Geodesic
Calculation formulas of linear geometrical spreading at ray tracing in a~3D block-inhomogeneous gradient medium
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 3, pp. 325-339

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In this paper, recurrent formulas, suitable for direct programming, to calculate a linear geometrical spreading of the central field of seismic rays in a 3D block-gradient medium needed for organizing shooting to area observation systems have been obtained. For recalculation formulas through the interface, a new representation using a special operator of nonorthogonal projection allowing an additive separation of terms depending only on the ray curvature, the boundary curvature, and the variable character of the velocities ratio along the boundary, has been found. Formulas for partial derivatives of the eikonal via linear and angular geometrical spreading are presented. 
@article{SJVM_2009_12_3_a7,
     author = {V. A. Tsetsocho and A. V. Belonosova and A. S. Belonosov},
     title = {Calculation formulas of linear geometrical spreading at ray tracing in {a~3D} block-inhomogeneous gradient medium},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {325--339},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2009_12_3_a7/}
}
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V. A. Tsetsocho; A. V. Belonosova; A. S. Belonosov. Calculation formulas of linear geometrical spreading at ray tracing in a~3D block-inhomogeneous gradient medium. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 3, pp. 325-339. http://geodesic.mathdoc.fr/item/SJVM_2009_12_3_a7/