When simulating a production scenario of a reservoir, it is fundamental to have a rock physics model, that allows to predict the variation of subsurface elastic properties, during the different production stages. The time lapse elastic inversion technique, aims to retrieve from seismic data, the variation in time of the elastic properties describing the subsurface. In this context, the presented project, aims to apply a Bayesian approach in order to solve a time lapse inversion problem. Furthermore, an analysis of the solution uncertainties and stability are also presented, for different initial conditions such as different noise level in the seismic data set, or the case in which the frequency of the ricker wavelet used in the forward operator during the inversion process doesn’t match the source wavelet frequency. The thesis belongs to the framework of a reservoir characterization process, and can be summarised in two stages: the first part, included the implementation of a rock physics model in a non clastic reservoir, in order to estimate the behaviour of the subsurface elastic properties for various reservoir fluid saturation conditions. The second part, consisted in exploiting such predicted properties, in order to compute the observed data for the selected reservoir saturation conditions, and hence applying an ensemble based Bayesian inversion in order to evaluate the subsurface elastic properties and their associated uncertainties at different time. More in depth, during the first phase of the project four well logs dataset belonging to the same reservoir have been provided, as well as information relative to the fluid in place, and reservoir conditions. The geological setting, is represented by an isolated carbonate platform. Therefore, being a carbonate environment, characterised by a strong heterogeneity, it was crucial to set up a rock physics model that allowed to differentiate the presence of different pore types and map their distribution in order to obtain more reliable results when performing the fluid substitution workflow using Gassmann equation. Such task has been successfully accomplished by combining the Xiu Payne method for RPM and the Eshelby-Walsh theory: through the use of the relationship between P sonic and effective porosity log curves, the model here presented, allows to discriminate between three different pore types (interconnected refence pores, stiff pores and microcracks) and assigns a specific pore aspect ratio to each of them. Furthermore, a modified Gassmann equation, (Elshby-Walsh theory) has been applied, which includes a rock-matrix term m, function of the pore aspect ratio distribution, hence enabling a more realistic modelling of the bulk rock elastic properties for different fluid saturation conditions. It must be remarked here, that in reservoir characterisation, the use of a reliable RPM is fundamental, in order to build a reliable static model of the subsurface otherwise, there may be strong misleading estimations when using such static model as starting point for the dynamic reservoir simulations. The fluid replacement workflow allowed to simulate five different reservoir saturation scenarios, going from the in situ conditions, 100% gas saturation, to the fully brine saturated reservoir, with a constant increment of 25% in brine saturation. Batzle Wang equations have been used to model gas and brine elastic properties, and finally, Reuss-Voigt-Hill average has been applied when estimating the density and bulk modulus of the two phases fluid. Having obtained the values for P velocity, S velocity and Density at each simulated scenario, the corresponding 3D reservoir models of the subsurface have been created, using Jason software. Such models, have been exported as seg-y files and used as starting point for the second part of the thesis. In the second phase of the project, due to com