A Question of Adequacy of Observations in Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 120

5.5 Discussion and Conclusions

With the natural appeal that comes from application of variational data assimilation
to weather analysis and prediction problems, the experiments conducted warn that
an iterative process leading to a reduction in the cost functional is not necessarily
heading for a minimum of the functional. In the low-order systems similar to the
one we have investigated, where calculation of the eigenvalues of the Hessian about
the terminal point of iteration is not overly challenging, the presence or absence of
a minimum at the terminal point can be definitively determined.
In the more realistic systems associated with NWP, a meaningful structure of the
true Hessian is difficult to determine despite a variety of innovative methods that
have been developed to capture this structure.
In the absence of knowing the Hessian, this research has presented another view
of the problem that holds promise for improving the chances of finding optimal
correction to control through reliance on the forward sensitivity method (FSM)—
knowledge of forward evolving sensitivity of model variable to elements of control
in the context of a variational data assimilation scheme. Generally, this requires
straightforward yet computationally demanding integration of the equations of
sensitivity [described in detail in Lakshmivarahan and Lewis ( 2010 )]. The FSM
identifies those locations in space and time where observations are most likely to
have little impact on the assimilation process—observations that generally lead to
an ill-posed variational adjustment problem. Although the methodology does not
give a recipe for an ideal or optimal placement of observations, valued locations are
identified through their sensitivity to the various elements of control.
The methodology developed in this paper has application to the more-realistic
NWP models used in operations. For example, in a post-mortem examination of
the biased forecasts associated with return flow over the Gulf of Mexico, the FSM
can identify those observational locations in space and time where the model
variables exhibit sensitivity to elements of control. Calculation of error at these
points, assuming availability of instrumented buoys near these locations, along with
knowledge of sensitivity, provide the means to make corrections to control. Under
the assumption that the model is faithful to the event (i.e., inclusion of reasonable
representations of major physical processes), those elements that require the largest
relative corrections are candidates for producing the bias.

Acknowledgements The authors thank Carlisle Thacker for a copy of his notes on variational
data assimilation associated with his 1989 invited lectures at the Forschungscentrum in Hamburg,
Germany. His discussion of adequacy/inadequacy of observations in variational analysis was
especially lucid. We also thank research meteorologists Geoff Manikin and Zavisa Janjic of
NCEP/EMC (National Center for Environmental Prediction/Environmental Modeling Center) for
valued discussions related to the possible sources of bias in forecasts of return flow over the Gulf
of Mexico.
We thank Joan O'Bannon, former graphics specialist at National Severe Storms Laboratory, for
drafting Fig. 5.1 , and we thank the National Geographic Society, particularly Eric Lindstrom, the
Society's senior map editor and map library director, for permission to reproduce the bathymetric