2.3 Spherical harmonics and leakage

Spherical harmonic masks are used to separate the contributions from modes of different degree and azimuthal order into complex time series, which can then be transformed to acoustic Fourier spectra.

The radial component of the velocity at the solar surface from a mode with a given degree l, azimuthal order m, and radial order n is given by

|m| imϕ Vnlm (ϕ,𝜃,t) = Re[anlm(t)Pl (cos𝜃)e )], (4 )
where Re[...] denotes the real part, ϕ is longitude, and 𝜃 is latitude (see, for example, Schou and Brown 1994aJump To The Next Citation Point). The masks used separate the different spherical harmonics take the form
Ml,m ∝ Yl,m(𝜃,ϕ )A(ρ), (5 )
where A is an apodization function and ∘ -------------------- ρ ≡ cos2𝜃 + sin2𝜃 sin2 ϕ represents the fractional distance from disk center in the solar image. The line-of-sight projection factor is √ -----2 V = 1 − ρ.

Because only part of the solar surface is visible at any time, the masks are not completely orthogonal and the modes “leak” into neighboring spectra. This leakage complicates the analysis and can cause systematic errors in the measured frequencies if it is not correctly taken into account. For a detailed discussion of the calculation of the so-called “leakage matrix,” see Schou and Brown (1994a) and Hill and Howe (1998). Briefly, the leakage matrix element s(l,m, l′,m ′) for leakage from the l′,m ′ mode to the l,m spectrum can be computed as

′ ′ 1 ∫ 1∫ π∕2 m m′ ′ s(l,m, l,m ) = π- Pl (x)Pl′ (x)cos(m ϕ)cos(m ϕ)V (ρ)A(ρ)dxd ϕ. (6 ) −1 − π∕2
Symmetry properties in this expression lead to some simple exclusion rules; for example, leaks with odd |δl + δm | (where δm ≡ m − m ′ and δl ≡ l − l′) are not allowed.

One example of the importance of the leakage is in the contribution of the so-called m-leaks (δl = 0,δm = ±2) to the estimation of low-degree splittings. As pointed out, for example, by Howe and Thompson (1998), these leaks are strongest for small |m |; they are also asymmetrical, especially for |m | = l, where the m = l peak has an m = l − 2 leak on one side and no counterbalancing m = l + 2 leak on the other. Especially for l = 1, this can introduce a serious systematic error into the estimate of the splitting if not properly accounted for.

Leakage also means that integrated-sunlight instruments (which effectively use the l = 0 mask) can detect modes of 0 ≤ l ≤ 5, though the sensitivity falls off rapidly for l > 1. All these modes appear in a single acoustic spectrum; the instruments are sensitive to odd-m modes for odd l and to even-m modes for even l, with the sectoral, or |m | = l, modes most strongly detected.

In general, the leakage has effects throughout the acoustic spectrum, but the most deleterious effects arise when the leaks cannot be resolved from the target peaks. This occurs for m-leaks at frequencies above about 2 mHz; for higher-degree modes the leakage between modes of adjacent l becomes a problem, as the ridges become both broader, and more closely spaced in frequency, at around l = 150. Beyond this point the peaks cannot be fitted independently, and some modeling of the leakage is essential in order to estimate the mode parameters.

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Figure 9: a1 (top) and a3 (bottom) coefficients for (left) 1986 BBSO observations, (middle) 108 days of GONG observations in 1996, (right) the mean of 35 consecutive 108-day periods of GONG observations from 1995 – 2005, plotted as a function of phase speed with the turning point radius marked on the upper axis.

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