A study of sunspots with phase time and travel time of p-mode waves in acoustic imaging

Solar acoustic signals constructed with the technique of acoustic imaging contain both intensity and phase information. The phase of constructed signals can be studied by computing the cross-correlation function between time series constructed with ingoing and outgoing waves. The envelope peak of the cross-correlation function provides information about wave travel time, associated with the group velocity along the wave path. The phase time of the cross-correlation function provides information about phase changes along the wave path, including the phase change at boundaries of the mode cavity and boundaries of flux tubes. We study the active region NOAA 7978 using the data taken with the Michelson Doppler Imager (MDI) instrument. We construct two-dimensional phase-shift maps and envelope-shift maps of the active region at the surface by determining the phase time and envelope time of the cross-correlation function at each point. Both phase time and envelope time in magnetic regions are smaller than those in the quiet Sun. The phase shift and envelope shift at each point are computed by subtracting the average values in the quiet Sun. The envelope shift is greater than the phase shift by about 2.5 minutes inside sunspots. Both the envelope shift and the phase shift in sunspots increase approximately linearly with frequency. The difference between the envelope shift and phase shift is the same at different frequencies. There is evidence for an additional phase shift in sunspots relative to plages. We discuss inferences about the structure of sunspots from these phenomena.