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Extra info for Advances in Wave Propagation in Heterogenous Earth
The history function for controlled initial-value ray tracing usually does not include the KMAH index, because the asymptotic summation of Gaussian beams, the asymptotic summation of Gaussian packets, and usually also the interpolation within ray cells do not require sorting rays according to the KMAH index. In controlled initial-value ray tracing for the interpolation within ray cells, the secondary ray-parameter metric tensor may take into account geometrical spreading along the whole ray. In controlled initial-value ray tracing for the asymptotic summation of Gaussian beams, the secondary ray-parameter metric tensor should control the discretization error, see Eq.
Ray tracing system (28) has found useful applications in heterogeneous anisotropic media. , 1988). Ray tracing system (28) can also be expressed in an alternative form. We define subdeterminants 1 Dij = εikl εj rs ( kr − δkr )( ls − δls ) (29) 2 of the Christoffel matrix. Here εij k is the Levi-Civita symbol (ε123 = ε312 = ε231 = 1, ε321 = ε213 = ε132 = −1, εij k = 0 otherwise). It is not difficult to prove that, for a selected elementary wave for which Gm = 1 and Dss = 0, the factor Dij /Dss can be expressed in terms of the eigenvector of the Christoffel matrix corresponding to the considered wave as follows: Dij (m) (m) = gi gj .
In controlled initial-value ray tracing for the asymptotic summation of Gaussian beams, the secondary ray-parameter metric tensor should control the discretization error, see Eq. (171). The secondary ray-parameter metric tensor then corresponds to the shape of Gaussian beams optimized for the propagation from the source to the reference surface. In controlled initial-value ray tracing for the asymptotic summation of Gaussian packets, the secondary ray-parameter metric tensor should also be used to control the discretization error.