In general the linear forward problem can be written as
There have been numerous efforts to approximate the kernel functions for time-distance travel times. These efforts will be described in Section 4.3.5. The kernel functions for ring diagrams have typically been approximated as constant within the area the ring is measured over, with a depth dependence derived from normal mode theory (see Section 4.2.3).
The linear forward problem for normal-mode frequencies is quite well known (see the upcoming Living Reviews paper on global helioseismology, Thompson (2005)). For example, Gough and Thompson (1990), Goldreich et al. (1991), and Dziembowski and Goode (2004) applied first-order perturbation theory to estimate the effect of magnetic fields on normal mode frequencies. This work may be helpful for computing the effects of magnetic fields on local helioseismic measurements.
In local helioseismology, the sensitivity functions may be computed using the Born approximation. The first Born approximation gives the lowest order approximation for estimating scattering amplitudes (Born, 1926); this method appears to have been first used by Strutt (1881). As explained below, the Born approximation is essentially an equivalent-source description of wave interaction. For small perturbations to a solar model, it is used to compute the first-order change in the wavefield. The first-order change in any particular helioseismic parameter can then be obtained from the first-order, as well as the zero-order, wavefields. To compute the first-order change in the wavefield we begin from Equation (7) and Equations (19, 20, 21). To first order we have
An alternative to the Born approximation is the Rytov approximation. In the Rytov approximation one computes the first order correction to the phase of the wavefield rather than the correction to the wavefield itself. For applications in the context of helioseismology, see Brüggen (2000) and Jensen and Pijpers (2003). The Born and Rytov approximations have been compared by Keller (1969).
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