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There are several ways to describe the velocity model by a set of parameters.
A layered parameterization described by global basis functions can be used, i.e. the velocity is estimated for regions and layers in the subsurface defined by the user. However, this parameterization constrains the possible range of solutions. Parameterization based on local basis functions is a more flexible description of the velocity model. The model can be described by cells or grid-points at which the values of the parameters (i.e. the values of the velocity field) are defined.
In the algorithm for tomographic inversion of focusing operators, Delaunay triangulations are used to construct cells (triangles) between grid-points (Fig.4a,b). The velocity is defined at the grid-points and the velocity within the triangles is calculated by a linear interpolation between the three grid-points defining each triangle (Fig.4b). In this way, every point can adopt an optimum velocity, and any kind of subsurface can be described. The focus points are defined independently of the velocity points. The focus points are related to positions where reflection energy is available. By parameterizing them independently of the velocity grid-points, the velocity changes are not dependent on, or constrained to, the reflectors.

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Figure 4 Parameterization of the velocity model.
(a) The velocity is defined at grid-points (black dots) in the subsurface.
(b) The grid-points are connected by Delaunay triangulation. The velocity within the triangles is calculated by linear interpolation.

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Stanford Exploration Project

9/18/2001