基于谱元法的层状介质探地雷达频散曲线正演模拟研究

探地雷达激发的电磁波在层状波导结构中传播时会产生频散现象，其频散曲线包含了地层的电性结构信息，通过反演其频散曲线可以获得地层的介电常数和层厚。目前针对探地雷达频散的研究较少且存在一些局限，已有的频散曲线理论计算方法忽略了电导率，即把地层看成无损介质来简化计算。并且，前人研究中并没有对频散模式进行细致的研究和区分。探地雷达频散和地震波频散具有相似性，因此本文将基于谱元法的地震波频散曲线理论计算方法推广到探地雷达频散研究中，得到了一套考虑电导率的探地雷达频散曲线计算新方法。该方法通过透射边界条件和半无限元法处理电磁波场的边界条件，得到了易于求解的线性特征值问题，因而可以直接利用特征值分解高效地计算频散曲线。

本文针对水平层状模型进行了数值实验，对于经典的低速波导模型和高速波导模型，通过与前人算法的对比，验证了本文方法在求解经典模型时的准确性。对于无损介质模型和有损介质模型，通过模式分析，将探地雷达频散分为导波模式和泄漏模式两种，并将不同频率下的导波模式和泄漏模式频散点与频散函数进行对比，证明了计算的不同模式频散点的准确性。对合成的探地雷达道集记录进行的频散分析表明，道集记录同时包含了导波模式和泄漏模式，并且其频散能量谱与本文方法计算的频散曲线高度一致，这证明本文方法可以成为研究探地雷达频散的有效手段。

Dispersion occurs when electromagnetic waves excited by ground penetrating radar (GPR) propagate in a layered waveguide. The dispersion curves contain information about the electrical structure of the formation. By inverting the dispersion curves, the dielectric constant and layer thickness of the formation can be obtained. At present, there is relatively limited research on the dispersion of GPR and there are some limitations. The existing theoretical calculation methods for dispersion curves ignore conductivity, considering the medium as non-conductive to simplify calculations. Moreover, previous studies have not conducted detailed research and classification on dispersion modes. The dispersion of GPR is similar to that of seismic wave. Therefore, this article utilizes the theoretical calculation method of seismic wave dispersion curves based on the spectral element method (SEM) and extends it to the study of GPR dispersion. We introduce a new method for calculating GPR dispersion curves while considering conductivity. This method addresses the boundary conditions of electromagnetic wave fields by utilizing transmission boundary conditions and the semi-infinite element method. It results in a linear eigenvalue problem that is easily solvable. Therefore, the dispersion curves can be efficiently calculated directly using eigenvalue decomposition.

This article presents numerical experiments conducted on horizontal layered models. We compare the accuracy of our method in solving the classical low-speed waveguide model and high-speed waveguide model with previous algorithms. For both lossless and lossy media models, GPR dispersion is divided into guided wave mode and leaky mode through mode analysis. The dispersion points of guided mode and leaky mode at different frequencies are compared with the dispersion function, demonstrating the precision of the calculated dispersion points for different modes. The dispersion analysis of the synthesized GPR gathers reveals the presence of both guided modes and leaky modes. In addition, the dispersion spectra of synthesized GPR gathers are found to be highly consistent with the calculated dispersion curves, which proves the effectiveness of the proposed method.

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