Higher mode identification

In sections 5.1.1 and 5.1.2, the modes are supposed to be correctly identified before proceeding with the inversion. In many real cases when dealing with the vertical component, the apparent dispersion curve with the lowest velocity is usually interpreted as the fundamental mode of Rayleigh waves. For active source experiments measured at high frequencies (above 10 Hz), higher modes may predominate (Gabriels et al. (1987); Socco and Strobbia (2004); Forbriger (2003b); Xia et al. (2003)). For ambient vibrations, which commonly yield dispersion curve at low frequencies, higher modes are less studied but their presence is sometimes suspected. According to the array resolution power, it is not always possible to separate modes and an intermediate velocity may be observed. In this last case, no post-processing can be considered on the observed apparent velocity values because there are too much parameters to play with (array geometry, source distance, energy partition between co-existing modes, ...). A prior knowledge of the ground structure or other geophysical acquisitions are necessary to detect anomalies on the supposed fundamental dispersion curve. This case is not analysed in this section.

For other cases, a bad identification of modes may ruin all inversion results as demonstrated by Zhang and Chan (2003) and by the following example. The same soil structure as in section 4.2 is used here. In figure 4.1(b), the fundamental and the first higher modes for Rayleigh waves are very close to each other around 9 Hz (osculation point). Depending on experimental conditions, it may be possible to select a branch below 9 Hz corresponding to fundamental mode and another branch above 9 Hz following the first higher mode. This situation is depicted in figure 5.7 where the observed apparent velocity is marked by black dots. At first glance, the obtained curve may be interpreted as a single fundamental mode. This curve is inverted as the fundamental Rayleigh mode with a prior information that the depth of the basement is situated between 95 and 105 m like in the inversion plotted in figure 4.10. The results of five runs are summarized in figure 5.8. The black lines in figures 5.8(a) and 5.8(b) are the theoretical ground model. The difference is especially strong on the first 20 metres where the velocity profiles are usually well retrieved. There are more than 50% of bias in the obtained results. For real sites, this phenomenon can be detected only if external data or a prior knowledge are also available. Indeed, there is no argument to reject the interpretation of figure 5.8 from the dispersion curve itself.

Figure 5.7: Composite dispersion curve. The black dots represent the dispersion as it can be observed. The grey line are the theoretical dispersion curves of the fundamental (plain line) and the first higher (dotted line) modes.

l0cm

If the results of the inversion with the fundamental Rayleigh mode are far from the expected profiles, the inversion with other Rayleigh modes can be tested with an inversion algorithm we developed to automatically identify higher modes. The inversion with this option requires only one data curve and the assumption of the number modes ($n_m$ ) that are encountered by the data curve. For each generated model and for each frequency sample of the data curve, $n_m$ modes are simultaneously calculated. Compared to usual inversions, the misfit is computed in a completely different way. The velocity difference ( $\delta v_i=v_{di}-v_{ci}$ ) at each frequency between the data velocity and the theoretical Rayleigh velocities of each mode (up to $n_m$ ) is calculated. Only the minimum value is kept in the summation of equation 3.38. Virtually, the best fitting mode may be different for each frequency sample. However, these kinds of oscillations are rarely observed due to the curve smoothness which naturally restricts the number of mode changes to one or two on the available frequency range. This method effectively adds one or two pseudo degrees of freedom to the inversion problem and it is sometimes necessary to use more restricted parameterized model.

Figure 5.8: Inversion of the composite curve assuming fundamental mode. (a) Resulting $V_p$ profiles. (b) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (c) Fundamental mode dispersion curves corresponding to models of figures (a) and (b). The black dots are the composite dispersion curves used as the target curve during inversion.

Figure 5.9: Inversion of the composite curve with mode identification. (a) Resulting $V_p$ profiles. (b) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (c) Fundamental and first higher mode dispersion curves corresponding to models of figures (a) and (b). The black dots are the composite dispersion curves used as the target curve during inversion.

Figure 5.10: Inversion of the composite curve with mode identification, splitting model families. (a), (d), (g), and (j) Resulting $V_p$ profiles. (b), (e), (h), and (k) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (c), (f), (i), and (l) Fundamental (below) and first higher (above) mode dispersion curves corresponding to models of the other figures. The black dots are the composite dispersion curves used as the target curve during inversion (see text for details).

The inversion method is tested on the dispersion curve displayed in figure 5.7 with the assumption that two modes may be present in the experimental curve. Tests with more than two modes have not been carried out so far. The frequency range of the dispersion curve is similar to the range used in figure 4.7^5.1 where it is clear that no information below 10 m is recovered. The parameterization used in figure 4.10 (table 4.4) offers a slightly better constraint and is chosen for the inversion with automatic mode identification. The results of the five inversion runs (5*15100 models) are gathered in figure 5.9. In figure 5.9(c), two modes are plotted for each model in figures 5.9(a) and 5.9(b).

Four families of curves with low misfit values can be distinguished. For clarity, these four categories are shown individually in figure 5.10. In the first category (figures 5.10(a) to 5.10(c)), the data curve is considered as being entirely the first higher mode. The minimum achieved misfit is higher (0.055) than for other groups, but it does not automatically mean that models are to be discarded. Valid arguments to reject them would be that superficial measurements revealed a lower $V_s$ or that a strong contrast between 40 and 60 m is not geologically admissible. The second category is the same as our first hypothese (all data considered as fundamental mode). Lower misfit values are obtained (0.025). Here again, complementary acquisitions about the superficial $V_s$ or depth criteria help to discard those models. In the third family of models, the data curves is also likened to first higher mode but in a different way than the first category. Here the difference with the theoretical model in terms of $V_s$ and depth is more subtile. The measurement of the dispersion curve on a larger frequency band, for instance if Love modes can be observed, may help the interpretation. And finally in the last category, a mode jump is noticed around 9 Hz and the velocity profiles correspond to the theoretical ground model. The parameter space sampling is certainly not exhaustive for depth below 100 m. Further model generation can be conducted with a shallow depth restricted around 10 m to get a more complete and confirmed model uncertainty (not done here). Tests were conducted with the parameterization of table 4.3 but nothing could be retrieved due to the insufficient level of constraint.

This algorithm allows a great flexibility to scan the various modes possibly contained in the observed dispersion curve. However, it adds at least one more degree of freedom, increasing then the non-uniqueness of the problem. The prior information is here, probably more than elsewhere, of prime importance to select the right model family.

Exactly the same technique has also been tested on synthetics to identify Love and Rayleigh modes (not shown here).