基于全局笛卡尔坐标系的高斯束地震波场模拟

对于射线中心坐标系下的高斯束，求解地下每一个点的波场值都要进行坐标变换。为此，Leung等采用拟设理论推导了在全局笛卡尔坐标系下求解高斯束的数学理论方法，此拟设可理解为沿中心射线对旅行时求导，从而增加一个虚部项作为中心射线在其邻域Taylor展开的二阶项，以此构建一个近似旅行时函数，传统高斯束的近似旅行时函数可由其近似旅行时函数通过局部射线中心坐标变换得到。首次将全局笛卡尔坐标系下求解高斯束的数学理论应用于地震波场模拟，分别对一条高斯束、格林函数、连续介质的单频波场以及复杂介质的波场进行模拟，并对比射线中心坐标系、全局笛卡尔坐标系的高斯束精度。结果表明：在均匀介质中，由全局笛卡尔坐标系高斯束、射线中心坐标系高斯束积分所计算的格林函数均很好地近似了解析格林函数；对连续介质模型而言，全局笛卡尔坐标系高斯束能很好地处理焦散问题，较好地模拟了复杂介质的波场传播过程，但高斯束的精度会降低，横向能量较弱。

Seismic modeling by Gaussian beams in the global Cartesian coordinate system

For the Gaussian beam in the ray-centered coordinate system, the coordinate transformation is needed for solving wave-field values at every point in the underground.To this end, Leung et al proposed a mathematical theory method to solve Gaussian beam in a global Cartesian coordinate system using the ansatz theory, which can be understood as deriving the traveling time along the central ray and adding an imaginary part as two-order Taylor expansion of central ray in its neighborhood, to construct an approximate travel time function.The approximate travel time function of conventional Gaussian beam can be obtained from the new approximate travel time function with a local ray-centered coordinate transformation.For the first time, the mathematical theory of Gaussian beam in the global Cartesian coordinate system is applied to seismic wave propagation.One Gaussian beam, Green's function, the single-frequency wave-field of continuous medium and wave-field of complex medium are simulated respectively.And the accuracy of Gaussian beam in the ray-center coordinate system and the global Cartesian coordinate system are compared.Results show that Green's function calculated by Gaussian beams in both of the coordinate systems approximates the analytic Green's function in a homogeneous medium.For continuous medium, Gaussian beam in the global Cartesian coordinate system can handle caustics problem and simulate wavefield propagation of complex media, but the accuracy will be reduced and the lateral energy is weak.