منابع مشابه
Non-oriented solutions of the eikonal equation
We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R. Instead of a vector field ∇u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with divP ∈ L. We prove existence and uniqueness for solutions of the equation P divP = 0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only ...
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S U M M A R Y In this paper, several methods for the solution of the eikonal equation are investigated. Using the method of characteristics, traveltimes are computed along rays. Approximate solutions of the eikonal equation can also be obtained. From Fermat’s principle, first-order changes in the traveltime can be found by integrating along initial ray trajectories. Alternatively, linearized ra...
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In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geomet...
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The transformation of the eikonal equation from depth coordinates (z, x) into verticaltraveltime coordinates (τ ,ξ ) enables the computation of reflections traveltimes independent of depth-mapping. This separation allows the focusing and mapping steps to be performed sequentially even in the presence of complex velocity functions, that otherwise would “require” depth migration. The traveltimes ...
متن کاملHopf maps as static solutions of the complex eikonal equation
We demonstrate that a class of torus-shaped Hopf maps with arbitrary linking number obeys the static complex eikonal equation. Further , we explore the geometric structure behind these solutions, explaining thereby the reason for their existence. As this equation shows up as an integrability condition in certain non-linear field theories, the existence of such solutions is of some interest.
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ژورنال
عنوان ژورنال: Journal of Differential Equations
سال: 2013
ISSN: 0022-0396
DOI: 10.1016/j.jde.2013.08.002