An adaptive‐grid formalism for traveltime tomography

Summary The use of an adaptive‐grid formalism for seismic, non‐linear traveltime tomography is proposed. The method is based on a parametric representation of the velocity model and involves the simultaneous inversion for both velocity and position of the grid points of the model discretization mesh. Therefore, the method seeks the optimal grid configuration to define the model. Cubic B‐spline basis functions have been used for model representation as they are particularly versatile in the reproduction of geologically complex structures. The traveltimes are calculated using a new initial‐value ray tracer that calculates the ray trajectories directly in the parametric domain. The method is tested against synthetic data generated for various cross‐hole geometries. It is found that, in parts of the model having good ray coverage, the method accurately retrieves velocity anomalies of arbitrary shape using a generally small number of grid points of the inversion discretization mesh. With the exception of initial meshes that are too coarse to describe accurately the complexity of the true structure, the method retrieves nearly identical final models regardless of the predefined node spacing and node configuration. Therefore, the method can avoid very fine discretization, and matrix sparseness, one of the main sources of indeterminacy, is similarly avoided. When compared with standard methods entailing a similar total number of inversion parameters, our results show the pitfalls that may derive from the a priori assigning of a fixed‐grid mesh. Overall, the parametric representation leads to some saving in the total number of inversion parameters when the structure consists of sparse distributions of irregular features. Other styles of model may be better recovered by regular grids for a given number of inversion parameters.