We derive various expressions for the amplitude of the ray-theory approximation of elastic waves in heterogeneous anisotropic media, and show their mutual relations. The amplitude of a wavefield with general initial conditions can be expressed in terms of two paraxial vectors of geometrical spreading in Cartesian coordinates, and in terms of the 2×2 matrix of geometrical spreading in ray-centred coordinates. The amplitude of the Green tensor can be expressed in six different ways: (a) in terms of the paraxial vectors corresponding to two ray parameters in Cartesian coordinates, (b) in terms of the 2×2 paraxial matrices corresponding to two ray parameters in ray-centred coordinates, (c) in terms of the 3×3 upper right submatrix of the 6×6 propagator matrix of geodesic deviation in Cartesian coordinates, (d) in terms of the 2×2 upper right submatrix of the 4×4 propagator matrix of geodesic deviation in ray-centred coordinates, (e) in terms of the 3×3 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver Cartesian coordinates, and (f) in terms of the 2×2 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver ray-centred coordinates. The step-by-step derivation of various equivalent expressions, both known or novel, elucidates the mutual relations between these expressions.

中文翻译：

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以笛卡尔或射线为中心的坐标计算各向异性介质中弹性波的振幅

我们推导出各种各向异性介质中弹性波的射线理论近似振幅的各种表达式，并显示它们的相互关系。具有一般初始条件的波场的振幅可以用笛卡尔坐标中的两个几何扩展的近轴矢量表示，也可以用以射线为中心的几何中的2×2矩阵表示。Green张量的振幅可以用六种不同的方式表示：（a）对应于笛卡尔坐标中两个射线参数的近轴矢量；（b）对应于x坐标中两个射线参数的2×2傍轴矩阵（c）以笛卡尔坐标系中测地线偏移的6×6传播矩阵的3×3右上子矩阵表示，（d）根据射线中心坐标中测地线偏移的4×4传播矩阵的2×2右上子矩阵，（e）根据射线的混合二阶空间导数的3×3矩阵相对于源和接收方笛卡尔坐标的特征函数，以及（f）相对于源和接收方射线中心坐标的特征函数的混合二阶空间导数的2×2矩阵。各种等效表达式（已知的或新颖的）的逐步推导阐明了这些表达式之间的相互关系。（e）根据特征函数相对于源和接收者笛卡尔坐标的混合二阶空间导数的3×3矩阵，以及（f）根据混合二阶坐标的2×2矩阵，相对于源和接收器的射线中心坐标，特征函数的阶空间导数。各种等效表达式（已知的或新颖的）的逐步推导阐明了这些表达式之间的相互关系。（e）根据特征函数相对于源和接收者笛卡尔坐标的混合二阶空间导数的3×3矩阵，以及（f）根据混合二阶坐标的2×2矩阵，相对于源和接收器的射线中心坐标，特征函数的阶空间导数。各种等效表达式（已知的或新颖的）的逐步推导阐明了这些表达式之间的相互关系。