杨积忠，刘玉柱，董良国



基于Born敏感核函数的速度、密度双参数全波形反演

杨积忠，刘玉柱*，董良国

同济大学海洋地质国家重点实验室，上海200092

摘要速度、密度之间的相互耦合使得密度在多参数全波形反演中较难获得.本文将截断高斯-牛顿法用于声介质速度、密度双参数全波形反演，通过考虑近似Hessian矩阵中反映速度、密度相互作用的非主对角块元素，有效解决了多参数全波形反演中速度、密度之间的耦合问题，在不采用反演策略的情况下，仍能够获得精度较高的速度、密度反演结果.常规的截断牛顿类全波形反演通常利用一阶伴随状态法求取目标函数对模型参数的梯度，利用二阶伴随状态法或有限差分法求解Hessian-向量乘，在每一步内循环迭代过程中需要额外求解两次正演问题，计算量较大.本文基于Born近似，将梯度计算中的核函数-向量乘表示为具有明确物理意义的向量-标量乘的累加运算，同时将Hessian-向量乘转化为两次核函数-向量乘，无需额外求解正演问题，有效降低了计算量.数值实验证明了本文提出的方法的有效性.

关键词全波形反演； 敏感核函数； 多参数； 速度； 密度； 截断高斯-牛顿法

In conventional truncated Newton algorithms,the gradients of the objective function with respect to model parameters are calculated with the first-order adjoint-state method,where the forward propagated source wavefields are zero-lag cross-correlated with the back-propagated residual wavefields.The model update is computed through a matrix-free conjugate gradient (CG) solution of the Newton linear system.This operation requires the computation of the Hessian-vector products,which can be acquired by the second-order adjoint-state formulas or the finite difference method.In such a way,two additional forward wave-propagation problems have to be solved in each internal CG loop,which increase the computational cost dramatically.In this study,we calculate the gradient based on the Born sensitivity kernels,where the kernel-vector products are transformed to the accumulation of vector-scalar products.Meanwhile,the Hessian-vector products are implemented by means of two nested kernel-vector products,thus the forward simulations are no longer required.

To demonstrate the effectiveness of the proposed method,we apply it to a canonical inclusion model and the SEG/EAGE Marmousi-2 model to simultaneously determine velocity and density.The results show that this method can obtain reasonable results for both velocity and density without using any inversion strategy.The vertical profiles extracted from the reconstructed models depict good consistency with the true ones.

Accurate consideration of the inverse Hessian is of great importance in the context of multi-parameter FWI,which helps to mitigate the coupling effects and rescale the magnitudes of different parameters.The Hessian-vector products can be efficiently computed using kernel-vector products without solving any additional forward simulation problems,thus the computational cost of the truncated Gauss-Newton FWI is reduced.Numerical experiments prove the capability of the proposed method on recovering multiple parameters simultaneously.

1引言

全波形反演(Full Waveform Inversion，FWI)是在最优化理论框架下，通过拟合模拟数据与观测数据，使残差达到最小，从而获取具有高分辨率的地下物性参数(如纵横波速度、密度、阻抗或各向异性参数等)的反演方法.

Lailly(1983)和Tarantola(1984)最早在时间域建立了基于最小二乘目标泛函……