4.5 Direct modeling

Woodard (2002

) introduced the idea of estimating subsurface flows from direct inversion of the correlations seen in the wavefield in the Fourier domain. The central notion is that for horizontally homogeneous steady models with no flow Fourier components of the physical wavefield are uncorrelated. Departures from horizontal homogeneity or time-dependence in general introduce correlations into the wavefield. Thus observations of the correlations in the Fourier domain can be used to estimate flows in the interior. Woodard (2002

) gave a practical demonstration of the ability of the technique to recover near-surface flows from the f-mode part of the spectrum.

4.5.1 Forward problem

The fact that a correlation in the Fourier components of the wavefield is introduced by a perturbation to a translation invariant model can be seen from the Born approximation of the perturbed wavefield, $dP$ . Rewriting Equation (30 ) in terms of Fourier coordinates gives

integral dP(k, w) = - (2p)3 d3rdt d2k'dw'dz e- ik.x+iwtGi(k, w,z) { [ s ]} × dL Gj(k', w',z,z )eik'.x-iw't S0 (k',w',z ), (83) (r,t) s i j s

where summation is assumed over the repeated indices $i$ and $j$ , $r = (x, z)$ is the scattering location, and the Green’s functions $G$ and $G$ are defined in Section 3.3. The perturbations to the source function $dS$ were ignored for the sake of simplicity. To obtain Equation (83

) we used the fact that the operator $dL$ , which describes the perturbation to the model at $(r, t)$ , is a linear operator. The equation shows that the perturbation to the model in general causes a response at wavevector $k$ and frequency $w$ to a source at $k'$ and $w'$ . This response is thus correlated with the unperturbed wave created by the same source. Notice that for a perturbation that depends both on space and time, the operator $dL$ explicitly depends on $r$ and $t$ . Using Equation (83

) the correlations of the wavefield in the Fourier domain can be approximated, to first order in the strength of the perturbation, as

* ' ' 6 integral 3 ' - ik.x+iwt i E[dP(k, w)P (k ,w )] = -(2p) d r dtdzs dzs e G (k, w,z) { [ j ' ' ik'.x-iw't]} × dL(r,t) G (k ,w ,z,zs)e i ' ' ' k* ' ' ' × mjk(k ,w ,zs,zs)G (k ,w ,zs), (84)

where the source covariances

express the assumptions of stationarity and horizontal spatial homogeneity (the functions $mij$ are equal to the functions $Mij$ within a multiplicative factor, see Section 3.2). In Equation (84

) summation over the repeated indices, $i$ , $j$ , and $k$ , is assumed. This expression is accurate to first order in the strength of the perturbation to the background model, which appears in the operator $dL$ .

4.5.2 An example calculation

Let us now connect the result of the previous section with the results of Woodard and Fan (2005 ). In real space, Woodard and Fan (2005 ) use

$dL(r,t) = 2r0(z)U (r,t) . \~/ r@t (86)$

to model the effect of a flow $U$ . This approximation neglects any possible effect of the flow on the wave sources and wave damping. Woodard and Fan (2005

) expand the velocity field in Fourier components, in the horizontal directions and time, and known functions of depth. For the sake of a relatively simple example, let us look at the special case of a vertical flow of the form

where the vector $q$ is a horizontal wavevector and $s$ is a temporal frequency. Under these assumptions, we have

Let us further assume that the sources are all located at a particular depth $zs$ and that only the $j = k = z$ component of $mjk$ is non-zero, again only for the sake of simplicity. This is done here only to avoid carrying the integrals over $zs$ and $' zs$ . Then Equation (84

) becomes

Equation (89

) is already quite informative. First of all, note that a velocity field with horizontal wavenumber $q$ only correlates components of the wavefield whose horizontal wavenumbers differ by $q$ . This is sensible. Likewise, velocity fields with harmonic time dependence with frequency $s$ only couple components of the wavefield whose frequencies differ by $s$ . The above result can be reduced to the equations of Woodard and Fan (2005

) by inserting the normal summation approximations for the Greens function’s (see Birch et al., 2004 , and Section 3.3.2) into the above result.

Woodard and Fan (2005 ) make the further approximation that oscillations of different radial orders are excited incoherently. This is certainly not the case in reality, as evidenced by the asymmetry in the power spectrum. It is not clear what effect this approximation will have on the final result, as most of the power is near the resonances, where line-asymmetry is not important.

Notice that we also need to compute the term $E [P(k, w)dP*(k', w')]$ as the final result that we wish to obtain is the first order change in $E [P(k, w)P*(k', w')]$ introduced by a small change in the model. This second term can be computed in exactly the same manner as the $* ' ' E [dP(k, w)P (k ,w )]$ term, which we have already done.

4.5.3 Inverse problem

So far we have only addressed the forward problem. The other half of the direct-modeling approach is the inverse problem, that is inferring the flows in the Sun from the Fourier domain correlations seen in the data. The inverse problem in direct modeling has been solved as a linear least-squares problem (Woodard, 2002 ; Woodard and Fan, 2005 ). A crucial observation is that a flow with horizontal wavenumber $q$ only couples modes with wavenumbers $k$ and $k'$ when $||k - k'|| = q$ . This is a result of the assumption that the background model is plane-parallel and translation invariant. As a result the different wavenumber components of the flow can be inferred one at a time, in the spirit of the MCD inversion scheme (Jacobsen et al., 1999 ). The same argument applies in the time domain, and flows can be inferred one frequency component at a time.

Woodard (2002 ) tested his direct-modeling approach on a quiet sun region using MDI data. The method was used to invert for near-surface supergranular-scale flow. Although many approximations were made in carrying out the inversion, general agreement (a correlation coefficient of 0.68) was found between the Doppler component of the seismically inferred flow in the photosphere and the directly observed surface Doppler signal (see Figure 29 ). Thus direct modelling appears to be a very promising technique.

Figure 29:

Line-of-sight velocity maps showing supergranular-scale flow, based on SOHO/MDI high spatial resolution Dopplergrams. Left panel: Line-of-sight projection of near-surface flows inferred from direct modeling of seismic data. Right panel: Velocity map obtained by averaging the $16 hr$ sequence of Dopplergrams used in the seismic analysis. From Woodard (2002 ).