tokens: and φ is longitude. This is an important property for the analysis to follow. Because the eigenfunctions are a complete set (with zero Lagrangian-pressure perturbation on the outer boundary, e.g. Christensen–Dalsgaard 2003 ), a general wavefield may be written as where ω is temporal frequency, a q are coefficients representing the phase and amplitude of the q th mode and we write for the sake of notational convenience. The governing equation for driven, undamped oscillations is given by where is the unperturbed source, the zeroth-order wavefield and is the unperturbed operator (whose properties are derived from a basic solar model such as model S; Christensen-Dalsgaard et al. 1996 ) as defined in equation ( 2 ). Note that the MDI and HMI missions provide time series of spherical-harmonic coefficients of oscillations, i.e. a k (ω), measured at the solar photosphere. In practice, because we only observe the side of the Sun that faces us, it is not possible to precisely spatially decompose the measured oscillation signals into individual spherical harmonics. This partial visibility leads to leakage between modes across l, introducing systematical errors in eventual inferences (e.g. Larson Schou 2015 ; Woodard 2016 ). Further, at a given frequency (between ridges for instance), many different radial orders contribute, leading to diminished ability in distinguishing between them. Temporal gaps in observations, due to instrumental or other issues, lead to mixing across temporal frequencies (and hence radial orders), thereby exacerbating this issue. To mitigate the problem of radial-order leakage, we use measurements at frequencies within a linewidth of resonance, ensuring that the associated radial order contributes a dominant fraction of observed mode power. Nevertheless, leakage across radial orders and spherical-harmonic degrees will systematically influence the results of mode-coupling analyses and needs to be carefully considered. 2.1 Perturbation theory The effects of rotation, flows, magnetic fields or thermal asphericities are modelled by adding a perturbation to the operator to in equation ( 5 ), in turn leading to a perturbation to the wavefield, . Ignoring perturbations to the source , we obtain for the wavefield in a perturbed Sun. For time-varying perturbations, behaves as a convolution operator. To the first order, we have therefore where the convolution over frequency captures the time variability of the perturbation operator (equation D4 of Appendix D ). Perturbing equation ( 4 ) while assuming that the eigenfunctions are unmodified by the perturbation, we may express the change to the wavefield as a linear combination of the original set of eigenfunctions, Using equation ( 2 ), the left hand side of equation ( 6 ) may be simplified thus Solar modes are subject to a small amount of attenuation γ k ≪ ω k that perturb the eigenfunctions and eigenfrequencies (obtained from equation 1 ) by introducing small imaginary components. However,
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