Localisation and inflation

The use of a finite ensemble size, and especially the use of an ensemble size much smaller than the state dimension, comes at a price. When M ⌧ n, the forecast sample covariance matrix ˆPt, whose rank is at mostM 1, is severely rank deficient and usually a poor substitute for the true, possibly full rank, covariance matrix Pt. In particular, ˆPt is known to su↵er from what is known as spurious correlations, which refers to overestimation of o↵-diagonal elements of Pt that are supposed to be close to zero. This undesirable behaviour is demonstrated in Figure 2(b) which shows the sample covariance matrix obtained fromM = 20 independent samples from a Gaussian distribution of dimensionn= 100 with zero mean and covariance matrix shown in Figure 2(a). Spurious correlations can lead to an updated ensemble with a too small spread which, when done sequentially, can result in filter divergence in the sense that the variability of the forecast and filtering ensembles become smaller and smaller and the ensemble mean eventually drifts away from the truth. Spurious correlations are a natural result of sampling errors and occur also forM > n, however whenM is sufficiently large the e↵ect is often small enough to avoid filter divergence. Two techniques proposed to correct for spurious correlations are localisation and inflation. Both techniques are commonly applied in practical applications, often in combination.

Localisation

Localisation relies on the fact that, in most spatial geophysical systems, corre-lations between variables decrease rapidly with the distance between them, often

exponentially. The correlation between two variablesxⁱ_tandx^j_t ofxtshould there-fore be close to zero if the indices i and j correspond to locations far apart in space. Two di↵erent localisation techniques have been proposed in the literature:

covariance localisation and domain localisation. A nice review of the two proce-dures can be found in Sakov and Bertino (2011), where also the relation between them is investigated.

Covariance localisation (Hamill and Whitaker, 2001; Houtekammer and Mitchell, 2001) seeks to increase the rank of the estimated forecast covariance matrix and to suppress spurious correlations by replacing the sample covariance matrix with its Schur (element-wise) product with some well chosen correlation matrix⇢2R^n⇥n,

Pˆt!⇢ Pˆt. (27)

The matrix⇢is chosen so that it reflects how the correlations between variables decrease with the distance between them as seen in real geophysical systems. The Schur product in Eq. (27) should then result in a regularised covariance matrix where the spurious correlations are dampened. A common approach is to use the Gaspari-Cohn function (Gaspari and Cohn, 1999),

G(r) = 8>

>>

<

>>

>:

1 ⁵₃r²+⁵₈|r|³+¹₂r^{4 1}₄|r|⁵, if 0 |r|<1, 4 5|r|+⁵₃r²+⁵₈|r|^{3 1}₂r⁴+ ₁₂¹|r|⁵ ₃²_|r|, if 1 |r|<2,

0, if|r| 2,

(28)

and define the entries of⇢as

⇢ij =G((i j)/L), (29)

where L is a so-called correlation length which determines the rate at which the correlations decrease towards zero. Figure 2(c) shows the covariance matrix obtained from the Schur product of the sample covariance matrix shown in Figure 2(b) and a correlation matrix⇢defined by Eq. (29) withL= 10. The downside of covariance localisation is that, brute force, it involves storage and computation ofn⇥nmatrices. A possible way to circumvent this, is to choose⇢as sparse.

Domain localisation (Ott et al., 2004; Hunt et al., 2007; Janjic et al., 2011), or local analysis, instead divides the state vector into several disjoint subsets and performs a ’local’ update for each subset. In each of the updates, only a corre-sponding local subset of the observation vector, containing observations within

some chosen cut-o↵ radius from the centre of the assimilation region, is considered.

Computationally, this approach is advantageous over covariance localisation as it can exploit the ensemble representations and avoid explicit maintenance ofn⇥n matrices. An issue of concern, however, is lack of smoothness in the updated realisations due to the division of the global domain into several subdomains.

Di↵erent techniques have been proposed to correct for this issue. For example, Hunt et al. (2007) propose to weight the observation error covariance matrix so that observations further away from the assimilation region are assigned larger variances.

Inflation

For high-dimensional models, localisation alone is often not sufficient to avoid filter divergence, and covariance inflation (Anderson and Anderson, 1999) is also applied to stabilise the filter. With inflation, the estimated forecast distribution is artificially broadened by multiplying the sample covariance matrix ˆPtby a factor

>1,

Pˆt! Pˆt,

or, equivalently, by multiplying the ensemble-anomaly matrixXtby a factorp , i.e. Xt! p

Xt. The inflation factor is usually only slightly larger than one and needs to be tuned to obtain satisfactory performance. Such tuning can be a burden computationally, but adaptive schemes have been proposed (e.g., Wang and Bishop, 2003; Anderson, 2007, 2009).

An illustrative example

Here, we present a simple simulation example which illustrates the poten-tial e↵ect of using covariance localisation in the EnKF. The example involves a linear-Gaussian state-space model, and results obtained using the stochastic EnKF scheme, with and without covariance localisation, are compared. For demonstration purposes, we also present output from a modified EnKF where the true forecast covariance matrices, computed with the Kalman filter, are used to update the ensemble in each iteration. Of course, such an approach is not something one would be able to run in practice, but for the purpose of this ex-periment it is convenient to use the output as a reference, since it reflects how an ensemble of M realisations ideally should look like. The dimension of the state vectorxtisn= 100, and for every fifth variable ofxt there is an observationy^j_t,

so that the dimension of the observation vectorytism= 20. The relatively small ensemble size M = 20 is used in all three schemes, and the correlation matrix

⇢used in the localisation procedure is defined by Eq. (29) with L= 10. (J. L.

Anderson & Anderson, 1999)

Figure 3(a) shows the true state vector xt and the observation vector yt at time step t = 40, and Figures 3(b)-(d) show the posterior ensemble members e

x⁽¹⁾_t , . . . ,ex^(M_t ⁾ obtained from the three di↵erent EnKF schemes described above at this time step. Notice in particular that the ensemble spread in Figure 3(b) obtained with the traditional EnKF, with no localisation, unmistakably is much too narrow compared to the spread in Figure 3(d) obtained with the scheme using the correct covariance matrices. The true values of the x^j_t’s are then also quite often far outside the spread of the ensemble. In Figure 3(c), which shows the results from the covariance localisation scheme, this e↵ect is considerably reduced, and the ensemble spread is more comparable to that in Figure 3(d).