基于曲率高斯束的地震数据稀疏分解与重构方法研究

The reconstruction of seismic wave field data is one of the hot topics in recent years. On one hand, a huge amount of seismic data is acquired; but on the other hand, seismic wave information is incomplete. Therefore, it is of great scientific significance to study the compression and reconstruction of seismic data. In this project, the sparse decomposition and reconstruction of seismic data based on curvature Gaussian beam basis functions are studied. By introducing curvature parameter, the sparsity of the decomposition is increased and the stability of the solution is improved. In the process of sparse decomposition, The L0 norm minimization model is established. The L0 norm minimization method is the most sparse and most direct method for solving the decomposition problem. Two fast algorithms will be used to solve the L0 norm optimization problem. One is the method of smooth approximation. The L0 norm minimization problem is translated into extremal problem of smooth function. The other is the method of reduction dimensions. The polynomial Radon transform is used to extract the seismic wavelet. It can greatly reduce the dimensions of sparse inversion by eliminating the zero in the unknown vector. Then, the sparse decomposition method proposed in the project is compared with the sparse decomposition method based on curvelet transform in terms of sparsity, reconstruction error, and so on. Finally, the decomposition method proposed in the project is applied to sparse Gaussian beam migration. The increase of sparsity decreases the number of Gaussian beams obtained by decomposition. It can improve efficiency of migration.

地震波场数据重构是近年来的热点研究问题之一，一方面地震采集数据量巨大，另一方面采集过程中存在信息不完整的问题，因此对地震数据压缩和重构研究具有重要的科学意义。本项目研究基于曲率高斯束基函数的地震数据稀疏分解与重构，引入曲率参数增加了分解的稀疏度，能够提高分解的稳定性。在稀疏分解过程中，建立L0范数最优化模型，L0范数极小化方法是数据分解问题最稀疏、最直接的求解方法。通过两种快速算法对L0范数极小化问题进行求解：（1）光滑逼近法，将L0范数极小化问题转化为光滑函数的极值问题进行求解；（2）降维法，通过多项式Radon变换提取地震子波，消除待求向量中的零值，大大降低稀疏反演的维度。接着把本项目提出的稀疏分解方法与基于curvelet变换的稀疏分解方法在稀疏度、重构效果等方面进行对比研究。最后把本项目提出的分解方法应用到稀疏高斯束偏移中，稀疏度的增加减少了分解得到的高斯束的数目，提高偏移效率。

随着油气资源勘探区向复杂构造地区转移，采集的地震数据质量越来越差，基于稀疏约束的地震数据分解与重构方法受到了广泛关注。在远场平面波假设条件下，高斯束稀疏分解方法只适用于曲率较小的地震数据。本项目研究了基于L0范数约束的曲率高斯束稀疏分解和重构地震数据的方法，发展了一套基于地震数据的L0范数实现方法，即多项式Radon变换和光滑逼近相互结合的方法。增加曲率参数提高了高斯束的稀疏性，重构地震数据减少约1/3高斯束的数目。通过与Curvelet变换对比研究，可以看出本项目所提方法在去噪和弱信号保存方面具有较强的优势，不同于Curvelet变换基函数，所有高斯束基函数在时间-空间域都是紧支的。. 把曲率高斯束分解地震数据的方法应用于稀疏高斯束偏移，能够进一步提高稀疏高斯束偏移的计算效率，以及小幅改善成像剖面的信噪比，这对于油气勘探具有重大的理论意义和实践价值。