Full Waveform Inversion (FWI) is a large-scale nonlinear ill-posed problem for which implementation of the Newton-type methods is computationally expensive. Moreover, these methods can trap in undesirable local minima when the starting model lacks low-wavenumber part and the recorded data lack low-frequency content. In this paper, the Gauss-Newton (GN) method is modified to address these issues. We rewrite the GN system for multisoure multireceiver FWI in an equivalent matrix equation form whose solution is a diagonal matrix, instead of a vector in the standard system. Then we relax the diagonality constraint, lifting the search direction from a vector to a matrix. This relaxation is equivalent to introducing an extra degree of freedom in the subsurface offset axis for the search direction. Furthermore, it makes the Hessian matrix separable and easy to invert. The relaxed system is solved explicitly for computing the desired search direction, requiring only inversion of two small matrices that deblur the data residual matrix along the source and receiver dimensions. Application of the Extended GN (EGN) method to solve the extended-source FWI leads to an algorithm that has the advantages of both model extension and source extension. Numerical examples are presented showing robustness and stability of EGN algorithm for waveform inversion.
翻译:完整波形 Inversion (FWI) 是一个大规模非线性的问题, 使用牛顿型方法的费用是计算成本昂贵的。 此外, 当初始模型缺少低波序部分, 记录的数据缺乏低频内容时, 这些方法会困在不受欢迎的本地迷你模式中。 在本文中, Gaus- Newton (GN) 方法被修改来解决这些问题。 我们重写GN 系统, 用于多soure 多重接收器 FWI, 以等量的矩阵方程式形式, 其解决方案是双向矩阵, 而不是标准系统中的矢量。 然后我们放松对二向限制, 将矢量的搜索方向从矢量提升到矩阵。 这种放松相当于在次表下偏偏偏偏偏偏偏偏偏偏重轴中引入额外程度的自由。 此外, 使赫西亚矩阵的矩阵相互连接, 容易倒置。 为了计算理想的搜索方向, 我们简单解决了宽松的系统, 只需要将两个小矩阵转换成双向源和接收方。 扩展的GNU(EGNGN) 模型的扩展演算法的扩展法的扩展法是显示源FWIFWIFWIV的扩展法的扩展法的扩展法。