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。

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.