Markov Chain Monte Carlo Methods: Theory and Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 76

cloud retrieval error, the characteristics of the error (e.g., changes in the shape of
the posterior distribution with changes in crystal type and the presence of discrete
multiple modes when all types are allowed) would have been difficult to determine
in advance. It is important to recognize that an MCMC-based retrieval method
provides the unapproximated joint posterior PDF of all retrieved quantities for
every pixel in the scene and thus can yield a much more robust estimate of the
scene dependent characteristics of the error. This topic is explored in more detail in
Posselt et al. ( 2008a ).

3.4.2 Model Parameter Estimation and Uncertainty Analysis

Errors and/or uncertainty in model physics parameterizations are increasingly
recognized to be an important source of forecast error in weather and climate
prediction ( Murphy et al. 2004 ; Palmer et al. 2005 ; Stainforth et al. 2005 ; Berner
et al. 2011 ; Jarvinen et al. 2010 , 2012 ; Laine et al. 2012 ). Specifically, empirically
specified parameters associated with simplifying assumptions about the form of the
particle size distribution of ice and liquid condensate have an important effect on the
details of cloud and precipitation development and feed back on the radiative fluxes,
heating rates, and thermodynamic environment ( Tao et al. 1995 ; Grabowski et al.
1999 ; Wu et al. 1999 ; Petch and Gray 2001 ; Gilmore et al. 2004 ; van den Heever and
Cotton 2004 ). It is reasonable to expect certain sets of parameters to produce model
trajectories that are consistent with observations. However, due to nonlinearity in
the parameter-state relationship and errors in observations, there may not exist
one optimal set of parameter values. The issue of how to quantitatively represent
parameterization uncertainties presents a significant challenge, and has implications
for the efficacy of ensemble weather and climate forecasting, data assimilation, and
model physics development.
In this section, we demonstrate how MCMC can be used to understand the
functional relationship between model physics parameters and model output
variables. The outcome is an estimate of the sensitivity of the simulation output
to the model formulation, as well as information on how to properly account
for parameter uncertainty in a data assimilation system. The parameters of interest
define the particle shape, density, and size distribution in a bulk cloud microphysical
parameterization ( Lin et al. 1983 ; Rutledge and Hobbs 1983 , 1984 ; Tao et al. 2003 ;
Lang et al. 2007 ), and are listed in Table 3.1 . To evaluate parameter uncertainty
in isolation from the complications introduced by feedback to the flow field
and thermodynamic state, the physical parameterization is driven with specified
vertical motion and water vapor tendencies that vary sinusoidally with height, and
change in magnitude with time. Particles are allowed to settle according to their
mass weighted fall speed and interact fully with long and shortwave radiation.
The thermodynamic environment, water vapor forcing, and vertical motion are
set consistent with a vertical column passing through a tropical deep convective
squall line. As such, the model demonstrates two distinct regimes: convective and