Travel time sensitivity kernels

We derive, following the standard first Born approximation approach used in the geophysics literature, an expression for the travel time perturbation caused by a perturbation to sound speed. In our simple model we employ a point source at one point and calculate the time taken for a wave packet created at the source to move to a second point. In the first Born approximation the travel time delay caused by a perturbation to the background model can be expressed as the integral over the whole sun of some function, called the travel time sensitivity kernel, multiplied by the perturbation. The sensitivity kernels are zero along the geometrical ray connecting the two points and have maximum weight in a tube around the ray; they are the solar equivalent of 'the banana-doughnut' kernels discussed in the geophysics literature. Calculating sensitivity kernels that are more accurate than those derived from ray theory is important for the future of inversions done with time-distance helioseismology data as they will allow greater confidence in the results as well as increased resolution.