基于数据驱动紧框架小波稀疏约束优化的地震数据重建

Due to physical and economic constraints, missing data is one of the common issues in seismic data acquisition. Reconstruction and interpolation of missing traces in seismic records is a critical element of the data processing chain, which is very important for subsequent processing steps including improvement of spatial resolution, multiple suppression, migration, imaging, and amplitude versus offset （AVO） analysis. In comparison with traditional POCS method and Spitz's prediction filters, the recent methods named sparsity-promoting L1-norm (SPL1) optimization much improved the reconstruction of missing traces. The methods shift the problem of seismic interpolation to a L1-norm regularized minimization. They include two stages: 1) how to find a suitable transform that can sparsely represent seismic data, 2) build a reasonable sparsity-promoting model and fast algorithm. Following this way, the principle investigator (J. Ma) also proposed improved curvelet SPL1 method and directional weighted shearlet SPL1 method. However, how to find a "best" sparse transform for a given field data is still a challenge problem. In this proposal, we will present data-driven adaptive tight frame (framelet), and a novel anti-aliasing SPL1 method for seismic data restoration. Furthermore, we will develop high-dimensional data-driven framelets for higher dimensional seismic data reconstruction (e.g., five dimensions). Finally, we will consider the combination of nuclear-norm regularization with SPL1 regularization. Basically, the rank-reduction nuclear norm is a global regularization while the framelet/shearlet/curvelet sparsity L1 norm is somewhat local constraint. The proposed method (without additional prediction filtering) should beat the traditional high-dimensional Fourier-based methods in terms of reconstruction performance.

不规则丢失道的重建以及稀疏采样数据的抗假频道加密重建是地震数据处理中非常重要的一个环节，其重建的效果严重关系到多次波去除、偏移、成像和AVO分析的质量。与传统的POCS方法和Spitz预测滤波方法比，目前比较有效的方法是稀疏约束优化的地震数据重建，它将重建问题转化为L1范数最小化问题来求解。其关键是两步：1）找到合理的稀疏变换，2）建立合理的稀疏约束模型和算法。申请人近期也提出了改进的Curvelet稀疏约束方法和方向加权的Shearlet 约束方法。但是如何针对实际数据来更好构造稀疏变换仍是一个挑战。本项目我们立足于地震勘探和应用数学的交叉研究，提出数据驱动的自适应紧框架小波，及新的抗假频稀疏约束优化方法，并开发基于高维数据驱动紧框架的地震数据重建方法，有望改进工业界中流行的高维Fourier数据重建方法。我们还将考虑与核范数降秩技术结合，建立全局约束和局部约束联合重建模型和快速算法。

在地震勘探数据采集过程中，地理环境等因素常常会造成采集数据缺失，会对后期地震数据处理和解释造成不利影响，因此在地震数据处理过程中，地震数据重建是不可或缺的步骤。我们研究了基于自适应稀疏变换的数据重构方法：数据驱动紧框架（DDTF）。DDTF能够根据数据自适应生成稀疏变换，通过稀疏促进方法可以重构完整数据。我们将DDTF拓展到高维情形并应用于三维和五维地震数据重构，得到优于传统算法的重构效果。另外，我们提出了四种新的字典学习算法，在计算效率、重构质量上有所改进。1）蒙特卡罗DDTF，通过蒙特卡罗块选择方法大幅度降低计算量并同时保持了重构效果。2）基于克罗内克的数据驱动紧框架字典学习方法。该字典学习方法利用了克罗内克结构，避免了向量化操作，有效的保持了数据之间的联系性。3）基于图正则化的字典学习方法。图正则化方法考虑了数据的局部和非局部相似性。4）结合非局部高斯混合尺度模型，提出了图结构字典学习模型。除此之外，我们还进行了与稀疏变换密切相关的低秩约束、非对称调频小波、几何模态分解的研究，进一步拓宽了研究内容。基于自适应稀疏变换的五维地震数据重构对于降低勘探成本、提高采集效率、提高成像结果的分辨率具有重要意义。

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