A connection between ensemble synchronisation and other data

The methodology described in the previous section has strong similarities with an Ensemble Smoother (van Leeuwen and Evensen, 1996; van Leeuwen, 1999; Evensen and van Leeuwen, 2000). Basically, the Ensemble Smoother update can be formulated as:

xj+1= f (xj) + X(H(X))T(H(X)(H(X))T + R)−1(Yj− S(xj)) (4.11) The main differences between Ensemble Smoothers and the Ensemble-based Synchronisation are the following:

1. Synchronisation ignores observation errors in the gain, i.e. it does not include the observation error covariance matrix R in this term. This allows for an inversion of H(X) directly, instead of (H(X)(H(X))T + R). The latter is a larger matrix, but efficient implementations are available, in which the matrix to be inverted in an ensemble smoother is of size Nens× Nens. Localisation, though, makes the matrices in both methods to have

similar sizes. The fact that the ensemble synchronisation ignores the observation error covariance matrix is maybe a disadvantage in the scheme, but this weakness is partly compensated for by a very useful tool, the tuning factor g which appears in equation (4.9). (In the experiments performed in this chapter and in the previous one, this powerful feature of the coupling constant g is not fully explored. However, the importance of this tuning parameter will be highlighted in the following chapters, where the ensemble synchronisation scheme will be effectively used as an important piece in a particle filter). 2. Observations are used multiple times in this time-embedded framework. That would be

considered a forbidden practice in an ensemble smoother, as using the same observations multiple times would introduce correlations between the errors of the state iterates and the observation errors, bringing significant complications to the scheme. Ensemble synchronisation, however, is not hampered by this issue, and observations inside the time embedding interval are used several times, helping to increase the observability of the system. This is a crucial advantage of the ensemble synchronisation scheme, which will be used when this new method is inserted in the context of proposal densities in a particle filter, following e.g. van Leeuwen (2009), who shows that observations can be used in a proposal density as often as one would like to.

methods that employ ensembles to avoid adjoint models, like, e.g. 4DEnsVar (Liu et al., 2008; Fairbairn et al., 2014; Gustafsson and Bojarova, 2014), and iterative Ensemble Smoothers like the IEnKS (Iterative Ensemble Kalman Smoother) of Bocquet and Sakov (2014), as these systems also explore the space-time correlations from ensembles, as pioneered by van Leeuwen and Evensen (1996).

The formulation of the 4DEnsVar is based on the use of an ensemble to estimate sensitivities to observations within a data assimilation window, i.e. replace the calculation of adjoint models with an ensemble of perturbations (Liu et al., 2008). The drawback of this strategy is that the method inherits the rank-deficiency issue, when applied to high-dimensional systems. Consequently, localisation is required as well, although a spatial localisation in a 4D environment is not that trivial, as one will need to localise covariances inside the data assimilation window between state variables and observations at different times (asynchronously). Additionally, regularisation in this case is not straight-forward, as explained in detail in Asch et al. (2016).

The IEnKS is a methodology derived from Bayes’ rule, which also estimates sensitivities by using an ensemble and so preventing adjoint calculations. Like the ensemble synchronisation scheme, this method also assimilates observations ahead in time, aiming for stabilisation in the direction of the unstable modes. However, differently from the scheme presented in this work, the IEnKS reuses all the observations inside the data assimilation window, whereas the ensemble synchronisation uses only the observations that appear at every τ time steps in the [j, j+(Dd−1)τ ]

window. Additionally, the IEnKS scheme uses observations multiple times, but with inflated covariances. In practice, this approach factorises the likelihood and assimilates the resulting sequence of likelihoods sequentially. Note that, differently from the ensemble synchronisation, this reuse of observations does not change the observability of the system.

It is important to keep in mind that the ensemble synchronisation scheme targets a different purpose, when compared to the formal data assimilation methodologies. While data assimilation methods aim to approximate the true posterior pdf, the ensemble synchronisation technique proposed here tries to find a model trajectory that follows the observations closely, to synchronise with the truth. Therefore, as mentioned earlier, ensemble synchronisation is not a complete data assimilation method, as uncertainties in model and observations are not incorporated explicitly in the formulation. An efficient use for this scheme will be shown later in following chapters, where it is included as part of a proposal density, in a more comprehensive data-assimilation method, a particle filter.

The results to be presented in the next sections of this chapter are a motivating engine to extend the use of this scheme to the particle filtering world, as it properly fits to the needs of this data assimilation methodology. Pre-existing data assimilation methods were, so far, not good enough to be joined as a proposal density in a particle filter. Authors like Browne (2016) did use a proper Ensemble Kalman smoother as a proposal, without obtaining satisfying results. In this

thesis, a new and exciting field is explored, in conjunction with a particle filter, a data assimilation method that receives an increasing attention in the geosciences field.