Exact local refinement using Fourier interpolation for nonuniform-grid modeling

Abstract

Numerical solver using a uniform grid is popular due to its simplicity and low computational cost, but would be unfeasible in the presence of tiny structures in large‐scale media. It is necessary to use a nonuniform grid, where upsampling the wavefield from the coarse grid to the fine grid is essential for reducing artifacts. In this paper, we suggest a local refinement scheme using the Fourier interpolation, which is superior to traditional interpolation methods since it is theoretically exact if the input wavefield is band limited. Traditional interpolation methods would fail at high upsampling ratios (say 50); in contrast, our scheme still works well in the same situations, and the upsampling ratio can be any positive integer. A high upsampling ratio allows us to greatly reduce the computational burden and memory demand in the presence of tiny structures and large‐scale models, especially for 3D cases.

Spatial- and temporal-interpolations for efficient hybrid wave numerical simulations

“…In the local simulation, the hybrid inputs are then interpolated/recovered to a suitable time step for a local target structure. However, the traditional spline interpolation method (Monteiller et al, 2021) would fail at high upsampling ratios, as shown by Zhang et al (2017). In this section, we introduce a new scheme for sampling and saving the hybrid inputs in the time domain using the Fourier interpolation method (Schafer and Rabiner, 1973).…”

“…This means that the maximum non-zero frequency component of band-limited wavefield would below the Nyquist limit to avoid the aliased spectral leakage; in other words, the amplitude spectrum must equal to zero around the Nyquist frequency. According to Zhang et al (2017), this requirement can be met by applying a smooth tapering window. Thus, the rough sampling series x[n] will be multiplied by a smooth window function (eg., Hanning window) to ensure that the wavefield is periodic, which will improve the accuracy of the Fourier interpolation even further.…”

“…In this article, we first introduced the Fourier interpolation into the hybrid simulation method, which outperforms the traditional spline interpolation method at high upsampling ratios (Figure 6). The requirement for the Fourier interpolation is that the data must be band-limited below the Nyquist limit (Zhang et al, 2017). This requirement can be readily fulfilled by introducing a smooth tapering window function.…”

“…Monteiller et al (2021) proposed that the cubic B-spline interpolation is efficient for upsampling the hybrid inputs because it only requires two additions and two multiplications per sample point. However, this method would fail to recover the original hybrid inputs at a very high upsampling ratio (Zhang et al, 2017).…”