SUMMARY Gradient computations in full-waveform inversion (FWI) require calculating zero-lag cross-correlations of two wavefields propagating in opposite temporal directions. Lossless media permit accurate and efficient reconstruction of the incident field from recordings along a closed boundary, such that both wavefields propagate backwards in time. Reconstruction avoids storing wavefield states of the incident field to secondary storage, which is not feasible for many realistic inversion problems. We give particular attention to velocity–stress modelling schemes and propose a novel modification of a conventional reconstruction method derived from the elastodynamic Kirchhoff–Helmholtz integral. In contrast to the original formulation (in a previous related work), the proposed approach is well-suited for velocity–stress schemes. Numerical examples demonstrate accurate wavefield reconstruction in heterogeneous, elastic media. A practical example using 3-D elastic FWI demonstrates agreement with the reference solution.