Assimilating Coherent Fluids Is Difficult

April 1, 2003

 

This figure shows the difficulty that unaccounted for coherence attributes (position, scale, shape) causes in optimal estimation in the presence of sparse, noisy observations. The analysis (red) gets deformed and distorted relative to the truth (dotted black line), when the prediction (green) has position error. We argue that coherent fluids must be treated as coherent patterns in addition to being random fields.
   
Environmental data assimilation is the methodology for combining imperfect model predictions with uncertain data in a way that acknowledges their respective uncertainties. However, data assimilation can only work when the estimation process properly represents all sources of error. The difficulties created by improperly represented errors are particularly apparent in coherent atmospheric phenomena such as thunderstorms, squall-lines, wild fires, hurricanes, precipitation, and fronts. They also show up elsewhere, for example in reservoir applications where permeability fields have strong contrast.

Errors in mesoscale models can arise in many ways but they often manifest themselves as errors in position. Please note that the term position here is local and thus for a gridded field this would be a deformation field. The results of position errors is then manifest in terms of shape or scale errors, for example. We typically cannot attribute position error to a single source and it is likely that they are the aggregate result of errors in parameter values, initial conditions, boundary conditions and others.

In the context of cyclones, operational forecasters resort to ad hoc procedures such as bogussing to fix the problem. A more sophisticated alternative is to use data assimilation methods. Unfortunately sequential, ensemble and variational state estimation methods used in data assimilation applications adjust amplitudes. Adjusting amplitudes does not really fix position error, and can produce unacceptably distorted estimates instead. As seen in the figure, a simulated “one dimensional front,” the true signal (dashed line) observed sparsely (blue dots) by a noisy sensor produces a reasonable estimate (red) from an initial guess (green) when the position error is negligible. It causes disaster (right image) when the error is relatively larger. That’s the problem.

 The issue captured in the above figure appears to occur irrespective of the nature of the contemporary estimation procedure used. Solutions have also been proposed, most notably as an optimization problem expressed in the space of grid position and field amplitude, with suitable constraints on the grid deformation. Rather than just adjusting amplitudes, this is a more general way of minimizing misfit between two spatial fields. Not surprisingly, it can be applied to many geophysical, meteorological and oceanographic problems.

Coherence implies pattern formation, which data assimilation must properly account for i.e. we need to think in terms of pattern data assimilation.  You can read the papers here and go to this blog entry to learn how pattern data assimilation problem is solved.