Title: Inspired by machine learning optimization: can gradient-based optimizers solve cycle skipping in full waveform inversion given sufficient iterations? Show Chinese title

Title: 受机器学习优化的启发：给定足够的迭代次数，基于梯度的优化器能否解决全波形反演中的循环跳过问题？

Abstract: Full waveform inversion (FWI) iteratively updates the velocity model by minimizing the difference between observed and simulated data. Due to the high computational cost and memory requirements associated with global optimization algorithms, FWI is typically implemented using local optimization methods. However, when the initial velocity model is inaccurate and low-frequency seismic data (e.g., below 3 Hz) are absent, the mismatch between simulated and observed data may exceed half a cycle, a phenomenon known as cycle skipping. In such cases, local optimization algorithms (e.g., gradient-based local optimizers) tend to converge to local minima, leading to inaccurate inversion results. In machine learning, neural network training is also an optimization problem prone to local minima. It often employs gradient-based optimizers with a relatively large learning rate (beyond the theoretical limits of local optimization that are usually determined numerically by a line search), which allows the optimization to behave like a quasi-global optimizer. Consequently, after training for several thousand iterations, we can obtain a neural network model with strong generative capability. In this study, we also employ gradient-based optimizers with a relatively large learning rate for FWI. Results from both synthetic and field data experiments show that FWI may initially converge to a local minimum; however, with sufficient additional iterations, the inversion can gradually approach the global minimum, slowly from shallow subsurface to deep, ultimately yielding an accurate velocity model. Furthermore, numerical examples indicate that, given sufficient iterations, reasonable velocity inversion results can still be achieved even when low-frequency data below 5 Hz are missing. Show Chinese abstract

Abstract: 全波形反演（FWI）通过最小化观测数据和模拟数据之间的差异来迭代更新速度模型。 由于全局优化算法相关的高计算成本和内存需求，FWI通常使用局部优化方法来实现。 然而，当初始速度模型不准确且缺乏低频地震数据（例如低于3赫兹）时，模拟数据和观测数据之间的差异可能超过半个周期，这种现象称为周期跳变。 在这些情况下，局部优化算法（例如基于梯度的局部优化器）容易收敛到局部极小值，导致反演结果不准确。 在机器学习中，神经网络训练也是一个容易陷入局部极小值的优化问题。 它通常使用具有相对较大学习率的基于梯度的优化器（超出通常由线搜索数值确定的局部优化理论极限），这使得优化过程表现出类似准全局优化器的行为。 因此，在经过数千次迭代训练后，我们可以获得具有强大生成能力的神经网络模型。 在本研究中，我们也为FWI采用具有相对较大学习率的基于梯度的优化器。 合成数据和实际数据实验的结果表明，FWI可能会最初收敛到局部极小值；然而，随着额外的足够迭代次数，反演可以逐渐接近全局极小值，从浅层地表缓慢向深层推进，最终得到一个准确的速度模型。 此外，数值例子表明，即使缺少低于5赫兹的低频数据，只要迭代次数足够，仍可以获得合理的速度反演结果。