Accurate representation and estimation of seismic sources is crucial to the joint medium-source full waveform inversion problem. We focus on the source estimation subproblem and its difficulties, where seismic sources are modeled by truncated series of multipoles. The source full waveform inversion formulation results in a highly ill-conditioned (potentially ill-posed) linear least squares problem which we attempt to solve iteratively via conjugate gradient. Our main contribution lies in developing a preconditioner to accelerate the performance of conjugate gradient on multipole source inversion. The proposed preconditioner consists of (fractional) time derivative/integral operators based on analytical solutions to the wave equation with multipole sources in an unbounded, homogeneous medium. Numerical results in 2D demonstrate that the conjugate gradient iterations are accelerated when incorporating the proposed preconditioning scheme.

地震源的准确表示和估算对于联合介质源全波形反演问题至关重要。我们集中于震源估计子问题及其困难，其中地震源是通过截断的多极级数序列建模的。源全波形反演公式会导致病态严重（潜在病态）的线性最小二乘问题，我们尝试通过共轭梯度来迭代求解。我们的主要贡献在于开发预处理器，以加快共轭梯度在多极源反演中的性能。拟议的预处理器由（分数）时间导数/积分算子组成，这些算子基于对无界均匀介质中具有多极源的波动方程的解析解。