  Seismic traveltime measurements are a crucial tool in the investigation of
the solar interior, particularly in the examination of fine-scale structure.
Traditional analysis of traveltimes relies on a geometrical ray picture of
acoustic wave propagation, which assumes high frequencies. However, it is
well-known that traveltimes obtained from finite-frequency waves are sensitive
to variations of medium parameters in a wide Fresnel zone around the ray path.
To address this problem, Fr\'{e}chet traveltime sensitivity kernels have
previously been developed. These kernels use a more realistic approximation of
the wave propagation to obtain a linear relationship between traveltimes and
variations in medium parameters. Fr\'{e}chet kernels take into account the
actual frequency content of the measured waves and, thus, reproduce the Fresnel
zone. Kernel theory has been well-developed in previous work on plane-parallel
models of the Sun for use in local helioseismology. Our primary purpose is to
apply kernel theory to much larger scales and in a spherical geometry. We also
present kernel theory in a different way, using basic functional analytic
methods, in the hope that this approach provides an even clearer understanding
of the theory, as well as a set of tools for calculating kernels. Our results
are very general and can be used to develop kernels for sensitivity to sound
speed, density, magnetic fields, fluid flows, and any other medium parameter
which can affect wave propagation.
