\subsection{Geostatistical methods}
In our approach, elastic properties (i.e.,$V_p$, $V_s$, density ($\rho$) and porosity ($\phi$)) are first simulated using Sequential Gaussian CoSimulation (SGCS) \citep{Deutsch1998,Doyen2007}. SGCS is a method that allows simulating continuous random variables and that requires only the knowledge of their variogram, histogram and coefficient of correlation. Starting from prior information available form well log data, the algorithm visits each node of the grid along a random path. At each step along this path, the algorithm co-simulates a series of values for each variable. At each node, the prior information and the previously simulated values are used to compute the kriging mean and variance of each variable. This feedback loop ensures that the simulation is spatially correlated. By construction, SGCS  will be conditioned to the well data, i.e.\ the simulations reproduce the well observations and thus have the same overall statistical properties. Multiple simulation are generated by using different random paths and random seeds in order to obtain independent sets of realizations. For an extended review about SGCS methods, refer to \citet{Deutsch1998} and \citet{Doyen2007}.
    
    
  