  Normal-mode helioseismic data analysis uses observed solar oscillation
spectra to infer perturbations in the solar interior due to global and
local-scale flows and structural asphericity. Differential rotation, the
dominant global-scale axisymmetric perturbation, has been tightly constrained
primarily using measurements of frequency splittings via "$a$-coefficients".
However, the frequency-splitting formalism invokes the approximation that
multiplets are isolated. This assumption is inaccurate for modes at high
angular degrees. Analysing eigenfunction corrections, which respect cross
coupling of modes across multiplets, is a more accurate approach. However,
applying standard inversion techniques using these cross-spectral measurements
yields $a$-coefficients with a significantly wider spread than the
well-constrained results from frequency splittings. In this study, we apply
Bayesian statistics to infer $a$-coefficients due to differential rotation from
cross spectra for both $f$-modes and $p$-modes. We demonstrate that this
technique works reasonably well for modes with angular degrees $\ell=50-291$.
The inferred $a_3-$coefficients are found to be within $1$ nHz of the frequency
splitting values for $\ell > 200$. We also show that the technique fails at
$\ell < 50$ owing to the insensitivity of the measurement to the perturbation.
These results serve to further establish mode coupling as an important
helioseismic technique with which to infer internal structure and dynamics,
both axisymmetric (e.g., meridional circulation) and non-axisymmetric
perturbations.
