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Integrated Asset Management and Optimization Workflows 221

of convergence efficiency (Efendiev et al., 2005), and the variant using
sampling from the streamline-sensitivity constrained proxy model
(Maucec et al., 2007, 2011, 2013a,b; Ma et al., 2006). The multistage
McMC approach still satisfies the necessary detailed-balance condition
to sample from the equilibrium (i.e., stationary or posterior) distribution
(Maucec et al., 2007) and significantly faster convergence, which can be
monitored via, for example, maximum entropy condition (Maucec et al.,
2007; Bratvold et al., 2010) or by using multivariate potential scale reduc-
tion factor (MPSRF) (Li and Reynolds, 2017).
The McMC algorithm is designed to rigorously sample from the poste-
rior distribution; its drawback lies in the reliance on the specification of the
prior model statistics and also the computational cost in exploring the pos-
terior distribution when the number of parameters is large. When realistic
field conditions are considered, the number of parameters of the prior model
6
expands dramatically (i.e., 10 ). Computation of the prior term of the
objective function then becomes highly demanding and time consuming,
particularly due to inversion of the prior covariance matrix C M . To maintain
acceptable computation effort, the AHM algorithms have to resort to model
parameterization techniques as outlined in Table 6.2. For example, Maucec
et al. (2007) introduce the approach where parameterization and model
reduction on a covariance matrix C M is performed using SVD, and new
model realizations are generated in the wave-number domain by a simple
convolution of zero-mean independently distributed entries (white noise)
with an appropriate Fourier linear filter. Jafarpour and McLaughlin
(2007) introduce the application of the discrete cosine transform (DCT),
an industry standard for image compression (e.g., the JPEG format) in res-
ervoir model inversion and history matching, while Maucec and Cullick
(2015) and Maucec (2016) develop a DCT-based approach for rapid gener-
ation of model updates in wave-number domain with applications to AHM
(Maucec et al., 2011, 2013a,b).

6.3.3 Data Assimilation

One of the most popular data assimilation approaches in petroleum and
groundwater applications is Ensemble Kalman Filter (EnKF), introduced
in the early 1990s (Evensen, 1994) for forecasting error statistics. The EnKF
was first deployed for solving history match reservoir simulation problems by
Nævdal et al. (2003). It approaches the AHM optimization (minimization)
problem by solving the Bayesian form of the objective function O(m) in the