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Local helioseismology has so far relied on semi-analytical methods to computethe spatial sensitivity of wave travel times to perturbations in the solarinterior. These methods are cumbersome and lack flexibility. Here we propose aconvenient framework for numerically solving the forward problem oftime-distance helioseismology in the frequency domain. The fundamental quantityto be computed is the cross-covariance of the seismic wavefield. We choosesources of wave excitation that enable us to relate the cross-covariance of theoscillations to the Green's function in a straightforward manner. We illustratethe method by considering the 3D acoustic wave equation in an axisymmetricreference solar model, ignoring the effects of gravity on the waves. Thesymmetry of the background model around the rotation axis implies that theGreen's function can be written as a sum of longitudinal Fourier modes, leadingto a set of independent 2D problems. We use a high-order finite-element methodto solve the 2D wave equation in frequency space. The computation is`embarrassingly parallel', with each frequency and each azimuthal order solvedindependently on a computer cluster. We compute travel-time sensitivity kernelsin spherical geometry for flows, sound speed, and density perturbations underthe first Born approximation. Convergence tests show that travel times can becomputed with a numerical precision better than one millisecond, as required bythe most precise travel-time measurements. The method presented here iscomputationally efficient and will be used to interpret travel-timemeasurements in order to infer, e.g., the large-scale meridional flow in thesolar convection zone. It allows the implementation of (full-waveform)iterative inversions, whereby the axisymmetric background model is updated ateach iteration.

Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flows