Fourier-Hankel spectral method
Wavefield decomposition
The Hankel spectral analysis was introduced by Braun et al. (1987) in order to study the relationship between inward and outward traveling waves around sunspots. Consider a spherical polar coordinate system (colatitude θ, azimuth ψ) with a sunspot situated on the polar axis θ = 0, and an annular region on the solar surface surrounding the sunspot (θmin < θ < θmax). The goal is to decompose the oscillation signal in the annular region, Φ(θ, ψ, t), into components of the form
$${\Phi _{Lm}}(\theta, \psi, t) = \left[ {{A_m}(L,\nu)H_m^{(1)}(L\theta) + {B_m}(L,\nu)H_m^{(2)}(L\theta)} \right]{e^{{\rm{i}}(m\psi + 2\pi \nu t)}},$$
(31)
where m is the azimuthal order, L = [l(l + 1)]1/2, l is the spherical harmonic degree, ν is the temporal frequency, and \(H_{m}^{(1,2)}\) are Hankel functions of the first and second kind. Hankel functions are used as numerical approximations,
$$H_m^{(1,2)}(L\theta) \simeq {(- 1)^m}{{(l - m)!} \over {(l + m)!}}\left[ {P_l^m(\cos \theta) \pm {{2i} \over \pi}Q_l^m(\cos \theta)} \right],$$
(32)
to the more exact combination of the Legendre functions P and Q used in spherical geometry. This approximation, valid in the limit l ≫ m, is always good in practice and is not a limitation of the technique (Braun, 1995). For reference, we recall that in the far-field approximation (θ ≫ 1/L),
$$H_m^{(1,2)}(L\theta) \simeq \sqrt {{2 \over {\pi L\theta}}} {e^{\pm {\rm{i}}(L\theta - m\pi/2 - \pi/4)}},$$
(33)
which makes clear that the functions Am(L, ν) and Bm(L, ν) represent the complex amplitudes of the incoming and outgoing waves respectively. The wave amplitudes are computed according to (Braun et al., 1992):
$${A_m}({L_i},{\nu _j}) = {{{C_i}} \over {2\pi T}}\int_0^T {dt} \int_0^{2\pi} {d\psi} \int_{{\theta _{\min}}}^{{\theta _{\max}}} \theta \;d\theta \;\Phi (\theta, \psi, t)H_m^{(2)}({L_i}\theta){e^{- {\rm{i}}(m\psi + 2\pi {\nu _j}t)}},$$
(34)
$${B_m}({L_i},{\nu _j}) = {{{C_i}} \over {2\pi T}}\int_0^T {dt} \int_0^{2\pi} {d\psi} \int_{{\theta _{\min}}}^{{\theta _{\max}}} \theta \;d\theta \;\Phi (\theta, \psi, t)H_m^{(1)}({L_i}\theta){e^{- {\rm{i}}(m\psi + 2\pi {\nu _j}t)}},$$
(35)
In these expressions, T is the duration of the observation and Ci ≃ πLi/(2Θ) is a normalisation constant, where Θ = θmax − θmin. The procedure to perform the numerical transforms is discussed by Braun et al. (1988). The time and azimuthal transforms use the standard fast Fourier transform algorithm, with azimuthal orders in the range |m| < 20. In particular, the frequency grid is given by νj = jΔν, where Δν = 1/T and j is an integer value. The spatial Hankel transform is computed for a set of discrete values Li, that provide an orthogonal set of Hankel functions:
$$\int_{{\theta _{\min}}}^{{\theta _{\max}}} {H_m^{(1)}} ({L_i}\theta)H_m^{(2)}({L_k}\theta)\;\theta \;d\theta = 0\quad \quad {\rm{for}}\quad \quad i \ne k.$$
(36)
This condition is approximately satisfied for Li = iΔL, where ΔL = 2π/Θ and i is an integer (Braun et al., 1988). The maximum degree is given by the Nyquist value Lmax = π/Δθ, where Δθ is the spatial sampling, while the minimum degree below which the orthogonality condition ceases to be valid is Lmin = m/θmin. We note that the outer radius of the annulus, θmax, must not be too large since waves should remain coherent over the travel distance 2R⊙θmax. This implies that θmax < uτ/(2R⊙), where u(L, ν) and τ (L, ν) are respectively the group velocity and the lifetime of the waves under study. In addition, the travel time across the annulus should be less than the duration of the observation T (waves must be observed first as incoming waves and later as outgoing waves). This implies θmax < uT/(2R⊙).
Absorption coefficient
The original motivation for the Hankel analysis was to search for wave absorption by sunspots. Braun et al. (1987) defined the absorption coefficient by
$${\alpha _m}(L,\nu) = {{{P_{{\rm{out}}}} - {P_{{\rm{in}}}}} \over {{P_{{\rm{out}}}}}} = 1 - {{\vert {B_m}(L,\nu){\vert ^2}} \over {\vert {A_m}(L,\nu){\vert ^2}}}.$$
(37)
Braun (1995) remarks that this definition of the absorption coefficient may not necessarily represent wave dissipation as it ignores mode mixing. In practice, some averaging must be done to reduce the noise level. For example, Braun et al. (1987) averaged |Am|2 and |Bm|2 over all azimuthal orders and over the frequency range 1.5 mHz < ν < 5 mHz. Braun (1995) obtained a reasonable signal-to-noise level only by averaging in frequency space over the width of a ridge (radial order n) at fixed L:
$${\alpha _m}(L,n) = 1 - {{{{\langle {\vert {B_m}(L,\nu){\vert ^2} - {\lambda _m}(L,\nu)} \rangle}_n}} \over {{{\langle {\vert {A_m}(L,\nu){\vert ^2} - {\lambda _m}(L,\nu)} \rangle}_n}}},$$
(38)
where the angle brackets denote a frequency average over the width of the n-th ridge around the frequency of the mode (L, n). The function λm(L, ν) is introduced to remove the background power (see Figure 8). Braun (1995) also made averages over all azimuthal orders, to obtain an absorption coefficient for each ridge and each wavenumber denoted by α(L, n). Hankel analysis revealed that sunspots are strong absorbers of incoming p and f modes. As a check, the same analysis applied to quiet-Sun data does not show significant absorption.
Phase shifts
Braun et al. (1992) and Braun (1995) were successful at measuring the relative phase shift between outgoing and ingoing waves. Phase shifts measurements require a lot of care. Braun et al. (1992) pointed out that the phases of Am and Bm represent averages over a resolution element with size (ΔL, Δν). For example, Braun (1995) has ΔL = 20 and Δν = 4 µHz.
The finite resolution in wavenumber introduces spurious phases. This can be seen by considering first the A-transform (Equation (34)) of an incoming wave with amplitude |A|, wavenumber L, azimuthal order m, frequency ν, and phase φin. Using the far-field approximation of the Hankel functions, we have
$${{A_m}({L_i},{\nu _j}) \simeq {{\vert A\vert {C_i}} \over {T\Theta}}\int_0^T {{e^{{\rm{i}}2\pi (\nu - {\nu _j})t + i{\phi _{{\rm{in}}}}}}dt} \int_{{\theta _{\min}}}^{{\theta _{\min}}} {{e^{{\rm{i}}{{(L - {L_i})}^\theta}}}d\theta}}$$
(39)
$$\matrix{{= \vert A\vert {\rm{sinc}}[\pi (\nu - {\nu _j})T]{\rm{sinc}}[(L - {L_i})\Theta/2]} \hfill \cr {\quad \times \exp [{\rm{i}}{\phi _{{\rm{in}}}} + {\rm{i}}\pi (\nu - {\nu _j}) - {\rm{i}}({L_i} - L)({\theta _{\min}}{+}{\theta _{{\rm{max}}}})/{\rm{2}}],} \hfill \cr}$$
(40)
where sinc x = sin x/x. The B-transform (Equation (35)) applied to an outgoing wave with amplitude |B|, wavenumber L, azimuthal order m, frequency ν, and phase φout, gives
$$\matrix{{{B_m}({L_i},{\nu _j}) \simeq \vert B\vert {\rm{sinc[}}\pi (\nu - {\nu _j})T{\rm{]sinc}}[(L - {L_i})\Theta/2]} \hfill \cr {\quad \quad \quad \quad \;\; \times \exp [{\rm{i}}{\phi _{{\rm{out}}}} + {\rm{i}}\pi (\nu - {\nu _j}) + {\rm{i}}({L_i} - L)({\theta _{\min}} + {\theta _{\max}})/2].} \hfill \cr}$$
(41)
Thus the phase difference between the outgoing and incoming waves measured at grid point (Li, νj) is given by
$$\Delta {\phi ^{{\rm{meas}}}}({L_i},{\nu _j}) \simeq \Delta \phi (L,\nu) + ({L_i} - L)({\theta _{\min}} + {\theta _{\max}}),$$
(42)
where Δφ = φout − φin is the correct phase difference at wavenumber L, and (Li − L)(θmax + θmin) is a spurious phase shift. (The notations here are not the same as those of Braun (1995).)
Let us now consider actual observations. The p modes are distributed along ridges corresponding to different radial orders n, whose frequency positions in the L-ν diagram are specified by known functions νn(L). There may be several unresolved modes in the resolution element (ΔL, Δν). The condition θmax < uT/(2R⊙) (Section 4.1.1) is equivalent to ΔL > 2Δν/νn′(L), where the prime denotes a derivative. This means that the spread in L of the ridge across the frequency step Δν is much smaller than the wavenumber resolution ΔL. In this case Braun et al. (1992) estimate the spurious phase shift from Equation (42) by letting L equal to the mean wavenumber L of the unresolved modes. Since \({\nu _j} - {\nu _n}({L_i}) \simeq \nu _n^{\prime}({L_i})(\bar L - {L_i})\), the frequency dependence of the spurious shifts is given by (Braun et al., 1992)
$$\Delta {\phi ^{{\rm{spur}}}}({L_i},{\nu _j}) \simeq - {{{\theta _{\max}} + {\theta _{\min}}} \over {\nu _n^{\prime}({L_i})}}[{\nu _j} - {\nu _n}({L_i})].$$
(43)
Figure 9 shows phase shifts measurements for quiet-Sun data (Braun, 1995). The predicted values of the spurious shifts, given by the above equation, agree with the observations. The corrected average phase shift is obtained from the phase of BmAm* exp(−iΔφspur) averaged over m and some frequency range. As expected for the quiet Sun, the average phase shift is zero once the spurious shift has been subtracted. This control experiment shows, in particular, that the values of νn (Li) and (Li) interpolated from modern p-mode frequency tables are precise enough for the analysis.
Mode mixing
Using a similar technique, Braun (1995) measured correlations between incoming and outgoing waves with different radial orders n and n′. He found that sunspots introduce non-zero m-averaged phase shifts for n = n′ ± 1. This observation gives hope for a full characterization of the sunspot-wave interaction, encapsulated in the scattering matrix Sij defined by
$${B_i} = \sum\limits_j {{S_{ij}}} {A_j},$$
(44)
where the indices i and j refer to individual modes (l, n, m), the Aj are the complex amplitudes of the incoming waves, and the Bi are the complex amplitudes of the outgoing waves. Mode mixing introduced by a scatterer shows as off-diagonal elements in the scattering matrix. All techniques of local helioseismology are based on some knowledge of the scattering matrix, implicitly or explicitly.
Ring-diagram analysis
Local power spectra
Ring-diagram analysis is a powerful tool to infer the speed and direction of horizontal flows below the solar surface by observing the Doppler shifts of ambient acoustic waves from power spectra of solar oscillations computed over patches of the solar surface (typically 15° × 15°). Thus ring analysis is a generalization of global helioseismology applied to local areas on the Sun (as opposed to half of the Sun).
The ring diagram method is based on the computation and fitting of local k − ω power spectra and was first introduced by Hill (1988). A small patch is tracked as it moves across the disk. In this process, images are remapped onto a projection grid, such as the cylindrical equal-area projection, Postel’s azimuthal equidistant projection, or the transverse cylindrical equidistant projection (see, e.g., Corbard et al., 2003). A series of tracked images form a data cube, i.e., the surface Doppler velocity as a function of the two spatial coordinates and time Φ(x, t). The data cube is Fourier transformed to obtain Φ(k, ω), where k is the horizontal wavevector and ω is the angular frequency (approximately a plane-wave decomposition). The three-dimensional power spectrum of the resulting data cube, P(k, ω) = |Φ(k, ω)|2, yields the basic input data for ring diagrams. As in the global power spectrum, the main features are the ridges corresponding to the normal modes of the Sun. As there are two wavenumber directions, the ridges now appear as rings when seen in a cut through the spectrum at constant frequency (Figure 10). Flows in regions, over which the power spectrum is computed, introduce Doppler shifts in the oscillation spectrum, and changes in sound speed alter the locations of the rings. Thus by fitting the positions and shapes of the rings in the power spectrum the subsurface flows and sound-speed can be estimated.
Measurement procedure
One method of fitting local power spectra was described by Schou and Bogart (1998) (see also Haber et al., 2002; Howe et al., 2004). The first operation is to construct cylindrical cuts at constant k = ∥k∥ in the 3D power spectrum, denoted by Pk(ψ, ω), where ψ gives the direction of k (the angle between \(\hat{x}\) and k) and ω is the angular frequency. The power spectrum is then filtered to remove all but the lowest azimuthal orders m in the expansion of the form
$${P_k}(\psi, \omega) = \sum\limits_m {{a_m}} (\omega)\cos m\psi + {b_m}(\omega)\sin m\psi.$$
(45)
For each p-mode ridge (radial order n), the filtered power spectrum is fitted at constant wavenumber using the maximum likelihood method (see, e.g., Anderson et al., 1990) with a model of the form
$${P_{{\rm{fit}}}}(\psi, \omega) = {A \over {1 + {{(\omega - {\omega _0} - k{U_x}\cos \psi - k{U_y}\sin \psi)}^2}/{\gamma ^2}}} + B{k^{- 3}}.$$
(46)
Here A is an amplitude, γ is the half linewidth, and ω0 is the frequency at resonance. The ring fit parameters Ux and Uy correspond to a depth-averaged horizontal flow U = (Ux, Uy) causing a Doppler shift k · U. The background noise is modeled by Bk−3.
Other fitting techniques are described by Patrón et al. (1997) and Basu et al. (1999). Essentially all ring-diagram fitting has been done assuming symmetric profiles for the ridges in the power spectrum. Basu and Antia (1999) concluded that including line asymmetry in the model power spectrum improves the fits, but does not substantially change the inferred flows.
The most important parameters used in ring-diagram analysis are the flow parameters Ux and Uy, and the central frequency ω0. For each radial order n and wavenumber k (or spherical harmonical degree l) corresponds a new set of values: It is convenient to use the notations \(U_{x}^{nl},U_{y}^{nl}\) for the flow parameters, and δwnl for the difference between the fitted mode frequency and the mode frequency calculated from a standard solar model.
Depth inversions
The goal of inverting ring fit parameters is to determine the sound speed, density, and mass flows in the region underneath the tile over which the local power spectrum was computed. Generally, the assumption is that the sound speed, density, and flows are only functions of depth within the region covered by a particular tile. The forward problem for ring diagrams has traditionally been done by analogy with global mode helioseismology.
Inversions for sound speed and density are done using the mode frequencies determined from the ring fitting (e.g., Basu et al., 2004). The changes in the mode frequencies can be related to changes in the sound speed and density according to (Dziembowski et al., 1990):
$${{\delta {\omega _{nl}}} \over {{\omega _{nl}}}} = \int {dz} \;K_{c,\rho}^{nl}(z){\delta \over c}(z) + \int {dz} \;K_{\rho, c}^{nl}(z){{\delta \rho} \over \rho}(z) + {{F({\omega _{nl}})} \over {{I_{nl}}}}.$$
(47)
Here the kernel \(K_{c,\rho}^{nl}\), is the sensitivity of the frequency of the mode (n, l) to a fractional change in the sound speed at constant density, and \(K_{\rho,c}^{nl}\) is the sensitivity to the fractional change in density at constant sound speed. The function F is a smooth function, I is the mode inertia, and F/I gives the effect of near-surface conditions and non-adiabatic effects on the mode frequencies. We recall that mode inertia is the total kinetic energy of the mode divided by the square of the mode velocity at the solar surface; it has the units of mass and is independent of the choice of normalization of the mode eigenfunction. The kernels K can be computed using normal mode theory (for details see the Living Reviews paper on global helioseismology, Thompson (2005)). The function F is determined as part of the inversion procedure. The inversion problem is then to determine δc/c(z) and δρ/ρ(z) given a set of observed δwnl.
Horizontal flows in the solar interior are related to a set of ring fit parameters \(U_{x}^{nl}\) and \(U_{y}^{nl}\), according to
$$U_x^{nl} = \int {dz} \;K_v^{nl}(z){v_x}(z),$$
(48)
$$U_y^{nl} = \int {dz} \;K_v^{nl}(z){v_y}(z),$$
(49)
where vx and vy are the two horizontal components of the flow velocity, assumed to be functions of depth only. As for the kernels for sound speed and density, the flow kernel \(K_{v}^{nl}(z)\) can be obtained from normal mode theory (flow parameters can be converted to frequency splittings and standard global mode methods applied). The inversion problem is then to use the observed \(U_{x}^{nl}\) and \(U_{y}^{nl}\) to estimate vx(z) and vy(z).
As we have seen, the general one-dimensional ring diagram inversion problem is to use a set of equations of the form
$${d_i} = \int {dz} \;{K^i}(z)f(z)$$
(50)
together with the knowledge of the kernels Ki and a set of observed data di to determine the model function f(z). The two main approaches that have been used for the inversion of ring diagram parameters are the regularized least squares (RLS) and the optimally localized averages (OLA) methods.
The RLS method (see, e.g., Hansen, 1998; Larsen, 1998) selects the model that minimizes a combination of the misfit between the observed data and the data predicted by the model and some function of the model, for example the integral of the square of the second derivative. In particular RLS minimizes a function of the form
$${\chi ^2} = \sum\limits_i {{1 \over {\sigma _i^2}}} {\left[ {{d_i} - \int {dz} \;{K^i}(z)f(z)} \right]^2} + \lambda {\cal R}f.$$
(51)
Here we have assumed a diagonal error covariance matrix with σi being the standard deviation of the error on the measurement di. The regularization operator ℛ takes the function f(z) and returns a scalar. The regularization parameter which controls the relative importance of minimizing the misfit to the data and smoothing the solution is λ. Example resolution kernels for RLS inversion are plotted in Figure 12.
The OLA method (Backus and Gilbert, 1968) attempts to produce localized averaging kernels while simultaneously controlling the error magnification. For a discussion of OLA in ring diagram inversions see Basu et al. (1999) and Haber et al. (2004). The essential idea in OLA methods is that the estimate f of the model f at a particular depth, can be written as a linear combination of the data
$$\tilde f({z_0}) = \sum\limits_i {{a_i}} ({z_0}){d_i},$$
(52)
so that we can write
$$\tilde f({z_0}) = \int {dz} \;\kappa (z,{z_0})f(z),$$
(53)
with the averaging kernels κ given by
$$\kappa (z,{z_0}) = \sum\limits_i {{a_i}} ({z_0}){K^i}(z).$$
(54)
The OLA method then attempts to choose the coefficients ai(z0) so as to produce averaging kernels κ that are well localized while at the same time controlling the error variance \(\sum\nolimits_{i}a_{i}^{2}(z_{0})\sigma_{i}^{2}\), on the estimate \(\tilde{f}(z_{0})\).
Time-distance helioseismology
The purpose of time-distance helioseismology (Duvall Jr et al., 1993) is to measure and interpret the travel times of solar waves between any two locations on the solar surface. A travel time anomaly contains the seismic signature of buried inhomogeneities within the proximity of the ray path that connects two surface locations. An inverse problem must then be solved to infer the local structure and dynamics of the solar interior (see, e.g., Jensen, 2003, and references therein).
Fourier filtering
The first operation in time-distance helioseismology is to track Doppler images at a constant angular velocity to remove the main component of solar rotation, as is done in ring-diagram analysis (see Section 4.2.1). Postel’s azimuthal equidistant projection is often used in time-distance helioseismology. The resulting data cube is Fourier transformed to obtain Φ(k, ω).
A filtering procedure is then applied to the data (Duvall Jr et al., 1997). First, frequencies below 1.5 mHz are filtered out in order to remove granulation and supergranulation noise. The data are further filtered to select parts of the wave propagation diagram.
In the case of p modes, a phase speed filter is applied to the data, and the f-mode ridge is filtered out. This choice of filter is based on the fact that acoustic waves with the same horizontal phase speed ω/k travel the same horizontal distance Δ (see Bogdan, 1997). Thus, to measure the travel time for acoustic waves propagating between two surface locations, it is appropriate to consider only those waves with the same phase speed. The choice of the phase speed depends on the travel distance. A list of Gaussian phase-speed filters,
$${F_i}(k,\omega) = \exp [ - {(\omega/k - {v_i})^2}/(2\delta v_i^2)],$$
(55)
is provided in Table 1 for various distances. Here, vi is the mean phase speed, δvi is the dispersion, and k = ∥k∥ is the wavenumber. The filtered signal is given by Ψ(k, ω) = Fi(k, ω)Φ(k, ω). The larger the phase speed, the deeper p-mode wavepackets penetrate inside the Sun.
Table 1: Parameters of Fourier filters used in p-mode time-distance helioseismology. The Gaussian phase-speed filters are defined by \({\rm exp}{{[-(\omega/k - v_i)}^{2}}/(2 \delta v_{i}^{2})]\). Courtesy of T.L. Duvall and S. Couvidat.
A separate filtering procedure is applied for surface gravity waves, which are used to probe the near surface layer. In this case the filter function F (k, w) is 1 if \(\vert\omega\pm\sqrt{gk}\vert < kU_{\rm cut}\) and 0 otherwise. The parameter Ucut controls the width of the region around the f-mode dispersion relation ω2 = gk. A reasonable choice is Ucut = 1 km s−1. This value allows for large Doppler frequency shifts introduced by flows, and does not let the p1 ridge through.
There is some freedom in the selection of the Fourier filters. For example one may construct filters that depend explicitly on the direction of the wavevector k, and not simply on the wavenumber (see, e.g., Giles, 1999).
Cross-covariance functions
The basic computation in time-distance helioseismology is the temporal cross-covariance between filtered signals at two points x1 and x2 on the solar surface,
$$C({x_1},{x_2},t) = {h \over {T - \vert t\vert}}\sum\limits_{{t^{\prime}}} \Psi ({x_1},{t^{\prime}})\Psi ({x_2},{t^{\prime}} + t),$$
(56)
where ht is the temporal sampling and T is the time duration of the observation. Times are evaluated at discrete values ti = iht. Multiplication by the temporal window function is already included in the definition of Ψ, and the sum over t′ is a short notation to mean the sum over all discrete times in the interval −T/2 ≤ t′ < T/2. The normalization factor 1/(T − |t|) is a correction that becomes significant only when T is small. We note that in practice it is faster to compute the time convolution in the Fourier domain. The positive time lags (t > 0) give information about waves moving from x1 to x2, and the negative time lags about waves moving in the opposite direction.
The cross-correlation is useful as it is a phase-coherent average of inherently random oscillations. It can be seen as a solar seismogram, providing information about travel times, amplitudes, and the shape of the wave packets traveling between any two points on the solar surface. Figure 13 shows a theoretical cross-covariance Δ as a function of distance between x1 and x2, and time lag t. This time-distance diagram was computed for a spherically symmetric solar model (see Sekii and Shibahashi, 2003). The first ridge corresponds to acoustic waves propagating between the two points without additional reflection from the solar surface. The next ridge corresponds to waves which arrive after one reflection from the surface, and the ridges at greater time delays result from waves arriving after multiple bounces. The backward branch associated with the second ridge corresponds to waves reflected on the far-side of the Sun. In most applications, only the direct (first-bounce) travel times are measured. Figure 14 shows the p-mode cross-covariance function at short distances, computed with the Fourier filters given in Table 1. In this example, cross-covariance functions were averaged over pairs of points at fixed distance to reduce noise. An example cross-covariance function for the f-mode ridge is plotted in Figure 15.
The travel times of the wave packets are measured from the (first-bounce) cross-covariance function. Local inhomogeneities in the Sun will affect travel times differently depending on the type of perturbation. For example, temperature perturbations and flow perturbations have very different signatures. Given two points x1 and x2 on the solar surface the travel time perturbation due to a temperature anomaly is, in general, independent of the direction of propagation between x1 and x2. However, a flow with a component directed along the direction x1 → x2 will break the symmetry in travel time for waves propagating in opposite directions: Waves move faster along the flow than against it. Magnetic fields introduce a wave speed anisotropy and will have yet another travel time signature (this has not been detected yet).
Travel time measurements
Cross-covariance functions can have a large amount of realization noise due to the schochastic nature of solar oscillations, and it has proved difficult to measure travel times between two individual pixels on the solar surface. Some averaging is often required. The time over which the cross-covariances are computed is usually T ≥ 8 hr (this puts a limit on the temporal scale of the solar phenomena that can be studied). In order to further enhance the signal-to-noise ratio, Duvall Jr et al. (1993) and Duvall Jr et al. (1997) suggested to average C(x1, x2, t) over points x2 that belong to an annulus or quadrants centered at x1. For instance, to measure flows in the east-west direction in the neighborhood of point x, cross-covariances are averaged over two quadrant arcs \({\cal A}_{\rm w}(x,\Delta)\) and \({\cal A}_{\rm e}(x,\Delta)\) that include points a distance Δ from x:
$${C_{{\rm{ew}}}}(x,t;\Delta) = \sum\limits_{{x^{\prime}} \in {{\cal A}_{\rm{w}}}} {C(x,{x^{\prime}},t)} + \sum\limits_{{x^{\prime}} \in {{\cal A}_{\rm{e}}}} {C({x^{\prime}},x,t)},$$
(57)
Figure 16 shows the geometry of the averaging procedure. An example cross-covariance Cew is shown in Figure 17 in the f-mode case. Correlations at positive times (t > 0) correspond to waves that propagate westward, and correlations at t < 0 to waves that propagate eastward. As shown in Figure 17, cross-covariances can be further averaged along lines of constant phase within a range of distances. Likewise an average cross-correlation Csn is constructed from south-north quadrants to measure flows in the meridional direction. Another average Cann is obtained by averaging over the whole annulus,
$${C_{{\rm{ann}}}}(x,t;\Delta) = \sum\limits_{{x^{\prime}} \in {\cal A}} {C(x,{x^{\prime}},t),} $$
(58)
where \({\cal A}(x,\Delta)\) is an annulus of radius Δ centered at x. This average can be used to measure separately the waves that propagate inward and outward from the central point.
At fixed x and Δ, the cross-covariance function oscillates around two characteristic (first-bounce) times t = ±tg. Center-to-annulus or center-to-quadrants travel times are often measured by fitting Gaussian wavelets (Duvall Jr et al., 1997; Kosovichev and Duvall Jr, 1997). This procedure distinguishes between group and phase travel times by allowing both the envelope and the phase of the wavelet to vary independently. The positive-time part of the cross-correlation is fitted with a function of the form
$${w_+}(t) = A\exp [ - {\gamma ^2}{(t - {t_{\rm{g}}})^2}]\cos [{\omega _0}(t - {\tau _+})],$$
(59)
where all parameters are free, and the negative-time part of the cross-correlation is fitted separately with
$${w_-}(t) = A\exp [ - {\gamma ^2}{(t - {t_{\rm{g}}})^2}]\cos [{\omega _0}(t - {\tau_-})].$$
(60)
The times τ+ and τ− are the so-called phase travel times. The basic observations in time-distance helioseismology are the travel time maps τ+(x, Δ) and τ−(x, Δ), measured for each of the three averaged cross-correlations Cew(x, t; Δ), Csn(x, t; Δ), and Cann(x, t; Δ). The travel time differences τdiff = τ+ − τ− are mostly sensitive to flows while the mean travel times τmean = (τ− + τ+)/2 are sensitive to wave-speed perturbations. Maps of measured travel times are shown in Figure 18: Most of the signal in these maps is due to supergranular flows (15–30 Mm length scales). We note that an alternative definition of travel time can be obtained by fitting a model cross-covariance function to the data (Gizon and Birch, 2002). This last definition is often used in geoseismology (see, e.g., Zhao and Jordan, 1998).
In order to maximize the potential resolution of time-distance helioseismology it is desirable to obtain travel times from cross-covariances measured with shorter T and with as little spatial averaging as possible. However conventional fitting methods will fail when the cross-covariance is too noisy. A new robust definition of travel time was introduced by Gizon and Birch (2004) to measure travel times between individual pixels and T as short as 2 hr. According to this definition, the point-to-point travel time for waves going from x1 to x2, denoted by τ+(x1, x2), and the travel time for waves going from x2 to x1, denoted by τ−(x1, x2), are given by
$${\tau_\pm}({x_1},{x_2}) = {h_t}\sum\limits_t {{W_\pm}} ({\bf{\Delta}},t)[C({x_1},{x_2},t) - {C^0}({\bf{\Delta}},t)],$$
(61)
with the weight functions W± given by
$${W_\pm}({\bf{\Delta}},t) = {{\mp f(\pm t)\partial {C^0}({\bf{\Delta}},t)} \over {{h_t}\sum\nolimits_{{t^{\prime}}} f (\pm {t^{\prime}}){{[{\partial _{{t^{\prime}}}}{C^0}({\bf{\Delta}},{t^{\prime}})]}^2}}}.$$
(62)
In this expression C0 is a (smooth) reference cross-covariance function computed from a solar model and Δ = x2 − x1. The window function f(t′) is a one-sided function (zero for t′ negative) used to separately examine the positive- and negative-time parts of the cross-correlation. The window f(t′) is used to measure τ+, and f(−t′) is used to measure τ−. A standard choice is a window that isolates the first-skip branch of the cross-covariance.
This definition has a number of useful properties. First, it is very robust with respect to noise. The fit reduces to a simple sum that can always be evaluated whatever the level of noise. Second, it is linear in the cross-covariance. As a consequence, averaging various travel time measurements is equivalent to measuring a travel time on the average cross-covariance. This is unlike previous definitions of travel time that involve non-linear fitting procedures. Third, the probability density functions of τ+ and τ− are unimodal Gaussian distributions. This means, in particular, that it makes sense to associate an error to a travel time measurement. The Born sensitivity kernels discussed in Section 4.3.5 were derived according to this definition of travel times.
Noise estimation
In global helioseismology, it is well understood that the precision of the measurement of the pulsation frequencies is affected by realization noise resulting from the stochastic nature of the excitation of solar oscillations (see, e.g., Woodard, 1984; Duvall and Harvey, 1986; Libbrecht, 1992; Schou, 1992). It is important to study these properties since the presence of noise affects the interpretation of travel time data. In particular, the correlations in the travel times must be taken into account in the inversion procedure.
An interesting approach, pioneered by Jensen et al. (2003a), consists of estimating the noise directly from the data by measuring the rms travel time within a quiet Sun region. The underlying assumptions are that the fluctuations in the travel times are dominated by noise, not by ‘real’ solar signals, and that the travel times measured at different locations can be seen as different realizations of the same random process. By ‘real’ solar signals we mean travel time perturbations due to inhomogeneities in the solar interior that are slowly varying over the time of the observations. Jensen et al. (2003a) studied the correlation between the center-to-annulus travel times as a function of the distance between the central points, at fixed annulus radius.
Gizon and Birch (2004) derived a simple model for the full covariance matrix of the travel time measurements. This model depends only on the expectation value of the filtered power spectrum and assumes that solar oscillations are stationary and homogeneous on the solar surface. The validity of the model is confirmed through comparison with MDI measurements in a quiet Sun region. Gizon and Birch (2004) showed that the correlation length of the noise in the travel times is about half the dominant wavelength of the filtered power spectrum. The signal-to-noise ratio in quiet-Sun travel time maps increases roughly like the square root of the observation time and is maximum for a distance near half the length scale of supergranulation.
Travel time sensitivity kernels
In this section we show example computations of travel time sensitivity kernels. Kernels are the functions Kα that connect small changes in the solar model with small changes in travel times
$${\delta _\tau}({x_1},{x_2}) = \sum\limits_\alpha {\int_\odot {{d^3}}} r\;{K_\alpha}({x_1},{x_2};r)\delta {q_\alpha}(r).$$
(63)
The sum over α is taken over all possible relevant types of perturbations to the model, for example local changes in sound speed, density, flows, magnetic field, or source properties. For each type of perturbation a there is a corresponding kernel Kα(x1, x2; r) that depends on a, the observation points x1 and x2 that the travel time is measured between, and a spatial location r that ranges over the entirety of the solar interior. In this section we will describe the various types of approximate calculations of the Kα that have been done.
Notice that kernels can be computed for quantities other than travel times. For example, in geophysics Dahlen and Baig (2002) computed the first-order sensitivity of the amplitude of the observed waveform to a small local change in sound speed. In particular it might be helpful to compute kernels for the mean frequency and amplitude of the cross-correlations, with the aim of using these quantities to help constrain inversions.
Ray approximation
The first efforts at computing the sensitivity of travel times to changes in the solar model (see, e.g., Kosovichev, 1996; D’Silva et al., 1996; D’Silva, 1996; Kosovichev and Duvall Jr, 1997) were based on the ray approximation. In the ray approximation the travel time perturbation is approximated as an integral along the ray path. Fermat’s principle (see, e.g., Gough, 1993) says that in order to compute the first order change in the travel time we do not need to compute the first order change in the ray path, i.e., we can simply take the integral over the unperturbed ray path. In particular, we have (Kosovichev and Duvall Jr, 1997):
$${\delta _\tau}({x_1},{x_2}) = - \int_\Gamma {ds} \left[ {{{\hat n \cdot U} \over {{c^2}}} + {1 \over {{v_{\rm{p}}}}}{{\delta c} \over c} + \left({{{\delta {\omega _{\rm{c}}}} \over {{\omega _{\rm{c}}}}}} \right){{\omega _{\rm{c}}^2{v_{\rm{p}}}} \over {{\omega ^2}{c^2}}} + {1 \over 2}\left({{{c_{\rm{A}}^2} \over {{c^2}}} - {{{{(k \cdot {c_{\rm{A}}})}^2}} \over {{k^2}{c^2}}}} \right)} \right],$$
(64)
where the ray path connecting x1 and x2 is denoted by Γ. The increment of path length is ds and \(\hat{n}\) is a unit vector directed along the path, in the sense of going from x1 to x2. The other quantities in Equation (64) are the change in the sound speed δc, the mass flow U, the Alfvén velocity \(c_{\rm A}=B/\sqrt{4\pi\rho}\), the acoustic cut-off frequency ωc, and the phase speed vp = ω/k.
Ray theory is based on the assumption that the perturbations to the model are smooth and that the wavepacket frequency bandwith is very large. Bogdan (1997) showed that the energy density of a realistic wavepacket was substantial away from the ray path. This result strongly suggested that perturbations located away from the ray path could have substantial effects on travel times. It is now well known that ray theory fails when applied to perturbations that are smaller that the first Fresnel zone (see, e.g., Hung et al., 2001; Birch et al., 2001).
Finite-wavelength kernels in the single-source approximation
The three approximations that have been used to treat small scale perturbations for time-distance helioseismology are the Born approximation, the Rytov approximation, and the Fresnel zone approximation. Here we describe some results that have been obtained using these methods in the single-source approximation. The single-source approximation, which was motived by the “Claerbout Conjecture” (Rickett and Claerbout, 1999) and by analogy with previous ray-theory based work, says that the one-way travel time perturbation can be found by looking at the time-shift of a wavepacket created at one observation location and then observed at the other. Gizon and Birch (2002) showed that the single-source approximation ignores a potentially important scattering process; this issue is discussed in detail in the paragraph on distributed source models.
Motivated by the use of the Born approximation to compute travel time sensitivities in the context of geophysics (see, e.g., Marquering et al., 1998), Birch and Kosovichev (2000) used the Born approximation to compute travel time kernels for time-distance helioseismology. Figure 19 shows an example result. The shape of the kernel is the characteristic “banana-doughnut” shape first described by Marquering et al. (1998). The kernel is hollow along the ray path, travel times are not sensitive to small-scale sound-speed perturbations located along the ray path. This has been explained in terms of wavefront healing (Nolet and Dahlen, 2000). The ringing in the kernel with distance from the ray path is a result of the finite bandwidth of the wavefield.
Another approach to producing finite-wavelength kernels is the Fresnel zone approximation (see, e.g., Sneider and Lomax, 1996; Jensen et al., 2000). The Fresnel zone approach makes a simple parametrized approximation to the kernels. In particular, in the Fresnel zone approximation, the travel time kernels, for fixed x1 and x2, are written as (Jensen et al., 2001)
$$K({x_1},{x_2};r) = {A \over {\tau ({x_2}\vert r)\tau (r\vert {x_1})}}\sin ({\omega _0}\;\Delta \tau)\exp \left[ {- {{\left({{{a\;\Delta \tau} \over {{T_0}/2}}} \right)}^2}} \right],$$
(65)
where τ(x|r) is the ray-theoretical travel time from x to r, and Δτ = τ(x1|r)+ τ(r|x2) − τ(x1|x2). The parameter ω0 is the central frequency of the wavepacket, T0 = 2π/ω0 is the corresponding period, and a is related to the frequency bandwidth of the wavepacket. The amplitude A of the kernel is determined by demanding that the total integral of the kernel be equal to τ(x1|x2), which is the ray theory value for the total integral of the kernel. Notice that the value of the kernel goes to infinity as r tends to x1 or x2. Jensen et al. (2000) showed, by comparison with direct simulation, that a two dimensional analogue of the Fresnel zone kernel (Equation (65)) was a reasonable approximation to the actual linear sensitivity of travel time to changes in the sound speed.
The Rytov approximation gives a first order correction to the phase of the wavefield, instead of a first order correction to the wavefield itself as one obtains from the Born approximation. Jensen and Pijpers (2003) used the Rytov approximation to write travel time kernels for the effects of sound-speed perturbations. The results were qualitatively similar to the Born approximation results of Birch and Kosovichev (2000). No comparison has been made, in the helioseismology literature, of the ranges of validity of the Rytov and Born approximations.
Distributed source models
Gizon and Birch (2002), motived by Woodard (1997), gave the first comprehensive treatment of the linear forward problem for time-distance helioseismology. Such a treatment must include a physical model of wave excitation which takes into account the fact that waves are stochastically excited by sources (granules) that are distributed over the solar surface (see Section 3.2). An example calculation (Figure 20) demonstrates that the single-source approximation is qualitatively incorrect, as it ignores a scattering process that can be dominant for some perturbations. Figure 20 compares an f-mode travel time kernel for local changes in the wave damping rate, computed in the single-source and distributed-source models. The hyperbolic feature in the distributed-source kernel is not present in the single-source model. As discussed by Gizon and Birch (2002), the single-source approximation ignores the scattered wave that is seen at the observation point that is treated as a source in the single-source approximation.
Birch et al. (2004) used the recipe derived by Gizon and Birch (2002) to obtain the linear sensitivity of p-mode travel times to changes in sound speed. The results showed that the sensitivity depends on the filtering that is done to the wavefield before the travel times are measured. Figure 21 shows sound-speed kernels computed for three different cases: a model with no filters, a model with an MTF (see Section 3.4) roughly like the one for MDI full-disk data, and a model with an MTF and also a phase-speed filter of the type employed in routine time-distance analysis. The three kernels are all strikingly different. In the unfiltered case, the kernels looks much like the traditional “banana-doughnut” kernel (see Figure 19). With the introduction of the MTF, the travel time becomes more sensitive to deeper sound-speed perturbations and less sensitive to the near-surface perturbations. This is because the MTF preferentially removes high wavenumbers, and thus modes with low phase speeds and shallow turning points. With the introduction of the phase-speed filter the kernel again changes drastically. The phase-speed filter removes all of the waves with lower turning point much below the turning point of the target ray. As a result, the sensitivity much below the lower turning point of the ray is greatly reduced. This example emphasizes the importance of including the filtering in models of travel time sensitivity kernels. The strong dependence of the shape of the kernels on the filtering suggests that the kernels could to be tailored by adjusting the filters that are used in the data analysis procedure.
Inversions of travel times
The inverse problem is to determine the perturbations δqα(r) to the solar model that are consistent with a particular set of observed travel times δτi. For the inverse problem, the kernel functions and the noise covariance are in general assumed to be known.
Unlike in the ring-analysis case, where a set of one dimensional inversions are performed (see Section 4.2.3), the time-distance inversion procedure is three-dimensional. So far, only RLS inversions have been implemented in time-distance helioseismology. In the RLS approach, the minimization problem reads
$${\chi ^2} = \sum\limits_i {{1 \over {\sigma _i^2}}} {\left[ {\delta {\tau _i} - \sum\limits_\alpha {\int_\odot {{d^3}} r\;K_i^\alpha (r)\delta {q_\alpha}(r)}} \right]^2} + \lambda {\cal R}\{\delta {q_\alpha}\},$$
(66)
where we have assumed, for simplicity of notation, a diagonal covariance matrix. The σi are the noise estimates for each of the travel times δ τi. The regularization parameter is λ and ℛ is the regularization operator. Notice that we have not assumed any particular parametrization of the perturbations to the solar model δqα(r). The ideal choice of regularization operator and the optimal method for choosing the regularization parameter are currently open questions.
The minimization of χ2, Equation (66), has been done using either the LSQR algorithm (Paige and Saunders, 1982), the multi-channel deconvolution (MCD; Jacobsen et al., 1999), or via singular value decomposition (Hughes and Thompson, 2003). The inputs to either of these methods are the travel time kernels, which result from the linear forward problem (see Section 3.5), the regularization operator, the choice of regularization parameter, and in general also the covariance matrix of the noise.
The central question about inversions is the degree to which any particular inversion method is able to retrieve the true Sun, e.g., the actual sound-speed variations and flows in the interior, from a set of observed travel times. The degree to which any inversion method succeeds will presumably depend on a number of things: (i) the accuracy of the forward model, (ii) the noise level in the travel times, (iii) the depths and spatial scales of the real variations in the Sun, (iv) the number and type of travel times that are available as input to the inversion, and (v) the accuracy of the travel time noise covariance matrix. The two main approaches that have been taken to study inversion methods are either to invert artificial data or to invert real data for perturbations that are relatively well known from global helioseismology or from surface measurements. It is also useful to compare the results of different inversion methods applied to the same set of observed travel times.
The approach of inverting real data is appealing as it is by definition “realistic”. This approach for time-distance was pioneered by Giles (1999) who did ray theory based inversions to measure differential rotation. The results were in approximate agreement with the results of global model inversions, which validated time-distance inversions applied to slowly-varying large scale flows. Gizon et al. (2000) inverted f-mode travel times for horizontal flows in the near-surface. Figure 22 shows a comparison between the projection of the inferred flows onto the line of sight and the directly observed line-of-sight Doppler velocity. The correlation coefficient between the two is 0.7 (Gizon et al., 2000), which shows that their inversion method, an iterative deconvolution, is able to approximately retrieve the horizontal flow velocities from the travel times.
There have been a number of efforts to validate inversions by the generation and inversion of artificial data. These tests are useful as they provide an intuitive understanding of various regularization schemes, the choice of regularization parameter, the effects of limited sets of input travel times, the effects of incorrect assumptions regarding the noise covariance, the potential resolution of any particular inversion, and the cross-talk between different model parameters. A drawback to testing inversions with artificial data is that one always wonders about the realism of the artificial data. Hopefully, in the not distant future quite realistic data will be available for the purpose of testing inversion schemes.
Couvidat et al. (2004) compared inversions done using ray-approximation kernels with inversions done with Fresnel-zone kernels. The results were quite similar for depths between the surface and the lower turning point of the deepest rays used in the inversion. Inversions done with the ray kernels cannot retrieve sound speed perturbations below the lower turning point of the deepest ray, while the Fresnel-zone kernels extend below the depth of the deepest ray. The results of Couvidat et al. (2004), as well as those of Giles (1999), suggest that even though the ray approximation overestimates travel times for small scale perturbations this effect does not seriously corrupt the large scale features found by inversion.
Another test of inversion methods was done by Zhao and Kosovichev (2003a). The two main results of this study, regarding inversions methods, were that there is cross-talk between upflows and convergent flows, and between downflows and divergent flows, and that the MCD and LSQR methods give essentially the same results when applied to quiet sun MDI data. Figure 23 shows an example of a test of the LSQR method. The initial model is shown in the top panel. Travel times were generated using the ray approximation. The travel times were then inverted using five iterations of LSQR to obtain the flow field shown in the second panel of Figure 23. Notice that there are small vertical flows near the surface at the centers and boundaries of the supergranulations cells. After 100 iterations of LSQR, these small vertical velocity features are removed (bottom panel of Figure 23).
The first end-to-end test of time distance helioseismology using artificial data was done by Jensen et al. (2003b). In this important study, artificial data were generated by numerical wave propagation through a three-dimensional stratified and horizontally inhomogeneous model. Waves were generated by stochastic sources distributed over a layer 50 km below the upper boundary of the surface of the model. The artificial wave field was “observed” at the spatial resolution and temporal cadence of MDI high-resolution observations. This artificial data set was then subjected to standard time-distance analysis, and the inversion was performed with the Fresnel-zone kernels described by Jensen and Pijpers (2003). Figure 24 shows a comparison of the original sound-speed model and the result of the inversion of the artificial data. Notice that the inversion recovers much of the original structure. The inversion result contains noise, which is to be expected from travel times computed from finite time series. The only noise source in this example is realization noise, noise resulting from the stochastic nature of the wave excitation.
Helioseismic holography
Helioseismic holography was introduced in detail by Lindsey and Braun (1990), although the basic concept was first suggested by Roddier (1975). For a recent theoretical introduction to helioseismic holography see Lindsey and Braun (2000a). The central idea in helioseismic holography is that the wavefield, e.g., the line-of-sight Doppler velocity observed at the solar surface, can be used to make an estimate of the wavefield at any location in the solar interior at any instant in time. In this sense, holography is much like seismic migration, a technique in geophysics that has been in use since the 1940’s (see, e.g., Hagedoorn, 1954, and references therein). In migration, the wave equation is used to determine the wavefield in the interior of the earth at past times, given the wavefield observed at the surface (see, e.g., Claerbout, 1985).
In Section 4.4.1 we introduce the basic computations of helioseismic holography, the ingression and egression. In Section 4.4.2 we review the various calculations that have been used to obtain the Green’s functions used in holography. In Section 4.4.3 we introduce the notion of the local control correlation. Acoustic power measurements are described in Section 4.4.4. Phase-sensitive holography is introduced in Section 4.4.5. Section 4.4.6 describes the technique of far-side imaging. In Section 4.4.7 we describe acoustic imaging (Chang et al., 1997), a technique closely related to holography.
Ingression and egression
The ingression and egression are attempts to the answer the following question: Given the wavefield observed at the solar surface, what is the best estimate of the wavefield at some point in the solar interior assuming that the observed wavefield resulted entirely from waves diverging from that point (for the egression) or waves converging towards that point (for the ingression)?
The development of holography has, historically, been motived by analogy with optics (e.g., Lindsey and Braun, 2000a) and as a result the holography literature often employs optical terminology. The target point in the solar interior at which we attempt to estimate the wavefield is termed the “focus point”. In practice, only data in a restricted area on the surface above the focus point is used to compute the egression and ingression. This region is called the “pupil”. The choice of pupil geometry depends on the particular application and will be described when we discuss particular applications of holography.
As described by Lindsey and Braun (2000a), a mathematical motivation for the ingression and egression is the Kirchoff integral solution to the wave equation (e.g., Jackson, 1975). Suppose that a scalar field Φ solves the equation
$${k^2}\Phi (r) + {{\bf{\nabla}} ^2}\Phi (r) = 0.$$
(67)
The Green’s function, G, for this problem is defined as the solution to
$${k^2}G(r) + {{\bf{\nabla}} ^2}G(r) = {\delta _{\rm{D}}}(r),$$
(68)
where δD is the 3D Dirac delta function. The Kirchoff integral equation for Φ is
$$\Phi (r) = \int_S {ds} \;[\{{\partial _n}G(r - s)\} \Phi (s) - G(r - s){\partial _{\rm{n}}}\Phi (s)].$$
(69)
The integration variable s ranges over any surface S enclosing the locations r. For a derivation see for example the introduction in Jackson (1975), noting that the Kirchoff integral equation is a special case of Green’s second identity. The normal derivative, outwards, is denoted by ∂n. Equation (69) is interesting as it says that given the value of the field on the surface of a region we can determine the value of the field anywhere inside this region. Holography is an attempt to do just that, determine the wavefield in the solar interior given the wavefield on the solar surface. As discussed in detail by Lindsey and Braun (2004), helioseismic holography is much more complicated than solving the simple scalar wave Equation (67) addressed by the Kirchoff integral.
Motivated by the Kirchoff identity (Equation 69), the egression \(H_{+}^{{\cal P}}\) and ingression \(H_{-}^{{\cal P}}\) are defined as
$$H_\pm ^{\cal P}(r,\omega) = \int_{\cal P} {{d^2}} {x^{\prime}}G_\pm ^{{\rm{holo}}}(x - {x^{\prime}},z,{\omega ^{\prime}})\Phi ({x^{\prime}},\omega).$$
(70)
The focus point is located at position r = (x, z) in the solar interior. The integration variable x′ denotes horizontal position on the solar surface and ranges over the surface region denoted by the pupil \({\cal P}\), typically an annular region, or a sector of an annular region, centered on the horizontal position of the focus point, x. For examples of pupils see Lindsey and Braun (2000a). The wavefield on the surface is Φ(x′, ω) and ω is temporal angular frequency. Recall that the egression, \(H_{+}^{{\cal P}}\), is an attempt to estimate the wavefield at the location r and frequency ω based on the waves that are seen, in the pupil P at the surface, diverging away x. Likewise, the ingression, H− is an attempt to estimate the wavefield at the location r and frequency ω based on the waves seen, within the pupil, converging towards the focus point. The holography Green’s functions, which depend on a horizontal displacement x, a focus depth z, and a temporal frequency u are denoted by \(G_{\pm}^{{\rm holo}}(r,\omega)\). In the time domain, the causal Green’s function \(G_{\pm}^{{\rm holo}}(r,t)\), obtained by temporal inverse Fourier transform from \(G_{\pm}^{{\rm holo}}(r,\omega)\), is zero for t < 0. In the time domain, the anti-causal Green’s function is zero for t > 0. The time-domain causal Green’s function and anti-causal Green’s function are related by \(G_{-}^{{\rm holo}}(r,t)=G_{+}^{{\rm holo}}(r,-t)\). The holography Green’s function are not defined as the solution to any particular set of equations, but rather we use the symbol \(G_{\pm}^{{\rm holo}}\) to denote whatever function is used in practice in the computation of the ingression and egression. There are a number of choices that have been used for these Green’s functions (see Section 4.4.2 for details).
An intuitive way to interpret Equation (70) is to view the Green’s functions as propagators. In this way, the egression can be seen as the result of using the anti-causal Green’s function \(G_{+}^{{\rm holo}}\) to propagate the surface wavefield backwards in time. Likewise, the ingression can be seen as the result of using the causal Green’s function \(G_{-}^{{\rm holo}}\) to propagate the observed surface wavefield forwards in time.
The egression and ingression are the essential quantities in helioseismic holography. There are many particular ways in which these quantities can be combined to learn about the solar interior. The main techniques are control-correlations (Section 4.4.3), acoustic power holography (Section 4.4.4), phase-sensitive holography (Section 4.4.5), and far-side imaging (Section 4.4.6) which is a special case of phase-sensitive holography. Of central importance to all of these methods is the choice of the holography Green’s function \(G_{\pm}^{{\rm holo}}(r,\omega)\), which we now describe.
Holography Green’s functions
The two approaches that have been used to construct the holography Green’s functions, \(G_{\pm}^{{\rm holo}}\), involve ray theory and wave theory.
Ray theory
Lindsey and Braun (2000a), motived by ray theory, prescribed the holography Green’s function as
$$G_\pm ^{{\rm{holo}}}(r,t) = \sum\limits_N {{A_N}} (x,z){\delta _{\rm{D}}}(t \pm {\tau _N}(x,z)),$$
(71)
where r = (x, z) and x = ∥x∥ is the horizontal distance. The role of the function \(G_{-}^{{\rm holo}}(G_{+}^{{\rm holo}})\) is to propagate the observed surface wavefield forwards (backwards) in time and into the interior of the solar model. Theses functions can also be seen as the “wavefield” response at a horizontal distance x, depth z, and time t to a surface source located at the origin at time t = 0. The observed surface wavefield can be thought of as a source, in the spirit of Huygen’s principle. The index N refers to the number of skips the ray path has taken off the solar surface. Each function AN(x, z) is the amplitude of a single skip component of the Green’s function and can be estimated from simple ray theory. In particular, in ray theory the amplitude functions AN are determined by the conservation of energy and the increase, with distance along the ray path, of the cross-sectional area of a ray bundle emerging from the source location. In Equation (71), the functions τn are the ray-theoretical travel times along the N-skip ray paths connecting a point on the surface with a point at depth z and a horizontal distance x away from the first point. The sum over skips N in Equation (71) is convenient as it can easily be truncated to estimate, for example, just the “one-skip” Green’s function, i.e., the Green’s function that contains only contributions from waves that have traveled once into the solar interior. The holography Green’s functions in the time domain (Equation 71) must then be Fourier transformed to obtain G±(r, ω) referred to in Equation (70).
Notice that there are a great many simplifications involved in writing the Green’s function in Equation (71). The substantial frequency dependence of the travel time (e.g., Jefferies et al., 1994) has been ignored. There is no account taken of damping, which is quite significant for high-degree modes (e.g., Duvall Jr et al., 1998). Furthermore, as the ray approximation requires the wavelength to be much smaller than the length scale of the variation of the background medium, the ray approximation is not expected to be valid near the solar surface. Further issues include the neglect of the buoyancy and cut-off frequencies in the calculation of τN, and also in the computation of the ray paths themselves (Lindsey and Braun 2000a). Recent work by Lindsey and Braun (2004) suggests that after empirical dispersion corrections have been applied (see Section 4.4.3), the results of ray theory calculations are quite similar to the results of wave theory calculations (see Figure 25).
Wave theory
An alternative to the ray calculation is to choose, somewhat arbitrarily, the holography Green’s functions to solve a wave equation for the vertical displacement of a fluid element. This was carried out by Lindsey and Braun (2004). The computation proceeds most simply in the 3D Fourier domain (k-ω domain). For each horizontal wavenumber and frequency component the Green’s function can be found by solving a one-dimensional boundary value problem. This can be seen by considering the equations of motion, written in the Fourier domain (Lindsey and Braun, 2004),
$${\omega ^2}{{dp} \over {dz}} = {k^2}gp + {\rho _0}\left({{\omega ^4} + {\omega ^2}{{dg} \over {dz}} - {k^2}{g^2}} \right){\xi _z},$$
(72)
$${\omega ^2}{{d{\xi _z}} \over {dz}} = \left({{k^2} - {{{\omega ^2}} \over {{c^2}}}} \right){p \over {{\rho _0}}} - {k^2}g{\xi _z}.$$
(73)
Here p and ξz are the pressure perturbation and vertical displacement associated with the Green’s function. The acceleration due to gravity is g and the background density and sound speed are ρ0(z) and c(z), respectively. Equations (72, 73) can be used to find the pressure and vertical displacement given two boundary conditions. The data, however, provide us with only one upper boundary condition. It is for this reason that the assumptions that all of the observed waves are up-going waves at the surface, when computing the egression, and down-going waves at the surface, when computing the ingression, are made (Lindsey and Braun, 2004). In this way the second boundary condition can be generated from the first. Notice that Equations (72, 73) do not include the effect of damping.
Local control correlations
The local control correlation is the correlation between the observed signal at a particular point on the solar surface and the holographic reconstruction of the signal at that same point computed from the data in a surrounding region. In particular the local control correlations are defined as (see, e.g., Lindsey and Braun, 2004)
$${C_-}(x,\omega) = {\langle {\Phi (x,\omega)H_- ^{\ast}(x,{z_{\rm{o}}},\omega)} \rangle _{\Delta \omega}},$$
(74)
$${C_+}(x,\omega) = {\langle {{H_+}(x,{z_{\rm{o}}},\omega){\Phi ^{\ast}}(x,\omega)} \rangle _{\Delta \omega}}.$$
(75)
Here the depth z0 denotes the z coordinate of the solar surface. The angle brackets denote the average over a frequency band of width Δω centered at frequency ω. Notice that we have not specified the pupil which goes into the ingression/egression calculations. The pupil that is used depends on the particular situation. Figure 26 shows quiet Sun local control correlations, measured using both ray theory and wave theory Green’s functions.
If the egression/ingressions were perfect reconstructions of the observed signal, then the phases of the C± would both, on average, be zero in the quiet Sun. In practice, this is not the case (Lindsey and Braun, 2004). The phases of the holography Green’s functions are typically corrected so that the phase of the average quiet Sun local control correlation is zero (e.g., Lindsey and Braun, 2000a). We refer to this correction as the empirical dispersion correction.
Acoustic power holography
The goal of acoustic power holography is to estimate the amount of wave power emitted from a particular region, either at particular time or a particular frequency. The estimate of the power emitted at a particular frequency, ω, at horizontal location x and depth z is
$$P(r,\omega) = \vert {H_+}(r,\omega){\vert ^2},$$
(76)
and the estimate of the power emitted at a particular time t is
$$P(r,t) = \vert {H_+}(r,t){\vert ^2}.$$
(77)
Here H+(r, t) is the temporal Fourier transform of H+(r, ω). Notice that these estimates depend on the estimate of the Green’s function and the pupil that are used to compute H+. For further discussions of acoustic power holography see, for example, Lindsey and Braun (1997) and Donea et al. (1999). Acoustic power holography has been used in studies of wave excitation by flares (e.g., Donea et al., 1999) and around active regions (e.g., Donea et al., 2000).
Phase-sensitivity holography
The basic computation in phase-sensitive holography is the ingression-egression correlation (see, e.g., Lindsey and Braun, 2000a, 2004),
$${C_{{\cal P},{{\cal P}^{\prime}}}}(r,\omega) = {\langle {H_- ^{\cal P}(r,\omega)H_+ ^{{\cal P}{^{\prime}_\ast}}(r,\omega)} \rangle _{\Delta \omega}},$$
(78)
where \({\cal P}\) and \({\cal P}^{\prime}\) are two pupils and the angle brackets denote averaging over a frequency range of width Δω centered on frequency ω. Phase-sensitive holography has been used to look for sound-speed perturbations and mass flows. In both cases the geometry is the quadrant geometry that is also used in time-distance measurements. In particular the pupil \({\cal P}\) is chosen to be a quarter of an annulus, let us call it ℒ for “left”, and \({\cal P}^{\prime}\) is chosen to be the quarter annulus located 180° away, call it ℛ for “right”. The symmetric phase is defined as
$${\phi _{\rm{s}}} = {1 \over 2}({\rm{Arg}}\;{C_{{\cal L},{\cal R}}} + {\rm{Arg}}\;{C_{{\cal R},{\cal L}}}).$$
(79)
Here the operator Arg returns the phase. The phase φs is mostly sensitive to sound-speed; this can be seen from the symmetry of the definition. In particular, φs does not change when the pupils ℒ and ℛ are interchanged. Notice that for a horizontally uniform sound-speed perturbation Arg Cℒ,ℛ = Arg Cℒ,ℛ. On the other hand, a uniform horizontal flow gives Arg Cℒ,ℛ = −Arg Cℒ,ℛ. Thus φs is zero for a horizontally uniform flow and non-zero for a horizontally uniform sound-speed perturbation. The anti-symmetric phase is defined as
$${\phi _{\rm{a}}} = {\rm{Arg}}\;{C_{{\cal L},{\cal R}}} + {\rm{Arg}}\;{C_{{\cal R},{\cal L}}}.$$
(80)
The symmetry of φa is such that φa changes sign under interchange of the pupils ℒ and ℛ and, thus, the anti-symmetric phase is sensitive to horizontal flows. The phases φs and φa are analogous to the mean and difference travel times commonly employed in time-distance helioseismology.
Far-side imaging
Far-side imaging is a special case of phase-sensitive holography (e.g., Lindsey and Braun, 2000b; Braun and Lindsey, 2001). The idea is to use the wavefield on the visible disk to learn about active regions on the far-side of the Sun. Figure 27 shows the geometry that was used by Braun and Lindsey (2001). In order to obtain full coverage of the far-side of the Sun, two different geometries were employed. For regions near the antipode of the center of the visible disk a two-skip geometry and two-skip Green’s functions were employed (panel (a) of Figure 27). For focus positions near the limb, the ingression/egression were computed using a single-skip pupil and corresponding Green’s functions, and then correlated with the egression/ingression computed using a three-skip pupil and Green’s functions (the geometry is shown in panel (b) of Figure 27).
Acoustic imaging
Acoustic imaging was first introduced by Chang et al. (1997); for a recent review see Chou et al. (2003). We have included acoustic imaging in the section on helioseismic holography as the definitions and philosophical motivation for the two techniques are quite similar.
The central computations in acoustic imaging (see, e.g., Chou et al., 1999, 2003) are wavefield reconstructions in the solar interior, Ψin and Ψout, defined by
$${\Psi _{{\rm{out}},{\rm{in}}}}(r,t) = \sum\limits_{\tau = {\tau _1}}^{{\tau _2}} {W(\tau, z)} \overline \Phi (\Delta (\tau, z),t \pm \tau).$$
(81)
The focus position is r = (x, z). The function \(\overline{\Phi}(\Delta,t)\) denotes the azimuthal average, around x, of the surface wavefield Φ measured a horizontal distance Δ away from x. The quantity τ represents the ray-theoretical travel times from the subsurface focus point at r to surface points a horizontal distance Δ away from the focus point. For a given τ, the distance Δ satisfies the time-distance relation established between the focus point and a surface point. This relation, Δ = Δ(τ, z), is computed from a standard solar model, using the ray approximation (see Figure 28). The function W is a smooth weight function of τ (or Δ), explicitly given by Chou et al. (1999). The sum in Equation (81) involves the observed wavefield inside an annulus with inner and outer radii specified by the range τ1 < τ < τ2.
In some sense, acoustic imaging is a special case of holography. Here, the pupil is an annulus with inner radius Δ1 and outer radius Δ2 given by the time-distance relation Δj = Δ(τj, z). The equivalent holography Green’s function is given by WδD(t − τ). The signal Ψout corresponds to the egression, i.e., is the signal reconstructed from the observations of waves diverging from the focal point, while corresponds to the ingression, the signal estimated from the waves seen converging towards the focus position.
As with holography, both the amplitudes and phases of Ψout, in are used to learn about the solar interior. The square of the amplitude of Ψout is an estimate of the power contained in the waves seen diverging from the focus position (Chang et al., 1997). In the terminology of holography, the squared of the amplitude of Ψout is called the “egression power.” Chen et al. (1998) introduced the correlation
$$C(r,t) = \int {d{t^{\prime}}{\Psi _{{\rm{in}}}}} (r,{t^{\prime}}){\Psi _{{\rm{out}}}}(r,{t^{\prime}} + t)$$
(82)
and then measured phase and group times by fitting a Gaussian wavelet to C(r, t) at fixed target point r. Changes in the phase between Ψin and Ψout result in changes in C(t) and thus shift the travel times. A phase shift between the two reconstructions, Ψout and Ψin, is evidence for local changes in sound speed (Chen et al., 1998). Chou and Duvall (2000) discuss the relationship between time-distance travel times and the travel times measured by acoustic imaging.
The forward problem that has received the most attention in acoustic imaging is the dependence of travel times on changes in the sound speed. Chou and Sun (2001) used the ray approximation to estimate the sensitivity of acoustic imaging phase travel times to changes in sound speed. The results showed that in general the horizontal resolution is greater than the vertical resolution and that the resolution decreases with increasing focus depth.
The inverse problem of determining the sound speed from a given set of travel times has been studied in the ray approximation as well. Sun and Chou (2002) tested ray-theory RLS inversions of phase-time measurements on artificial data. These tests showed that the RLS inversions were, for good choices of the regularization parameter, capable of reconstructing the model used to generate the artificial data. Inversions of travel times for sound speed were also discussed in detail by Chou and Sun (2001).
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.
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 (30) in terms of Fourier coordinates gives
$$\matrix{{\delta \Phi (k,\omega) = - {{(2\pi)}^3}\int {{d^3}r\;dt\;{d^2}{k^{\prime}}\;d{\omega ^{\prime}}\;d{z_{\rm{s}}}\;{e^{- {\rm{i}}k \cdot x + {\rm{i}}\omega t}}} {G^i}(k, \omega, z)} \hfill \cr {\quad \quad \quad \quad \;\; \times {{\left\{{\delta {L_{(r,t)}}\left[ {{G^j}({k^{\prime}},{\omega ^{\prime}},z,{z_s})\;{e^{{\rm{i}}{k^{\prime}} \cdot x - {\rm{i}}{\omega ^{\prime}}t}}} \right]} \right\}}_i}S_j^0({k^{\prime}},{\omega ^{\prime}},{z_{\rm{s}}}),} \hfill \cr}$$
(83)
where summation is assumed over the repeated indices i and j, r = (x, z) is the scattering location, and the Green’s functions \({\cal G}\) and G are defined in Section 3.3. The perturbations to the source function were ignored for the sake of simplicity. To obtain Equation (83) 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 u 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 (83) the correlations of the wavefield in the Fourier domain can be approximated, to first order in the strength of the perturbation, as
$$\matrix{{E[\delta \Phi (k,\omega){\Phi ^{\ast}}({k^{\prime}},{\omega ^{\prime}})] = - {{(2\pi)}^6}\int {{d^3}} r\;dt\;d{z_{\rm{s}}}\;dz_{\rm{s}}^{\prime}\;{e^{- {\rm{i}}k\cdot x + {\rm{i}}\omega t}}{{\cal G}^i}(k,\omega,z)} \hfill \cr {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \times {{\left\{{\delta {{\cal L}_{(r,t)}}\left[ {{G^j}({k^{\prime}},{\omega ^{\prime}},z,{z_s}){e^{{\rm{i}}{k^{\prime}}\cdot x + {\rm{i}}{\omega ^{\prime}}t}}} \right]} \right\}}_i}} \hfill \cr {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \times {m_{jk}}({k^{\prime}},{\omega ^{\prime}},{z_{\rm{s}}},z_{\rm{s}}^{\prime}){{\cal G}^{k\ast}}({k^{\prime}},{\omega ^{\prime}},z_{\rm{s}}^{\prime}),} \hfill \cr}$$
(84)
where the source covariances
$$E[S_i^0(k,\omega,{z_{\rm{s}}})S_i^{0\ast}({k^{\prime}},{\omega ^{\prime}},z_{\rm{s}}^{\prime})] = {m_{ij}}(k,\omega,{z_{\rm{s}}},z_{\rm{s}}^{\prime}){\delta _{\rm{D}}}(k - {k^{\prime}}){\delta _{\rm{D}}}(\omega - {\omega ^{\prime}})$$
(85)
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 δℒ.
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
$$\delta {{\cal L}_{(r,t)}} = 2{\rho _0}(z)U(r,t) \cdot {{\bf{\nabla}} _r}{\partial _t}$$
(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
$$U(r,t) = \hat z{U_z}(z){e^{{\rm{i}}q \cdot x - {\rm{i}}\sigma t}},$$
(87)
where the vector q is a horizontal wavevector and σ is a temporal frequency. Under these assumptions, we have
$$\delta {{\cal L}_{(r,t)}}\left[ {{G^j}({k^{\prime}},{\omega ^{\prime}},z,{z_{\rm{s}}}){e^{{\rm{i}}{k^{\prime}} \cdot x - {\rm{i}}{\omega ^{\prime}}t}}} \right] = - 2{\rm{i}}{\omega ^{\prime}}{\rho _0}(z){U_z}(z){\partial _z}{G^j}({k^{\prime}},{\omega ^{\prime}},z,{z_{\rm{s}}}){e^{{\rm{i}}({k^{\prime}} + q) \cdot x - {\rm{i}}({w^{\prime}} + \sigma)t}}.$$
(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 (84) becomes
$$\matrix{{E[\delta \Phi (k,\omega){\Phi ^{\ast}}({k^{\prime}},{\omega ^{\prime}})] = 2{\rm{i}}{\omega ^{\prime}}{{(2\pi)}^9}\delta (k - {k^{\prime}} - q)\delta (\omega - {\omega ^{\prime}} - \sigma){m_{zz}}({k^{\prime}},{\omega ^{\prime}}){{\cal G}^{z\ast}}({k^{\prime}},{\omega ^{\prime}},{z_{\rm{s}}})} \hfill \cr {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \times \int {dz} \;{\rho _0}(z){U_z}(z){{\cal G}^i}(k,\omega,z){\partial _z}G_i^z({k^{\prime}},{\omega ^{\prime}},z,z{_s}).} \hfill \cr}$$
(89)
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 a only couple components of the wavefield whose frequencies differ by σ. 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 \(E[\Phi(k,\omega)\delta\Phi^{\ast}(k^{\prime},\omega^{\prime})]\) as the final result that we wish to obtain is the first order change in \(E[\Phi(k,\omega)\Phi^{\ast}(k^{\prime},\omega^{\prime})]\) introduced by a small change in the model. This second term can be computed in exactly the same manner as the \(E[\delta\Phi(k,\omega)\Phi^{\ast}(k^{\prime},\omega^{\prime})]\) term, which we have already done.
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.