\section{Introduction} 
One of the major challenges limiting large-scale deployment of Carbon Capture and Storage (CCS) operations is the issue of evaluating and forecasting the fate of CO_2 in deep rock formations. The characteristics of the reservoir determine from the very beginning the design and operational conditions of most of the CCS chain.  Indeed, having a comprehensive knowledge of the physical characteristics of the storage site is crucial to determine the optimal rate of CO_2 injection, which influences the rate of capture, as well as the parameters of a proper monitoring strategy.  Let us recall that monitoring is an essential component of any CCS project and that most, if not all, jurisdictions require more or less elaborated monitoring, verification and accounting (MVA) programs (e.g.\ European CCS directive).

A common task in CCS projects is thus to monitor the spatial distribution of CO_2 over time. However, the spatial distribution of the CO_2 plume that is estimated through monitoring is often inconsistent, to varying degrees, with model-based simulations \citep{Ramirez2013}. This is mainly due to the lack or sparseness of direct measurements of petrophysical properties of the reservoir, to the low spatial coverage of geophysical surveys, to the underlying resolution of inversion techniques and to uncertainties that limits the understanding of the subsurface and therefore the ability to produce accurate reservoir models allowing reliable forecast of the CO_2 plume.

The process of optimizing a reservoir model to fit dynamic data is commonly known as history matching and is extensively used in the oil and gas industry. In conventional history matching, the target variables are reservoir engineering properties such as pressure, water-cut, rate of production of oil, and the like, all involving at least one injector and one pumping well. In CO_2 storage projects, such well configuration is not conceivable (pumping is not done) and only one injector well is usually available. In this context, only indirect data have the spatial coverage and the potential to improve the estimation of the CO_2 plume within the deep saline aquifers.

History matching requires the knowledge, at least in the stochastic sense, of the spatial distribution of the static properties of the reservoir (porosity, permeability). Due to the geological complexity and the scarcity of direct observations (i.e.\ well data), probabilistic methods appear to be the most suitable choice to build a reliable reservoir model. In addition, it is established that seismic measurements are well suited to constrain static reservoir property modeling as they provide indirect but correlated and spatially extensive information about reservoir properties \citep{Doyen2007}.

The estimation of static parameters from seismic data is a complex, ill-conditioned, nonlinear inverse problem due principally to the limited bandwidth and resolution of the seismic data, to noise, to measurement errors, and to the assumptions underlying to 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) multiple step inversion methods and (2) stochastic inversion methods \citep{Grana2012}.

In multiple step 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 through 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 algorithm based on prior information usually from available well log data and a spatial continuity pattern (e.g.\ variogram, training image) to create fine-scale reservoir models \citep{Bosch2009} . Then suitable rock-physics transforms 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. Several optimization methods exist to infer the elastic properties based on the measured traces. \citet{Gonzalez2008} performed a trace-by-trace deterministic optimization, \citet{Bosch2009} proposed an iterative optimization based on Newton's method; Markov chain Monte Carlo approach has been used successfully 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-scale reservoir models in a stochastic inversion. A 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 addition to the static information and in order to evaluate the performance of the reservoir in terms of CO_2 storage, reservoirs models need to be constrained by dynamic data obtained from the CO_2 injection operations. To be meaningful, reservoir models must match the observed dynamic behavior of the reservoir within some interval of tolerance. History matching is also an ill-posed problem and can be solved using the same algorithms as the one used for estimating the static properties, but then involving the injection data and forward fluid flow modeling.

In this paper we propose a stochastic inversion workflow using a modified 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 analyze and monitor the CO_2 injection and its 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 presents an the application of the approach to characterize a synthetic reservoir for the CO_2 injection in the St.\ Lawrence Lowlands, Quebec, Canada.

    
    
  