Geoscience Reference
In-Depth Information
The MCMC algorithm used to perform the retrieval was a straightforward
implementation of Metropolis-Hastings sampling with uncorrelated zero mean
Gaussian proposal distribution for each of five unknown retrieved parameters: the
cloud top and base height, effective radius, ice water path, and ice crystal shape.
Proposed values were generated for all parameters simultaneously and proposal
variance was tuned during the initial portion of the Markov chain to converge to
an acceptance rate of approximately 25 %. A Gaussian PDF was assumed for the
satellite brightness temperature uncertainty, and the error standard deviation was
assumedtobe1.5and1.0Kforthe11
m
brightness temperature difference. Cloud top and base height were obtained from
CloudSat reflectivity profiles and assumed to have a Gaussian uncertainty with
standard deviation of 2 km. Note that we also tested a log-Normal error distribution
for cloud top height uncertainty. These results are presented in
Posselt et al.
(
2008a
)
and will not be discussed here.
The prior distribution for all five retrieved parameters was assumed to be bounded
Uniform with bounds set to [0,100]
m brightness temperature and 11-13.3
m for the effective radius, [0,200] g m
2
for
IWP, and [0,15] km for cloud top base and height. Ice crystal shape was varied by
allowing the proposal to sample from all real numbers in the range [0.5,4.5] then
rounding to the nearest integer value. The ice crystal shape was corresponding to this
integer value was then used in the forward radiative transfer model (reordering the
crystal shape index was found to have no influence on the outcome of the retrieval).
Cloud base was constrained to lie below cloud top by treating any proposal with
cloud base
cloud top height as an automatic rejection.
Posterior PDFs of IWP and effective radius retrieved for the pixel of interest
are shown in Fig.
3.5
.InFig.
3.5
a, b, the ice crystal shape is assumed to be solid
columns, and the cloud top and base height are fixed (Fig.
3.5
a) and allowed to vary
(Fig.
3.5
b). It can be seen that the characteristics of the posterior solution do not
change significantly when cloud top and base are allowed to vary-the maximum
a posteriori estimate (mode) is unchanged and the functional relationship between
IWP and effective radius is consistent. The primary effect of variability in cloud
top and base location is to increase the variance in the solution. When ice crystal
shape is allowed to vary, the characteristics of the posterior PDF change markedly.
Solid columns and droxtals (Fig.
3.5
a, c) produce posterior PDFs with similar
characteristics, but with slightly larger retrieved IWP and Re for droxtals. The PDF
corresponding to bullet rosettes (Fig.
3.5
d) is bimodal with a primary mode that
is much more compact and circular in shape than for the other crystal shapes and
with a solution that returns far smaller IWP and effective radius. In contrast, when
aggregates are the assumed crystal shape (Fig.
3.5
e), the posterior mode is elongated
and centered at much larger values of IWP and effective radius. When the algorithm
is allowed to adaptively choose a crystal shape (Fig.
3.5
f), the result is bimodal with
the primary mode associated with aggregates and the secondary mode a combination
of bullet rosettes, droxtals, and columns.
The results demonstrate the utility of MCMC for examining the properties of
a retrieval with unknown uncertainty characteristics. In the above case, while it is
generally acknowledged that ice crystal shape is an important contributor to ice
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