Markov Chain Monte Carlo Methods: Theory and Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 72

reduction in the entropy of the estimate due to the addition of information from
observations. In discrete form, the entropy of state P is defined as:

X

S.P/ D p i log

2 .p i /;

(3.16)

i D 1

where is the number of discrete bins the PDF is

divided into. Shannon information content is then defined as the difference between
the entropy of the a priori state and the entropy of the retrieved state

p i is the discretized PDF and K

" # " #

N X N X

H D S. x a / S. x / D p i . x a / log 2 .p i . x a // p i . x / log 2 .pi. x // ;

i D 1 i D 1

(3.17)

which can be interpreted as the extent to which the number of allowable states is
reduced by the addition of information from the measurements. Because MCMC
algorithms return a sample of the full PDF, can be computed directly from the

above relationship. Select examples of application of MCMC to satellite retrievals
and model uncertainty evaluation are presented below.

H

3.4 Select Applications of MCMC in the Atmospheric

Sciences

3.4.1 Retrieval of Atmospheric State Variables from Satellite

Measurements

The surface-based Earth observing network is generally limited to land and to
regions close to major population centers and consequently does not sufficiently
sample the global atmosphere. For this reason, studies of the Earth's weather and
climate have increasingly depended on observations from satellites, which have
the potential to provide a more complete temporal and spatial observational record.
In contrast to many in-situ measurements, satellite-based observations are indirect
measurements of the quantities of interest. Extracting geophysical quantities from
satellite radiances requires the solution to an inverse problem: inference of the state
of the atmosphere from observations of radiative properties via some functional
relationship (e.g., a forward radiative transfer model, regression, etc.; Rodgers 2000 ;
Miller et al. 2000 and references therein). As such, the geophysical quantities
obtained from satellite are said to be retrieved from what the satellite actually
measures. The magnitude and characteristics of the uncertainty in a retrieval depend
on the characteristics of uncertainty in the observations, their sensitivity to the
parameters of interest, forward model accuracy, and the quality of available prior