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Team Science Investigations Hilbert Transform Analysis
A direct two-dimensional generalization of the procedure is not possible, because a field of randomly orientated waves of like frequency cannot be represented by a single harmonic function with a unique slowly varying phase gradient and amplitude. So instead we try to isolate pencils of waves by averaging perpendicular to a set of target directions. If the outcome were genuinely to isolate unidirectional waves, the result of subsequent one-dimensional Hilbert-transform analyses would be Radon transforms of the background inhomogeneity, which could then be inverted. However, the results are degraded by interference from laterally propagating waves, whose pattern propagates with zero group velocity and therefore cannot easily be distinguished from the nonuniformity of the background state. Work by Gough, Keith Julien and Toomre is under way to try to isolate that inference, in the hope that its angular coherence is insufficient to contribute significantly to that part of the inverse Radon transform with large horizontal (and vertical) scales, so that it can be removed subsequently by spatial filtering. As before, one should realize that interference by laterally propagating waves is suffered by any other procedure that has so far been tried, or even conceived, and is not an artefact of the Hilbert method. In ring analysis it is manifest as a distortion of the ring at angles displaced from the orientation of the offending wave, and, as emphasized by Tom Duvall, in time-distance analyses it appears as a contribution to the correlation arising from waves whose apparent origin is neither of the two points at which the signal is being cross-correlated.
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