5.6 Sampler of oceanic and atmospheric data assimilation 169
where c
0
is a prior for c, and σ
2
c
is the hypothesized variance of the prior error. Varying
K with respect to c and u(x, t) yields the extremal condition
ˆ
c = c
0
+ σ
2
c
∂
ˆ
u
∂x
• λ (5.6.9)
and the Euler–Lagrange equations for J as before. Note that
ˆ
u and λ depend upon
ˆ
c,
so (5.6.9) is highly nonlinear even though the dynamics of the toy model are linear.
However, iteration schemes for solving (5.6.9) are readily devised (Eknes and Evensen,
1997, who consider linear Ekman layer dynamics with an unknown eddy viscosity),
and interlaced with iterated representer algorithms in the case of nonlinear dynamics
(Muccino and Bennett, 2002, who consider the Korteveg–DeVries equation with un-
known parameters for phase speed, amplitude and dispersion). Convergence of these
schemes is in no way assured.
5.6.5
Monte Carlo smoothing and filtering
Consider the toy nonlinear model (3.3.1)–(3.3.3), where the random inputs f , i and b
have covariances C
f
, C
i
and C
b
respectively. The methods of §3.2.4 may be used to gen-
erate pseudo-random samples of the inputs consistent with their respective covariances.
A pseudo-random sample of the state u(x , t) is then obtained by integrating (3.3.1)–
(3.3.3). Sample estimates of the expectation Eu(x, t) and covariance C
u
(x, t, y, s)
follow from repeated generation and integration.
The sample moments of u may then be used for space–time optimal interpolation of
data collected in some time interval 0 < t < T , as outlined in §2.2.4. The prior for the OI
or best linear unbiased estimate (2.2.22) would not be u
F
(x, t), but rather the sample
estimate of Eu(x, t ), while the covariance C
q
(x, t, y, s) would be the sample estimate
of C
u
(x, t, y, s). As a consequence of the nonlinearity of (3.3.1), the OI estimate is not
an extremum of the penalty functional (1.5.9), even if W
f
were related to C
f
through
(1.5.11), (1.5.12), etc. That is, the OI estimate is not the solution of an inverse model.
Nevertheless the attraction of such “Monte Carlo smoothing” is obvious: there is no
need to linearize the dynamics, nor is it necessary to derive the adjoint dynamics.
Storing C
u
(x, t, x
m
, t
m
), where (x
m
, t
m
) is a data point, may not be feasible for
all x, t and for 1 < m < M, but it may be feasible to store C
u
(x, t
m
, y, t
m
), for all
x, y and for one time t
m
. Data collected at the time t
m
may thus be optimally in-
terpolated in space, provided it is assumed that the data errors are uncorrelated in
time. This Monte Carlo filtering method has become known as the “Ensemble Kalman
Filter” or EnKF (Evensen, 1994). For its application to operational forecasting of the
North Atlantic Ocean, see http://diadem.nersc.no/project; for application to
seasonal-to-interannual forecasting of the Tropical Pacific Ocean, see Keppenne (2000).
For a careful comparison of the computational efficiency of the EnKF with that of the
indirect iterated representer algorithm, in the context of an hydrological model and
satellite observations of soil moisture, see Reichle et al. (2001, 2002).