Hierarchical traveltime tomography

A new traveltime tomographic method was developed with hierarchical shape functions of the finite element method as slowness or velocity interpolation functions. The degree of the approximation of velocity modelling is adjusted by selecting a set of hierarchical shape functions in each element. The ray density parameter of each element controls the selection to make the approximation fine or coarse in the high- or low-ray-density area. The proposed method is applied to both synthetic traveltime data and actual data. The AIC is used to determine the number of model parameters. The result of the synthetic data shows that low-resolution model parameters can be eliminated by the ray density parameter. The result of the actual data shows that the velocity pattern is approximately the same in the fine approximation area and that the velocity fluctuation is suppressed in the coarse approximation area, compared with that obtained from a full set of hierarchical shape functions. The number of model parameters is drastically reduced. The resolution can be estimated by the checkerboard restoration test. The result of the real data set was compared with that of the linear velocity grid model.