Markov Chain Monte Carlo Methods: Theory and Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 78

Tabl e 3. 2 Observations used
in the MCMC-based
parameter sensitivity
experiments, along with their
units and error estimates

Observation Units Error

h 1 h 1

Precipitation rate mm 2.0 mm

Liquid water path mm 0.5 mm

Ice water path mm 1.0 mm

W m 2 10 W m 2 TOA LW radiative flux

W m 2 20 W m 2 TOA SW radiative flux

As in the retrieval problem, all ten parameters are perturbed simultaneously using
a Gaussian proposal distribution centered on the current parameter value and with
variance that is adaptively tuned early in each Markov chain so that the acceptance
rate is approximately 25 %. Parameter prior ranges are obtained from observations
of cloud particle properties ( Locatelli and Hobbs 1974 ; Mitchell 1996 ; Tokay and
Short 1996 ; Heymsfield et al. 2002 ; Roy et al. 2005 ). A set of specified parameters
is used to produce a true state, from which observations are drawn at 30, 60, 90,
120, 150, and 180 min of simulated time. Observations consist of precipitation rate,
liquid and ice water path, and outgoing longwave and shortwave radiative fluxes,
and measurement uncertainty is set equal to values consistent with error estimates
on Tropical Rainfall Measuring Mission (TRMM) retrievals. Each Markov chain in
the MCMC parameter estimation experiment was run for 4 10 6 iterations. Further

details of the parameter values, observations, and simulation output can be found in
Posselt and Vukicevic ( 2010 ).
Two dimensional marginal PDFs for select sets of parameters are shown in
Fig. 3.7 . The parameter sets depicted in this plot are chosen because they exhibited
multi-mode posterior PDFs and a non-trivial influence on each of the output vari-
ables of interest. Each row in Fig. 3.7 corresponds to assimilation of observations
with different characteristics and each demonstrates the utility of MCMC for
assessing observation impact as well as the vagaries of the assimilation algorithm.
Note that obtaining the results depicted in each row also required a new run of the
MCMC algorithm. It can be seen from Fig. 3.7 a-d that observations of precipitation
rate and liquid and ice water path are not sufficient to constrain the parameter
values; the mode of the joint PDF does not line up with the true parameter values.
When radiative flux observations are added to the likelihood function (Fig. 3.7 e-h),
the number of possible solutions is reduced and the mode in the PDF lies at
approximately the true value. However, there are clearly multiple modes in the
solution space. In an attempt to reduce the solution to a single set of most likely
parameter values, the algorithm is re-run under the assumption that more accurate
observations of outgoing long and shortwave radiation are available. This serves
only to exacerbate the multimodality (Fig. 3.7 i-l). In essence, in the presence of a
multimode solution, more accurate observations cannot serve to eliminate one of the
modes, they only serve to make the modes more distinct. This may actually make
the data assimilation problem more difficult, and Posselt et al. ( 2008a ) found that
this was also true of cloud property retrievals. Note that including more observation
times may help to constrain the problem via reduction in the number of possible
solutions ( Vukicevic and Posselt 2008 ).