  One goal of helioseismology is to determine the subsurface structure of
sunspots. In order to do so, it is important to understand first the
near-surface effects of sunspots on solar waves, which are dominant. Here we
construct simplified, cylindrically-symmetric sunspot models, which are
designed to capture the magnetic and thermodynamics effects coming from about
500 km below the quiet-Sun $\tau_{5000}=1$ level to the lower chromosphere. We
use a combination of existing semi-empirical models of sunspot thermodynamic
structure (density, temperature, pressure): the umbral model of Maltby et al.
(1986) and the penumbral model of Ding and Fang (1989). The OPAL equation of
state tables are used to derive the sound speed profile. We smoothly merge the
near-surface properties to the quiet-Sun values about 1mm below the surface.
The umbral and penumbral radii are free parameters. The magnetic field is added
to the thermodynamic structure, without requiring magnetostatic equilibrium.
The vertical component of the magnetic field is assumed to have a Gaussian
horizontal profile, with a maximum surface field strength fixed by surface
observations. The full magnetic field vector is solenoidal and determined by
the on-axis vertical field, which, at the surface, is chosen such that the
field inclination is 45$^\circ$ at the umbral-penumbral boundary. We construct
a particular sunspot model based on SOHO/MDI observations of the sunspot in
active region NOAA 9787. The helioseismic signature of the model sunspot is
studied using numerical simulations of the propagation of f, p$_1$, and p$_2$
wave packets. These simulations are compared against cross-covariances of the
observed wave field. We find that the sunspot model gives a helioseismic
signature that is similar to the observations.
