一阶Rytov近似有限频走时层析

Wave equation traveltime tomography using Rytov approximation

传统的波动方程走时核函数(或走时Fréchet导数)多基于互相关时差测量方式及地震波场的一阶Born近似导出,其成立条件非常苛刻.然而,地震波走时与大尺度的速度结构具有良好的线性关系,对于小角度的前向散射波场,Rytov近似优于Born近似.因此,本文基于Rytov近似和互相关时差测量方式,导出了基于Rytov近似的有限频走时敏感度核函数的两种等价形式:频率积分和时间积分表达式.在此基础之上,本文提出了一种隐式矩阵向量乘方法,可以直接计算Hessian矩阵或者核函数与向量的乘积,而无需显式计算和存储核函数及Hessian矩阵.基于隐式矩阵向量乘方法,本文利用共轭梯度法求解法方程实现了一种高效的Gauss-Newton反演算法求解走时层析反问题.与传统的敏感度核函数反演方法相比,本文方法在每次迭代过程中,无需显式计算和存储核函数,极大降低了存储需求.与基于Born近似的伴随状态方法走时层析相比,本文方法具有准二阶的收敛速度,且适用范围更广.数值试验证明了本文方法的有效性.

The conventional wave-equation traveltime sensitivity kernel (TSK) or traveltime Fréchet derivative is derived from the Born approximation and cross-correlation measurement,which has a very narrow valid condition.In fact,the seismic traveltime has a more linear relationship with the large-scale velocity structure.For small-angle forward scattered wavefield,Rytov approximation is proved to be superior to Born approximation.Based on the Rytov approximation and cross-correlation measurement,a new wave-equation traveltime sensitivity kernel is derived.Meanwhile,an implicit matrix-vector product method is proposed,which can directly calculate the product of a matrix (TSK) and a model-space vector as well as the product of a matrix transpose and a data-space vector,eliminating the need of calculating TSK explicitly.Based on the proposed implicit matrix-vector product method,traveltime tomography using the Gauss-Newton inversion algorithm is implemented efficiently by solving the normal equation iteratively using a conjugate gradient method.Compared with the conventional TSK method,the proposed inversion strategy is free of TSK calculation and storage,making it more practical for large-scale problem.Compared with the adjoint traveltime tomography,the proposed method has a quasi-second-order convergent rate and a broader valid condition.Numerical examples demonstrate the effectiveness of the proposed method.