This paper presents two approaches to mathematical modelling of a synthetic seismic pulse, and a comparison between them. First, a new analytical model is developed in two-dimensional Cartesian coordinates. Combined with an initial condition of sufficient symmetry, this provides a valuable check for the validity of the numerical method that follows. A particular initial condition is found which allows for a new closed-form solution. A numerical scheme is then presented which combines a spectral (Fourier) representation for displacement components and wave-speed parameters, a fourth-order Runge–Kutta integration method, and an absorbing boundary layer. The resulting large system of differential equations is solved in parallel on suitable enhanced performance desktop hardware in a new software implementation. This provides an alternative approach to forward modelling of waves within isotropic media which is efficient, and tailored to rapid and flexible developments in modelling seismic structure, for example, shallow depth environmental applications. Visual comparisons of the analytic solution and the numerical scheme are presented.

中文翻译：

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连续介质中地震波动方程的解析解和数值解

本文提出了两种合成地震脉冲数学建模方法，并比较了它们。首先，在二维笛卡尔坐标中开发了一种新的分析模型。结合充分对称的初始条件，这为后续数值方法的有效性提供了有价值的检查。找到了一个特定的初始条件，它允许一个新的封闭形式的解决方案。然后提出了一个数值方案，它结合了位移分量和波速参数的光谱（傅立叶）表示、四阶龙格-库塔积分方法和吸收边界层。在新的软件实现中，在合适的增强性能的桌面硬件上并行求解所得的大型微分方程系统。这提供了一种在各向同性介质中对波进行正向建模的替代方法，该方法是有效的，并且适用于地震结构建模的快速和灵活发展，例如浅层环境应用。给出了解析解和数值方案的视觉比较。