{'fraction_non_alphanumeric': 0.018645121193287758, 'fraction_numerical': 0.002486016159105034, 'mean_word_length': 5.793248945147679, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Fréchet derivatives for curved interfaces in the ray approximation

SUMMARY The sensitivity of ray and beam theoretical seismograms to changes in velocity models and curved interfaces is discussed in this paper. Previous results from Nowack & Lutter (1988) give the derivatives of travel-time and ray amplitude with respect to changes of smoothly varying velocities. These derivatives are required to perform linearized maximum likelihood inversions for structure. In the ray approximation, smooth interfaces are incorporated by applying Snell’s law locally, correcting wavefront curvature, and using local plane-wave reflection and transmission coefficients. The partial derivatives for travel-time are directly calculated along the original ray trajectory using Fermat’s principle. For perturbations of the ray amplitude of a reflected/transmitted ray, the ray shift of the perturbed two-point ray trajectory must be accounted for. The approach followed here is to calculate the approximately perturbed two-point ray using perturbation theory without additional ray tracing. The perturbed ray amplitudes are then computed directly, including modified reflection/transmission coefficients and geometric spreading, along this approximate twopoint ray. Several numerical experiments are conducted which invert for velocity and interface shape using both travel-time and amplitude in order to test the derived partial derivative operators. Travel-time and amplitude inversion results are also contrasted with amplitude being less sensitive to larger scale features and more sensitive to heterogeneity curvature.