基于面波相速度频散曲线的三维速度结构波动方程反演

  利用地震面波的频散特性进行反演成像是获取地球内部结构的主要手段之一。传统基于一维速度结构正演的地震面波层析成像方法只能获得近似的三维速度模型，而基于波动方程模拟的三维地震面波波动方程反演方法具备反演任意复杂三维结构的潜力，却面临两个难题：（1）需要存储每个台站的四维应变格林函数，存储量过大；（2）当前的面波波动方程反演方法只使用了面波群走时的信息，但面波的相走时相比于群走时具有更深的敏感核分布，可以更好地约束深部结构。如何利用面波相走时进行波动方程反演仍待研究。

  针对三维地震波动方程反演数据存储量过大的问题，本研究提出了一种基于多维数据压缩的波动方程反演方法，该方法基于Tucker张量分解可以在相同压缩精度下获得很高的压缩比，显著减小了四维应变格林张量的数据大小，提高了波动方程反演的计算效率。本文研究了应变张量压缩误差对波动方程反演敏感核的影响，并定量地研究了波动方程反演应用中所能允许的相对压缩误差控制参数。实际数据波动方程反演算例表明，在获得大约1 000倍的压缩比时，压缩误差不会影响反演结果。

  本研究对如何利用面波相走时进行波动方程反演进行了研究。首先，提出了一种计算面波相走时波动方程反演Fréchet敏感核的原理和方法。通过采用Kaiser滤波器对面波波形进行极窄带滤波，可以准确计算从多分量地震波形中提取的面波相走时波动方程反演Fréchet敏感核，其中包括Z分量上的基阶Rayleigh面波、T分量上的Love面波，以及从Z分量和R分量中分离出的一阶Rayleigh面波。这些三维Fréchet敏感核的准确计算为波动方程反演高分辨率速度模型奠定了基础。其次，提出了一种基于面波相速度频散曲线波动方程反演方法。通过利用面波相速度频散曲线，可以准确计算相走时差，有效地避免了周波跳跃问题，并且实现了直接使用面波相速度频散曲线进行三维速度结构波动方程反演。通过数值算例验证了该方法的可行性，结果显示，与传统面波走时层析成像和群走时波动方程反演相比，该方法获得的反演结果具有更高的分辨率，并且能够分辨更深的速度结构。此外，该方法还成功应用于龙门山地区的实际地震数据，获得的反演结果与前人研究结果总体一致，验证了该方法在反演实际地震数据方面的可行性。通过波动方程反演获得的高分辨率反演结果，为研究该地区地质构造、汶川大地震的孕震机制和动力学背景提供了重要的参考依据。

  本研究为快速高效地反演高精度地下速度模型提供了基础。所得速度模型在研究地球内部结构、地球演化过程、地震与火山成因等方面具有重要的实际意义。

    Using the dispersion characteristics of seismic surface waves for internal structure imaging is one of the primary means of obtaining information about the Earth's interior. Traditional seismic surface wave tomography methods, based on one-dimensional velocity structures, can only yield approximate three-dimensional velocity models. In contrast, the three-dimensional seismic surface wave equation inversion method based on wave equation simulation holds the potential to invert for any complex three-dimensional structure. However, it faces two challenges: (1) the need to store the four-dimensional strain Green's functions for each station, resulting in significant storage requirements; (2) the current surface wave equation inversion method only utilizes information from the group travel times, while the phase travel times of surface waves, compared to the group travel times, have a deeper sensitivity kernel distribution that could better constrain deeper structures. The exploration of how to effectively utilize phase travel times of surface waves in waveform inversion based on the seismic wave equation remains to be studied.

    To address the issue of excessive data storage in three-dimensional seismic wave equation inversion，this study introduces a wave equation inversion method based on multidimensional data compression. The method, based on Tucker tensor decomposition, achieves high compression ratios under the same compression accuracy, significantly reducing the data size of four-dimensional strain Green tensors and effectively addressing the pain points of wave equation inversion based on scattering integral methods, thus improving the computational efficiency of wave equation inversion. This study investigates the influence of strain tensor compression errors on wave equation inversion sensitivity kernels and quantitatively studies the allowable relative compression error control parameters in wave equation inversion applications. Numerical examples of actual data wave equation inversion demonstrate that compression errors do not affect the inversion results when achieving a compression ratio of approximately 1 000 times.

    This study investigates the utilization of phase travel times of surface waves in waveform inversion. Firstly, a principle and method for computing the wave equation inversion Fréchet sensitivity kernel using phase travel times of surface waves are proposed. By applying Kaiser filtering to the surface waveforms for ultra-narrowband filtering, accurate computation of the wave equation inversion Fréchet sensitivity kernel can be achieved. This includes the fundamental Rayleigh surface wave on the Z component, Love surface wave on the T component, and the first-order Rayleigh surface wave separated from the Z and R components. The accurate computation of these three-dimensional Fréchet sensitivity kernels forms the foundation for high-resolution velocity model inversion in wave equation inversion. Secondly, a method based on the phase velocity dispersion curve of surface waves for wave equation inversion is introduced. By utilizing the phase velocity dispersion curve of surface waves, accurate computation of phase travel time differences is achieved, effectively avoiding frequency jump issues. This method allows for direct three-dimensional velocity structure inversion using the phase velocity dispersion curve of surface waves. Feasibility of the proposed method is validated through numerical examples, demonstrating that compared to traditional surface wave travel time tomography and group travel time wave equation inversion, the proposed method yields higher resolution inversion results capable of distinguishing deeper velocity structures. Furthermore, the method is successfully applied to actual seismic data from the Longmenshan region, producing inversion results consistent with previous studies, thus validating the feasibility of the method in inverting real seismic data. The high-resolution results obtained through wave equation inversion provide important reference points for studying geological structures, the pre-seismic mechanism, and the dynamic background of the Wenchuan earthquake in this region.

    This study establishes the foundation for rapidly and efficiently inverting high-precision subsurface velocity models. The obtained velocity models hold significant practical importance for researching internal Earth structure, Earth evolution processes, and the causes of earthquakes and volcanoes.

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