  Normal mode seismology is a promising means of measuring rotation in gas
giant interiors, and ring seismology presents a singular opportunity to do so
at Saturn. We calculate Saturn's normal modes of oscillation and zonal gravity
field, using nonperturbative methods for normal modes in the rigidly rotating
approximation, and perturbative methods for the shifts that Saturn's deep winds
induce in the mode frequencies and zonal gravity harmonics. The latter are
calculated by solving the thermo-gravitational wind equation in an oblate
geometry. Comparing many such models to gravity data and the frequencies of
ring patterns excited by Saturn normal modes, we use statistical methods to
estimate that Saturn's cloud-level winds extend inward along cylinders before
decaying at a depth 0.125-0.138 times Saturn's equatorial radius, or
7,530-8,320 km, consistent with analyses of Cassini gravity and magnetic field
data. The seismology is especially useful for pinning down Saturn's poorly
constrained deep rotation period, which we estimate at 634.7 min (median) with
a 5/95% quantile range 633.8-635.5 min. Outstanding residuals in mode
frequencies at low angular degree suggest a more complicated deep interior than
has been considered to date. Smaller but still significant residuals at high
angular degree also show that our picture for the thermal, composition, and/or
rotation profile in Saturn's envelope is not yet complete.
