### 3.1 The inversion problem

The basic 2-dimensional rotation inversion problem can be stated as follows: we have a number of
observations , from which we wish to infer the rotation profile where is distance from the
center of the Sun, and is (conventionally) colatitude. Each datum is a spatially weighted average of the
rotation rate:
where is the solar radius, the error term corresponds to the noise and measurement error in the
data, and is a model-dependent spatial weighting function known as the kernel (Hansen
et al., 1977; Cuypers, 1980). For the two-dimensional rotation inversion, the radial part is related to the
eigenfunction of the mode and the latitudinal part to the associated Legendre polynomial; Schou
et al. (1994) give the expression for the kernel as
where
, , is the radial displacement for the eigenfunction of the mode, is
the horizontal displacement, and is the density (see Figure 10 for illustrations of sample
kernels).
The aim of the inversion is to find

where is the location at which the inferred rotation rate is to be found and the are the
coefficients to be used to weight the data; the inversion process can be thought of as the search for the best
values for these coefficients.