Geography Reference
In-Depth Information
Hemisphere, but they are not incorporated easily into traditional objective analysis
schemes.
Deficiencies of the traditional approach have led the major forecast centers to
devise assimilation procedures in which the separate processes of objective analy-
sis and data initialization are combined into a continuous cycle of data assimilation,
which is referred to as
four-dimensional variational assimilation
(4DVAR). The
objective of this approach is to utilize all available information during an assim-
ilation window period of several hours to obtain a best estimate of the state of
the atmosphere at the analysis time. The information utilized includes a first guess
based on a short-term forecast from the previous analysis time and all observations
acquired during the window period. These various pieces of information, weighted
by their statistical errors, are used to derive a
cost function
, expressing the misfit
between the observations and the analysis. The cost function is then minimized in
order to achieve an analysis that is the most likely estimate of the true state of the
atmosphere at the analysis time.
A simple example of the cost function approach was given by Kalnay (2003).
Suppose that we have two independent observations of temperature, T
1
and T
2
,
which are unbiased but have error variances of σ
1
and σ
2
, respectively. The cost
function can then be defined as
(T
T
1
)
2
σ
1
T
2
)
2
σ
2
1
2
−
(T
−
J (T )
=
+
(13.69)
where T is the analysis temperature that we wish to determine. The minimum of
J is determined by evaluating ∂J /∂T
=
0, which yields the best estimate for T :
σ
2
σ
1
T
1
+
σ
1
σ
1
T
2
T
=
(13.70)
σ
2
σ
2
+
+
Thus, an observation that has small error variance is weighted more heavily than
an observation with large error variance.
An example of a 4DVAR procedure, as used at ECMWF, is shown schematically
in Fig. 13.8. The ECMWF scheme utilizes a 12-h assimilation window extending
from 9 h prior to 3 h following the analysis times, which are 00Z and 12Z. An
assimilation cycle is begun by using a forecast from the initial state at the previous
analysis time as the first guess, or background state. All available data within the
assimilation window are used to update the analysis defined by the first guess, with
weightings determined by the error statistics for the particular type of data. The
update process is subject to dynamical constraints to assure a balanced state at the
analysis time. Forecasts are then carried out for a 10-day period once each day
starting from the initialized state for 12Z.
