Smooth detectors of linear phase

Functions with (linear) phase depend on (linear) combinations of the independent variables. For example plane waves are functions of a single linear combination of the variables. Detection of phase is an important problem in seismic velocity analysis and ocean acoustic signal processing, amongst other applications. Robust estimation of phase by local optimization requires the construction of smooth objective functionals. For this purpose it is useful to characterize those quadratic functionals of functions with linear phase which are smooth in the phase: all such smooth phase detectors are pseudodifferential. Some of these pseudodifferential quadratic forms are globally convex in the phase, hence permit phase estimation using local smooth optimization methods.