Full-waveform inversion (FWI) is a highly nonlinear and nonconvex problem. To mitigate the dependence of FWI on the quality of starting model and on the low frequencies in the data, we apply the gradient sampling algorithm (GSA) introduced for nonsmooth, nonconvex optimization problems to FWI. The search space is hugely expanded to have more freedom to accommodate large velocity errors in the starting model. The original implementation of GSA requires explicit calculation of the gradient at each sampled vector, which is prohibitively expensive. Based on the observation that a slight perturbation in the velocity model causes a small spatial shift of the wavefield, we have approximated the sampled gradients by crosscorrelating the space-shifted source- and receiver-side wavefields. Theoretical derivation suggests that the two wavefields should be shifted in the same direction to obtain reasonable low-wavenumber updates. The final descent search direction is obtained by summing all the shifted gradients. For practical implementation, we only take one random space shift at each time step during the gradient calculation. This simplification provides an efficient realization in which the computational costs and memory requirements are the same as conventional FWI. Multiple numerical examples demonstrate that the proposed method alleviates the cycle-skipping problem of conventional FWI when starting from very crude initial velocity models without low-frequency data.

中文翻译：

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通过随机梯度采样缓解全波形反演的跳频

全波形反演（FWI）是一个高度非线性且非凸的问题。为了减轻FWI对初始模型质量和数据低频的依赖性，我们将针对非平稳，非凸优化问题引入的梯度采样算法（GSA）应用于FWI。搜索空间被极大地扩展以具有更大的自由度来适应起始模型中的大速度误差。GSA的原始实现要求显式计算每个采样矢量处的梯度，这是非常昂贵的。基于观测到的速度模型中的微小扰动会引起波场的空间偏移，我们通过将空间偏移的源侧和接收器侧波场进行互相关来近似采样的梯度。理论推导表明，两个波场应在同一方向上移动以获得合理的低波数更新。最终的下降搜索方向是通过求和所有偏移的梯度得出的。对于实际的实现，我们在梯度计算过程中的每个时间步长只进行一次随机空间移位。这种简化提供了一种有效的实现方式，其中计算成本和内存需求与常规FWI相同。多个数值算例表明，从没有频率数据的粗略初始速度模型开始时，所提出的方法减轻了常规FWI的循环跳跃问题。对于实际的实现，我们在梯度计算过程中的每个时间步长只进行一次随机空间移位。这种简化提供了一种有效的实现方式，其中计算成本和内存需求与常规FWI相同。多个数值算例表明，从没有频率数据的粗略初始速度模型开始时，所提出的方法减轻了常规FWI的循环跳跃问题。对于实际的实现，我们在梯度计算过程中的每个时间步长只进行一次随机空间移位。这种简化提供了一种有效的实现方式，其中计算成本和内存需求与常规FWI相同。多个数值算例表明，从没有频率数据的粗略初始速度模型开始时，所提出的方法减轻了常规FWI的循环跳跃问题。