation of this mode is stopped. The same test as for the fundamental mode is performed before definitively accepting the curve.