The finite-difference scheme is popular in the field of seismic exploration for numerical simulation of wave propagation; however, its accuracy and computational efficiency are restricted by the numerical dispersion caused by numerical discretization of spatial partial derivatives using coarse grid. The constant-coefficient optimization method is widely used for suppressing the numerical dispersion by tuning the finite-difference weights. While gaining a wider effective bandwidth under a given error tolerance, this method undoubtedly encounters larger errors at low wavenumbers and accumulates significant errors. We proposed an approach to reduce the error accumulation. First, we construct an objective function based on the L1 norm, which can better constrain the total error than the L2 and L∞ norms. Second, we translated our objective function into a constrained L1 norm minimization model, which can be solved by the alternating direction method of multipliers. Finally, we perform theoretical analyses and numerical experiments to illustrate the accuracy improvement. The proposed method is shown to be superior to the existing constant-coefficient optimization methods at the low-wavenumber region; thus, we can obtain higher accuracy with less error accumulation, particularly at longer simulation times. The widely used objective functions, defined by the L2 and L∞ norms could handle a relatively wider range of accurate wavenumbers, compared with our objective function defined by the L1 norm, but their actual errors would be much larger than the given error tolerance at some azimuths rather than axis directions (e.g., about twice at 45°), which greatly degrade the overall numerical accuracy. In contrast, our scheme can obtain a relatively even 2D error distribution at various azimuths, with an apparently smaller error. The peak error of the proposed method is only 40~65% that of the L2 norm under the same error tolerance, or only 60~80% that of the L2 norm under the same effective bandwidth.

NanoSIMS images are usually affected by random noises because of various types of sources, which degrade the quality of ion images and increase the uncertainty of the geochemical interpretations. Here, we applied the weighted nuclear norm minimization (WNNM) method to reduce the random noise in the NanoSIMS image. The low‐rank property of the image is fully considered to suppress random noise while retaining reliable details of weak signals. Numerical experiments on four different kinds of NanoSIMS ion images show that the denoising ability of the WNNM method is superior to that of the median filter, no matter the size of the filtering windows used (eg, 3 × 3, 5 × 5, and 7 × 7). The WNNM method can reduce random noise while preserving the most critical details in the original NanoSIMS observations, which can significantly enhance reliability when distinguishing critical boundaries and structures.

The finite-difference (FD) method is widely used in numerical simulation; however, its accuracy suffers from numerical spatial dispersion and numerical anisotropy. The single-direction optimization methods, which optimize the FD coefficients along a single spatial direction, can suppress numerical spatial dispersion but are suboptimal for mitigating numerical anisotropy on rectangular grids. We propose a multi-direction optimization method that penalizes approximation errors among all propagation angles on rectangular grids with the minimum norm (i.e., L1 norm) to mitigate numerical spatial dispersion and numerical anisotropy simultaneously. Given maximum absolute error tolerance and grid-spacing ratio, we first determine the optimal order of the FD operator in each spatial direction. Then, we penalize approximation errors within the wavenumber-azimuth domain to obtain the optimized FD coefficients. Theoretical analysis and numerical experiments show that the proposed method is superior to single-direction optimization methods in suppressing numerical spatial dispersion and mitigating numerical anisotropy for square and rectangular grids. For homogenous models with grid-spacing ratios of 1.0 (i.e., square grids), 1.2, and 1.4, the root-mean-square (RMS) errors obtained by the proposed method are 77%, 80%, and 72% that of the single-direction optimization method adopting the L1 norm, respectively. For the Marmousi model with a grid-spacing ratio of 1.4, the RMS error of the proposed method is 36% that of the single-direction optimization method based on the L1 norm. Such an evident improvement on error suppression is critical for numerical simulations adopting more flexible grid-spacing ratios.