Multidimensional Fourier transform on an irregular grid is a useful tool for various seismic forward problems caused by complex media and wavefield distributions. Using a shape-function-based strategy, we have developed four different algorithms for 1D and 2D nonuniform Fourier transforms, such as two high-accuracy Fourier transforms (the linear shape-function-based Fourier transform [LSF-FT] and the quadratic shape-function-based Fourier transform [QSF-FT]) and two nonuniform fast Fourier transforms (NUFFTs) (the linear shape-function-based NUFFT [LSF-NUFFT] and the quadratic shape-function-based NUFFT [QSF-NUFFT]), respectively, based on their linear and quadratic shape functions. The main advantage of incorporating shape functions into the Fourier transform is that triangular elements can be used to mesh any complex wavefield distribution in the 2D case. Therefore, these algorithms can be used in conjunction with any irregular sampling strategies. The accuracy and efficiency of the four nonuniform Fourier transforms are investigated and compared by applying them in frequency-domain seismic wave modeling. All of the algorithms are compared with the exact solutions. Numerical tests indicate that the quadratic shape-function-based algorithms are more accurate than those based on the linear shape function. Moreover, LSF-FT/QSF-FT exhibits higher accuracy but much slower calculation speed, whereas LSF-NUFFT/QSF-NUFFT is highly efficient but has lower accuracy at near-source points. In contrast, a combination of these algorithms, by using QSF-FT at near-source points and LSF-NUFFT/QSF-NUFFT at others, achieves satisfactory efficiency and high accuracy at all points. Although our tests are restricted to seismic models, these improved NUFFT algorithms may also have potential applications in other geophysical problems, such as forward modeling in complex gravity and magnetic models.

中文翻译：

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用于不规则网格地震建模的基于形状函数的非均匀傅立叶变换

不规则网格上的多维傅里叶变换是解决由复杂介质和波场分布引起的各种地震前向问题的有用工具。使用基于形状函数的策略，我们为一维和二维非均匀傅里叶变换开发了四种不同的算法，例如两种高精度傅里叶变换（基于线性形状函数的傅里叶变换 [LSF-FT] 和二次形状-基于函数的傅立叶变换 [QSF-FT]) 和两个非均匀快速傅立叶变换 (NUFFT)（基于线性形状函数的 NUFFT [LSF-NUFFT] 和基于二次形状函数的 NUFFT [QSF-NUFFT]） ，分别基于它们的线性和二次形状函数。将形状函数合并到傅立叶变换中的主要优点是三角形单元可用于在二维情况下对任何复杂的波场分布进行网格划分。因此，这些算法可以与任何不规则采样策略结合使用。通过将四种非均匀傅立叶变换应用于频域地震波建模，研究并比较了它们的精度和效率。所有算法都与精确解进行了比较。数值试验表明，基于二次形函数的算法比基于线性形函数的算法更准确。此外，LSF-FT/QSF-FT 具有更高的精度，但计算速度要慢得多，而 LSF-NUFFT/QSF-NUFFT 效率高，但在近源点精度较低。相比之下，这些算法的组合，通过在近源点使用QSF-FT，在其他点使用LSF-NUFFT/QSF-NUFFT，在所有点上都实现了令人满意的效率和高精度。尽管我们的测试仅限于地震模型，但这些改进的 NUFFT 算法也可能在其他地球物理问题中具有潜在应用，例如复杂重力和磁模型中的正演建模。