弹性波全波形反演中的四种优化方法对比

弹性波全波形反演（EFWI）是一种高精度成像方法。由于EFWI本质是一个强非线性问题，因此常采用局部优化算法进行求解，不同优化算法的反演结果差异很大。在较为常用的共轭梯度法（CG）、L-BFGS法（Limited-memory Broyden-Fletcher-Goldfarb-Shanno Algorithm）的基础上使用伪Hessian矩阵作为梯度预条件算子，实现了预条件共轭梯度法（P-CG）和预条件L-BFGS （P-L-BFGS）反演方法。文中首先对这四种优化算法的原理及实现流程进行了介绍；然后通过绕射体模型和Marmousi Ⅱ模型对四种算法进行测试。结果表明：①近似Hessian预条件算子可以对深部能量进行补偿，并加快反演的收敛速度；②CG法、P-CG法实现较为简单，但由于仅使用了一阶梯度信息，无法对多参数耦合效应进行压制，对于较为复杂的MarmousiⅡ模型，P-CG法可得到略差于L-BFGS法的反演结果；③L-BFGS法和P-L-BFGS法的实现更复杂，但由于在反演过程中使用了近似Hessian矩阵，对于多参数耦合效应具有一定的压制效果；④对于MarmousiⅡ模型，L-BFGS法和P-L-BFGS法都能反演出精度较高的纵、横波速度模型，但密度反演会出现过拟合现象。

Comparison of four optimization methods in elastic full-waveform inversion

Elastic full-waveform inversion(EFWI)is a high-precision imaging method.Since it is a highly nonlinear problem in nature,local optimization algorithms are often used to solve it,but the inversion results of different optimization algorithms are very different.On the basis of the commonly used conjugate gradient(CG)method and limited-memory Broyden-Fletcher-Goldfarb-Shanno(L-BFGS)method,the pseudo-Hessian matrix is adopted as the gradient preconditioning operator to devise a preconditioning CG(P-CG)method and a preconditioning L-BFGS(P-L-BFGS)method.The principles and implementation processes of these four optimization algorithms are expounded.Then,the diffraction model and the MarmousiⅡmodel are used to test the four algorithms.The following conclusions can be drawn from the results:①The pseudo-Hessian preconditioning operator can compensate for the deep energy and accelerate the convergence of inversion;②The CG method and the P-CG method are easy to implement,but they cannot suppress the multi-parameter coupling effect because they only use first-order gradients.However,the P-CG method can deliver inversion results that are slightly inferior to those of the L-BFGS method for the complex MarmousiⅡmodel.③The implementation of the L-BFGS method and the P-L-BFGS method is more complicated.Nevertheless,because the pseudo-Hessian matrix is calculated in the inversion process,the multi-parameter coupling effect is suppressed to a certain extent.④As for the MarmousiⅡmodel,both the L-BFGS method and the P-L-BFGS method can obtain compressional wave(P-wave)and shear wave(S-wave)velocity models with high inversion accuracy,but density inversion will be subject to the overfitting phenomenon.