Adaptive multiple subtraction (AMS) is a critical and challenging task for multiple attenuation. Recently, pattern-learning-based AMS approaches, including pattern coding and decoding steps based on the ${\ell }_2$ norm, have achieved impressive results. However, the traditional pattern-learning approach might hurt the primaries under the minimum energy condition of the ${\ell }_2$-norm-based decoding. Thus, we adopt a hybrid ${\ell }_2$-${\ell }_1$ pattern-learning-based AMS approach, which includes the ${\ell }_2$-norm-based multiband coding and the ${\ell }_1$-norm-based decoding steps. In the coding stage, the pattern dictionary is obtained by using the ${\ell }_2$-norm-based principal component analysis, which has been proven to be an effective feature extraction method. Subsequently, in the decoding stage, the learned patterns from the predicted multiple are used to estimate the multiples in the recorded data, whereas the primaries are treated as additive noise. In general, the primaries are better represented by a Laplacian than by a Gaussian distribution. Consequently, the proposed method uses the ${\ell }_1$ norm to decode the multiples contained in the recorded data and then uses an alternating split Bregman algorithm to solve the decoding problem. We validate the approach on synthetic and field data sets, and our method yields better results compared with the ${\ell }_1$-norm-based matching filter and the traditional pattern-learning approach.

中文翻译：

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使用混合[数学]-[数学]模式学习方法的自适应多次减法

自适应多重减法（AMS）是多重衰减的一项关键且具有挑战性的任务。近来，基于模式学习的AMS方法包括基于模式学习的模式编码和解码步骤。${\ell }_2$规范，取得了骄人的成绩。然而，传统的模式学习方法可能会在最小能量条件下伤害初等人。${\ell }_2$-基于规范的解码。因此，我们采用混合动力${\ell }_2$--${\ell }_{\mathrm{1个}}$ 基于模式学习的AMS方法，其中包括 ${\ell }_2$基于规范的多频带编码和 ${\ell }_{\mathrm{1个}}$-基于规范的解码步骤。在编码阶段，通过使用${\ell }_2$基于规范的主成分分析，已被证明是一种有效的特征提取方法。随后，在解码阶段，从预测倍数中学习的模式用于估计记录数据中的倍数，而基元被视为加性噪声。通常，用拉普拉斯算子而不是用高斯分布更好地表示原初。因此，建议的方法使用${\ell }_{\mathrm{1个}}$规范来解码包含在记录数据中的倍数，然后使用交替拆分Bregman算法来解决解码问题。我们在综合和现场数据集上验证了该方法，与之相比，我们的方法产生了更好的结果${\ell }_{\mathrm{1个}}$-norm的匹配过滤器和传统的模式学习方法。