Efficient frequency-domain Full Waveform Inversion (FWI) of long-offset/wide-azimuth node data can be designed with a few discrete frequencies. However, 3D frequency-domain seismic modeling remains challenging since it requires solving a large and sparse linear indefinite system per frequency. When such systems are solved with direct methods or hybrid direct/iterative solvers, based upon domain decomposition preconditioner, finite-difference stencils on regular Cartesian grids should be designed to conciliate compactness and accuracy, the former being necessary to mitigate the fill-in induced by the Lower-Upper (LU) factorization. Compactness is classically implemented by combining several second-order accurate stencils covering the eight cells surrounding the collocation point, leading to the so-called 27-point stencil. Accuracy is obtained by applying optimal weights on the different stiffness and consistent mass matrices such that numerical dispersion is jointly minimized for several number of grid points per wavelength ($G$). However, with this approach, the same weights are used at each collocation point, leading to suboptimal accuracy in heterogeneous media. In this study, we propose a straightforward recipe to improve the accuracy of the 27-point stencil. First, we finely tabulate the values of $G$ covering the range of wavelengths spanned by the subsurface model and the frequency. Then, we estimate with a classical dispersion analysis in homogeneous media the corresponding table of optimal weights that minimize dispersion for each $G$ treated separately. We however apply a Tikhonov regularization to guarantee smooth variation of the weights with $G$. Finally, we build the impedance matrix by selecting the optimal weights at each collocation point according to the local wavelength, hence leading to a wavelength-adaptive stencil.

中文翻译：

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使用波长自适应 27 点有限差分模板进行精确的 3D 频域地震波建模：全波形反演工具

长偏移距/宽方位角节点数据的高效频域全波形反演 (FWI) 可以用几个离散频率来设计。然而，3D 频域地震建模仍然具有挑战性，因为它需要解决每个频率的大型稀疏线性不定系统。当使用直接方法或混合直接/迭代求解器求解此类系统时，基于域分解预处理器，应设计常规笛卡尔网格上的有限差分模板以协调紧凑性和准确性，前者对于减轻由下-上 (LU) 分解。紧凑性通常是通过组合几个二阶精确模板来实现的，这些模板覆盖了搭配点周围的八个单元格，从而产生了所谓的 27 点模板。通过在不同的刚度和一致的质量矩阵上应用最佳权重来获得精度，这样对于每个波长的多个网格点 ($G$) 共同最小化数值色散。然而，使用这种方法，在每个搭配点使用相同的权重，导致异构媒体中的准确性不理想。在这项研究中，我们提出了一个简单的方法来提高 27 点模板的准确性。首先，我们精细地列出了 G$ 的值，涵盖了地下模型和频率所跨越的波长范围。然后，我们用均匀介质中的经典色散分析来估计相应的最佳权重表，这些表使每个单独处理的 G$ 的色散最小化。然而，我们应用 Tikhonov 正则化来保证权重随 $G$ 的平滑变化。