\subsection{Model Validation}
A statistical analysis is needed to validate the results of our stochastic inversion approach. Figure \ref{fig:correlation} shows the correlation between observed data and one initial SGS realization chosen at random among the set of 100, as well as the correlation between observed data and the final inversion result. The correlation between observed data and final inversion result is slightly improved, though the initial set of SGS realizations are already well correlated with the observed data.  It is also important to obtain a result that honor the original distribution (i.e.\ the observed data). The quantile-quantile (q-q) plot is best suited to study how much two different distributions are similar. Figure \ref{fig:qqplot} shows the q-q plot for the reference model and the inversion result for $V_p$, $V_s$, $\rho$ and $\phi$. A \num{45}-reference line is also plotted in red. Since almost all the points fall along the reference line, we consider that the inversion results distributions are the same as the observed distributions.
    
    
  