Nudging Methods: A Critical Overview - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 27

2.1 Introduction

To avoid confusion and repetition, we begin by establishing basic notation and
definitions. Let x 2 R n refer to the state vector of the forecast model, and

M W R n ! R n denote the one-step transition map. A discrete time nonlinear
dynamic model is given by

x.k C 1/ D M.x.k// (2.1)

with x.0/ the initial condition. Given x.0/ , the sequence of states f x.k/ g k 0 is

called the model forecast.
Let z 2 R m and h W R n ! R m where

z .k/ D h.x.k// C V.k/ (2.2)

denote the observation at time k ,where x.k/ is the “true” unknown state of the

system captured by the model in ( 2.1 ), V.k/ is the Gaussian white noise sequence

2 R m m is a known symmetric and positive definite V.k/ N.0; R / where R

covariance matrix of V.k/ . It is assumed that the unknown true state evolves

according to the dynamics

N M.x.k//

x.k C 1/ D (2.3)

with x.0/ as the initial condition. The differences

M.x/ D M.x/ M.x/

(2.4)

and

x.0/ D x.0/ x.0/

denote the model error and the error in the initial condition, respectively.
A modern version of the standard dynamic data assimilation problem ( Lewis
et al. ( 2006 )) may be stated as follows: Given a set

f z .k/ W 1 k N g of N

x .0/

observations, find the optimal initial condition that minimizes the cost

functional

N X

J 1 .x.0/ D 1
2

R 1 e.k/ >

< e.k/;

(2.5)

k D 1

where

e.k/ D z .k/ h.x.k// (2.6)

<a;b> D a T b is the forecast error and is the standard inner product of two vectors

b 2 R n where

a denotes the transpose. The importance of this problem stems

from the fact that the model forecast starting from

, T

x .0/

“best fits” the observation

that in turn is a surrogate of the truth.