Full-waveform inversion (FWI) is for most geophysical applications an ill-posed inverse prob-16lem, with non-unique solutions. We examine its non-uniqueness by exploring the nullspace17shuttle, which can efficiently generate an ensemble of data-fitting solutions. We construct18this shuttle based on a quasi-Newton method, the square-root variable-metric (SRVM)19method. The latter provides access to the inverse data-misfit Hessian in FWI for large-scale20applications. Combining the SRVM method with a randomised singular value decomposi-21tion, we obtain the eigenvector subspaces of the inverse data-misfit Hessian. Its primary22eigenvalue and eigenvector are considered to determine the null space of inversion result.23Using the SRVM-based nullspace shuttle, we can modify the inverted result a posteriori24in a highly efficient manner without corrupting the data misfit. Also, because the SRVM25method is embedded through elastic FWI, our method can be extended to multi-parameter26problems. We confirm and highlight our approach with the elastic Marmousi example.