Love wave full waveform inversion via Pseudo-Hessian gradient pre-conditioning operator
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摘要: 构建近地表横波速度模型是煤田多分量地震资料处理的重要环节。相较于面波多道分析法，全波形反演在构建近地表横波模型中具有更高的分辨率。然而，在基于梯度的全波形反演中，由于地震记录频带有限、波场的非均匀覆盖以及双重散射等原因导致梯度算子不随深度的增加而缩放，模型深部参数得不到明显更新。目标函数的Hessian算子包含曲率信息，可清晰预测梯度算子中的焦散现象及双重散射产生的伪影，因此，逆Hessian算子则可作为反卷积算子实现对梯度的预处理，加强对模型深部的照明能力。然而Hessian算子具有巨大维度，对其显式计算十分困难。基于此，借鉴逆散射理论的思想，给出勒夫波全波形反演目标函数的拟Hessian算子的表达式，并提出一种梯度预处理的全波形反演方法。将该方法分别应用于断层模型、凹陷模型以及起伏界面模型的重构试验，反演结果表明：与传统的共轭梯度全波形反演方法相比，基于拟Hessian算子的预处理共轭梯度方法可加快收敛速度，提升成像质量。Abstract: The construction of near surface shear wave velocity is an important step in multi-component seismic data processing in coalfield. Compared with the multichannel analysis of surface wave, the full waveform inversion(FWI) has higher resolution in the construction of near surface shear wave velocity model. However, in the gradient-based FWI, the gradient operator is not scaled with increasing depth due to the narrow frequency band of seismic records, the non-uniform coverage of the wavefield, and the double scattering. The parameters of the deep model cannot be updated significantly. The Hessen operator of the objective function contains curvature information, which can clearly predict the defocusing phenomenon and the artifacts generated by double scattering in the gradient operator. The inverse Hessen operator can be used as a deconvolution operator to realize gradient pre-conditioning and enhance the illumination ability of the deep model. However, the explicit calculation of Hessian operator is very difficult because it has huge dimensions. Based on this, inverse scattering theory is referred to, the expression of the pseudo-Hessian operator of the objective function of full-waveform inversion is given, and a pre-conditioned gradient-based FWI method is developed. The proposed method was applied to the reconstruction tests of the fault model, subsidence model, and undulating interface model, respectively. The inversion results show that, compared with the classic conjugate gradient-based FWI, the pre-conditioned conjugate gradient method based on the pseudo-Hessian operator can accelerate the convergence rate and improve the inversion accuracy.
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图 1 断层模型重构测试中的真实模型、初始模型及反演结果
Fig. 1 The real model, initial model and inversion results in the fault model reconstruction test
图 2 断层模型重构测试的第6炮地震记录
Fig. 2 The seismic record of shot 6 in the fault model reconstruction test
图 4 凹陷模型重构测试中的真实模型、初始模型及反演结果
Fig. 4 The real model, initial model and inversion results in the subsidence model reconstruction test
图 5 凹陷模型重构测试的第6炮地震记录
Fig. 5 The seismic record of shot 6 in the subsidence model reconstruction test
图 7 起伏界面模型重构测试中的真实模型、初始模型及反演结果
Fig. 7 The real model, initial model and inversion results in the undulating interface model reconstruction test
图 8 起伏界面模型重构测试的第6炮地震记录
Fig. 8 The seismic record of shot 6 in the undulating interface model reconstruction test
图 9 界面起伏模型重构测试的评价曲线
Fig. 9 Evaluation curves of undulating interface model reconstruction test
表 1 断层模型重构测试中PCG与CG算法的性能对比评价
Table 1 Comparison and evaluation of the PCG and CG algorithms in fault model reconstruction test
寻优算法 迭代次数 归一化目标函数终值 模型对比误差初值 模型对比误差终值 CG 39 0.026 6 0.080 1 0.047 2 PCG 17 0.011 6 0.080 1 0.022 9 
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表 2 凹陷模型重构测试中PCG与CG算法性能对比评价
Table 2 Comparison and evaluation of the PCG and CG algorithms in subsidence model reconstruction test
寻优算法 迭代次数 归一化目标函数终值 模型对比误差初值 模型对比误差终值 CG 40 0.052 1 0.102 8 0.088 7 PCG 40 0.005 8 0.102 8 0.030 6
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表 3 界面起伏模型重构测试中PCG与CG算法性能对比评价
Table 3 Comparison and evaluation of the PCG and CG algorithms in undulating interface model reconstruction test
寻优算法 迭代次数 归一化目标函数终值 模型对比误差初值 模型对比误差终值 CG 21 0.117 3 0.089 3 0.081 2 PCG 19 0.011 5 0.089 3 0.078 5
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表 4 PBGJ-PCG与PCG算法性能对比评价
Table 4 Comparison and evaluation of the PBGJ-PCG and PCG algorithms
寻优算法 迭代次数 归一化目标函数终值 模型对比误差初值 模型对比误差终值 PCG 19 0.011 5 0.089 3 0.078 5 PBGJ-PCG 40 0.004 8 0.089 3 0.033 4
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