Inference of subsurface magnetic field of sunspots from absorption coefficients of p-modes in sunspots

The interaction of p-mode waves with an active region can be described by a phenomenological parameter, hereafter called the interaction parameter. The measured absorption coefficients of p-modes relate to the interaction parameter by Chou et al.'s an integral equation. We assume that the sunspot is axisymmetric and the interaction parameter is uniform in the horizontal direction inside the sunspot. The depth dependence of the sunspot radius is given in our study. Thus, the problem becomes one-dimensional: to determine the interaction parameter versus depth. We use the optimally localized averages method to invert the absorption coefficients, measured with the Taiwan Oscillation Network data and South Pole data, to infer the interaction parameter versus depth. For two sunspots we have studied, the depth dependences of the interaction parameter are similar: the interaction parameter increases rapidly, as depth increases, to a maximum value at a depth of about of 7 Mm, and then gradually drops to zero at a depth of about 28-35 Mm. The distribution of the interaction parameter also depends on the depth variation of sunspot radius in such a way that the interaction parameter anticorrelates with the sunspot radius.