  Phase perturbations due to inclined surface magnetic field of active region
strength are calculated numerically in quiet Sun and simple sunspot models in
order to estimate and compare the direct and indirect (thermal) effects of the
fields on helioseismic waves. It is found that the largest direct effects occur
in highly inclined field characteristic of penumbrae, and scale roughly
linearly with magnetic field strength. The combined effects of sunspot magnetic
and thermal anomalies typically yield negative travel-time perturbations in
penumbrae. Travel-time shifts in umbrae depend on details of how the thermal
and density structure differs from the quiet Sun. The combined shifts are
generally not well approximated by the sum of the thermal and magnetic effects
applied separately, except at low field strengths of around 1 kG or less, or if
the thermal shift is small. A useful rule-of-thumb appears to be that
travel-time perturbations in umbrae are predominantly thermal, whereas in
penumbrae they are mostly magnetic.
