DATA ASSIMILATION IN BIOGEOCHEMICAL MODELLING
535
n to n + 1 via x n+1 = Ax n , A t is its transpose, and e is some intrinsic
uncertainty of the model that at each time step increases the forecast
error. This equation is written in vector form to point out the main
computational burden of the Kalman Filter: for an N -dimensional state
vector x, σ 2 is a N × N matrix. This means that the computation of
σ 2
n+1 is as expensive as stepping forward the state vector 2N times. In
many applications, updating the covariance matrix will dominate the
computational effort and often render it impractical.
Note, that, in addition to computational constraints, the above
proach will provide optimal solutions only for linear systems and for
Gaussian error distributions. Both conditions are not generally met for
marine biogeochemical systems, and a number of adaptations to the
original Kalman Filter approach have been developed. To cope with
nonlinear systems, the so-called Extended Kalman Filter (EKF) steps
forward the error covariance equation by the tangent linear operator
(Jacobian) of the full model operator [Evensen, 1992]. While this works
well for weakly nonlinear systems, the Ensemble Kalman Filter (EnKF)
approach [Evensen, 2003] can also be applied to strongly nonlinear systems. It uses Monte-Carlo generated state vectors to initialise an
semble of model forecast runs from which the error distribution of the
model forecast is estimated. More recent developments include the Singular Evolutive Extended Kalman (SEEK) filter that has been applied
to assimilate ocean colour data by Carmillet et al. [2001]. A sequential
method that is not directly related to the Kalman Filter is the Sequential
Importance Resampling (SIR) filter [Bertino et al., 2003].
A common feature of all sequential methods is the generation of a
model trajectory that is “only” piecewise self-consistent. Whenever observations become available, merging the respective model forecast and
observation into a new analysed state generates unsteady “jumps” in
the state vector trajectory. Special care has to be taken if one wants,
for example, to ensure mass conservation across these jumps. Similarly,
any analysis of output from an assimilation experiment has to account
for fluxes or perturbations associated with the assimilation steps.
5.2
Variational methods
While sequential assimilation methods attempt to estimate a “best”
state vector at each instant observations become available, variational
methods attempt to find a “best” model trajectory. The strong-constraint
variational method ensures that over the entire time interval considered,
the “best” model trajectory exactly obeys the model dynamics. Minimising a model-data misfit thereby becomes a constrained optimisa
ap
en
,
tion
535
n to n + 1 via x n+1 = Ax n , A t is its transpose, and e is some intrinsic
uncertainty of the model that at each time step increases the forecast
error. This equation is written in vector form to point out the main
computational burden of the Kalman Filter: for an N -dimensional state
vector x, σ 2 is a N × N matrix. This means that the computation of
σ 2
n+1 is as expensive as stepping forward the state vector 2N times. In
many applications, updating the covariance matrix will dominate the
computational effort and often render it impractical.
Note, that, in addition to computational constraints, the above
proach will provide optimal solutions only for linear systems and for
Gaussian error distributions. Both conditions are not generally met for
marine biogeochemical systems, and a number of adaptations to the
original Kalman Filter approach have been developed. To cope with
nonlinear systems, the so-called Extended Kalman Filter (EKF) steps
forward the error covariance equation by the tangent linear operator
(Jacobian) of the full model operator [Evensen, 1992]. While this works
well for weakly nonlinear systems, the Ensemble Kalman Filter (EnKF)
approach [Evensen, 2003] can also be applied to strongly nonlinear systems. It uses Monte-Carlo generated state vectors to initialise an
semble of model forecast runs from which the error distribution of the
model forecast is estimated. More recent developments include the Singular Evolutive Extended Kalman (SEEK) filter that has been applied
to assimilate ocean colour data by Carmillet et al. [2001]. A sequential
method that is not directly related to the Kalman Filter is the Sequential
Importance Resampling (SIR) filter [Bertino et al., 2003].
A common feature of all sequential methods is the generation of a
model trajectory that is “only” piecewise self-consistent. Whenever observations become available, merging the respective model forecast and
observation into a new analysed state generates unsteady “jumps” in
the state vector trajectory. Special care has to be taken if one wants,
for example, to ensure mass conservation across these jumps. Similarly,
any analysis of output from an assimilation experiment has to account
for fluxes or perturbations associated with the assimilation steps.
5.2
Variational methods
While sequential assimilation methods attempt to estimate a “best”
state vector at each instant observations become available, variational
methods attempt to find a “best” model trajectory. The strong-constraint
variational method ensures that over the entire time interval considered,
the “best” model trajectory exactly obeys the model dynamics. Minimising a model-data misfit thereby becomes a constrained optimisa
ap
en
,
tion