Including Poisson's ratio

None of the described methods offers a really uniform prior distribution like the one obtained for the arbitrary profiles (section 4.3.1 and figure B.1). And the physical limits, like the limits on the Poisson's ratio, are not handled. However, with a fixed $V_p$ profile (not random), it is possible to generate pseudo $V_s$ profiles between 0 and 1 m/s with one of the available methods. In a second step, $V_s$ in each layer is scaled to $[V_{min}, V_{i,max}]$ , the maximum values being calculated from the increasing and fixed $V_p$ profile. The minimum value must be the same for all layers to avoid any LVZ when scaling the pseudo profile. The effect of the scaling transformation applied to the diagonal method (section B.7) is shown in figure B.7 for a fixed $V_p$ profile equal to the one of the theoretical model (figure 4.1(a)). By comparison the scaling transformation is also applied to the interpole method, starting from last layer (section B.4). If $V_p$ is also variable, the $V_s$ prior density of probability is less uniform than in figure B.7. With the interpole method, the maximum $V_s$ profile represented by the black line has almost no chance to be generated by the inversion algorithm. The diagonal offers a more uniform prior distribution with an similar probability for all model.

Figure B.7: Prior information carried by parameterization: scaled diagonal. The black lines are the minimum and maximum admissible velocity profiles.

Figure B.8: Prior information carried by parameterization: scaled interpole. The black lines are the minimum and maximum admissible velocity profiles.