  Purpose: Inter-scan motion is a substantial source of error in $R_1$
estimation, and can be expected to increase at 7T where $B_1$ fields are more
inhomogeneous. The established correction scheme does not translate to 7T since
it requires a body coil reference. Here we introduce two alternatives that
outperform the established method. Since they compute relative sensitivities
they do not require body coil images.
  Theory: The proposed methods use coil-combined magnitude images to obtain the
relative coil sensitivities. The first method efficiently computes the relative
sensitivities via a simple ratio; the second by fitting a more sophisticated
generative model.
  Methods: $R_1$ maps were computed using the variable flip angle (VFA)
approach. Multiple datasets were acquired at 3T and 7T, with and without motion
between the acquisition of the VFA volumes. $R_1$ maps were constructed without
correction, with the proposed corrections, and (at 3T) with the previously
established correction scheme.
  Results: At 3T, the proposed methods outperform the baseline method.
Inter-scan motion artefacts were also reduced at 7T. However, reproducibility
only converged on that of the no motion condition if position-specific transmit
field effects were also incorporated.
  Conclusion: The proposed methods simplify inter-scan motion correction of
$R_1$ maps and are applicable at both 3T and 7T, where a body coil is typically
not available. The open-source code for all methods is made publicly available.
