Time-distance helioseismology: The forward problem for random distributed sources

The forward problem of time-distance helioseismology is computing travel-time perturbations that result from perturbations to a solar model. We present a new and physically motivated general framework for calculations of the sensitivity of travel times to small local perturbations to solar properties, taking into account the fact that the sources of solar oscillations are spatially distributed. In addition to perturbations in sound speed and flows, this theory can also be applied to perturbations in the wave excitation and damping mechanisms. Our starting point is a description of the wave field excited by distributed random sources in the upper convection zone. We employ the first Born approximation to model scattering from local inhomogeneities. We use a clear and practical definition of travel-time perturbation, which allows a connection between observations and theory. In this framework, travel-time sensitivity kernels depend explicitly on the details of the measurement procedure. After developing the general theory, we consider the example of the sensitivity of surface gravity wave travel times to local perturbations in the wave excitation and damping rates. We derive explicit expressions for the two corresponding sensitivity kernels. We show that the simple single-source picture, employed in most time-distance analyses, does not reproduce all of the features seen in the distributed-source kernels developed in this paper.