in $m$. However, asphericities in the solar interior, such as rotation, temperature variations and the presence of magnetic fields, lift the degeneracy and measurements of the differences in the frequencies of the $m$ components allow inferences to be made concerning the effects responsible for the asphericities.

The frequency of a mode, $\nu_{n,l,m}$, can be expressed in terms of the $m=0$ central frequency, $\nu_{n,l}$, and a polynomial expansion of splitting (or $a$) coefficients, $a_j(n,l)$: $$\label{equation[splitting]}
    \nu_{n,l,m}=\nu_{n,l}+\sum^{j_{\textrm{\scriptsize{max}}}}_{j=1}a_j(n,l)\mathcal{P}_j^{(l)}(m),$$ where $\mathcal{P}_j^{(l)}(m)$ are the Ritzwoller-Lavely formulation of the Clebsch-Gordon expansion [@1991ApJ...369..557R] and are given by $$\label{equation[RL]}
    \mathcal{P}_j^{(l)}(m)=\frac{l\sqrt{(2l-j)!(2l+j+1)!}}{(2l)!\sqrt{2l+1}}C^{lm}_{j0lm}.$$ Here, $C^{lm}_{j0lm}$ are the Clebsch-Gordon coefficients. The odd-order $a$ coefficients express the difference in mode frequency caused by the modes’ interaction with the rotation profile of the solar interior. The even-order coefficients are sensitive to all other departures from spherical symmetry. These could include local variations in the sound speed or cavity size [@1996MNRAS.278..437B], temperature variations [@1988ApJ...331L.131K], second order rotation effects from the differential rotation [@1992ApJ...394..670D], and the presence of magnetic fields [@1988IAUS..123..175G]. Isolating the contributions of all these effects to the even-order coefficients is extremely difficult.

As sound waves travel from their excitation point near the surface into the solar interior they are refracted because of increasing 