We propose a radial basis function collocation method (RBF method) to solve fractional Laplacian visco-acoustic wave equation for the Earth media having heterogeneous velocity model and complex geometry. Unlike the fractional Laplacian wave equation proposed in Zhu and Harris (2014), the wave equation we consider has a different definition for the fractional Laplacian. Specifically, spectral and Riesz fractional Laplacians are considered in Zhu and Harris (2014) and the present paper, respectively. Accordingly, the Fourier pseudospectral method (FPS method) and the RBF method are employed to solve the spectral and the Riesz fractional Laplacian wave equations. The two wave equations are observed to produce obviously different wavefields. We demonstrate the validity and flexibility of the proposed RBF method by considering five benchmarks of seismic forward modeling: (1) two-dimensional Earth media with four types of velocity models (homogeneous, two-layer, homogeneous but complex-geometry, and heterogeneous models) and (2) a spherical medium with homogeneous velocity model. We make a three-way comparison among numerical solutions to the Riesz fractional Laplacian, the spectral fractional Laplacian, and the integer-order visco-acoustic wave equations, and observe that when wave attenuation is weak the Riesz wave equation yields more similar wavefield to that of the integer-order wave equation than the spectral wave equation does. Furthermore, uniform and quasi-uniform layouts for collocation points of the RBF method are considered, and the latter layout turns out to be economical since it can preserve the solution accuracy with the minimum number of collocation points. The RBF method is truly mesh-free and dimension-free and can easily handle high-dimensional, irregular domains. Additionally, the method is easier to implement than element-based methods, such as finite element and spectral element methods, for discretizing the Riesz fractional Laplacian.

我们提出了一种径向基函数配置方法（RBF方法）来解决具有非均质速度模型和复杂几何形状的地球介质的分数拉普拉斯粘声波方程。与Zhu和Harris（2014）提出的分数拉普拉斯波动方程不同，我们认为波动方程对分数拉普拉斯方程有不同的定义。具体而言，分别在Zhu和Harris（2014）和本论文中考虑了谱分数和Riesz分数拉普拉斯算子。因此，采用傅里叶伪谱方法（FPS方法）和RBF方法来求解光谱和Riesz分数拉普拉斯波方程。观察到两个波动方程产生明显不同的波场。通过考虑地震正演模拟的五个基准，我们证明了所提出的RBF方法的有效性和灵活性：（1）具有四种速度模型（均质，两层，均质但复杂的几何模型和非均质模型）的二维地球介质）和（2）具有均匀速度模型的球形介质。我们对Riesz分数拉普拉斯算子，频谱分数Laplacian算子和整数阶粘声波方程的数值解进行了三项比较，并观察到，当波衰减较弱时，Riesz波方程产生的波场与该方程更相似。比频谱波方程的整数阶波动方程 此外，考虑了RBF方法的搭配点的统一和准统一布局，后者布局很经济，因为它可以用最少的并置点数来保持求解精度。RBF方法是真正无网格且无尺寸的，并且可以轻松处理高维，不规则区域。此外，该方法比离散化Riesz分数拉普拉斯算子的基于元素的方法（如有限元和光谱元素方法）更易于实现。