Operational Dust Prediction - Mineral Dust: A Key Player in the Earth System - page 250

consistency in the analysis (Courtier et al. 1994 ). This so-called “incremental”
approach is employed at ECMWF. Another key aspect of the variational methods
is the use of the adjoint model to calculate the gradient of the cost function needed
in the minimisation. Coding an adjoint of highly nonlinear parameterisations can be
involved, and the parameterisations may need to be linearised before an adjoint can
be constructed.
Kalman Filter and Ensemble Kalman Filter Methods (MRI/JMA, NRL)
Another data assimilation method, the Kalman filter (KF), has been well known
since the 1960s (Kalman 1960 ). KF, which is based on the linear minimum variance
estimation approach, evolves the error covariance matrix temporally. The KF
calculation requires neither tangent linear models nor adjoint models. Despite these
advantages, KF requires the inverse calculation of the matrices with the dimensions
of the model state space. The size of the model state space in geosciences is often
of the order of millions: for such large systems, KF cannot be adopted. In order to
exploit the advantages of KF and reduce the computational burden, the ensemble
Kalman filter (EnKF) was developed (Evensen 1994 , 2007 ). The basic concept
of EnKF is that the ensemble of the forward model forecasts is able to represent
the probability distribution function (PDF) of the system state and approximate
the error covariance distribution. The EnKF is mathematically equivalent to the
original Kalman filter, under the ideal conditions where the simulation model
is linear, and the EnKF employs an infinite ensemble size. In the MRI/JMA
aerosol assimilation system, a 4D expansion of the EnKF (4D-EnKF) is adopted
to assimilate asynchronous observations at the appropriate times. Using the 4D-
EnKF aerosol assimilation system, the surface emission intensity distribution of
dust aerosol is estimated (Sekiyama et al. 2010 , 2011 ). The vector augmentation
mentioned above enables EnKF to estimate the parameters through the background
error covariance between dust emissions and observations. Consequently, EnKF
simultaneously estimated the aerosol concentrations (as model variables) together
with the dust aerosol emission intensity (as model parameters). The MRI/JMA
aerosol assimilation system employs the local ensemble transform Kalman filter
(LETKF), which is one of the EnKF implementation schemes (Hunt et al. 2007 ).
The LETKF uses the ensemble transform approach (Bishop et al. 2001 ) to obtain
the analysis ensemble as a linear combination of the background ensemble forecasts.
The LETKF handles observations locally in space, where all the observations are
assimilated simultaneously.
It is important to note that 4D-Var and ensemble Kalman filter methods approx-
imately converge, when 4D-Var is run over a long assimilation window (e.g.
24 h) and model error is included, as they are both based on the Bayes theorem
which postulates that the probability distribution of the analysis errors is a linear
combination of the probability distribution of the observations and background
errors (Fisher et al. 2005 ).