1D relation for a₁-coefficients

Some experiments like GONG and MDI produce a small number of the so-called a-coefficients of splittings expansions on a set of orthogonal polynomials [Ritzwoller & Lavely1991]. Assuming that the relation between individual splittings and these coefficients is linear, an equation similar to Equation 1 can be established by computing the appropriate kernels related to each a_j^nl-coefficients for odd indices j (see e.g. [Pijpers1997]). Furthermore it has been shown by [Ritzwoller & Lavely1991] that the expansion of the splittings in orthogonal polynomials corresponds to an expansion of such that:

where are the Legendre polynomials. This forms the so called 1.5D problem where each a_j-coefficient is related to the expansion function of the same index through a 1D integral. Therefore the first term of the expansion Equation 5 which do not depend on the latitude can be related to the a₁-coefficients through:

The radial kernel is the same as in Equation 2 but, from Equation 5, the function obtained by inverting a₁-coefficients corresponds to the searched rotation rate only where the rotation do not depend on the latitude. Otherwise it corresponds to some average over latitudes that can be estimated by looking at the corresponding 2D averaging kernel (cf. Section 4 and Figure 3).