Phase Time and Envelope Time in Time‐Distance Analysis and Acoustic Imaging

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Abstract
Time-distance analysis and acoustic imaging are two related techniques for probing the local properties of the solar interior. In this study, we discuss the relation of phase time and envelope time between the two techniques. The location of the envelope peak of the cross-correlation function in time-distance analysis is identified as the travel time of the wave packet formed by modes with the same horizontal phase velocity. The phase time of the cross-correlation function provides information on the phase change accumulated along the wave path, including the phase change at the boundaries of the mode cavity. The acoustic signals constructed with the technique of acoustic imaging contain both phase and intensity information. The phase of constructed signals can be studied by computing the cross-correlation function between time series constructed with ingoing and outgoing waves. We use a simple theory of wave packets to obtain two predictions about the cross-correlation function of constructed ingoing and outgoing time series. First, if the envelope time measured in time-distance analysis is used to construct signals in acoustic imaging, the envelope time of the cross-correlation is zero. Second, the phase time of the cross-correlation is twice the difference between the phase time and envelope time measured in time-distance analysis. In this study, we use data taken with the Taiwan Oscillation Network (TON) instrument and the Michelson Doppler Imager (MDI) instrument. The analysis is carried out for the quiet Sun. We use the relation of envelope time versus distance measured in time-distance analysis to construct the acoustic signals in acoustic imaging analysis. The phase time of the cross-correlation function of constructed ingoing and outgoing time series is twice the difference between phase time and envelope time in time-distance analysis, as predicted. The envelope peak of the cross-correlation function between constructed ingoing and outgoing time series is located at zero time, as predicted for one-bounce results at 3 mHz for all four data sets and two-bounce results at 3 mHz for two TON data sets, but it is different from zero for other cases. The deviation of the envelope peak from zero has the same sign for all these cases. The cause is not known.