A comment on the relationship between the modal and time-distance formulations of local helioseismology

The relationship between the time-distance and modal-decomposition approaches to solar active region seismology is clarified through the consideration of the oscillations of a plane-parallel, isentropic polytrope. It is demonstrated by direct construction that a wave packet formed through the superposition of neighboring p-modes interferes constructively along a ray bundle that follows the appropriate WKBJ ray path obtained by using the eikonal approximation. Because the actual power envelope of the solar 5 minute oscillations restricts the excited p-modes to rather low radial orders, the ray bundles are diffuse and sample portions of the solar envelope that are some approximate to 10-30 Mm distant from the nominal WKBJ ray path. This behavior is consistent with the fact that the eikonal approximation becomes valid only in the limiting case of large radial orders (n much greater than 1). The p-mode wave packets that are isolated by employing the time-distance methods must therefore be described either as a superposition of individual p-modes (a wave packet), or as a sum of ray paths (a ray bundle), depending upon which representation proves to be optimal for the given circumstances.