Nudging Methods: A Critical Overview - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 48

and

u . 1;t/ D u .1;t/ D 0. boundary condition / (2.103)

the parameter is the constant phase velocity of the sinusoidal curve.

A useful way to characterize the solution of ( 2.101 ) is that the time derivative of
u along the characteristics is zero, that is,

c

along the characteristic W dt
dx

d u

dt

D 1
c

D 0 (2.104)

We use this latter property to illustrate the back and forth nudging scheme and its
properties.
The forward nudged dynamics in continuous time is given by

d u

dt

D g. z u / (2.105)

and where g>0 . The corresponding backward dynamics is given by

( Auroux ( 2011 ))

d

w
dt

D g. z w / (2.106)

Discrete form of ( 2.105 ) using Euler scheme is

u .k C 1/ D .1 g/ u .k/ C g z .k/ (2.107)

where u is the initial condition. Similarly, the discrete form of the backward

dynamics is given by

.0/

w .k/ D .1 ˛/ w .k C 1/ C ˛ z .k/ (2.108)

g
1 C g >0

where is the starting condition for the backward integration.

We use the same set of observations

˛ D and w .N/

f z .j/ W 0 j N 1 g (2.109)

in the nudging analysis, where it is tacitly assumed that z .j/ is model generated

starting from a true initial state u T .0/ .Let

u T .t/
dt

d

D 0 (2.110)

be the true model whose solution is given by

u T .t/ D u T .0/

(2.111)