Green's function traveltime interpolation

David E. Lumley

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ABSTRACT

Green's function traveltime fields are useful for Kirchhoff depth migration and inversion applications, and for tomographic velocity inversion. However, in 2-D, and especially 3-D, traveltimes become increasingly computationally expensive. Based on the physical continuity of (first arrival) traveltimes, I derive a least squares expression from the ray theoretic eikonal equation, which allows for the interpolation of known Green's function traveltime functions from several surface source positions, to one or more Green's function traveltime functions at intermediate arbitrary source positions. The least squares equation is parameterized in terms of the traveltime gradient, and is solved by standard weighted least-squares techniques and Singular Value Decomposition, and then integrated up to traveltime directly. For a 2-D (3-D) geometry, at least two (three) traveltime fields must be known in order to interpolate the unknown source traveltime field. I test the theory and algorithm on a constant velocity model, constant velocity gradient model, and the complex 2-D structural velocity model of Marmousi. The results are very encouraging to date, although some sparse numerical instability in the calculation of the traveltime gradient fields must be overcome in order to be of practical use.