基于L0范数约束的稀疏拟正交Radon变换方法及其应用研究

Radon transform is a useful and fundamental tool in exploration seismology with a core issue of exactly estimating its spectrum. Generally, this issue is cast as an inverse problem, and the transform from the spectrum to data domain is considered as a forward problem. To successfully solve this inverse problem, we must take the precision of its forward problem into account. However, the implicit assumption that predict data from Radon inverse transform is plane wave with constant amplitude and a fixed integral path obscures its precision and limit its validity regimes. In order to overcome the limitation of the conventional Radon transform, both amplitude and traveltime calibration operators are introduced to conserve the waveform of predict data and suit for the arbitrary integral path. In this sense, we approximately implement the orthogonal mapping from the forward to inverse Radon transforms, and vice versa, and thus improve the predict precision. We name the calibrated transforms the pseudo-orthogonal Radon transforms (PORT). Next, we will use the sparsity of Radon spectrum in time domain as a constraint to do a sparse inversion under L0 norm. Combining the PORT with matching pursuit (MP) algorithms, a self-adaptive orthogonal MP algorithm is proposed to solve the inverse sparse Radon spectrum problem. Consider of strong adaptability of arbitrary integral path and energy preservation, the PORT has wide potential applications such as body-surface wave mode separation for irregular surfaces, sparse plane wave decomposition and automatic stacking velocity estimation, and so on, with more improved effects and better efficiency than conventional methods.

Radon变换广泛应用于勘探地震中，其核心为Radon谱估计，通常解法将其构造为一个反问题。相应的，将Radon谱投影到数据域的变换定义为正问题。正问题的精度是反问题成功的基本保障。然而，基于反Radon变换的预测数据隐含了常振幅与固定积分路径假设，导致正问题的精度较低且适用范围受限。为突破此局限，本项目在常规反Radon变换中引入振幅估计以及走时校正算子，构造了一种保波形的反Radon变换算子，近似实现了正反Radon变换的正交化，提高了正问题的预测精度。接着，利用时间域Radon谱的稀疏性作为约束，将Radon谱反演构造为L0范数下的稀疏反问题。结合正交Radon变换对及匹配追踪算法，提出一种自适应正交匹配追踪算法求解稀疏Radon谱反问题。在应用方面，本项目将稀疏正交Radon变换应用于起伏地表面波分离，稀疏平面波分解以及自动叠加速度估计等方面，试图改进传统方法的效果并提高计算效率。

高分辨率Radon变换在地震信号处理中有广泛应用。传统的基于L2范数的Radon变换方法，难以得到最稀疏的反演结果。传统的基于L1/L0范数的Radon变换方法，只考虑模型空间解的稀疏性，无法兼顾数据逼近的精度。本项目从信号建模的角度出发，首先提出一种新的信号预测模型。进而在压缩感知框架下，通过匹配追踪算法及算子正交变换，实现了高分辨率的Radon谱反演方法。在保证模型空间解的稀疏性的同时，兼顾了数据空间的信号重构精度，因而适用性更广。.在应用研究方面，将本项目中的稀疏Radon谱反演用于高分辨率双曲Radon变换，可以实现全自动的叠加速度分析。因此，无需在进行速度谱的人工拾取，可以提高初始速度建模的效率并降低人工成本。同时，通过在高维空间中进行高维局部线性Radon变换，可以实现地震信号的特征表达（如斜率，走时，子波等），可以为后续的层析反演提供输入数据，无需人工交互。