Nudging Methods: A Critical Overview - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 53

S D X p 1

p 1 j

. j / .0/

jD 0 ˇ

(2.140)

It can be verified that S is a mean zero Gaussian random variable whose variance is
given by

/ D Va r . j / .0/ X p 1

jD 0 ˇ 2 .p 1 j

Va r . S / (2.141)

D Va r . j / .0/ 1 ˇ 2 p
Œ1 ˇ 2

Thus, for a fixed number N of observations,

Va r 2 .0/

Va r . S / ! (2.142)

Œ1 ˇ 2

as the number, p of back and forth iterations increase.

2.6 Discussion and Conclusions

There is an ever-growing literature on the applications of nudging as a simple
viable method for dynamic data assimilation. It is attractive to the geophysical
science community because of its ease of implementation and its intuitive appeal—
in essence, the use of the earlier known error in prediction to alter subsequent
prediction appeals to common sense. Yet, in its earliest stage of development where
empiricism was the theme, search for a suitable nudging coefficient exhibited great
computational demand through numerous simulations and validation against the
evolution of dynamical systems. And the final choice of the nudging coefficient was
always subject to debate—linked to the question: isn't there a better coefficient?
It naturally led to an effort to find a coefficient that exhibited optimality under
a specific form of the cost functional that forced the coefficient toward an a
priori estimate. And again, this brought up other questions concerning the “heavy
handedness” by producing a cost function that was forced to remain close to the a
priori estimate. Further, these methods have unintentionally omitted an important
aspect of the nudging problem—nudging dynamics carries with it the presence of a
serially correlated forecast error and this error must be accounted to find the optimal
coefficient. It is computationally demanding to find the structure of this correlated
error. For sure, its influence on the optimal nudging process is an important area of
investigation that remains open. Our review also indicates that the well-established
theory of observer design (as a practice in the contemporary control theory) deserves
further attention from those involved in data assimilation for numerical prediction
in the geophysical sciences. And the “back-and-forth” nudging offers promise for
application to operational prediction, but where attention must be focused on the
results for irreversible processes that are ubiquitous in the ocean-atmosphere system.