The inadequate sampling of seismic data in the spatial dimension results in migration artifacts. Conventional least-squares reverse time migration (LSRTM) could improve the image quality. However, even LSRTM will not work in some inadequately sampling situations. To mitigate the impact of migration artifacts, we have developed a new LSRTM method with a sparse regularization, which takes advantage of the effective sparse representation of the subsurface reflectivity model in the 2D undecimated wavelet transform (UWT) domain. Different from other sparse regularizations, a sparseness constraint in the 2D UWT domain is applied on the angle slices of the image. To efficiently solve the least-squares inversion problem, we employ an inversion scheme using the conjugate gradient method that uses a soft threshold method to achieve sparse constraint in updating the conjugate gradient direction. Compared with the sparse constraint based on the discrete wavelet transform (DWT), the threshold in this method is angle-dependent and is determined according to the energy distribution of each angle slice. To avoid overregularization that can lead to instability and increase the number of iterations, we also apply an exponential threshold strategy. Numerical tests on synthetic datasets demonstrate that our method is capable of improving the image quality by enhancing the resolution and suppressing migration artifacts caused by inadequately sampled seismic data. The method can converge more rapidly than conventional LSRTM. Because this method performs sparse regularization on several slopes, it achieves better performance on enhancing complex structures with discontinuities such as the steeply dipping faults compared to DWT-based regularization.

中文翻译：

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二维小波域中具有稀疏正则化的最小二乘逆时偏移

在空间维度上对地震数据的采样不足会导致迁移伪影。传统的最小二乘逆时偏移（LSRTM）可以提高图像质量。但是，即使LSRTM在某些采样不足的情况下也不起作用。为了减轻迁移伪影的影响，我们开发了一种具有稀疏正则化的新LSRTM方法，该方法利用了二维未抽取小波变换（UWT）域中地下反射率模型的有效稀疏表示。与其他稀疏正则化不同，将2D UWT域中的稀疏约束应用于图像的角度切片。为了有效解决最小二乘反演问题，我们采用共轭梯度法的反演方案，该方法使用软阈值方法在更新共轭梯度方向时实现稀疏约束。与基于离散小波变换（DWT）的稀疏约束相比，该方法的阈值取决于角度，并根据每个角度切片的能量分布确定。为了避免过度规范化会导致不稳定并增加迭代次数，我们还应用了指数阈值策略。对合成数据集的数值测试表明，我们的方法能够通过提高分辨率并抑制由于采样数据不足而引起的偏移伪影来改善图像质量。该方法可以比常规LSRTM更快地收敛。