  This study aims at introducing two conservative thermodynamic variables
(moist-air entropy potential temperature and total water content) into a
one-dimensional variational data assimilation system (1D-Var) to demonstrate
the benefit for future operational assimilation schemes. This system is
assessed using microwave brightness temperatures from a ground-based radiometer
installed during the field campaign SOFOG3D dedicated to fog forecast
improvement.
  An underlying objective is to ease the specification of background error
covariance matrices that are highly dependent on weather conditions when using
classical variables, making difficult the optimal retrievals of cloud and
thermodynamic properties during fog conditions. Background error covariance
matrices for these new conservative variables have thus been computed by an
ensemble approach based on the French convective scale model AROME, for both
all-weather and fog conditions. A first result shows that the use of these
matrices for the new variables reduces some dependencies to the meteorological
conditions (diurnal cycle, presence or not of clouds) compared to usual
variables (temperature, specific humidity).
  Then, two 1D-Var experiments (classical vs. conservative variables) are
evaluated over a full diurnal cycle characterized by a stratus-evolving
radiative fog situation, using hourly brightness temperatures.
  Results show, as expected, that analysed brightness temperatures by the
1D-Var are much closer to the observed ones than background values for both
variable choices. This is especially the case for channels sensitive to water
vapour and liquid water. On the other hand, analysis increments in model space
(water vapour, liquid water) show significant differences between the two sets
of variables.
