Studies on the Impacts of 3D-VAR Assimilation of Satellite Observations on the Simulation of Monsoon Depressions over India - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 635

(OI) in which the weights given to observations were related to observation errors
( Gandin 1963 ).
Furthermore, the OI method considered and utilized the importance of the back-
ground field information and its error characteristics as useful source of information.
The OI method, when first implemented in operational centers worldwide in late
1970s and early 1980s, had to invoke major approximations in order to meet the
calculations feasible. The advent of variational methods for data assimilation in the
mid 1980s saw the emergence of an important breakthrough in data assimilation
research.

26.2.1 Variational Data Assimilation

The underlying physical principle of variational data assimilation schemes is that
the analysis x a is the optimum state vector that minimizes a global “cost function”
J, the latter providing a measure of the mismatch between a model state vector x
and the background state x b and observation y . This cost function as utilized in
three-dimensional variational (3D-VAR) method is given as

x x b

T B 1 Œ

x x b

T R 1 Œ

J D 0:5ΠC 0:5Πy H . x / y H . x / (26.1)

where B is error covariance of background state, R is error covariance of observation
(including the representativeness errors) and T indicates transpose. The 3D-VAR
method was implemented operationally in mid 1990s at the National Centre for
Environmental Prediction (NCEP) first and later at European Centre for Medium
Range Weather Forecast (ECMWF). The minimization of cost function J in 3D-VAR
is usually performed in “control space”. The error covariance of the background
state B is usually estimated from the difference between pairs of forecasts that verify
at the same time ( Parrish and Derber 1992 ), the so called “NMC” method. The
observations are assumed to have no bias and no serious errors associated with the
malfunctioning of instruments. The observation errors are assumed to be Gaussian.
The 3D-VAR method assumes that all observations are valid at the same time and
further assumes that the background errors and observation errors are not correlated.
An important advantage of the variational method is that one can utilize observation
variables which are different from the model state variables.
An important extension of the 3D-VAR method is called the “four-dimensional
variational method” (4D-VAR) in which the cost function minimization is per-
formed over a time window, the latter accounting for observations spread over time
and lasting typically 6 h or 12 h for operational weather forecasts. The cost function
in the 4D-VAR method is as follows

N X

T R 1 Œ

x o x o

T B o Œ

x o x o C 0:5

J D 0:5ΠΠy i H . x i / y i H . x i / (26.2)

i D 1