稀疏反褶积正则化策略优选：数学模型构建与性能评估

Optimal Selection of Sparse Deconvolution Regularization Strategy: Mathematical Model Construction and Performance Evaluation

摘要: 在地震资料反演中，反褶积是一种重要的压缩地震子波、提高薄层纵向分辨率的地震数据处理方法。由于地层为层状结构，反射系数可视作稀疏的脉冲序列，因此地震反褶积可以描述为稀疏求解问题。然而，反褶积问题通常是病态的，需要引入正则化约束以获得稳定和准确的解。本研究介绍了几种不同的正则化方法，包括L1正则化、L2正则化、Cauchy正则化以及结合L1和L2正则化的方法，给出了它们的数学模型，并着重比较了Cauchy正则化与结合L1和L2正则化的方法。通过简单的一维模型和复杂的Marmousi2 (二维)模型的实验，我们评估了这些正则化方法在稀疏脉冲反褶积中的性能表现。结果表明，结合L1和L2正则化的联合方法在噪声抑制和分辨率提升方面表现优异，能够更准确地恢复地下结构的真实反射特性。本文的研究为选择适当的正则化策略以优化地震数据的反褶积处理提供了理论支持和实用指导。

Abstract: In seismic data inversion, deconvolution is an important seismic data processing method that compresses seismic wavelets and improves the vertical resolution of thin layers. Due to the layered structure of the strata, the reflection coefficient can be regarded as a sparse pulse sequence, so seismic deconvolution can be described as a sparse solution problem. However, deconvolution problems are often pathological and require the introduction of regularization constraints to obtain stable and accurate solutions. This study introduces several different regularization methods, including L1 regularization, L2 regularization, Cauchy regularization, and methods combining L1 and L2 regularization. Their mathematical models are given, and the comparison between Cauchy regularization and methods combining L1 and L2 regularization is emphasized. We evaluated the performance of these regularization methods in sparse pulse deconvolution through experiments using a simple one-dimensional model and a complex Marmousi2 (two-dimensional) model. The results show that the joint method combining L1 and L2 regularization performs well in noise suppression and resolution improvement, and can more accurately restore the true reflection characteristics of underground structures. This study provides theoretical support and practical guidance for selecting appropriate regularization strategies to optimize the deconvolution processing of seismic data.