All-Sky Satellite Radiance Data Assimilation: Methodology and Challenges - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 470

Fig. 19.4 Discontinuous
all-sky radiative transfer
operator defined by ( 19.19 ).
The left area of the figure
represents clear-sky
conditions and the right area
corresponds to cloudy
conditions. In principle, the
function value and its
derivatives all can experience
a discontinuity

hybrid variational-ensemble methods since often the variational component is used
for adjustment of dynamical variables, while the ensemble component is primarily
used for cloud variables. In this situation cloud variables will have a much better
preconditioning than dynamical variables eventually creating unbalanced analysis
that can also act to remove adjusted clouds and precipitation in the forecast.
The above examples illustrate the important role of Hessian preconditioning in
assimilation of all-sky radiances, and indirectly suggest that preconditioning method
should be related to dynamics.
Although nonlinearity of all-sky radiance operator has been generally acknowl-
edged, non-differentiability is typically not discussed and thus requires attention.
The definition of observation operator becomes an issue in the case of all-sky

radiances. This is because the radiative transfer operator has an on-off switch to
decide if it should go through the cloudy branch that normally includes scattering
effects, or not in the case of clear-sky radiances. Since this decision depends on the
atmospheric parameters such as cloud mixing ratios and temperature, the forward
radiative transfer operator has a discontinuity in the function value and derivative
implying discontinuity of the cost function. Therefore, one can write the radiative
transfer operator for all-sky radiances as

h

c.x/ x 2 C
s.x/ x … C

h.x/ D

(19.19)

where C represents the state subspace corresponding to clear-sky conditions, c is

the clear-sky component and is the cloudy component of the observation operator.

The point where the state can cross between clear and cloudy conditions is the
discontinuity point, and thus the operator

s

has two branches. This is visualized
in Fig. 19.4 , indicating that the function value and all its derivatives can have a
discontinuity.

h.x/