The linearisation of maps in data assimilation

For the purpose of linearising maps in data assimilation, the tangent-linear approximation is often used. We compare this with the use of the ‘best linear’ approximation, which is the linear map that minimises the mean square error. As a benchmark, we use minimum variance filters and smoothers which are non-linear generalisations of Kalman filters and smoothers. We show that use of the best linear approximation leads to a filter whose prior has first moment unapproximated compared with the benchmark, and second moment whose departure from the benchmark is bounded independently of the derivative of the map, with similar results for smoothers. This is particularly advantageous where the maps in question are strongly non-linear on the scale of the increments. Furthermore, the best linear approximation works equally well for maps which are non-differentiable. We illustrate the results with examples using low-dimensional chaotic maps.