3.1 The inversion problem

The basic 2-dimensional rotation inversion problem can be stated as follows: we have a number M of observations di, from which we wish to infer the rotation profile Ω (r,πœƒ) where r is distance from the center of the Sun, and πœƒ is (conventionally) colatitude. Each datum is a spatially weighted average of the rotation rate:
∫ R βŠ™∫ π di = 0 0 Ki(r,πœƒ)Ω (r,πœƒ)drd πœƒ + πœ–i, (11 )
where RβŠ™ is the solar radius, the error term πœ– corresponds to the noise and measurement error in the data, and K 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. (1994Jump To The Next Citation Point) give the expression for the kernel as
( [ ] m--{ -2 m 2 Knlm (r,πœƒ) = Inl(ξnl(r) ξnl(r ) − L ηnl(r) Pl (x) (12 )
⌊ ⌋) ηnl(r)2 (dP m )2 dP m m2 } + ----2--⌈ ---l- (1 − x2) − 2Pml ---l-x + -----2P ml (x)2⌉ ρ(r)rsinπœƒ, L dx dx 1 − x )
∫ R βŠ™ 2 2 2 Inl = [ξnl(r) + ηnl(r )]ρ(r)r dr, (13 ) 0
x = cosπœƒ, 2 L = l(l + 1), ξnl is the radial displacement for the eigenfunction of the mode, −1 L ηnl is the horizontal displacement, and ρ(r) is the density (see Figure 10View Image for illustrations of sample kernels).

The aim of the inversion is to find

M∑ Ω¯(r0,πœƒ0) = ci(r0,πœƒ0)di, (14 ) i=1
where (r,πœƒ ) 0 0 is the location at which the inferred rotation rate ¯Ω is to be found and the c i 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.
View Image

Figure 10: Sections through rotation kernels for selected azimuthal orders for l = 3,n = 9 (top), and l = 20, n = 5 (bottom).

  Go to previous page Go up Go to next page