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\section{Introduction}
One of the major challenges limiting large-scale Carbon Capture and Storage (CCS) operations, is the issue of CO_2 storage. As the reservoir often determines all design and operational conditions of the complete CCS from the very beginning, having a sound knowledge of the physical characteristics of a storage site is crucial to determine the optimal rate of CO_2 injection, which influence the rate of capture, as well as to assess a proper monitoring strategy to prevent the migration of the CO_2 up to the surface. \\
A common goal in CCS projects is to use the monitoring data to verify the CO_2 distributions that are predicted by simulations produced for geological models of the reservoir. However, CO_2 distributions that are imaged through monitoring are often inconsistent with the model-based simulations to varying degrees \citep{Ramirez2013}. This is mainly due to the lack of data and uncertainties that limits the understanding of the subsurface and therefore the ability to produce accurate reservoir models. \\
To build a numerical reservoir model, the spatial distribution of reservoir properties (e.g., lithology, porosity, permeability) first needs to be described. Because of the geological complexity and the scarcity of direct observation (i.e. well data) the probabilistic methods appears to be the most appropriate choice for reservoir modeling. Seismic measurements are well suited in reservoir modeling as provide indirect, but nevertheless spatially extensive information about reservoir properties that are not available form well data alone. In addition to the static information and in order to evaluate the performance of the reservoir in term of CO_2 storage, reservoirs models need to constrained to dynamic data obtained from the CO_2 injection operations. Ideally, reservoir models should match the observed dynamic behavior of the reservoir to within some accepted tolerance. To check the model's consistency with dynamic data, flow simulation is required. The process to adjusting/perturbing an initial reservoir model is commonly know as history matching and extensively used in the oil and gas industry.\\
The estimation of model parameters from seismic data is a complex, ill conditioned, nonlinear inverse problem due to the intrinsic limitations of the geophysical method: the limited bandwidth and resolution of the seismic data, noise, measurement errors, and physical assumptions about the involved forward models \citep{Tarantola_2005}. Seismic inverse problems may be developed following deterministic or probabilistic approaches and can be divided into two main categories: (1) multistep inversion methods and (2) stochastic inversion methods \citep{Grana2012}.\\
In multistep inversion methods, the problem of estimating reservoir properties from seismic data is split into two or more subproblems; elastic properties are first derived from partial stacked seismic data by elastic inversion; then, reservoir properties are classified by statistical techniques, such as Bayesian classification \citep{Avseth2001,Mukerji2001,Buland_Omre_2003}.\\
Iterative stochastic inversion methodologies solve the seismic inverse problem using deterministic or stochastic optimization techniques. First, a set of equivalent earth models is simulated using a stochastic sequential algorithm based on prior information usually from available well-log data and a spatial continuity pattern (e.g, variogram, training image) to create fine-scaled reservoir models \citep{Bosch2009} . Then suitable rock-physics transform are applied to generate the corresponding volumes of the elastic properties. Finally, synthetic seismic volumes are computed and compared to real seismic data to evaluate the mismatch. The final model is found by applying a suitable optimization method. \citet{Gonzalez2008} performed a trace-by-trace deterministic optimization, \citet{Bosch2009} propose an iterative optimization based on Newton's method; Markov chain Monte Carlo approach has been used succesfully for the stochastic exploration of the model space \citep{Eidsvik2004,Larsen2006,Gunning2007,Rimstad2010,Ulvmoen2010,Hensen2012}. \citet{Grana2012} show the efficiency of the probability perturbation method \citep{Caers2006} to estimate fine-scaled reservoir models in a stochastic inversion. Multidimensional scaling technique was successfully applied by \citet{Azevedo_2013} to asses how the parameter model space is explored by global elastic inversion algorithm. \\
In this paper we propose a stochastic inversion workflow using the gradual deformation parametrization method \citep{Roggero_1998} as a stochastic optimization technique that integrates geophysical and geological logs, seismic reflection data and CO_2 flow simulations in order to analyzing and monitor the CO_2 injection and propagation within a saline aquifer.
This paper is organized as follows. In the first section we focus on each step of the seismic inversion algorithm. The second section focus on the application of our approach to characterize a synthetic reservoir for the CO_2 injection in the St. Lawrence Lowlands, Quebec, Canada.
