The Monte Carlo technique has been coupled to the Kalman filter to overcome most of the problems encountered with EKF. By this approach, called ensemble Kalman filter (EnKF; Evensen, 1994), an ensemble of model state vectors is employed to estimate the background error covariance. In combination to localization (see Section 3.2.3) it provides an approximation to the Kalman filter which is feasible for operational atmospheric data assimilation (Houtekamer et al., 2005). Moreover, since EnKF allows to account for the nonlinear growth of the background error, it may be able to provide more accurate analyses than the EKF in situations where nonlinearity is pronounced (Hamill, 2006).
Several variants of the EnKF have been proposed after its first introduction by Evensen (1994). The forecast step is the same for each of the various EnKF formulations while differences concern the analysis step. Regarding the former, an ensemble of Nens analyses states x
a(i)
k−1 available at time tk−1, where i is the index
of the ensemble member ranging from 1 to Nens, is propagated to the next analysis
time tk by using the nonlinear model M:
xb(i)k = M xa(i)k−1, i = i, ..., Nens (3.24)
The ensemble of background states generated with Eq. (3.24) is then employed to estimate the background error covariance Pb
k. Defining the background ensemble
mean as its sample mean
¯ xbk = 1 Nens Nens X i=1 xb(i)k , (3.25)
the background covariance is estimated with the sample covariance of the ensemble, that is:
Pbk= 1
Nens− 1
Xbk(Xbk)T (3.26)
where Xb
kis the matrix which i -th column is the perturbation from the mean for the
i -th member, i.e. xb(i)k − ¯xb
k. Compared to the background error covariance of the
EKF in Eq. (3.20), the estimation of Pb
k in the EnKF assumes a much easier and
less costly form since it is not necessary to linearise the model and to compute its adjoint. However, the model error is not taken into account. Covariance inflation
can be employed to deal with it, as described in Section 3.2.3.
Regarding the analysis step, it can be formulated following two approaches which define two different types of EnKF schemes: stochastic filters and determin- istic filters. For the rest of the chapter, only quantities at analysis time tk will be
considered and so the subscript k is dropped hereafter. Stochastic update algorithms
The main feature of these algorithms is that each member of the ensemble is up- dated to a different set of observations perturbed with random noise. Accordingly, the analysis state for the i -th member is given by
xa(i)= xb(i)+ K yo(i) − Hxb(i)
(3.27) where yo(i) = yo+ y0(i) is the vector of perturbed observations, defined such that
y0(i) ∼ N (0, R) and 1 Nens Nens X i=1 y0(i) = 0 (3.28)
The Kalman gain K has the same form of that defined in Eq. (3.21) for the EKF, but in this case Pb is estimated from the background ensemble via Eq. (3.26). Finally, the analysis error covariance is given by the sample covariance of the analysis ensemble
Pa= 1
Nens− 1
Xa(Xa)T (3.29)
where Xa, in the same fashion as Xb, is the matrix which i -th column is the perturbation from the mean for the i -th member of the ensemble, i.e. xa(i)− ¯xa.
Without modifications to the algorithm, the use of perturbed observations is necessary. In fact, due to the limited size of real case ensembles, the use of unperturbed observations would determine an underestimation of Pa which, in
turns, would lead to a severe filter divergence (Burgers et al., 1998), as described in Section 3.2.3. In other words, the use of perturbed observations ensures that the analysis error covariance is the same, or at least close, to that defined by Eq. (3.12). However, spurious correlations between the background ensemble and perturbed observations may arise, leading to a degradation of analysis quality (Houtekamer
and Zhang, 2016).
Deterministic update algorithms
Algorithms in which random noise is not added to observations are referred to as deterministic, so named because if the background ensemble and the associated error statistics are known, the ensemble of analysis states will be completely known as well. In fact, according to Hamill (2006), they update in a way that generates the same analysis error covariance that would be obtained from the Kalman filter, i.e. by Eq. (3.12), assuming that the background error covariance Pb is modelled from
the background ensemble via Eq. (3.26). Depending on how the analysis ensemble is constructed, several implementations of deterministic EnKF filters have been proposed, such as the ensemble transform Kalman filter (ETKF; Bishop et al., 2001), the ensemble square root filter (EnSRF; Whitaker and Hamill, 2002) and the ensemble adjustment Kalman filter (EAKF; Anderson, 2001). A general review is provided in Tippett et al. (2003) while a detailed description of the deterministic filter named local ensemble transform Kalman filter (LETKF; Hunt et al., 2007) is provided in Section 3.2.4.
Since spurious correlations between observations and background ensemble are avoided, deterministic filters are more accurate than stochastic filters (Whitaker and Hamill, 2002). However, they are more vulnerable to errors in the estimation of Pb due to the direct relationship, via Eq. (3.12), between background and analysis error covariances (Houtekamer and Zhang, 2016).