ABSTRACT The complex‐valued first‐arrival traveltime can be used to describe the properties of both velocity and attenuation as seismic waves propagate in attenuative elastic media. The real part of the complex‐valued traveltime corresponds to phase arrival and the imaginary part is associated with the amplitude decay due to energy absorption. The eikonal equation for attenuative vertical transversely isotropic media discretized with rectangular grids has been proven effective and precise to calculate the complex‐valued traveltime, but less accurate and efficient for irregular models. By using the perturbation method, the complex‐valued eikonal equation can be decomposed into two real‐valued equations, namely the zeroth‐ and first‐order traveltime governing equations. Here, we first present the topography‐dependent zeroth‐ and first‐order governing equations for attenuative VTI media, which are obtained by using the coordinate transformation from the Cartesian coordinates to the curvilinear coordinates. Then, we apply the Lax–Friedrichs sweeping method for solving the topography‐dependent traveltime governing equations in order to approximate the viscosity solutions, namely the real and imaginary parts of the complex‐valued traveltime. Several numerical tests demonstrate that the proposed scheme is efficient and accurate in calculating the complex‐valued P‐wave first‐arrival traveltime in attenuative VTI media with an irregular surface.

中文翻译：

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计算具有不规则表面的衰减垂直横向各向同性介质中的复值 P 波初至旅行时间

摘要 复值初至走时可用于描述地震波在衰减弹性介质中传播时的速度和衰减特性。复值走时的实部对应于相位到达，虚部与由于能量吸收引起的振幅衰减相关。用矩形网格离散化的衰减垂直横向各向同性介质的特征方程已被证明可以有效和精确地计算复值走时，但对于不规则模型的准确度和效率较低。利用微扰法，可以将复值特征方程分解为两个实值方程，即零阶和一阶走时控制方程。这里，我们首先提出了衰减 VTI 介质的与地形相关的零阶和一阶控制方程，这些方程是通过使用从笛卡尔坐标到曲线坐标的坐标变换获得的。然后，我们应用 Lax-Friedrichs 扫描方法来求解依赖于地形的旅行时控制方程，以逼近粘度解，即复值旅行时的实部和虚部。一些数值试验表明，所提出的方案在计算具有不规则表面的衰减 VTI 介质中的复值 P 波初至旅行时方面是有效和准确的。我们应用 Lax-Friedrichs 扫描方法来求解依赖于地形的旅行时控制方程，以逼近粘度解，即复值旅行时的实部和虚部。一些数值试验表明，所提出的方案在计算具有不规则表面的衰减 VTI 介质中的复值 P 波初至旅行时方面是有效和准确的。我们应用 Lax-Friedrichs 扫描方法来求解依赖于地形的旅行时控制方程，以逼近粘度解，即复值旅行时的实部和虚部。一些数值试验表明，所提出的方案在计算具有不规则表面的衰减 VTI 介质中的复值 P 波初至旅行时方面是有效和准确的。