  Improving methods for determining the subsurface structure of sunspots from
their seismic signature requires a better understanding of the interaction of
waves with magnetic field concentrations. We aim to quantify the impact of
changes in the internal structure of sunspots on local helioseismic signals. We
have numerically simulated the propagation of a stochastic wave field through
sunspot models with different properties, accounting for changes in the Wilson
depression between 250 and 550 km and in the photospheric umbral magnetic field
between 1500 and 3500 G. The results show that travel-time shifts at
frequencies above approximately 3.50 mHz (depending on the phase-speed filter)
are insensitive to the magnetic field strength. The travel time of these waves
is determined exclusively by the Wilson depression and sound-speed
perturbation. The travel time of waves with lower frequencies is affected by
the direct effect of the magnetic field, although photospheric field strengths
below 1500 G do not leave a significant trace on the travel-time measurements.
These results could potentially be used to develop simplified travel-time
inversion methods.
