We present a time-domain matrix-free elastic Gauss–Newton full-waveform inversion (FWI) algorithm. Our algorithm consists of a Gauss–Newton update with a search direction calculated via elastic least-squares reverse time migration (LSRTM). The conjugate gradient least-squares (CGLS) method solves the LSRTM problem with forward and adjoint operators derived via the elastic Born approximation. The Hessian of the Gauss–Newton method is never explicitly formed or saved in memory. In other words, the CGLS algorithm solves for the Gauss–Newton direction via the application of implicit-form forward and adjoint operators which are equivalent to elastic Born modelling and elastic reverse time migration, respectively. We provide numerical examples to test the proposed algorithm where we invert for P- and S-wave velocities simultaneously. The proposed algorithm performs positively on mid-size problems where we report solutions of slight improvement than those computed using the conventional non-linear conjugate gradient method. In spite of the aforementioned limited gain, the theory developed in this paper contributes to a better understanding of time-domain elastic Gauss–Newton FWI.

中文翻译：

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时域弹性高斯-牛顿全波形反演：无矩阵方法

我们提出了一种时域无矩阵弹性高斯-牛顿全波形反演（FWI）算法。我们的算法包括一个高斯-牛顿更新，其搜索方向是通过弹性最小二乘反向时间偏移（LSRTM）计算的。共轭梯度最小二乘（CGLS）方法使用通过弹性Born近似推导的前向和伴随算子来解决LSRTM问题。高斯-牛顿法的Hessian从未明确形成或保存在内存中。换句话说，CGLS算法通过应用隐式形式的前向和伴随算符来求解高斯-牛顿方向，这分别相当于弹性Born建模和弹性反向时间偏移。我们提供了数值示例来测试所提出的算法，其中我们将P-和S取反波速。所提出的算法在中等大小的问题上表现出积极的作用，在该问题中，我们报告的解决方案比使用常规非线性共轭梯度法计算的解决方案略有改进。尽管上述增益有限，但本文开发的理论有助于更好地理解时域弹性高斯牛顿FWI。