Explicit solution of the inverse kinematic problem in the non-Herglotz case

Abstract

The inverse kinematic problem is solved in the half space R ^ν+1₊ ={(x,z)|z⩾0,x∈R^ν, ν⩾1 under the assumption that the index of refraction can be represented in the form

$$n^2 (x,z) = K^2 (z) + \sum\limits_{j = 1}^\nu {\Phi _j^2 (x_j ),} n_z< 0.$$

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The solution obtained is a generalization of the Herglotz-Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.