College of Earth Sciences, Guilin University of Technology

受采集环境和仪器性能的影响，实测探地雷达(GPR)剖面中不可避免会出现部分信号缺失和坏道现象，易造成目标体产生的反射波和绕射波同相轴不连续，严重降低后续处理与成像精度和分辨率。为此，将图像处理中广泛应用的凸集投影(POCS)算法与具有良好稀疏特性的曲波变换相结合，提出了一种基于曲波域POCS算法的缺失GPR信号高精度重建方法。从压缩感知理论出发，建立了离散曲波变换基下缺失信号重建的目标函数，并采用POCS算法详细推导了缺失GPR信号重建的时间域迭代公式。其中，线性和指数迭代阈值模型用于更新曲波变换系数，从而高精度重建时间域缺失信号；平均绝对误差、信噪比、峰值信噪比用于定量评价GPR信号重建精度。模拟与实测GPR信号的重建试验表明：POCS算法可有效重建GPR剖面中的缺失信号；与线性阈值模型的POCS算法相比，指数阈值模型的POCS算法重建精度更高；与指数阈值模型的频率域POCS算法相比，指数阈值模型的曲波域POCS算法用于重建GPR剖面中连续多道缺失信号的误差更小、纵向伪影能量更弱，且对复杂GPR结构模型的缺失信号重建具有较强的适用性；与线性和指数阈值模型的频率域POCS算法、线性阈值模型的曲波域POCS算法相比，指数阈值模型的曲波域POCS重建方法的重建精度更高、平均绝对误差下降45%~99%、信噪比和峰值信噪比提高1~20 dB，其重建结果可为后续处理与解释提供高质量GPR信号。

Influence by the acquisition environments and instrument performance, missing signals and destructed channels are inevitable in measured ground-penetrating radar (GPR) profiles. They can cause event discontinuity of reflected and diffracted waves generated by targets, severely impairing the accuracy and resolution of subsequent processing and imaging. Hence, by combining the projection onto convex sets (POCS) algorithm extensively used in image processing with the curvelet transform exhibiting high sparsity, this study proposed a high-accuracy reconstruction method for missing GPR signals based on curvelet-domain POCS. Building on the compressive sensing theory, the objective function for missing signal reconstruction based on discrete curvelet transform was established, and the time-domain iterative equation for reconstructing missing GPR signals was derived in detail using POCS. The curvelet transform coefficients were updated using linear and exponential iterative threshold models for high-accuracy reconstruction of missing signals in the time domain. The reconstruction accuracy of missing GPR signals was quantitatively evaluated using mean absolute errors (MAEs), signal-to-noise ratios (SNRs), and peak SNRs (PSNRs). The reconstruction experiments of simulated and measured GPR signals show that POCS can effectively reconstruct the missing signals in GPR profiles. Contrasting with the POCS of the linear threshold model, the POCS of the exponential threshold model yielded higher reconstruction accuracy. In the exponential threshold model, compared to frequency-domain POCS, curvelet-domain POCS exhibited smaller reconstruction errors and weaker longitudinal artifact energy during the reconstruction of continuous multi-channel missing signals in GPR profiles, and higher applicability to the reconstruction of missing signals in complex structural models. In contrast to the frequency-domain POCS of both linear and exponential threshold models and the curvelet-domain POCS of the linear threshold model, the curvelet-domain POCS reconstruction method of the exponential threshold model manifested higher reconstruction accuracy, average absolute errors reduced by 45%‒99%, and SNRs and PSNRs enhanced by 1‒20 dB, providing high-quality GPR signals for subsequent processing and interpretation.

国家自然科学基金项目(42364010)；广西自然科学基金项目(2022GXNSFAA035595，2020GXNSFAA159121)

吴祺铭，王洪华，席宇何，等. 基于曲波域凸集投影算法的缺失GPR信号高精度重建[J]. 煤田地质与勘探，2024，52(3)：130−143.

WU Qiming，WANG Honghua，XI Yuhe，et al. High-accuracy reconstruction of missing ground-penetrating radar signals based on projection onto convex sets in the curvelet domain[J]. Coal Geology & Exploration，2024，52(3)：130−143.