End-point check

For the fundamental mode, the velocity has to be lower than $V_{s,max}$ , even for Love waves. When it is equal, it generally results from a mode jumping taking place at a higher frequency. For the higher modes, the $V_{s,max}$ value is obtained when reaching the frequency threshold below which the mode does not exist. For all modes, the last point (at the lowest frequency) is checked by searching a hypothetical additional mode at a lower velocity (in the reverse direction of the initial search with the same step). If any root is encountered between the higher bound of the preceding mode and the lower bound of the last sample of the current mode, it means that at least one mode is missing. Unlike Herrmann's code, the root search and root refinement are always preserving the upper and lower limits of the roots. In this way, there is absolutely no risk to confuse the search result with the previously calculated modes. This check assumes that the distance between modes is changing along the frequency axis. When there are two osculation points with one located at the lowest frequency of the user range, this algorithm may however fail to detect any mode jumping. For the Rayleigh fundamental mode, there is absolutely not risk of such phenomena, if the user frequency range extends to a sufficiently low frequency, for instance, below the threshold frequency of the first higher mode.