### 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., 1977Cuypers, 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.