论文标题
某些结构化速度模型的最佳基于运输的波形反演的凸度
The convexity of optimal transport-based waveform inversion for certain structured velocity models
论文作者
论文摘要
完整的波形反转（FWI）是一种从表面上的信息中确定地球特性的方法。我们使用平方的瓦斯坦距离（平方$ W_2 $距离）作为目标函数，以将地震波的速度反转为地球中位置的函数，我们讨论了其相对于速度参数的凸度。在一个维度上，我们将恒定，分段的增加和线性增加速度模型作为位置的函数，并且在源函数是一个可能性测量时，我们在速度参数到速度参数的间隔上相对于速度参数到速度参数的真实值的速度参数相对于速度参数的凸度。此外，我们考虑了一个二维模型，其中，速度随着深度的函数的函数而线性增加，并证明了速度参数中包含真实值的速度参数中平方$ W_2 $距离的凸度。我们讨论了平方$ W_2 $距离的凸度，而平方$ l^2 $ norm的凸度，我们讨论了这些相应距离的频率与凸度之间的关系。我们还通过首先将波数据转换为概率度量来讨论多种非概率措施的最佳运输方法。
Full--waveform inversion (FWI) is a method used to determine properties of the Earth from information on the surface. We use the squared Wasserstein distance (squared $W_2$ distance) as an objective function to invert for the velocity of seismic waves as a function of position in the Earth, and we discuss its convexity with respect to the velocity parameter. In one dimension, we consider constant, piecewise increasing, and linearly increasing velocity models as a function of position, and we show the convexity of the squared $W_2$ distance with respect to the velocity parameter on the interval from zero to the true value of the velocity parameter when the source function is a probability measure. Furthermore, we consider a two--dimensional model where velocity is linearly increasing as a function of depth and prove the convexity of the squared $W_2$ distance in the velocity parameter on large regions containing the true value. We discuss the convexity of the squared $W_2$ distance compared with the convexity of the squared $L^2$ norm, and we discuss the relationship between frequency and convexity of these respective distances. We also discuss multiple approaches to optimal transport for non--probability measures by first converting the wave data into probability measures.
