4.1 Fourier-Hankel spectral method

4.1.1 Wavefield decomposition

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 $h$ , azimuth $y$ ) with a sunspot situated on the polar axis $h = 0$ , and an annular region on the solar surface surrounding the sunspot ( $h < h < h min max$ ). The goal is to decompose the oscillation signal in the annular region, $P(h, y, t)$ , into components of the form

where $m$ is the azimuthal order, $L = [l(l + 1)]1/2$ , $l$ is the spherical harmonic degree, $n$ is the temporal frequency, and $H(m1,2)$ are Hankel functions of the first and second kind. Hankel functions are used as numerical approximations,

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, 1995

). For reference, we recall that in the far-field approximation ( $h» 1/L$ ),

which makes clear that the functions $Am(L, n)$ and $Bm(L, n)$ represent the complex amplitudes of the incoming and outgoing waves respectively. The wave amplitudes are computed according to (Braun et al., 1992

integral integral integral -Ci-- T 2p hmax (2) -i(my+2pnjt) Am(Li, nj) = 2pT 0 dt 0 dy hmin hdh P(h, y,t)H m (Lih)e , (34) integral T integral 2p integral hmax Bm(Li, nj) = -Ci-- dt dy hdh P(h, y,t)H(1m)(Lih)e-i(my+2pnjt). (35) 2pT 0 0 hmin

In these expressions, $T$ is the duration of the observation and $Ci -~ pLi/(2Q)$ is a normalisation constant, where $Q = hmax - hmin$ . The procedure to perform the numerical transforms is discussed by Braun et al. (1988

). 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 $nj = jDn$ , where $Dn = 1/T$ 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:

This condition is approximately satisfied for $Li = iDL$ , where $DL = 2p/Q$ and $i$ is an integer (Braun et al., 1988

). The maximum degree is given by the Nyquist value $Lmax = p/Dh$ , where $Dh$ is the spatial sampling, while the minimum degree below which the orthogonality condition ceases to be valid is $Lmin = m/hmin$ . We note that the outer radius of the annulus, $hmax$ , must not be too large since waves should remain coherent over the travel distance $2Ro . hmax$ . This implies that $hmax < ut /(2Ro . )$ , where $u(L, n)$ and $t (L, n)$ 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 $hmax < uT/(2Ro . )$ .

4.1.2 Absorption coefficient

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

Braun (1995

) 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. (1987

) averaged $|Am |2$ and $|Bm |2$ over all azimuthal orders and over the frequency range $1.5 mHz < n < 5 mHz$ . Braun (1995

) obtained a reasonable signal-to-noise level only by averaging in frequency space over the width of a ridge (radial order $n$ ) at fixed $L$ :

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 $cm(L, n)$ is introduced to remove the background power (see Figure 8

). Braun (1995

) also made averages over all azimuthal orders, to obtain an absorption coefficient for each ridge and each wavenumber denoted by $a(L, n)$ . 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.

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 $c(L, n)$ . From Braun (1995

4.1.3 Phase shifts

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 $Am$ and $Bm$ represent averages over a resolution element with size $(DL, Dn)$ . For example, Braun (1995 ) has $DL = 20$ and $Dn = 4$ $m$ Hz.

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

integral T integral hmin Am(Li, nj) - ~ |A|Ci- ei2p(n-nj)t+ifindt ei(L-Li)hdh (39) TQ 0 hmin = |A |sinc [p(n - nj)T ] sinc [(L - Li)Q/2] × exp[ifin + ip(n - nj) - i(Li- L)(hmin + hmax)/2], (40)

where $sincx = sin x/x$ . The $B$ -transform (Equation (35

)) applied to an outgoing wave with amplitude $|B |$ , wavenumber $L$ , azimuthal order $m$ , frequency $n$ , and phase $f out$ , gives

Thus the phase difference between the outgoing and incoming waves measured at grid point $(Li,nj)$ is given by

where $Df = fout- fin$ is the correct phase difference at wavenumber $L$ , and $(Li- L)(hmax + hmin)$ is a spurious phase shift. (The notations here are not the same as those of Braun (1995

).)

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 (1995

Let us now consider actual observations. The p modes are distributed along ridges corresponding to different radial orders $n$ , whose frequency positions in the $L$ - $n$ diagram are specified by known functions $nn(L)$ . There may be several unresolved modes in the resolution element $(DL, Dn)$ . The condition $hmax < uT /(2Ro . )$ (Section 4.1.1) is equivalent to $DL > 2Dn/n'n(L)$ , where the prime denotes a derivative. This means that the spread in $L$ of the ridge across the frequency step $Dn$ is much smaller than the wavenumber resolution $DL$ . In this case Braun et al. (1992

) estimate the spurious phase shift from Equation (42

) by letting $L$ equal to the mean wavenumber $L$ of the unresolved modes. Since $nj - nn(Li) -~ n'n(Li)(L - Li)$ , the frequency dependence of the spurious shifts is given by (Braun et al., 1992

)

Figure 9

shows phase shifts measurements for quiet-Sun data (Braun, 1995

). 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 $BmA*m exp(- iDfspur)$ 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 $nn(Li)$ and $n'(L ) n i$ interpolated from modern p-mode frequency tables are precise enough for the analysis.

4.1.4 Mode mixing

Using a similar technique, Braun (1995 ) 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

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.