INVERSION SCHEMES

For a medium that can be described (or approximated) by elliptically anisotropic velocities (with vertical and horizontal symmetry axes), the traveltime between two points separated by a vertical distance z and a horizontal distance x is given by  

$$t^2 = M^x \: x^2 + M^z \: z^2,$$ (1)

where M^x and M^z are the squared horizontal and vertical slownesses. If the medium is heterogeneous, this equation is valid either within each cell or within each layer (if a layered description is appropriate) of the model. The inversion problem can be formulated as the search for the model (M^x_j,M^z_j), for all layers j, that minimizes the objective function  

$${\cal F} = \sum_{i=1}^N \left( t_{i} - t^{\prime}_{i} \right)^2,$$ (2)

where N is the number of source-receiver pairs, t_i is the measured traveltime corresponding to source-receiver index i, and $t^{\prime}_{i}$is the traveltime predicted by the perturbed model.