Geoscience Reference
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where, the objective function
O
m
combines prior and likelihood terms:
1
1
T
T
1
0
1
M
0
O
md g m
C
d gmmmmm
C
(7)
D
2
2
md
pdf corresponds to the minimum of
The
Maximum A Posteriori
(MAP)
p
|
(|)
O
m
,
md
with the set of parameters,
m
that minimizes
O
m
as the most probable estimate. The HM
minimization algorithm renders multiple plausible model realizations and the consequence
of non-linearity is that it requires an
iterative
solution. When considering realistic field
conditions, the number of parameters of the prior model expands dramatically (
i.e.,
order of
10
6
) and computation of the prior term of the objective function becomes highly demanding
and time consuming. A variety of model parameterization and reduction techniques have
been implemented in HM workflows, ranging from methods based on linear expansion of
weighted eigenvectors of the specific block covariance matrix
C
(Rodriguez
et al.,
2007;
Jafarpour and McLaughlin, 2009; Le Ravalec Dupin, 2005) to methods, where expensive
covariance matrix computations are avoided by generating model updates in wave-number
domain (Maučec
et al.,
2007; Jafarpour and McLaughlin, 2009; Maučec, 2010) that do not
require specifying the model covariance matrix,
C
, and performing expensive inversions.
M
2.5.1 Sampling from the posterior distribution
Two methods have been proposed to sample parameters of posterior distribution, for
example, sequential Markov chain Monte Carlo (MCMC) algorithms (Neal, 1993) and
approximate sampling methods, such as Randomized Maximum Likelihood (RML)
(Kitanidis, 1995), both with some inherent deficiencies. Traditional Markov chain Monte
Carlo (MCMC) methods attempt to simulate direct draws from some complex statistical
distribution of interest. MCMC techniques use the previous sample values to randomly
generate the next sample value in a form of a chain, where the transition probabilities
between sample values are only a function of the most recent sample value. The MCMC
methods arguably provide, statistically, the most rigorous and accurate basis for sampling
posterior distribution and uncertainty quantification but they come at high computational
costs. On the other hand, the approximate, but faster, RML methods are, in practice,
applicable mostly to linear problems.
In an attempt to improve computational efficiency and mixing for the MCMC algorithm,
Oliver
et al
.
,
1996, proposed a two-step approach in which (1) model and data variables were
jointly sampled from the prior distribution and (2) the sampled model variables were
calibrated to the sampled data variables, with Metropolis-Hastings sampler (Hastings, 1970)
used as the acceptance test. The method works well for linear problems, though it does not
hold for non-linear problems, such as HM studied here. To improve on that, Efendiev
et al.
,
2005, proposed a
rigorous
two-step MCMC approach to increase the acceptance rate and
reduce the computational effort by using the sensitivities calculated from tracing
streamlines (Datta-Gupta and King, 2007). When the sensitivities are known, the solution of
the HM
inverse problem is greatly simplified. One of the most important advantages of the
streamline approach is the ability to analytically compute the sensitivity of the streamline
Generalized-Travel-Time (GTT) with respect to reservoir parameters,
e.g.
porosity or
permeability. The GTT is defined as an optimal time-shift
at each well, so as to minimize
the production data misfit function
J
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