Generating increasing velocity profiles

As reported in section 4.3.1, ensuring an increasing velocity across a ground structure is necessary to retrieve information from the inversion of dispersion curves. Methods of parameterizations to achieve this requirement are proposed in this appendix. The parameterization may introduce prior information into the inversion by preferring some classes of models to others. The best method is the one that provides an equal chance to all models to be generated at random.

Figure B.1: Prior information carried by parameterization: LVZ ($V_s$ profile). The black lines are the minimum and maximum admissible velocity profiles.

L0cm

The various methods are tested in terms of prior information brought by the parameterization itself. For doing so, 10000 models are randomly generated with each type of parameterization. At each depth, an histogram is constructed counting the number of occurrences in each velocity class (100 classes from 0 to the maximum velocity allowed by the parameterization). All histograms are summarized in a velocity-depth plot with the number of occurrence indicated by grey scales. A first example is shown in figure B.1 for the $V_s$ profile of the inversion of section 4.3.1. The distribution at each depth is not perfectly uniform which is prone to introduce some uncontrolled prior information if $V_s$ is not well constrained by the dispersion curve. On the contrary, for the same case, the $V_p$ profile has a perfect uniform distribution (not shown here). This fact is unavoidable when making a variable transformation to obtain the physical parameters of the ground model (section 4.2).