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4.4 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 (2000aJump To The Next Citation Point). 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 1940s (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., 1997Jump To The Next Citation Point), a technique closely related to holography.

4.4.1 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, 2000aJump To The Next Citation Point) 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 (2000aJump To The Next Citation Point), a mathematical motivation for the ingression and egression is the Kirchoff integral solution to the wave equation (e.g., Jackson, 1975Jump To The Next Citation Point). Suppose that a scalar field Φ solves the equation

2 2 k Φ (r) + ∇ Φ(r) = 0. (67 )
The Green’s function, G, for this problem is defined as the solution to
k2G(r ) + ∇2G (r) = δ (r ), (68 ) D
where δD is the 3D Dirac delta function. The Kirchoff integral equation for Φ is
∫ Φ(r ) = ds [{∂nG (r − s)} Φ(s) − G (r − s)∂nΦ (s)]. (69 ) S
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 (69View Equation) 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 (2004Jump To The Next Citation Point), helioseismic holography is much more complicated than solving the simple scalar wave Equation (67View Equation) addressed by the Kirchoff integral.

Motivated by the Kirchoff identity (Equation 69View Equation), the egression š’« H+ and ingression š’« H− are defined as

∫ H š’«± (r,ω) = d2x ′ Gh±olo(x − x′,z,ω)Φ (x′,ω). (70 ) š’«
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 š’«, 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 (2000aJump To The Next Citation Point). The wavefield on the surface is Φ(x ′,ω ) and ω is temporal angular frequency. Recall that the egression, H š’«+, is an attempt to estimate the wavefield at the location r and frequency ω based on the waves that are seen, in the pupil š’« 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 ω are denoted by Gho±lo(r,ω ). In the time domain, the causal Green’s function Gholo(r,t) ±, obtained by temporal inverse Fourier transform from holo G ± (r,ω ), 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 Gho−lo(r,t) = Gh+olo(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 Gho±lo 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 (70View Equation) 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 Gholo + to propagate the surface wavefield backwards in time. Likewise, the ingression can be seen as the result of using the causal Green’s function holo G − 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 holo G ± (r,ω ), which we now describe.

4.4.2 Holography Green’s functions

The two approaches that have been used to construct the holography Green’s functions, Gholo ±, involve ray theory and wave theory.

4.4.2.1 Ray theory.
Lindsey and Braun (2000aJump To The Next Citation Point), motived by ray theory, prescribed the holography Green’s function as

Gholo(r, t) = ∑ A (x, z)δ (t ± τ (x, z)), (71 ) ± N N D N
where r = (x,z) and x = āˆ„x āˆ„ is the horizontal distance. The role of the function holo G − (holo G +) 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 (71View Equation), 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 (71View Equation) 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 71View Equation) must then be Fourier transformed to obtain G (r,ω ) ± referred to in Equation (70View Equation). Notice that there are a great many simplifications involved in writing the Green’s function in Equation (71View Equation). 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 2000aJump To The Next Citation Point). Recent work by Lindsey and Braun (2004Jump To The Next Citation Point) 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 25View Image).
View Image

Figure 25: Comparison of the amplitudes (left panel) and phases (right panel) of ingression-egression correlations (Section 4.4.5) at a focus depth of 4.2 Mm computed with ray theory (“eikonal”, Equation (71View Equation)) and with the wave theory (“hydromechanical”, Equations (72View Equation, 73View Equation)). In both cases, an empirical dispersion correction has been applied (see Section 4.4.3). The black points correspond to focus points where the photospheric magnetic field is weaker than 10 G and the red points correspond to focus points where the surface magnetic field is larger than 100 G. There is, in general, good correspondence between the results of the two methods, though the scatter in the phases appears to increase with increasing phase shifts. From Lindsey and Braun (2004Jump To The Next Citation Point).

4.4.2.2 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 (2004Jump To The Next Citation Point). 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, 2004Jump To The Next Citation Point),

dp ( dg ) ω2 ---= k2gp + ρ0 ω4 + ω2---− k2g2 ξz, (72 ) dz ( ) dz 2dξz 2 ω2 p 2 ω ----= k − -2- ---− k gξz. (73 ) dz c ρ0
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 ρ (z) 0 and c(z ), respectively. Equations (72View Equation, 73View Equation) 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, 2004Jump To The Next Citation Point). In this way the second boundary condition can be generated from the first. Notice that Equations (72View Equation, 73View Equation) do not include the effect of damping.

4.4.3 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, 2004Jump To The Next Citation Point)

∗ C − (x, ω) = āŸØΦ(x, ω)H − (x, zo,ω)āŸ©Δω, (74 ) C+ (x, ω) = āŸØH+ (x,zo,ω )Φ∗(x, ω)āŸ©Δω. (75 )
Here the depth zo denotes the z coordinate of the solar surface. The angle brackets denote the average over a frequency band of width Δ ω 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 26View Image shows quiet Sun local control correlations, measured using both ray theory and wave theory Green’s functions.
View Image

Figure 26: Spatial averages of quiet Sun local control correlations. (a) Results obtained using the wave theory Green’s functions. (b) Results of using ray theory. The numbers along each curve denote the cyclic frequency ω āˆ•2π in units of mHz. In both cases, notice that the phase is not zero. From Lindsey and Braun (2004Jump To The Next Citation Point).

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, 2004Jump To The Next Citation Point). 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, 2000aJump To The Next Citation Point). We refer to this correction as the empirical dispersion correction.

4.4.4 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

2 P (r,ω) = |H+ (r,ω )|, (76 )
and the estimate of the power emitted at a particular time t is
P (r, t) = |H (r,t)|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 (1997Jump To The Next Citation Point) and Donea et al. (1999Jump To The Next Citation Point). Acoustic power holography has been used in studies of wave excitation by flares (e.g., Donea et al., 1999Jump To The Next Citation Point) and around active regions (e.g., Donea et al., 2000).

4.4.5 Phase-sensitivity holography

The basic computation in phase-sensitive holography is the ingression-egression correlation (see, e.g., Lindsey and Braun, 2000a2004),

š’« š’« ′∗ C š’«,š’« ′(r,ω ) = āŸØH− (r,ω )H + (r,ω )āŸ©Δω, (78 )
where š’« and š’«′ are two pupils and the angle brackets denote averaging over a frequency range of width Δ ω 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 š’« is chosen to be a quarter of an annulus, let us call it ā„’ for “left”, and š’« ′ is chosen to be the quarter annulus located 180° away, call it ā„› for “right”. The symmetric phase is defined as
1 Ļ•s = --(Arg C ā„’,ā„› + Arg Cā„›,ā„’) . (79 ) 2
Here the operator Arg returns the phase. The phase Ļ•s is mostly sensitive to sound-speed; this can be seen from the symmetry of the definition. In particular, Ļ•s does not change when the pupils ā„’ and ā„› are interchanged. Notice that for a horizontally uniform sound-speed perturbation Arg C ā„’,ā„› = Arg C ā„’,ā„›. On the other hand, a uniform horizontal flow gives Arg C ā„’,ā„› = − Arg Cā„’,ā„›. Thus Ļ•s is zero for a horizontally uniform flow and non-zero for a horizontally uniform sound-speed perturbation. The anti-symmetric phase is defined as
Ļ•a = Arg C ā„’,ā„› − Arg C ā„›,ā„’. (80 )
The symmetry of Ļ•a is such that Ļ•a changes sign under interchange of the pupils ā„’ and ā„› and, thus, the anti-symmetric phase is sensitive to horizontal flows. The phases Ļ•s and Ļ•a are analogous to the mean and difference travel times commonly employed in time-distance helioseismology.

4.4.6 Far-side imaging

Far-side imaging is a special case of phase-sensitive holography (e.g., Lindsey and Braun, 2000bJump To The Next Citation PointBraun and Lindsey, 2001Jump To The Next Citation Point). The idea is to use the wavefield on the visible disk to learn about active regions on the far-side of the Sun. Figure 27View Image shows the geometry that was used by Braun and Lindsey (2001Jump To The Next Citation Point). 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 27View Image). 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 27View Image).

View Image

Figure 27: Geometry for far-side imaging. (a) Two-skip correlation scheme. (b) One-skip/three-skip correlation. In far-side imaging the data on the visible disk are used to estimate the wavefield at focal points on the far-side of the Sun. From Braun and Lindsey (2001Jump To The Next Citation Point).

4.4.7 Acoustic imaging

Acoustic imaging was first introduced by Chang et al. (1997Jump To The Next Citation Point); for a recent review see Chou et al. (2003Jump To The Next Citation Point). 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., 1999Jump To The Next Citation Point2003) are wavefield reconstructions in the solar interior, Ψin and Ψout, defined by

∑τ2 -- Ψout,in(r,t) = W (τ,z) Φ(Δ (τ,z),t ± τ). (81 ) τ= τ1
The focus position is r = (x, z). The function -- Φ(Δ, 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 28View Image). The function W is a smooth weight function of τ (or Δ), explicitly given by Chou et al. (1999Jump To The Next Citation Point). The sum in Equation (81View Equation) involves the observed wavefield inside an annulus with inner and outer radii specified by the range τ1 < τ < τ2.
View Image

Figure 28: Time-distance relations at various focal depths, computed from a standard solar model, based on ray theory, at 3 mHz. Each curve corresponds to different target depths, as indicated. For each curve, the dots correspond to modes whose l values are multiples of 10, starting with l = 80 at the right end of each curve. The highest-l mode which can reach the target depth is marked by an open circle on each curve. The l value decreases in either direction away from the open circle along the curve. From Chou et al. (1999Jump To The Next Citation Point).

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 W δD(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 Ψ in 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., 1997Jump To The Next Citation Point). In the terminology of holography, the squared of the amplitude of Ψout is called the “egression power.” Chen et al. (1998Jump To The Next Citation Point) introduced the correlation

∫ C(r, t) = dt′ Ψin(r,t′)Ψout(r, t′ + 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 Ψ in and Ψ out 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., 1998Jump To The Next Citation Point). Chou and Duvall Jr (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 (2001Jump To The Next Citation Point) 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).


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