Nudging Methods: A Critical Overview - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 29

formally, the forecast error e.k/ in ( 2.6 ) is used as an artificial forcing to the model

as follows:

x.k C 1/ D M.x.k// C G 0 .k/e.k/ (2.7)

G 0 .k/ 2 R n m is called the time varying nudging coefficient matrix. Since
the correction term in ( 2.7 ) is proportional to

where

e.k/ 2 R m (in the observation space),
this form of nudging is called observation nudging where G 0 .k/

is the associated

x a .k/ 2 R n be the state vector on the computational

nudging coefficient. Instead, let

grid obtained from z .k/ using any one of the DA schemes. Then,

x.k C 1/ D M.x.k// C G a .k/Œx a .k/ x.k/ (2.8)

G a .k/ 2 R n n is a time varying analysis nudging
coefficient. In either form, an appropriate measure of the forecast error is used
to force the model state towards the observation. The nudging method has also
been viewed as a case of Newtonian relaxation or “repeated insertion of data”
( Macpherson ( 1991 )).
The notion of using the error to drive a model towards a desired state is
a basic principle underlying the design of feedback control systems. Refer to
Bennett ( 1996 ), Bryson ( 1996 ), and Sussmann and Willems ( 1997 ) for historical
overviews of these techniques.
The literature on nudging covers nearly four decades (since 1974) and can be
broadly divided into parts or divisions as follows:

is called analysis nudging where

1. The nudging coefficient is empirically determined through examination of
dynamical simulation over a broad range of coefficients. The coefficient is

positive and may be time varying, but its magnitude is controlled in part by the
smallest time scale of the typical multi-scale phenomenon captured by the model.
2. The coefficient matrix

G

is optimally determined through minimization of a
functional that combines the standard fit (equation 2.5 ) augmented by a term
that fits the coefficient to an apriori estimate of that coefficient. The resulting
constrained minimization is solved by the 4D-Var method mentioned earlier.
3. A class of methods that exploit the similarity between nudged dynamics ( 2.7 -
2.8 ) and feedback control in observer theory ( Luenberger 1964 ).
4. A process labeled “back-and-forth nudging” that uses the same model in a
forward and backward mode to obtain a good match between the forecast model
and the observations ( Auroux ( 2009 )).

G

Nudging based dynamic data assimilation has been applied to a variety of problems
including the following:

1. Initialization of a dynamic model as originally proposed by Anthes ( 1974 )
and Hoke ( 1976 )where Hoke ( 1976 ) recommended an analysis-based nudging
process [as found in ( 2.8 )] as opposed to observation-based nudging [as found in
( 2.7 )]. Application has been made to forecast of the Indian Monsoon [ Krishna-
murti et al. ( 1991 ), Ramamurthy and Carr ( 1987 , 1988 )].