$V_s$ inversion without LVZ

In common geological situations, $V_s$ increases with depth: rock weathering, sediment compaction,...  (Scherbaum et al. (2003); Bachrach et al. (2000)). However, the velocity may decrease with depth in some cases: saturated layers, clays overlied by sandy formations, hard ground above unconsolidated sediments, lava flows,...From the above example (section 4.3.1), if the soil structure is made of thin intercalations of soft and rigid layers, the dispersion curve inversion cannot resolve the properties of each individual layer. Consequently, a limited number of LVZs can be tolerated in the model when the geological structure of the area justifies it. Between two particular LVZs, the velocity must be constant or must increase with depth. Taking these conditions into account during inversion is capital but not straightforward. There are numerous ways of implementing such prior information, we developed some of them, described in appendix B.

Figure 4.15: Inversion with a N-layer model rejecting LVZ by the diagonal method. (a) Resulting $V_s$ profiles. The black lines are the theoretical velocity profiles. (b) Dispersion curves corresponding to models of figure (a). The black dots are the theoretical dispersion curve used as the target curve during inversion.
Figure 4.16: Comparison of a three-layer and N-layer inversions. The minimum and maximum $V_s$ for models with a misfit lower than 0.02 are reported for each inversion case: three-layer inversion (plain lines), N-layer with LVZs (dotted lines), and N-layer without LVZs (dashed lines). Figure (b) is a zoom on the first ten metre for clarity.

Theoretically, the parameterization must ensure that any ground model included in the parameter space has an equal chance to be generated by the neighbourhood algorithm. If this is not verified, the inversion algorithm itself introduces prior information, prefering particular classes of models to others. For instance, in section 4.3.1, all $V_p$ profiles have the same chance to be generated, but the $V_s$ profiles are calculated by the mutliplication of two random variables and have not a uniform probability (figure B.1). The prior distributions of the proposed methods are detailed in appendix B.

The inversion of the broad band dispersion curve is started with five distinct random seeds, using the scaled diagonal parameterization for $V_s$ profiles and a fixed $V_p$ profile (section B.8). 50 iterations are launched per inversion process generating a total of 25500 models. Among them, 14000 have a misfit lower than 0.1. The results are shown in figure 4.15. In the same conditions, the scaled interpole method tested in figure B.8 produces only 285 models with misfit lower than 0.1. In this case, the choice of the method for generating models has a strong influence on the global efficiency of the inversion algorithm.

In figure 4.16, the inversion with a three-layer model and with a N-Layer model accepting LVZs (figure 4.13) or rejecting LVZs (figure 4.15) are compared. The misfit are calculated on the same data curves in the three cases. Only the minimum and maximum $V_s$ observed at each depth for each case is reported in the figures. The inversion which accepts LVZ always results with quite large uncertainties compared to the inversions assuming an increase of the velocity with depth. The three-layer inversion gives more information about the depth uncertainty, compared to other cases, whereas it under-estimates the uncertainty on the velocity, especially below the velocity contrasts (between 10 and 70 m, and between 100 and 180 m) and near the surface (between 0 and 5 m).

In conclusion, inverting with a very simple model made of uniform layers does not provide the complete uncertainty about the ground structure. In contrast, the inversion with a great number of layers requires the introduction of relationships between the velocities of adjacent layers, to avoid generating lot of low velocity zones. Those relationships can be translated into parameterization rules for a simple structure where $V_p$ is constant or increasing.