The second-order derivative information plays an important role for large-scale full waveform inversion problems. However, exploiting this information requires massive computations and memory requirements. In this study, we develop two inexact Newton methods based on the Lanczos tridiagonalization process to consider the second-order derivative information. Several techniques are developed to improve the computational performance for our proposed methods. We present an effective stopping condition and implement a nonmonotone line search method. A method based on the adjoint-state method is used to efficiently compute Hessian-vector products. In addition, a diagonal preconditioner using the pseudo-Hessian matrix is employed to accelerate solving the Newton equation. Furthermore, we combine these two inexact Newton methods to improve the computational efficiency and the resolution. 2D and 3D experiments are given to demonstrate the convergence and effectiveness of our proposed methods. Numerical results indicate that, compared with the inversion methods based on the first-order derivative, both methods have good computational efficiency. Meanwhile, the method based on MINRES solver performs better than the method with Lanczos_CG due to its ability of utilizing the negative eigenvalue information when solving strongly nonlinear and ill-posed problems.

二阶导数信息对于大规模全波形反演问题起着重要作用。但是，利用此信息需要大量的计算和内存需求。在这项研究中，我们基于Lanczos三对角化过程开发了两种不精确的牛顿方法，以考虑二阶导数信息。开发了几种技术来改善我们提出的方法的计算性能。我们提出了一个有效的停止条件，并实现了非单调线搜索方法。使用基于伴随状态方法的方法来有效地计算Hessian向量乘积。此外，采用伪Hessian矩阵的对角预处理器可加快牛顿方程的求解速度。此外，我们将这两种不精确的牛顿方法结合起来，以提高计算效率和分辨率。给出了2D和3D实验，以证明我们提出的方法的收敛性和有效性。数值结果表明，与基于一阶导数的反演方法相比，两种方法都具有良好的计算效率。同时，基于MINRES求解器的方法在处理强非线性和不适定问题时具有利用负特征值信息的能力，因此其性能优于Lanczos_CG。两种方法都具有良好的计算效率。同时，基于MINRES求解器的方法在处理强非线性和不适定问题时具有利用负特征值信息的能力，因此其性能优于Lanczos_CG。两种方法都具有良好的计算效率。同时，基于MINRES求解器的方法在处理强非线性和不适定问题时具有利用负特征值信息的能力，因此其性能优于Lanczos_CG。