基于带状Pseudo-Hessian矩阵的弹性波全波形反演方法研究

Elastic Full Wave-form Inversion (EFWI) is the most accurate elastic parameter inversion method at present, which can provide P-, S-wave velocities and density for inner Earth investigation and oil-gas exploration. Coupled effects between different parameters produce cross-talk artifacts in EFWI. Inversion strategies used for removing these cross-talks depend on data quality and prior information. The Hessian matrix describes the coupled effects and can be used to correct the corresponding cross-talk artifacts. However, the Hessian matrix is normally avoided in large inversion problems due to excessive computational cost. Conventional diagonal-approximate Hessian matrix ignores the coupled effects and cannot mitigate the cross-talks in EFWI. In order to correct the cross-talk artifacts from the coupled effects, we propose to develop a new EFWI method using banded pseudo-Hessian matrix in this proposal, based on block-diagonal pseudo-Hessian matrix we developed. We first construct the banded pseudo-Hessian matrix with those elements describing significant coupled effects. Then we incorporate the banded pseudo-Hessian matrix into quasi-Newton optimization method. This new EFWI can overcome the drawback in computational cost and memory requirement of the Hessian computation. Meanwhile, it effectively corrects cross-talk artifacts from the coupled effects and provide high-accuracy elastic parameter models for large-scale inversion problems.

弹性波全波形反演(EFWI)是目前理论上精度最高的弹性参数反演方法，可以为探索地球内部属性和油气勘探提供纵横波速度和密度。不同参数对地震波的耦合效应会在EFWI中产生交叉假象。用于消除这些交叉假象的反演策略依赖于数据质量和先验信息。Hessian矩阵能估计耦合效应并用于消除相应的交叉假象，但是巨量的计算需求阻碍其在大规模反演问题上的应用。而常规的对角近似Hessian矩阵无法消除参数耦合效应引起的影响。为了校正参数耦合效应引起的交叉假象，本课题基于申请人已经发表的块对角近似Hessian矩阵，拟发展一套基于带状近似Hessian矩阵的二维EFWI方法。需要使用代表强烈参数耦合效应的元素来构建带状近似矩阵，并将其整合到拟牛顿局部最优化方法中。该方法能克服引入Hessian有效信息时的计算和存储问题，同时又能显著地削弱多参数间的交叉脚印，可为大规模反演问题提供高精度的弹性参数模型。

不同参数对地震波的耦合效应会在弹性波全波形反演中产生交叉假象。Hessian矩阵能估计耦合效应并用于消除相应的交叉假象，但是巨量的计算需求阻碍其在大规模反演问题上的应用。针对以上问题，本项目展开了Hessian矩阵近似研究以及基于近似Hessian矩阵的弹性波全波形反演研究。.基于以上研究内容，本项目提出了一种带状Pseudo-Hessian矩阵，发展了利用该矩阵的二维弹性波全波形反演方法和技术。通过计算Gauss-Newton Hessian并分析其中元素的分布规律，获取参数间耦合效应最突出的区域。根据分析结果，抽取相距一个波长距离不同空间点所对应的Hessian矩阵元素，形成带状矩阵。并采用Pseudo-Hessian计算近似，形成带状Pseudo-Hessian矩阵。该矩阵是带状稀疏的，从而极大缓解了Hessian矩阵不便于直接计算和存储的问题，并且可以弥补多参数反演中传统对角近似Hessian无法预测参数耦合效应的问题。对于所提出的带状Pseudo-Hessian矩阵，采用共轭梯度法迭代地求取其逆矩阵，并将其作为预条件算子，整合到l-BFGS方法中，形成了使用带状Pseudo-Hessian预条件的l-BFGS方法。该方法能对Hessian的逆矩阵进行更准确的估计，从而预测不同参数对地震波的耦合效应。数值实验结果表明该方法能高效经济地去除反演结果中普遍存在的多参数交叉假象噪音，能为大规模地球物理反演问题提供更高精度的弹性参数模型。