  Sunspots are prominent manifestations of solar magnetoconvection and imaging
their subsurface structure is an outstanding problem of wide physical
importance. Travel times of seismic waves that propagate through these
structures are typically used as inputs to inversions. Despite the presence of
strongly anisotropic magnetic waveguides, these measurements have always been
interpreted in terms of changes to isotropic wavespeeds and flow-advection
related Doppler shifts. Here, we employ PDE-constrained optimization to
determine the appropriate parameterization of the structural properties of the
magnetic interior. Seven different wavespeeds fully characterize helioseismic
wave propagation: the isotropic sound speed, a Doppler-shifting flow-advection
velocity and an anisotropic magnetic velocity. The structure of magnetic media
is sensed by magnetoacoustic slow and fast modes and Alfv\'{e}n waves, each of
which propagates at a different wavespeed. We show that even in the case of
weak magnetic fields, significant errors may be incurred if these anisotropies
are not accounted for in inversions. Translation invariance is demonstrably
lost. These developments render plausible the accurate seismic imaging of
magnetoconvection in the Sun.
