An orthogonal projection and regularization technique for magnetospheric radio tomography

Abstract

A challenging problem in ill‐posed inverse problems is incorporating prior knowledge of the solution into reconstruction techniques. This problem is particularly important in magnetospheric radio tomography where the path integrated measurements of the target region may be sparse. We present in this paper an orthogonal projection and regularization (OPR) technique that incorporates prior knowledge of magnetospheric parameters from existing models or past measurements into a direct reconstruction algorithm. The OPR scheme extracts first an optimal orthonormal basis containing the main features of the unknowns from an ensemble of modeled or measured snapshots through the proper orthogonal decomposition (POD) and then projects the line‐of‐sight equations onto the subspace spanned by the empirical orthonormal basis. The resulting low‐dimensional model is well‐conditioned, its coordinates are uncorrelated, and it contains prior knowledge of the solution. The magnetospheric parameters in the transformed coordinate are reconstructed from the low‐dimensional model and quantities in the physical coordinate are easily recovered from the POD transformation. On the basis of magnetohydrodynamic (MHD) simulations and hypothetical satellite constellations, we demonstrate that the POD‐based method may perform significantly better than the regularized direct method with sparse path‐integrated measurements, combined with a few (5–10) model snapshots.