A first discussion towards a clearer equivalence between the concepts of synchronisation and data assimilation was pointed out by Duane et al. (2006). The main goal of synchronisation is the convergence between the model and the truth in the future, allowing a predictable behaviour between chaotic systems. Generally speaking, data assimilation aims to synchronise the model evolution with the true evolution of a system, finding the best estimate of the state or its evolution, including its uncertainty. This best state will then produce the best prediction, assuming that the model is unbiased.
To reach their goals, both methodologies make use of a coupling term, which in general is unidirectional (a characteristic of the generalised synchronisation, as described previously in
this chapter). In the data assimilation field, this coupling occurs from the truth to the model system. Additionally, it is incomplete, as observations (which are the real knowledge we have from the truth) are typically sparse and contain errors, likewise the model. Typically, as described previously, the model is extended with a nudging or relaxation term that forces it towards the observations. In synchronisation, this coupling has pretty much the same characteristics, but is specifically designed to control the spread of a probability distribution, centered on the synchronisation manifold xS= xM.
Duane et al. (2006), on a first attempt to find equivalences between synchronisation and data assimilation methods, have presented derivations in which they concluded that, in linear cases, synchronisation could be considered equivalent to 3D-Var or the Kalman filtering, depending on the background error used. For the nonlinear case, they have performed a hypothetical experiment with a three-variable system that switched, periodically, between the Lorenz set of equations and that system with the variables X and Z reversed:
˙ X = σ(Y − X) X = −βX + ZY˙ ˙ Y = ρX − Y − XZ Y = ρZ − Y − ZX˙ ˙ Z = −βZ + XY Z = σ(Y − Z)˙ (3.9)
A comparison was made between a Kalman filtering approach and a synchronisation scheme. The latter used an alternate coupling in the slave sub-systems as follows:
˙ X1 = σ(Y1− X1) + k(X − X1) X˙1 = −βX1+ Z1Y1 ˙ Y1 = ρX1− Y1− X1Z1+ k(Y − Y1) Y˙1 = ρZ1− Y1− Z1X1+ k(Y − Y1) ˙ Z1 = −βZ1+ X1Y1 Z˙1= σ(Y1− Z1) + k(Z − Z1) (3.10)
while the Kalman filter used observations of the X, Y and Z master system to drive the slave. For synchronisation to happen, the coupling parameter had a threshold value, i.e. synchronisation did not increase monotonically with k. Their results also showed that Kalman filtering presented bursts of desynchronisation during the transitions between Lorenz and the reversed system that were not happening in the coupling set used for synchronisation, although the latter presented greater errors on average. They suggested that the use of a time-dependent synchronisation coupling would lead to a reduction on the average error and so represent an optimal synchronisation scheme. They also argued that the Kalman filter method tended to be better than any other coupling scheme described in the synchronisation literature by that time. It is interesting to note though, that these methodologies, the Kalman filtering and synchronisation, have different purposes. The Kalman filter approach aims to compute an optimal analysis for a better departure of the model evolution, through a linear combination of an a priori state (forecast) and a weighted difference between observations and the predicted observations. Synchronisation
tries to find an optimal forecast, a model trajectory that follows the observations as closely as possible, aiming for the fastest convergence of the model with the true future trajectory. This is done by combining the forecast with a weighted difference between a set of observations and a set of forecasts.
Still on the equivalences between synchronisation and data assimilation methods, Yang et al. (2006) have also tested synchronisation using a three-variable Lorenz system and a similar type of nudging as described in (3.10) for the slave system, finding what they called an optimal coupling strength, which could generate comparable results to the ones produced by 3D-Var. That was considered indeed a remarkable result, considering the simplicity of the nudging scheme used, in comparison with the 3D-Var framework. They also tested synchronisation with bred vectors and singular vectors, suggesting that learning the relationship between the coupling parameter and the bred vector growth rate could help to reduce the error growth. Their overall conclusion at that time was that the development of a hybrid synchronisation-data assimilation methodology in the future would improve weather forecasts. Interestingly, that is exactly the aim of this thesis.
The use of time embeddings or time delays has also allowed significant advances in the synchronisation field. It consists of incorporating in the synchronisation coupling dynamics the existing information from different time delays. Note that here the term ”delay” can refer to information contained in the past and/or in the future. This concept comes from Taken’s delay embedding theorem (Kantz and Schreiber, 1997), which explains that, in the absence of enough observations, the delayed information can be used to reconstruct the topology of the attractor. Work from Abarbanel et al. (1994); Zhan et al. (2003); Dhamala et al. (2004); Feng et al. (2007); Chen and Kurths (2007); Huang et al. (2011); Ghosh et al. (2012); Paz´o et al. (2016) and other researchers related to different fields, from engineering to neural dynamics, focused on reaching synchronisation and/or predicting physical variables by using the concept of time-delay embeddings. The main structure of the time embeddings shares similarities with the one found in a strong constraint 4D-Var, as the main idea of these systems is to use a sliding observation window, bringing back information from time-advanced observations. Some crucial differences between these methods are noted, though. In synchronisation, the analysis is propagated in small time increments dt, while in 4D-Var this is done throughout the observation window. Additionally, in synchronisation the observations are re-used as we slide the window (note that not all observations inside this window are used, only specific observations are chosen at once). Another noticeable difference is that these time embedding synchronisation frameworks usually utilize a truncated singular value decomposition, while 4D-Var uses a background term to perform regularization.
Among these time embedding formulations, Rey et al. (2014a,b) have achieved remarkable synchronisation results with the coupling term designed in their framework. They performed tests considering a twin experiment using the Lorenz96 chaotic model (Lorenz, 1995), with 20 state variables to be estimated and only 1 observed state. Among their results, they were able
to find synchronisation errors with magnitudes of the order of 10−15, i.e. close to machine precision. Furthermore, the authors achieved successful prediction ranges of around 2,000 time steps after an estimation period of 10,000 time steps. It should be mentioned, however, that these results assumed that the observations contained no errors, which is unrealistic in the data assimilation context. In their setting, even small observational errors can have a large impact on the synchronisation results. Furthermore, their method is far too expensive to be used in realistic high-dimensional geophysical problems as it needs the propagation of the Jacobian matrix of the forward model, with size the model dimension squared. Despite these issues, their methodology has inspired the research in this thesis, given the powerful convergence between the model and the truth obtained. This framework will be described in more details next.