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频率域全波形反演充分利用全波场的振幅、相位以及频率信息,采用较少的频率便能反演得到精度很高的速度模型。本文以有限单元法为基础,对起伏地形条件下二维声波频率域全波形反演进行了研究。在正演算法中,针对截断边界问题,并考虑多频率联合反演中计算区域采用同一套剖分网格的需求,提出了一种适用于起伏地形的衰减边界条件算法。该算法的核心思想是在控制方程波数项中引入衰减因子,通过一定方式调节衰减因子使得声波在衰减层中充分衰减,达到压制截断边界影响的目的。根据指数衰减规律,文中推导出了一种新的衰减因子计算公式,并给出了不同频率条件下衰减层厚度计算公式;在反演算法中,采用共轭梯度法求解高斯牛顿反演迭代方程组,避免直接求解雅克比矩阵和HeSSian矩阵带来的巨额计算量,并采用相同的反演模型,对比分析了不同初始模型和频率组合对全波形反演结果的影响。起伏地形模型数值模拟和全波形反演数值试验表明,本文提出的指数衰减边界条件算法和基于该算法的全波形反演算法具有很好的应用效果。
The full waveform inversion in the frequency domain makes full use of the amplitude, phase and frequency information of the full-wave field, and can be inversed to obtain the high-accuracy velocity model with less frequency. In this paper, based on the finite element method, the full waveform inversion of two-dimensional acoustic wave frequency domain under undulating terrain condition is studied. In the forward algorithm, aiming at the problem of truncating the boundary and considering the need of using the same set of meshed meshes in the computational domain of multi-frequency joint inversion, an algorithm of decay boundary conditions for undulating terrain is proposed. The core idea of this algorithm is to introduce the attenuation factor into the wave number of the control equation, and to adjust the attenuation factor in a certain way so that the sound wave can be sufficiently attenuated in the attenuation layer to achieve the effect of suppressing the influence of the truncated boundary. According to the law of exponential decay, a new formula for calculating the attenuation factor is deduced, and the formulas for calculating the thickness of the attenuation layer under different frequencies are given. In the inversion algorithm, the conjugate gradient method is used to solve the iterative equation This method avoids directly solving the huge computational load caused by Jacobi and HeSSian matrices, and uses the same inversion model to compare and analyze the influence of different initial models and frequency combinations on the results of full waveform inversion. The numerical simulation of undulating terrain model and the numerical experiment of full waveform inversion show that the exponential decay boundary condition algorithm proposed in this paper and the full waveform inversion algorithm based on this algorithm have a good application effect.