Field Coalescence is a technique to find the mean of coherent fields. In the animations above, fields are shown to be “coalescing” over iterations of a search procedure for the mean field. You should look carefully on the ones on the right which is coalescing two fields, and easily on the left, which is coalescing multiple signals. 

What is actually shown in the animation is the best estimate of the mean at every iteration (dotted black line on the left, the 2D field you see on the right). As the structures come together from different fields the mean estimate improves.

The optimization uses the Expectation Maximization algorithm, which emerges from the mutual joint amplitude and position minimization for N bodies (field). In this framework the conditional mean is estimated from the ensemble, then ensemble members are deformed to maximize (minimize) error relative to mean, and this continues to convergence. This can equivalently seen as a the solution to an "N-body" problem between multiple fields.   

In the example below, you see on the left an uncoalesced ensemble of coherent fields and on the right the coalesced version. An animation of the evolution through the optimization is shown in another example at the top of this page; notice how well defined the mean gets over iterations. Please note that features are never detected (though obviously if this is reliable the problem becomes easy as Carl Wunsch once pointed out). To emphasize this point, the signals above are sampled only every third grid point for providing the "forcing" terms. Meaning, this works well when features cannot really be detected which is an issue if the features are complicated shapes (e.g. sheared structures) or broken (e.g. long fronts) or others. The radar picture of coalescence communicates that.

Coalescence is an incredibly useful tool for finding an ensemble mean.  Coalescence is a non-parametric and unsupervised way to discover the mean field in the space of positions (deformations) and amplitudes. Below is an animation of the search that leads to coalescence (more examples in the gallery). Preliminary ideas on coalescence were proposed by Ravela et al. in 2005-06 (EGU 2006) and realized using SCA and stochastic optimization (see here) With coalescence, you don't have to worry about losing your mind!

Coalescence is the first step in quantifying uncertainty and we use it for calculating the Principal Appearance and Geometry modes, or Principal Coherent Features. They are used in deformable-ROMs and, further, are useful for what are known as blending problems, discussed elsewhere in this website and in papers found here