  Inferring electromagnetic propagation characteristics within the marine
atmospheric boundary layer (MABL) from data in real time is crucial for modern
maritime navigation and communications. The propagation of electromagnetic
waves is well modeled by a partial differential equation (PDE): a Helmholtz
equation. A natural way to solve the MABL characterization inverse problem is
to minimize what is observed and what is predicted by the PDE. However, this
optimization is difficult because it has many local minima. We propose an
alternative solution that relies on the properties of the PDE but does not
involve solving the full forward model. Ducted environments result in an EM
field which can be decomposed into a few propagating, trapped modes. These
modes are a subset of the solutions to a Sturm-Liouville eigenvalue problem. We
design a new objective function that measures the distance from the
observations to a subspace spanned by these eigenvectors. The resulting
optimization problem is much easier than the one that arises in the standard
approach, and we show how to solve the associated nonlinear eigenvalue problem
efficiently, leading to a real-time method.
