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until the final two observation times are used-the posterior PDF that results from
assimilation of observations at 30, 60, 90, and 120 min has a single mode. The fact
that multimodality arises suddenly in the solution upon incorporation of a single
additional observation time is interesting, and has implications for the performance
of ensemble data assimilation methods. Note that the sample of the joint posterior
distribution produced by MCMC, in addition to its utility for parameter optimization
and uncertainty quantification, can also be used as a benchmark for examining
the characteristics of approximate solutions to inverse problems. A comparison of
the posterior PDF produced by MCMC with that obtained using deterministic and
stochastic versions of an Ensemble Transform Kalman Filter (ETKF,
Bishop et al.
2001
)ispresentedin
Posselt and Bishop
(
2012
).
3.5
Concluding Remarks
MCMC is now widely used for Bayesian inference in the statistical research
community, and is gaining popularity in the physical and social sciences. The large
dimensionality of Earth system datasets and complexity of process models present a
significant computational and algorithmic challenge. This is particularly true in the
case of global Earth system models and high resolution process models, which may
require weeks of compute time for a single integration. For these models, running
tens of thousands of integrations in a Markov chain is simply not feasible. Nev-
ertheless, the development of efficient sampling algorithms, judicious application
of simplified models, and widespread availability of multicore computing make
application of MCMC to problems in atmospheric, oceanic, and hydrologic sci-
ences increasingly feasible. As demonstrated above, significant progress is already
possible in the areas of model uncertainty quantification and satellite retrievals. In
particular, use of MCMC to evaluate simpler (and computationally more efficient)
satellite retrieval algorithms and data assimilation schemes appears to be particularly
promising. In addition, the PDF returned by MCMC can be effectively used to assess
the information content in new and future observing systems, and to examine which
types of observations might be used to constrain uncertain parameters in numerical
models. It is likely that the next few years will see continued development of
adaptive and hybrid algorithms, as well as innovative uses of MCMC for exploration
of processes in the physical system. Experiments with sequential MCMC will
allow more sophisticated evaluation of ensemble filters, as well as advance the
development of nonlinear, non-Gaussian data assimilation techniques.
On a final note, though it is still common for scientists and statisticians to write
their own MCMC software, open source codes are becoming more widely available.
These include (among others) the R package “mcmc” (
http://www.stat.umn.edu/
geyer/mcmc/
)
, Bayesian inference Using Gibbs Sampling (BUGS,
Lunn et al. 2009
,
http://www.openbugs.info
)
, the Delayed Reject Adaptive Metropolis code of
Haario
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