The seismology of sunspots: A comparison of time-distance and frequency-wavenumber methods

A pair of formulae are developed that relate the absorption coefficient and partial-wave phase shift concepts of frequency-wavenumber local helioseismology to the center-annulus cross-correlation function of time-distance helioseismology, under the general circumstances that both induced and spontaneous sunspot oscillations may be present. These formulae show that spontaneous emission of p-modes by magnetic and Reynolds stresses within the spot and the mode mixing between incoming and outgoing p-modes affect only the outgoing center-annulus cross-correlation time tau(+), and they caution that real or spurious phase lags of the umbral oscillation signal lead to differences in the incoming and outgoing correlation times, resulting in tau(-)not equal tau(+). The application of these methods to actual helioseismic data obtained by the Global Oscillation Network Group (GONG) project is carried out in order to provide a tangible illustration of how time-distance and frequency-wavenumber ideas can profitably be combined to yield deeper insight into the seismic probing of sunspots. By using the helioseismic GONG data in conjunction with concurrent observations of Doppler velocities and vector magnetic fields obtained by the High Altitude Observatory/National Solar Observatory (HAO/NSO) Advanced Stokes Polarimeter (ASP) for the 1995 October disk passage of active region NOAA 7912, we demonstrate that the inferred GONG umbral signal actually originates from the umbra-penumbra boundary about 6 Mm distant from the center of the spot. Further, the ASP observations show that the 5 minute oscillations at the umbra-penumbra boundary lag behind those in the center of the umbra by approximately 1 minute, which is precisely the difference between the incoming and outgoing correlation times for NOAA 7912 recently determined by Braun. This remarkable result underscores the perils of using umbral oscillations in time-distance helioseismology, and it calls into question previous claims that correlation time differences constitute direct evidence for the existence of a steady downflow in and around sunspots. Taken together, the observational and theoretical evidence suggest that the p-mode forcing of the spot leads to the generation of upwardly propagating slow magnetoatmospheric waves. These waves are in turn responsible for the decreased amplitudes of the outwardly propagating p-modes in the surrounding quiet Sun, and the dispersion in their travel times between the hidden subsurface layer where they are forced and the overlying level where the Doppler signals originate leads to the observed phase lag between the umbral and penumbral oscillations and the corresponding correlation time differences.