The Hankel spectral analysis was introduced by Braun et al. (1987) in order to study the relationship between inward and outward traveling waves around sunspots. Consider a spherical polar coordinate system (colatitude , azimuth ) with a sunspot situated on the polar axis , and an annular region on the solar surface surrounding the sunspot (). The goal is to decompose the oscillation signal in the annular region, , into components of the formet al., 1992): et al. (1988). The time and azimuthal transforms use the standard fast Fourier transform algorithm, with azimuthal orders in the range . In particular, the frequency grid is given by , where and is an integer value. The spatial Hankel transform is computed for a set of discrete values , that provide an orthogonal set of Hankel functions: et al., 1988). The maximum degree is given by the Nyquist value , where is the spatial sampling, while the minimum degree below which the orthogonality condition ceases to be valid is . We note that the outer radius of the annulus, , must not be too large since waves should remain coherent over the travel distance . This implies that , where and are respectively the group velocity and the lifetime of the waves under study. In addition, the travel time across the annulus should be less than the duration of the observation (waves must be observed first as incoming waves and later as outgoing waves). This implies .
The original motivation for the Hankel analysis was to search for wave absorption by sunspots. Braun et al. (1987) defined the absorption coefficient byet al. (1987) averaged and over all azimuthal orders and over the frequency range . Braun (1995) obtained a reasonable signal-to-noise level only by averaging in frequency space over the width of a ridge (radial order ) at fixed :
Braun et al. (1992) and Braun (1995) were successful at measuring the relative phase shift between outgoing and ingoing waves. Phase shifts measurements require a lot of care. Braun et al. (1992) pointed out that the phases of and represent averages over a resolution element with size . For example, Braun (1995) has and Hz.
The finite resolution in wavenumber introduces spurious phases. This can be seen by considering first the -transform (Equation (34)) of an incoming wave with amplitude , wavenumber , azimuthal order , frequency , and phase . Using the far-field approximation of the Hankel functions, we have
Let us now consider actual observations. The p modes are distributed along ridges corresponding to different radial orders , whose frequency positions in the - diagram are specified by known functions . There may be several unresolved modes in the resolution element . The condition (Section 4.1.1) is equivalent to , where the prime denotes a derivative. This means that the spread in of the ridge across the frequency step is much smaller than the wavenumber resolution . In this case Braun et al. (1992) estimate the spurious phase shift from Equation (42) by letting equal to the mean wavenumber of the unresolved modes. Since , the frequency dependence of the spurious shifts is given by (Braun et al., 1992)
Using a similar technique, Braun (1995) measured correlations between incoming and outgoing waves with different radial orders and . He found that sunspots introduce non-zero -averaged phase shifts for . This observation gives hope for a full characterization of the sunspot-wave interaction, encapsulated in the scattering matrix defined by
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