The Pattern Data Assimilation technique called Data Assimilation by Field Alignment solves a Bayesian problem of the form for a solution in deformation and amplitude adjustment space. Read the papers, read them directly here or watch a show here. In the above example, an ensemble of vortices, marked by contours, automatically deform in response to noisy measurements at sparse locations. The evolving ensemble mean shows the fidelity with which the assimilated measurements can help reconstruct truth. Our primary contributions are to show that:

1. Although non-Gaussian amplitude errors may result from position errors, we show that the covariance is always weakened (for non-zero displacements) and this is the principal problem plaguing effective estimation. The corrupted amplitude covariance, especially when it is flow-dependent, can no longer achieve the Cramer-Rao bound that was achieved in a world without position errors.

2. Error between spatial fields are quantified and reduced in position and amplitude subspaces without explicit detection of feature positions. However, if feature positions are available, they can be easily incorporated. 

3. Because feature positions need not explicitly be specified, situations where sparse observations are present or feature positions are difficult to specify (e.g. fronts, complex storm shapes) can be directly addressed. 

4. Because the deformation solution is dense, non-trivial alignments can be easily represented (as opposed to a strictly Lagrangian approach), including local skews, shears, rotations, divergence, convergence and higher modes. In fact there is no explicit parameterization of the deformation.

5. The error between spatial fields can be jointly reduced in position and amplitude subspaces optimally if and only if the sensitivity of the amplitude error covariance to position errors is accounted. 

6. This is a non-trivial problem except when (a) the amplitude error covariance is assumed to be static or (b) when position and amplitude errors are assumed to be independent or (c) jointly Gaussian. These assumptions will produce poor results when reality does not match them (which is almost always), but they are what have been assumed in earlier work. We do not make such an assumption and were the first to show, instead, that the joint error-reduction problem can be solved using sampling (ensemble) methods.

7. We were also the first to show that the error between spatial fields can be reduced in position and amplitude in two steps in a systematic way (though ad-hoc approaches were considered contemporarily). This approximation yields two numerical processes: a Field Alignment process and an amplitude adjustment process, the latter being familiar as contemporary data assimilation e.g. OI/KF/EKF/EnKF/3DVAR etc. The Field Alignment process becomes a pre-processor, which makes practical application feasible and with broad utility. See examples below and in the gallery.

8. Dynamical Balance is better preserved when data is assimilated by field alignment, which is easily extended to multivariate fields and vector fields.

9. Constraints on the displacement field can be simple “fluid-like” ones, such as smoothness and non-divergence, but they need to be specified at an appropriate length scale to be effective but this is often difficult to do. Otherwise, they produce an inappropriate, aliased explanation of the true deformation. An improved technique is Scale-Cascaded Alignment.

The techniques that are used to minimize jointly in the space of positions (deformations) and amplitudes have other applications too; in velocimetry, tracking, nowcasting, pattern recognition, downscaling and texture transfer and super-resolution to name a few. You can see some of them in the gallery.

Data Assimilation by Field Alignment

(some examples)

On the left -- a domain with an observed region (inner square) and a halo (outer square). In the middle observations at sparse locations (white dots) and on the right, the estimate by Field Alignment; notice a seamless border. 

Sai: where is this image? When i look at original source there is no file there, and I can't right-click and save it

Alignment is jointly done over multivariate fields, shown here are velocity and pressure from a GSI/GFS output from NCEP. On left measurements (red) at few spots (green) and FA analysis (black). On right, forecast (blue) and FA analysis overlaid (black). Shown below is the pressure field.