The linear sensitivity of helioseismic ring diagrams to local flows

Ring-diagram analysis is a technique of local helioseismology used to infer plasma flows in the solar convection zone which generates intermediate data products known as ring-fitting parameters. Knowing the sensitivity of ring-fitting parameters to actual flows in the Sun is important for interpreting these measurements. Working in plane-parallel geometry, we compute the linear sensitivity of ring-fitting parameters to small changes in the local power spectrum and then compute the sensitivity of the power spectrum to time-independent weak local flows. We combine these two results to obtain the three-dimensional Frechet kernels that give the linear sensitivity of ring-fitting parameters to both vertical and horizontal local mass flows. We find that ring measurements are essentially only sensitive to flows that are within the spatial region for which the ring diagram is computed. In addition, we find that the depth dependence of the sensitivity is essentially given by the mode kinetic energy density, as has traditionally been assumed. We show that the exact form of the sensitivity of ring measurements depends on the details of the fitting procedure.