A Question of Adequacy of Observations in Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 116

Following correction to control based on this first-order Taylor expansion, a
revised forecast is made and associated errors calculated. Iteratively the corrections
to control are made until satisfaction of some empirical criteria — a criteria such as
the norm of the incremental correction vector is smaller than a value commensurate
with the expected error norm of control. However, in the numerical experiments to
follow, corrections to control are found in a single step.

5.4 Numerical Experiments

5.4.1 Prelude

The problem we investigate assumes availability of surface air temperature observa-
tions along the known trajectory and an estimate of average SST along the trajectory.
Observations from the moored buoys operated by the National Data Buoy Center
(NDBC) and the U. S. Coast Guard serve as our guide in establishing a realistic
distribution of surface observations over the Gulf of Mexico for the numerical
experiments. For reasons related to economy of operation and maintenance of
instruments, most buoys over the Gulf of Mexico are located in the shelf waters—
roughly 50-100 km of the shoreline ( Hamilton 1986 ). Given the position of NDBC
buoys in the vicinity of the trajectory shown in Fig. 5.1 , it is reasonable to assume
that there are four instrumented buoys neighboring the trajectory (M D 4

).
Our premise is that differential placement of observations along the trajectory is
key to understanding the condition for sufficiency/insufficiency of observations—
the condition that makes it possible/impossible to minimize the cost function.
The mechanics for correction to control by FSM have been discussed in Sect. 5.3 .
The standard procedure for finding r J in 4D-Var is backward integration of the

model's adjoint. Under the simplified constraint of air/sea interaction, however, r J
is found by straightforward differentiation of the cost function using knowledge of
the sensitivities found in Table 5.1 . Further, the conjugate gradient algorithm is used
to determine search direction and step size ( Lewis et al. 2006 ).

5.4.2 Forecast Errors

In our experiments, we assume observations are true—derived from the strong
constraint ( 5.2 ) with true control (

Y

). The forecast error stems from incorrect control

Y 0 ). The error vector E

( is displayed in Table 5.2 and the systematic nature of the

error is obvious—an under-forecast the order of .0:01 1 ı C/ up to t D 5 handan

over-forecast the order of C .0:1 1 ı C/ from t D 6 h until the end of the forecast

at t D 18

h.