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4.1 Fourier–Hankel spectral method

4.1.1 Wavefield decomposition

The Hankel spectral analysis was introduced by Braun et al. (1987Jump To The Next Citation Point) in order to study the relationship between inward and outward traveling waves around sunspots. Consider a spherical polar coordinate system (colatitude πœƒ, azimuth ψ) with a sunspot situated on the polar axis πœƒ = 0, and an annular region on the solar surface surrounding the sunspot (πœƒmin < πœƒ < πœƒmax). The goal is to decompose the oscillation signal in the annular region, Φ (πœƒ,ψ, t), into components of the form

[ ] ΦLm (πœƒ,ψ,t) = Am (L, ν)H (1)(Lπœƒ) + Bm (L, ν)H (2)(Lπœƒ) ei(mψ+2πνt), (31 ) m m
where m is the azimuthal order, L = [l(l + 1)]1βˆ•2, l is the spherical harmonic degree, ν is the temporal frequency, and H (1,2) m are Hankel functions of the first and second kind. Hankel functions are used as numerical approximations,
(l − m )! [ 2i ] H (m1,2)(L πœƒ) ≃ (− 1)m-------- Pml (cosπœƒ) ± --Qml (cos πœƒ) , (32 ) (l + m )! π
to the more exact combination of the Legendre functions P and Q used in spherical geometry. This approximation, valid in the limit l ≫ m, is always good in practice and is not a limitation of the technique (Braun, 1995Jump To The Next Citation Point). For reference, we recall that in the far-field approximation (πœƒ ≫ 1βˆ•L),
∘ ----- H (1,2)(Lπœƒ) ≃ --2-e± i(Lπœƒ−m πβˆ•2− πβˆ•4), (33 ) m πL πœƒ
which makes clear that the functions Am (L,ν ) and Bm (L, ν) represent the complex amplitudes of the incoming and outgoing waves respectively. The wave amplitudes are computed according to (Braun et al., 1992Jump To The Next Citation Point):
∫ ∫ ∫ -Ci-- T 2π πœƒmax (2) −i(m ψ+2πνjt) Am (Li,νj) = 2πT 0 dt 0 dψ πœƒmin πœƒdπœƒ Φ (πœƒ,ψ, t)H m (Liπœƒ)e , (34 ) C ∫ T ∫ 2π ∫ πœƒmax Bm (Li,νj) = --i-- dt dψ πœƒdπœƒ Φ (πœƒ,ψ, t)H (1m)(Liπœƒ)e−i(m ψ+2πνjt). (35 ) 2πT 0 0 πœƒmin
In these expressions, T is the duration of the observation and Ci ≃ πLiβˆ•(2Θ ) is a normalisation constant, where Θ = πœƒmax − πœƒmin. The procedure to perform the numerical transforms is discussed by Braun et al. (1988Jump To The Next Citation Point). The time and azimuthal transforms use the standard fast Fourier transform algorithm, with azimuthal orders in the range |m | < 20. In particular, the frequency grid is given by νj = jΔ ν, where Δν = 1βˆ•T and j is an integer value. The spatial Hankel transform is computed for a set of discrete values Li, that provide an orthogonal set of Hankel functions:
∫ πœƒmax (1) (2) H m (Li πœƒ)H m (Lkπœƒ) πœƒdπœƒ = 0 for i ⁄= k. (36 ) πœƒmin
This condition is approximately satisfied for Li = iΔL, where ΔL = 2πβˆ•Θ and i is an integer (Braun et al., 1988Jump To The Next Citation Point). The maximum degree is given by the Nyquist value Lmax = πβˆ•Δ πœƒ, where Δ πœƒ is the spatial sampling, while the minimum degree below which the orthogonality condition ceases to be valid is Lmin = m βˆ•πœƒmin. We note that the outer radius of the annulus, πœƒmax, must not be too large since waves should remain coherent over the travel distance 2R βŠ™πœƒmax. This implies that πœƒmax < uτβˆ•(2R βŠ™ ), where u (L, ν) and τ (L, ν) 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 T (waves must be observed first as incoming waves and later as outgoing waves). This implies πœƒmax < uT βˆ•(2R βŠ™).

4.1.2 Absorption coefficient

The original motivation for the Hankel analysis was to search for wave absorption by sunspots. Braun et al. (1987Jump To The Next Citation Point) defined the absorption coefficient by

Pout −-Pin |Bm-(L,-ν)|2 αm (L,ν) = P = 1 − |A (L, ν)|2 . (37 ) out m
Braun (1995Jump To The Next Citation Point) remarks that this definition of the absorption coefficient may not necessarily represent wave dissipation as it ignores mode mixing. In practice, some averaging must be done to reduce the noise level. For example, Braun et al. (1987Jump To The Next Citation Point) averaged |Am |2 and |Bm |2 over all azimuthal orders and over the frequency range 1.5 mHz < ν < 5 mHz. Braun (1995Jump To The Next Citation Point) obtained a reasonable signal-to-noise level only by averaging in frequency space over the width of a ridge (radial order n) at fixed L:
⟨|Bm (L, ν)|2 − λm (L, ν)⟩n αm (L,n ) = 1 − -----------2-------------, (38 ) ⟨|Am (L, ν)| − λm (L, ν)⟩n
where the angle brackets denote a frequency average over the width of the n− th ridge around the frequency of the mode (L,n). The function λm (L,ν ) is introduced to remove the background power (see Figure 8View Image). Braun (1995Jump To The Next Citation Point) also made averages over all azimuthal orders, to obtain an absorption coefficient for each ridge and each wavenumber denoted by α(L, n). Hankel analysis revealed that sunspots are strong absorbers of incoming p and f modes. As a check, the same analysis applied to quiet-Sun data does not show significant absorption.
View Image

Figure 8: The m-averaged net power (incoming plus outgoing) as a function of frequency for two representative values of degree l. These power spectra were obtained from the analysis of 1988 quiet-Sun south pole data. The p-mode ridges are labeled by the value of the radial order n. The dashed lines show the background power λ(L, ν). From Braun (1995Jump To The Next Citation Point).

4.1.3 Phase shifts

Braun et al. (1992Jump To The Next Citation Point) and Braun (1995Jump To The Next Citation Point) were successful at measuring the relative phase shift between outgoing and ingoing waves. Phase shifts measurements require a lot of care. Braun et al. (1992Jump To The Next Citation Point) pointed out that the phases of Am and Bm represent averages over a resolution element with size (ΔL, Δ ν). For example, Braun (1995Jump To The Next Citation Point) has ΔL = 20 and Δ ν = 4 μHz.

The finite resolution in wavenumber introduces spurious phases. This can be seen by considering first the A-transform (Equation (34View Equation)) of an incoming wave with amplitude |A|, wavenumber L, azimuthal order m, frequency ν, and phase Ο•in. Using the far-field approximation of the Hankel functions, we have

∫ ∫ A (L ,ν ) ≃ |A-|Ci- T ei2π(ν−νj)t+iΟ•indt πœƒmin ei(L−Li)πœƒdπœƒ (39 ) m i j T Θ 0 πœƒmin = |A| sinc [π(ν − ν )T] sinc [(L − L )Θ βˆ•2] j i × exp [iΟ•in + iπ(ν − νj) − i(Li − L )(πœƒmin + πœƒmax)βˆ•2 ], (40 )
where sinc x = sinx βˆ•x. The B-transform (Equation (35View Equation)) applied to an outgoing wave with amplitude |B |, wavenumber L, azimuthal order m, frequency ν, and phase Ο•out, gives
B (L ,ν ) ≃ |B| sinc [π (ν − ν )T ] sinc[(L − L )Θ βˆ•2] m i j j i × exp[iΟ•out + iπ(ν − νj) + i(Li − L )(πœƒmin + πœƒmax)βˆ•2]. (41 )
Thus the phase difference between the outgoing and incoming waves measured at grid point (Li,νj) is given by
meas Δ Ο• (Li,νj) ≃ Δ Ο•(L,ν ) + (Li − L )(πœƒmin + πœƒmax), (42 )
where Δ Ο• = Ο•out − Ο•in is the correct phase difference at wavenumber L, and (Li − L)(πœƒmax + πœƒmin) is a spurious phase shift. (The notations here are not the same as those of Braun (1995Jump To The Next Citation Point).)
View Image

Figure 9: The total m-summed power as a function of frequency across the p-mode ridge corresponding to n = 5 and l = 205 (top panel) and the raw phase measurements across the same range in frequency (bottom panel). The solid lines indicate the predicted value of the spurious phase shifts. 1988 south pole quiet-Sun data. From Braun (1995Jump To The Next Citation Point).

Let us now consider actual observations. The p modes are distributed along ridges corresponding to different radial orders n, whose frequency positions in the L-ν diagram are specified by known functions νn(L). There may be several unresolved modes in the resolution element (ΔL, Δ ν). The condition πœƒ < uT βˆ•(2R ) max βŠ™ (Section 4.1.1) is equivalent to ΔL > 2Δ νβˆ• ν′(L) n, where the prime denotes a derivative. This means that the spread in L of the ridge across the frequency step Δν is much smaller than the wavenumber resolution ΔL. In this case Braun et al. (1992Jump To The Next Citation Point) estimate the spurious phase shift from Equation (42View Equation) by letting L equal to the mean wavenumber L¯ of the unresolved modes. Since νj − νn(Li) ≃ ν′n(Li)(¯L − Li ), the frequency dependence of the spurious shifts is given by (Braun et al., 1992Jump To The Next Citation Point)

spur πœƒmax + πœƒmin Δ Ο• (Li,νj) ≃ − ---ν′(L-)--[νj − νn(Li)]. (43 ) n i
Figure 9View Image shows phase shifts measurements for quiet-Sun data (Braun, 1995Jump To The Next Citation Point). The predicted values of the spurious shifts, given by the above equation, agree with the observations. The corrected average phase shift is obtained from the phase of B A ∗ exp(− iΔ Ο•spur) m m averaged over m and some frequency range. As expected for the quiet Sun, the average phase shift is zero once the spurious shift has been subtracted. This control experiment shows, in particular, that the values of νn (Li) and ν′n(Li) interpolated from modern p-mode frequency tables are precise enough for the analysis.

4.1.4 Mode mixing

Using a similar technique, Braun (1995Jump To The Next Citation Point) measured correlations between incoming and outgoing waves with different radial orders n and n′. He found that sunspots introduce non-zero m-averaged phase shifts for n = n′ ± 1. This observation gives hope for a full characterization of the sunspot-wave interaction, encapsulated in the scattering matrix Sij defined by

∑ Bi = SijAj, (44 ) j
where the indices i and j refer to individual modes (l,n,m ), the A j are the complex amplitudes of the incoming waves, and the Bi are the complex amplitudes of the outgoing waves. Mode mixing introduced by a scatterer shows as off-diagonal elements in the scattering matrix. All techniques of local helioseismology are based on some knowledge of the scattering matrix, implicitly or explicitly.
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