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the observation error R and the model error Q (associated with the temporal evolution of
the model) is often tuned or adaptively updated (e.g., Desroziers et al. 2005 ; Reichle et al.
2008 ).
In general, in data assimilation, errors are assumed to be Gaussian. The most funda-
mental justification for assuming Gaussian errors, which is entirely pragmatic, is the rel-
ative simplicity and ease of implementation of statistical linear estimation under these
conditions. Because Gaussian probability distribution functions are fully determined by
their mean and variance, the solution of the data assimilation problem becomes compu-
tationally practical. Note that the assumption of a Gaussian distribution is often not jus-
tified in land data assimilation applications.
Typically, there are biases between different observations, and between observations
and model (see, e.g., M ยด nard 2010 ). These biases are spatially and temporally varying, and
it is a major challenge to estimate and correct them. Despite this, and mainly for pragmatic
reasons, in data assimilation it is often assumed that errors are unbiased. For NWP many
assimilation schemes now incorporate a bias correction, and various techniques have been
developed to correct observations to remove biases (e.g., Dee 2005 ); these methods are
now being applied to land data assimilation (De Lannoy et al. 2007a , b ).
4.4 Advantages and Disadvantages of Assimilation Methods
The feasibility of 4D-Var has been demonstrated in NWP systems (see, e.g., Simmons and
Hollingsworth 2002 ). Its main advantage is that it considers observations over a time
window that is generally much longer than the model time step, that is, it is a smoothing
algorithm. This allows more observations to constrain the system and, considering satellite
coverage, increases the geographical area influenced by the data. For nonlinear systems (as
is generally the case for the land surface), this feature of 4D-Var, together with the non-
diagonal nature of the adjoint operator which transfers information from observed regions
to unobserved regions, reduces the weight of the background error covariance matrix in the
final 4D-Var analysis compared to the KF analysis (for linear systems, the general
equivalence between 4D-Var and the KF implies that the same weight is given to all data in
both systems).
In contrast to the above advantages of 4D-Var, three weaknesses must be mentioned.
First, its numerical cost is very high compared to approximate versions of the KF or
ensemble methods. Second, its formalism cannot determine the analysis error directly;
rather, it has to be computed from the inverse of the Hessian matrix (again, this procedure
is prohibitive in both computation time and memory). Finally, its formalism requires the
calculation of the adjoint model, which is time-consuming and may be difficult for a
system such as the land surface which exhibits nonlinearities and on-off processes (e.g.,
presence or lack of snow).
The EKF is capable of handling some departure from Gaussian distributions of model
errors and nonlinearity of the model operator. However, if the model becomes too non-
linear or the errors become highly skewed or non-Gaussian, the trajectories computed by
the EKF will become inaccurate.
The EnKF is attractive as, for example, it requires no derivation of a tangent linear
operator or adjoint equations and no integrations backward in time, as for 4D-Var (see
Evensen 2003 ). The EnKF also provides a cost-effective representation of the background
error covariance matrix, P f . Several issues need to be considered in developing the EnKF:
(1) ensemble size; (2) ensemble collapse; (3) correlation model for P f , including locali-
zation (see, e.g., Kalnay 2010 ); and (4) specification of model errors.
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