First higher mode alone

The first higher mode curve in figure 4.1(b) is resampled with 30 points and a constant frequency step on a log scale between 2.75 and 20 Hz. This curve is inverted with the parameterization described in table 4.3.

Figure 5.1: Inversion of first higher mode alone: no prior information. (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 fundamental curve, not used for the misfit computation

The results are shown in figure 5.1. The fundamental mode is required to calculate the first higher mode (section 3.1.5). This is why both modes are plotted in figure 5.1(c), clearly visible with two families of curves, the highest velocity values being the first higher mode. The high misfit obtained (compared to less than 0.02 in figure 4.6) is due to the bad fit of the first higher mode between 6 and 12 Hz. When comparing the theoretical fundamental curve (dotted black line) with the calculated fundamental mode (second family of curves, the lowest), a clear gap is observed. In figure 5.1(b), almost no model is generated with a depth of the first interface above 20 m. Contrary to fundamental mode, with this parameterization, the inversion seems to be trapped in a secondary minimum of the parameter space with a misfit around 0.1.

Figure 5.2: Inversion of first higher mode alone: depth between 1 and 20 m/s. (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 fundamental curve, not used for the misfit computation.

To force the algorithm to explore other regions of the parameter space, the inversion is done again with the interval for the first thickness reduced to $[1,20]$  m. The results are displayed in figure 5.2 in the same way as in figure 5.1. A minimum misfit less than 0.01 is found with a depth and a fundamental model that better fit the theoretical model. From 4 Hz and below, the calculated fundamental curve does not follow the theoretical curve. This indicates that the solution is not completely investigated by the neighbourhood algorithm and that the first higher mode does not carry exactly the same information as the fundamental mode. Intensive inversion runs would generate good fitting models with a fundamental mode around the theoretical curve (not done here). In this case, we know that a better solution exists for depths lower than 20 m. But even for real cases, this kind of operation is adviced to check the validity of the obtained profiles.