L-BFGS法时间域全波形反演中初始矩阵的选择方法

Selection strategy of the initial matrix for L-BFGS method in time domain full waveform inversion

有限内存BFGS(Limited-memory Broyden-Fletcher-Goldfarb-Shanno,L-BFGS)方法是地震全波形反演(Full waveform inversion,FWI)中应用最广泛的优化方法之一。该方法需要提供Hessian逆矩阵的一个初始近似矩阵,初始近似矩阵可以在每次迭代中都进行更新,也可以固定不变。L-BFGS法的收敛性能与初始矩阵的选择以及更新方式紧密相关。在分析了全波形反演中Hessian逆矩阵的几种不同近似方式之后,分别将它们作为初始矩阵应用到了时间域L-BFGS法全波形反演中,详细比较了不同近似方式对反演精度和效率的影响,为LBFGS法全波形反演中初始矩阵的选择提供了依据。数值试验结果表明,在不同的初始矩阵选择以及不同更新方式下,L-BFGS法的收敛速度表现出明显差异。当初始矩阵固定不变时,以虚震源为基础的两种方式收敛最快,采用波场能量矩阵的两种方式要慢一些,其中包含有记录残差反传波场能量的方式收敛最慢。初始矩阵每次迭代更新时,各种方式的收敛速度都慢于固定不变时的情形。此外,同种方式分别作为L-BFGS法的初始矩阵和共轭梯度法的预条件矩阵时,L-BFGS法的性能要优于预条件共轭梯度法。

L-BFGS method is one of the most widely used optimization methods for full waveform inversion (FWI). An initial approximate matrix to the inverse Hessian matrix is needed to be provided, which can be updated or be fixed during iteration. The convergence performance of L-BFGS method is closely related to the selection of the initial matrix and its updating manner. After the analysis of several approximation ways to the inverse Hessian matrix, they are used as the initial matrix of L-BFGS method which is then applied to the time domain FWI respectively. Their impacts on the inversion accuracy and efficiency are fully compared to provide basis for the initial matrix selection of L-BFGS method in FWI. Experimental results show that the performances of L-BFGS method with different initial matrix and different updating manner are quite distinct. When the initial matrix is fixed, the two ways based on pseudo sources converge most rapidly. Another two ways with wave energy are slower than the preceding two ones. More specifically, the way consisting of the energy of recorded residual back-propagated wavefield converges slowest. When updated during iteration, the convergence rate corresponding to every way is slower than that in the fixed case. In addition, L-BFGS outperforms the preconditioned conjugate gradient (CG) method, when the same way is used respectively to play the role of the initial matrix in L-BFGS and the precondition matrix in CG method.