SOI Team Science Investigations Hilbert Transform Analysis

Detailed Specifications

The main objectives of the team:

The principles of the procedure are most easily comprehended in the one-dimensional case. It is assumed that the background state of the sun is independent of time and that modes of oscillation exist with well defined frequencies, though the spectrum need not be discrete. It is assumed that some projection procedure has been applied to the data to isolate a narrow band in (temporal) frequency and to isolate waves propagating along a single great circle. It is recognized, however, that because the data set in not infinite in duration, because the entire surface of the sun cannot be observed simultaneously, and because, in view of the asphericity of the sun, wave propagation is not precisely along great circles, neither frequency bandwidth nor purely parallel propagation can be isolated. Consequently, one must be content with studying wave packets.

One way of analyzing the wave packet is to take a spatial Fourier transform of the disturbance along the presumed directions of propagation and measure in some way the characteristic wave number of the two peaks in the power spectrum corresponding to waves propagating in opposite directions associated with a given ridge in a k-omega diagram. Coupled with the temporal frequency, the difference and the sum of those two wave numbers provide information about the mean propagation speed and the mean Doppler shift produced by horizontal flow. That mean is an average over the horizontal window of observations and over the depth sampled by the mode. This is the essence of a one-dimensional version of ring analysis. It is presumed that a standard helioseismic inversion technique can be used to invert the results over a range of temporal frequencies to determine the variation with depth.

The aim of the Hilbert analysis is to go further by estimating the horizontal variation of propagation speed and advection velocity. It is hoped that, by relating the wave-like disturbance over the region without which a localized average is sought to the properties of the disturbance before it enters and after it leaves that region, a more reliable representation of the wave can be obtained, and consequently a more reliable estimation of the background state. It is presumed that Nature has more sense than to carry out spatial Fourier transforms, and that the entire wave packet approximately satisfies a single equation that closely resembles the spatial part of the spatially and temporally separated wave equation, namely a Helmholtz equation whose solution can be represented by a sinusoidal function with slowly varying amplitude and wave number. There are numerous methods that one might concoct to achieve that separation of the signal into two slowly varying functions, and here we choose to do so via Hilbert transformation, by a procedure which would have been exact had the disturbance been a genuine single one-dimensional wave with a unique frequency. The spatial variation of the wave number so obtained is presumed to relate to the variation of the background state in precisely the same way as for a simple wave, so once the wave number is determined it is presumed that the rest of the inversion is in principle complete.

Last updated by Alexander Kosovichev Sun Aug 6 19:21:56 PDT 1995
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