Fundamental mode alone

To estimate the quality of the information carried by each mode, it is necessary to visit again the fundamental mode inversion. The inversion plotted in figure 4.6 is relaunched with the simultaneous computation of the first higher mode. The results are shown in figure 5.3. Comparing

Figure 5.3: Inversion of the fundamental mode alone. (a) Resulting $V_p$ profiles. (b) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (c) Dispersion curves corresponding to models of figures (a) and (b). The grey curves are the calculated fundamental mode (lowest curves) and the first higher mode (highest curves). The black dots are the theoretical dispersion curve used as the target curve during inversion. The dotted line is the first higher mode, not used for the misfit computation.

figures 5.2(c) and 5.3(c), where all models with a misfit less than 0.1 are selected, the deviations around the target curves (black dots) are similar. Obviously, the fundamental curve offers a weaker constraint over the depth of the second layer than the inversion of the first higher mode (figures 5.2(b) and 5.3(b)). However, the fundamental curve inversion does not tolerate $V_s$ greater than 3400 m/s just below 100 m, whereas for the first higher mode, many models with $V_s$ greater than 3200 m/s are found with a low misfit. In figure 5.3, the average curve calculated for the first higher mode fits perfectly the theoretical curve for all frequencies below 5 Hz. Between 5 and 15 Hz, the fundamental mode does not constrain the first higher mode, in a similar way that the first higher mode cannot constrain the fundamental mode below 5 Hz (figure 5.2). From these observations, the fundamental curve seems to be necessary below 5 Hz and the first higher mode is mandatory above 5 Hz, other parts are carrying redundant information. These threshold frequencies are valid only for this case and do not have a general meaning.

Figure 5.4: Inversion of the fundamental and the first higher mode. (a) Resulting $V_p$ profiles. (b) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (c) Dispersion curves corresponding to models of figures (a) and (b). The grey curves are the calculated fundamental mode (lowest curves) and the first higher mode (highest curves). The black dots are the theoretical dispersion curves used as the target curve during inversion. The dotted lines are the fundamental and first higher mode not used for the misfit computation.