Characterise the non-uniqueness in full-waveform inversion by using the square-root variable metric based null-space shuttle

Full-waveform inversion is an ill-posed inverse problem, with non-unique solutions. We examine its non-uniqueness by exploring the null-space shuttle, which can generate an ensemble of data-fitting solutions efficiently. We construct this shuttle based on a quasi-Newton method, the square-root variable-metric (SRVM) method. This method enables a retrieval of the inverse data-misfit Hessian after the SRVM-based elastic FWI converges. Combining SRVM with randomised singular value decomposition (SVD), we obtain the eigenvector subspaces of the inverse data-misfit Hessian. The first one among them is considered to determine the null space of the elastic FWI result. Using the SRVM-based null-space shuttle we can modify the inverted result a posteriori in a highly efficient manner without corrupting data misfit. Also, because the SRVM method is embedded through elastic FWI, our method can be extended to multi-parameter problems. We confirm and highlight our methods with the elastic Marmousi example.