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4.5 Direct modeling

Woodard (2002Jump To The Next Citation Point) 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 (2002Jump To The Next Citation Point) 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, δΦ. Rewriting Equation (30View Equation) in terms of Fourier coordinates gives

∫ δΦ (k,ω) = − (2π)3 d3r dtd2k′dω ′dz e− ik⋅x+iωt𝒢i(k, ω,z) { [ s ]} × δℒ Gj (k′,ω′,z,z )eik′⋅x− iω′t S0 (k′,ω′,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 δS were ignored for the sake of simplicity. To obtain Equation (83View Equation) we used the fact that the operator δℒ, 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 ω to a source at k′ and ω′. 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 δℒ explicitly depends on r and t. Using Equation (83View Equation) the correlations of the wavefield in the Fourier domain can be approximated, to first order in the strength of the perturbation, as
∗ ′ ′ 6∫ 3 ′ − ik⋅x+iωt i E [δΦ(k, ω)Φ (k ,ω )] = − (2π) d r dt dzsdzs e 𝒢 (k, ω,z) { [ j ′ ′ ik′⋅x−iω′t]} × δℒ (r,t) G (k ,ω ,z, zs)e i ′ ′ ′ k∗ ′ ′ ′ × mjk (k ,ω ,zs,zs)𝒢 (k ,ω ,zs), (84 )
where the source covariances
E[S0(k, ω,zs)S0∗(k′,ω′,z′)] = mij (k, ω,zs,z′)δD(k − k ′)δD(ω − ω ′) (85 ) i j s s
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 (84View Equation) 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 δℒ.

4.5.2 An example calculation

Let us now connect the result of the previous section with the results of Woodard and Fan (2005Jump To The Next Citation Point). In real space, Woodard and Fan (2005Jump To The Next Citation Point) use

δℒ(r,t) = 2ρ0(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 (2005Jump To The Next Citation Point) 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
U (r,t) = zˆU (z)eiq⋅x−iσt, (87 ) z
where the vector q is a horizontal wavevector and σ is a temporal frequency. Under these assumptions, we have
[ ] δℒ (r,t) Gj (k′,ω′,z,zs)eik′⋅x−iω′t = − 2iω′ρ0(z )Uz (z)∂zGj (k′,ω′,z,zs)ei(k′+q)⋅x− i(ω′+σ)t. (88 )
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 (84View Equation) becomes
∗ ′ ′ ′ 9 ′ ′ ′ ′ z∗ ′ ′ E[δΦ (k,ω )Φ (k ,ω )] = 2iω∫ (2π) δ(k − k − q)δ(ω − ω − σ )mzz(k ,ω )𝒢 (k ,ω ,zs) i z ′ ′ × dz ρ0(z)Uz(z)𝒢 (k,ω, z)∂zG i(k ,ω ,z, zs). (89 )
Equation (89View Equation) 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 σ only couple components of the wavefield whose frequencies differ by σ. The above result can be reduced to the equations of Woodard and Fan (2005Jump To The Next Citation Point) 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 (2005Jump To The Next Citation Point) 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 [Φ (k, ω)δΦ ∗(k′,ω′)] as the final result that we wish to obtain is the first order change in E [Φ (k,ω )Φ∗(k′,ω′)] introduced by a small change in the model. This second term can be computed in exactly the same manner as the ∗ ′ ′ E [δΦ (k,ω )Φ (k ,ω )] 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, 2002Jump To The Next Citation PointWoodard 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 (2002Jump To The Next Citation Point) 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 29View Image). Thus direct modelling appears to be a very promising technique.

View Image

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).


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