The radial component of the velocity at the solar surface from a mode with a given degree , azimuthal order , and radial order is given by
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 for leakage from the mode to the spectrum can be computed as
One example of the importance of the leakage is in the contribution of the so-called -leaks () to the estimation of low-degree splittings. As pointed out, for example, by Howe and Thompson (1998), these leaks are strongest for small ; they are also asymmetrical, especially for , where the peak has an leak on one side and no counterbalancing leak on the other. Especially for , 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 mask) can detect modes of , though the sensitivity falls off rapidly for . All these modes appear in a single acoustic spectrum; the instruments are sensitive to odd- modes for odd and to even- modes for even , with the sectoral, or , 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 -leaks at frequencies above about 2 mHz; for higher-degree modes the leakage between modes of adjacent becomes a problem, as the ridges become both broader, and more closely spaced in frequency, at around . 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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