3.5 Effects of small steady perturbations

In order to do linear inversions of helioseismic data, it is necessary to first solve the linear forward problem. The linear forward problem is to compute the first-order effect of small perturbations to the solar model. By first-order we mean first order in the strength of the perturbations. Essentially all inversions that have been done have assumed that the perturbations to the solar model are time-independent over the time duration during which the observations are made.

In general the linear forward problem can be written as

The $ddi$ are the first-order changes in the observed helioseismic parameters, for example changes in travel times, ring-fit parameters, or changes in ingression-egression correlations (which will be described in Section 4). The kernel functions $Ki (r) a$ depend on the three-dimensional position vector $r$ . The first-order changes to the solar model are given by the functions $dqa(r)$ , where the index $a$ refers to the type of the perturbation, for example we can look at the effect of flows, changes in sound speed, and changes in wave excitation and damping. For each type of perturbation $a$ , corresponds a particular kernel function $Ka$ . Notice that we have assumed the functions $qa$ to be constant in time. The goal of the linear forward problem is to compute the functions $Ki a$ for any particular type of measurement $di$ .

There have been numerous efforts to approximate the kernel functions $K$ 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 $Ka$ 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

This is the equation for the first Born approximation. The first-order correction to the observed wavefield is therefore given by

integral 3 ' ' 3 i ' '{ j ' ' } 0 dP(x, t) = - d r dt d rsdts G (x,t;r ,t) dLG (r ,t;rs,ts) iSj (rs, ts) integral 3 i + d rs dts G (x, t;rs,ts)dSi(rs, ts), (30)

where $j {dLG }i$ is the $i- th$ component of the vector $j dLG$ and summation on repeated indices is implied. From Equation (30

) we can predict the first-order change in any quantity in local helioseismology that will result from a small change in the model. We can look at, for example, first-order changes in the cross-covariance, travel times, power spectrum, or holographic measurements.

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 ).