Issue 
A&A
Volume 620, December 2018



Article Number  A136  
Number of page(s)  8  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201833825  
Published online  10 December 2018 
Signal and noise in helioseismic holography
^{1}
MaxPlanckInstitut für Sonnensystemforschung, JustusvonLiebigWeg 3, 37077
Göttingen, Germany
email: gizon@mps.mpg.de
^{2}
Institut für Astrophysik, GeorgAugustUniversität Göttingen, FriedrichHundPlatz 1, 37077
Göttingen, Germany
^{3}
Center for Space Science, NYUAD Institute, New York University Abu Dhabi, PO Box 129188
Abu Dhabi, UAE
^{4}
Magique3D, Inria Bordeaux SudOuest, E2S UPPA, 64000
Pau, France
Received:
11
July
2018
Accepted:
7
October
2018
Context. Helioseismic holography is an imaging technique used to study heterogeneities and flows in the solar interior from observations of solar oscillations at the surface. Holographic images contain noise due to the stochastic nature of solar oscillations.
Aims. We aim to provide a theoretical framework for modeling signal and noise in Porter–Bojarski helioseismic holography.
Methods. The wave equation may be recast into a Helmholtzlike equation, so as to connect with the acoustics literature and define the holography Green’s function in a meaningful way. Sources of wave excitation are assumed to be stationary, horizontally homogeneous, and spatially uncorrelated. Using the first Born approximation we calculated holographic images in the presence of perturbations in soundspeed, density, flows, and source covariance, as well as the noise level as a function of position. This work is a direct extension of the methods used in timedistance helioseismology to model signal and noise.
Results. To illustrate the theory, we compute the holographic image intensity numerically for a buried soundspeed perturbation at different depths in the solar interior. The reference Green’s function is obtained for a sphericallysymmetric solar model using a finiteelement solver in the frequency domain. Below the pupil area on the surface, we find that the spatial resolution of the holographic image intensity is very close to half the local wavelength. For a soundspeed perturbation of size comparable to the local spatial resolution, the signaltonoise ratio is approximately constant with depth. Averaging the image intensity over a number N of frequencies above 3 mHz increases the signaltonoise ratio by a factor nearly equal to the square root of N. This may not be the case at lower frequencies, where large variations in the holographic signal are due to the contributions from the longlived modes of oscillation.
Key words: Sun: helioseismology / Sun: interior / Sun: oscillations
© ESO 2018
1. Introduction
Local helioseismology aims at studying the solar interior in three dimensions by exploiting the information contained in the waves observed at the solar surface (see, e.g., Gizon & Birch 2005 and references therein). Helioseismic holography is one particular approach of local helioseismology, which images subsurface scatterers by backpropagating the surface wave field to target points in the interior. Helioseismic holography is also known as LindseyBraun (LB) holography (Lindsey & Braun 1997, 2000a and references therein). It has been used to study solar convection (Braun et al. 2004, 2007), active region emergence (Birch et al. 2013, 2016), sunspot subsurface structure (Braun & Birch 2008b; Birch et al. 2009), to image wave sources (Lindsey et al. 2006), to study sunquakes caused by solar flares (Zharkov et al. 2013; BesliuIonescu et al. 2017), and to detect active regions on the far side of the Sun (Lindsey & Braun 2000b; Liewer et al. 2014).
In acoustics, a wellestablished version of holography in a medium that contains sources is PorterBojarski (PB) holography (Porter & Devaney 1982). PB holography uses both the wave field and its normal derivative at the surface to produce holographic images (Porter 1969). PB holography was introduced in helioseismology by Skartlien (2001, 2002), where deterministic sources and scatterers were recovered in a solar background. Yang (2018) recently studied PB holographic images in a homogeneous medium permeated by localized deterministic sources to study ghost images near the observational boundary.
In this paper we apply PB holography in a realistic helioseismological setting. First we rewrote the wave equation in Helmholtz form, in order to properly define the Green’s functions that are involved in the definition of PB holographic data. The background density and soundspeed are taken from a standard solar model. The model of wave excitation is described by a reasonable source covariance function, which leads to a solarlike power spectrum for acoustic oscillations.
The signal is defined as the expectation value of the holographic image intensity that results from perturbations in sound speed, density, and flows with respect to the reference solar model. The corresponding sensitivity kernels were computed in the firstorder Born approximation (Gizon & Birch 2002; Birch & Gizon 2007; Braun et al. 2007; Birch et al. 2011). This signal must take into account the correlations between incident and scattered wave fields, which are both connected to the sources of excitation (turbulent convection).
Random noise in holographic images is due to the stochastic nature of the sources of excitation. While noise can sometimes be estimated from the data (Lindsey & Braun 1990; Braun & Birch 2008a), a theoretical understanding is useful to design holographic experiments. In this paper we have extended the noise model developed in timedistance helioseismology to holography (Gizon & Birch 2004; Fournier et al. 2014). We did not attempt to image individual sources as in Skartlien (2002), which in our view is not a wellposed problem (see also Lindsey et al. 2006), except in the case of sunquakes. Instead we considered sources to be specified through a source covariance function.
2. Reduced wave equation
At angular frequency ω and spatial position r in the computational domain V, the propagation of acoustic waves in a 3D heterogeneous moving medium is described by the displacement vector ξ(r, ω), solution to
where ρ(r) and c(r) are the density and sound speed, and u(r) is a steady vector flow. Wave attenuation is included through the function γ(r, ω). The source term F(r, ω) is a realization from a random process; it describes the stochastic excitation of the waves by turbulent convection. Following Lamb (1909) and Deubner & Gough (1984), we consider the scalar variable
to recast the wave equation into a Helmholtzlike equation
where S = ρ^{1/2}c^{2}∇⋅F is a scalar source term. The local wavenumber k(r, ω) is given by
where the squared acoustic cutoff frequency is
In obtaining Eq. (3), we ignored gravity terms and assumed slow variations of c, u, and γ compared to the wavelength (Gizon et al. 2017). The advection term is such that the corresponding operator is Hermitian symmetric for the inner product under the conditions that the flow conserves mass and that it does not cross the boundary (u_{n} = 0 on ∂V).
The stochastic sources of excitation are assumed to be stationary and spatially uncorrelated. They are described by a source covariance function of the form
To solve Eq. (3), one needs to specify a boundary condition at the computational boundary. Here we apply an outgoing radiation boundary condition
We applied the boundary condition (Atmo RBC 1) from Barucq et al. (2018), which assumes an exponential decay of the background density at the boundary of the domain but neglects curvature. Then, the local wavenumber k_{n} from Eq. (7) is given by
where H = −1/(d ln ρ/dr) is the density scale height at the boundary. The last term in Eq. (8) is connected to the cutoff frequency for an isothermal atmosphere (Lamb 1909), thus k_{n} is an approximation of the wavenumber k from Eq. (4). Fournier et al. (2017) discuss several of the boundary conditions derived in Barucq et al. (2018).
3. Holographic image intensity
The following calculations are done at constant ω, thus we drop ω from the list of function arguments when not explicitly needed. The Porter–Bojarski integral is defined by Porter & Devaney (1982):
where H_{α} is an acoustic wave propagator for the reference medium and A is a surface on the Sun where ψ and ∂_{n}ψ are observed. The role of H_{α} is to propagate the wave field backward (or forward) in time, which leads to the concept of egression (or ingression) in LB holography (Lindsey & Braun 2000a).
Several choices for the propagators have been proposed in the literature, as detailed in Table 1. These depend on the outgoing (G_{0}) and incoming () Green’s functions defined in a reference medium with ρ_{0}, c_{0}, γ_{0}, and u_{0} = 0:
Possible wave propagators.
with
where k_{0} is k in the reference medium and k_{n} is from Eq. (8). The Green’s functions or are backward propagators (cf. egression), while ( is a forward propagator (cf. ingression). When the surface A is closed, it is equivalent to use and Im G_{0} (Devaney & Porter 1985). Tsang et al. (1987) proposed as a backward propagator to correct for wave attenuation.
If the observations are made at the computational boundary and the wave field satisfies the same boundary condition as the Green’s function, then Eq. (9) reads
When , we have
which corresponds to the egression as defined by Lindsey & Braun (2000a). Thus LB and PB integrals are closely related, at least for the upper boundary condition employed here.
In LB holography, information is extracted from the egressioningression correlation (wavespeed perturbations and flows) and from the egression power (source covariance). Analogously, we define the PB image intensity (or covariance) as
For the case α = β we define
Different choices of pupils and propagators will provide sensitivity to different quantities as shown in Table 2. Scatterers are detected by correlating the forward and backward propagated wavefields. Different pupil shapes give access to different components of the flow (Table 2 and Fig. 1).
Fig. 1.
Pupil geometries used to compute soundspeed or source kernels (P) and flow kernels (Qs), also see Table 2. 
Proposed propagators and associated pupils (H_{α}, A) and (H_{β}, A′) for different types of perturbations.
4. Firstorder perturbations with respect to a reference solar model
We wish to study how perturbations to the background medium affect the holographic images. Using the Born approximation, the first step is to express the perturbations to the wavefield and use this expression in Eq. (9) to obtain the perturbations to the PB integral and the image intensity.
4.1. Perturbations to the wavefield
We considered perturbations δc, δρ, δγ, u with respect to the reference medium defined by ρ_{0}, c_{0}, γ_{0}, and u_{0} = 0. The perturbations to the sources of excitation are described through the perturbations to the source covariance,
where
Using the Born approximation up to first order, we write the wave field as
where the zeroth and firstorder wave fields are given by
The perturbed wave operator is
with
In terms of the Green’s function G_{0}, we have
where the scattering location r_{s} spans the entire solar volume V.
4.2. Perturbations to the PB integral
For convenience, we introduce the source kernels
such that the PB integral is given by
We denote by K_{α, 0} the source kernel in the reference medium (when G is replaced by G_{0}).
Replacing the wavefield by its first order expansion (Eq. (24)) in the definition of the PB integral (Eq. (9)), one obtains
where Φ_{α, 0} is the value for the reference medium and δΦ_{α} is due to perturbations in the medium:
where
4.3. Perturbations to the image intensity
We write the holographic image intensity to first order in the form
The expectation values of the zeroth and firstorder image intensities are
Using the definition of the source kernels, we obtain
where
is the expectation value of the crosscovariance function and, for any function F(r, r′), the double brackets mean
The perturbation to the image intensity is given by
where
The kernels for δk^{2} and δk^{2*} can be combined using Eq. (23) to obtain kernels for soundspeed , density and attenuation . For example,
with
4.4. Choice of the source covariance
In order to compute the above kernels, we still need to choose the source covariance function M_{0} in order to define the reference crosscovariance C_{0} using Eq. (34). One possibility is to place the sources at a single depth, a few hundred kilometers below the solar surface.
Another possibility is to choose a source covariance of the form
where Π(ω) is the source power spectrum (see Gizon et al. 2017). This choice implies
The surface term depends on the boundary condition. It vanishes for a Dirichlet boundary condition (free surface), while it remains for radiative boundary conditions (e.g., Sommerfeld). Below the acoustic cutoff frequency, modes are trapped well below the observational and computational boundaries and the surface term is negligible. In this paper we use Eq. (42) in the convection zone and switch off the sources above the photosphere. By doing so, the surface term in Eq. (43) vanishes.
5. Noise
To compute the noise level, we computed the variance of the image intensity in the absence of scatterers:
where
Under the (very reasonable) assumption that S is a realization drawn from a Gaussian random process, the fourthorder moment is the sum of products of the secondorder moments:
The first term in M_{4} cancels out the squared term in Eq. (44). The third term is zero as the frequencies are uncorrelated. Thus,
When α = β, the standard deviation of I_{α, 0} is equal to its expectation value. This is because the probability density function of I_{α, 0} is a χ^{2} with two degrees of freedom, in other words, an exponential distribution.
6. Average over frequencies
In order to increase the signaltonoise ratio (S/N), one usually averages the image intensity over a range of frequencies [ω_{0} − Δω/2, ω_{0} + Δω/2]. For an observation duration T, this interval will contain N = Δω T/2π independent frequencies.
The frequencyaveraged perturbation to the image intensity (i.e. the signal) is denoted by
The variance of the noise in the average image intensity is then given by
since the noise in the data at different frequencies is uncorrelated.
The expected S/N is thus
The number of available frequencies N within a fixed frequency band Δω is proportional to the observation duration T, hence the noise level goes like T^{−1/2}. Provided that the frequency interval Δω is small with respect to the variations of the signal, then the S/N will increase as T^{1/2}.
7. Example computations
In order to illustrate the theory, we have computed holographic images in the presence of soundspeed perturbations at different depths and calculate the corresponding S/Ns.
7.1. Reference Green’s function
The main input quantity required to compute PB integrals is the reference Green’s function (Eq. (10)). Here it is computed in the frequency domain using the sphericallysymmetric standard solar Model S (ChristensenDalsgaard et al. 1996). The wave attenuation model is taken from Gizon et al. (2017). Below 5.3 mHz, we have γ = γ_{0}ω/ω_{0}^{5.77}, where γ_{0}/2π = 4.29 μHz and ω_{0}/2π = 3 mHz. Above 5.3 mHz, γ/2π = 125 μHz is kept constant. The radiation boundary condition defined by Eq. (7) is applied at the computational boundary located 500 km above the solar surface with the local wavenumber k_{n} (where H = 105 km). The wave equation is solved using the finiteelement solver Montjoie (Duruflé 2006; Gizon et al. 2017).
The reference Green’s function only depends on the angular distance Θ between the two points at radii r and r′. To speed up the computations, we place one of the points on the polar axis and compute the axisymmetric component of the Green’s function at each spherical harmonic degree l, to obtain:
where we used an approximate equality because the sum is truncated at l_{max} = 300.
7.2. Soundspeed kernels
The soundspeed kernel is computed using Eq. (41) and the definition of . We need to evaluate two surface integrals, which can be computed analytically via a decomposition of all quantities into spherical harmonic coefficients (Fournier et al. 2018).
Figure 2 shows a soundspeed kernel at a single frequency of 3 mHz. The pupil P is a polar cap of angular size 120° and the wave propagators H_{α} and H_{β} are given in Table 2. As expected the kernel peaks around the scatterer position at z = 0.7 R_{⊙}. A cut along the polar axis is shown in Fig. 3; the kernel width is about half the local wavelength. In addition, we find ghost values at the antipode.
Fig. 2.
Meridional slice through the soundspeed kernel (x, r_{s}) computed in Model S at a single frequency of 3 mHz, in units of 10^{−34} kg m^{−3} s^{−3}. Both the real (left panel) and imaginary (right panel) parts of the kernel are shown. The scatterer at z_{s} = 0.7 R_{⊙} is indicated by the crosses. The observation pupil P is a polar cap of full angular size 120°. We notice the “ghost values” at the antipode. 
Fig. 3.
Cut along the z axis through the soundspeed kernel from Fig. 2. The scatterer is at z_{s} = 0.7 R_{⊙}. 
7.3. Signal
At position along the polar axis, we consider a localized increase in sound speed of 10% over a volume V_{s}, such that the signal (Eq. (40)) may be written as
The volume V_{s} is that of a ball of diameter λ(r_{s})/2 = π/[Re k(r_{s}, ω_{0})] with ω_{0}/2π = 3 mHz. This is an approximate but much faster way to compute the effect of a perturbation of volume comparable to the highest possible holographic resolution. We checked that the answer does not differ significantly from the one obtained by integrating numerically the kernel over the ball of volume V_{s}. For reference, we note that λ/2 = 38 Mm at r = 0.7 R_{⊙} and λ/2 = 20 Mm at r = 0.9 R_{⊙}.
Figure 4 shows the signal 𝔼[δI_{αβ}(x)] for soundspeed perturbations located at two different depths z_{s} = 0.7 R_{⊙} (red curve) and 0.9 R_{⊙} (blue curve). The pupil P and the wave propagators are the same as those of Fig. 2. The left panel of Fig. 4 shows the results at a single frequency of 3 mHz. With only one frequency, the signal peaks close to the scattering location but the spatial resolution is relatively poor, with a ghost on the far side. To demonstrate the benefits of averaging, the right panel shows the signal after averaging over 101 frequencies uniformly distributed in the interval 2.75–3.25 mHz. The frequency resolution 5 μHz corresponds to an observation duration T = 55.5 h. Averaging over frequencies improves the spatial resolution which approaches λ/2 and the ghost is suppressed. A horizontal line segment is plotted on the right panel at each depth to mark half the local wavelength.
Fig. 4.
Left panel: image intensity 𝔼[δI_{αβ}(x)] at a single frequency of 3 mHz, as a function of position x along the zaxis. The soundspeed perturbation (see Eq. (54)) is placed at two different positions along the zaxis, z_{s} = 0.7 R_{⊙} (red) and 0.9 R_{⊙} (blue). The standard deviation of the noise is given by the black curve. We note that the jagged aspect of the curves is not due to numerical inaccuracies. Right panel: Image intensity and noise level after averaging over 101 frequencies uniformly distributed in the interval from 2.75 to 3.25 mHz. The frequency resolution is 5 μHz, implying an observation duration of T = 55.5 h. A horizontal line segment is plotted at each depth to mark half of the local wavelength. 
As seen in Fig. 5 the spatial extent of the frequencyaveraged kernels is approximately λ/2 in both the radial and horizontal directions, for all scattering points in the range 0.6 < z_{s}/R_{⊙} < 0.98. Thus helioseismic holography is a diffractionlimited imaging technique as suggested by Lindsey & Braun (1997).
Fig. 5.
Radial and horizontal widths of the frequencyaveraged soundspeed kernel ⟨𝒦^{c}⟩ as functions of scattering position. These are close to half the local wavelength at 3 mHz (dotted line). 
7.4. Noise
The noise is obtained from Eq. (47), which requires the computation of 𝔼[I_{α, 0}] and 𝔼[I_{β, 0}] using Eq. (33). The reference crosscovariance C_{0} is precomputed. The double surface integral is evaluated in a similar way as for the kernel computations.
For a frequency of 3 mHz the left panel of Fig. 4 (black curve) shows the noise level, together with the signal described in the previous section. The jagged aspect of the noise variations with position is not due to numerical inaccuracies but to the details of the Green’s function. As shown in the right panel of Fig. 4, the noise level goes down by a factor of about ten after averaging over 101 frequencies, and varies more smoothly with depth.
Braun & Birch (2008a) studied the noise level in observed travel times measured from LB holography. These measurements, however, include contributions from supergranulation and so are not directly comparable to what is shown in Fig. 4.
7.5. Signaltonoise ratio
Figure 6 shows the S/N as a function of scatterer location. We recall that the soundspeed perturbation is specified by Eq. (54) and is the same as in Sect. 7.3. The results are shown at a single frequency of 3 mHz and after averaging over 101 frequencies in the interval 2.75–3.25 mHz. After averaging, the S/N is above two and is roughly independent of depth in the range 0.6 < z_{s}/R_{⊙} < 0.98 for a pupil of angular size 120°. We note that the ghost at −z_{s} is much below the noise level.
Fig. 6.
S/N in PB image intensity for a 10% soundspeed perturbation over a volume V_{s}(z_{s}) placed at z_{s} along the polar axis (Eq. (54)). The results are shown at a single frequency of 3 mHz (solid curve) and after averaging over 101 frequencies in the interval from 2.75 to 3.25 mHz (dashed curve). 
We find that both signal and noise vary rapidly with frequency for deep located soundspeed scatterers. Figure 7 shows an example of a soundspeed scatterer located at z_{s} = 0.7 R_{⊙}. Strong frequency variations in signal and noise are evident for frequencies below 3.5 mHz. This can be understood as follows. Lowfrequency modes have narrowlypeaked power spectra due to their long lifetimes. At these low frequencies, the amplitude of the kernel function and the noise will change rapidly when the frequency coincides with a particular mode frequency. Additionally, the kernel function may not peak exactly at the soundspeed scatterer position when only a few modes contribute to the kernel function. Figure 8 shows the S/N as a function of frequency for a soundspeed scatterer located at z_{s} = 0.9 R_{⊙}. For this target depth closer to the surface, the rapid variations disappear above 3 mHz, due to the larger contribution of highdegree modes which are not resolved in frequency space because of their short lifetimes.
Fig. 7.
Signal, noise, and S/N as function of frequency for a soundspeed scatterer located at z_{s} = 0.7 R_{⊙}. Here we show the result for a frequency range of 2–7 mHz. The rapid changes are not due to numerical inaccuracies. The red dots indicate the values at frequency 2.4000 mHz. 
Fig. 8.
S/N as in Fig. 7, but for a soundspeed scatterer located closer to the surface at z_{s} = 0.9 R_{⊙}. 
The kernel function at frequency 2.4000 mHz for z_{s} = 0.7 R_{⊙} is shown in Fig. 9; this particular frequency corresponds to the peak indicated in Fig. 7 with a red dot. We see that this kernel is much less localized around the scattering point than the kernel at 3 mHz (Fig. 2).
Fig. 9.
Meridional slice of the soundspeed kernel with z_{s} = 0.7 R_{⊙} computed at the low frequency of 2.4000 mHz, which corresponds to the spike with a red dot in Fig. 7. This kernel displays oscillations and is not peaked as much as the 3 mHz kernel from Fig. 2. 
8. Conclusion
We derived a framework for computing the expected signal and the noise level in PB helioseismic holography. The same framework could be used to interpret LB data, including phasesensitive data.
PorterBojarski holography requires knowledge of the wave field, ψ = ρ^{1/2}c^{2}∇⋅ξ, and its normal derivative on the solar surface, ∂_{n}ψ. With this definition of ψ, the Green’s function used in the definition of PB integrals solves a welldefined Helmholtzlike equation, which we solve numerically (Gizon et al. 2017). The need for finitefrequency Green’s functions was demonstrated in LB holography by Pérez Hernández & González Hernández (2010). In the numerical examples shown in the previous section, we assumed that we have full knowledge of ∂_{n}ψ on the surface. In practice, we do not observe directly the normal derivative of the wave field; it must be approximated. According to complementary simulations (not shown here), this can be achieved by using the approximate outgoing radiation condition ∂_{n}ψ = ik_{n}ψ derived by Barucq et al. (2018).
We found that, for a sufficiently large pupil, scatterers can be imaged at a resolution that is very close to half the local wavelength, λ/2. This confirms the claim by Lindsey & Braun (1997, 2000a) that helioseismic holography is diffractionlimited. In that sense, helioseismic holography is superior to deepfocusing timedistance helioseismology, which gives lower spatial resolution (Munk 2001; Pourabdian et al. 2018). For large pupils, we found that the S/N in PB images does not vary much with depth in the convection zone, when a perturbation in soundspeed fills a volume corresponding to the holographic resolution.
Averaging over frequencies improves the S/N. For a scatterer at the bottom of the convection zone, the signal and the noise vary smoothly with frequency above 4 mHz (see Fig. 7). At lower frequencies, however, the signal varies rapidly with frequency (due to contributions from individual longlived p modes) and it is not obvious how the signal should be averaged. A specific analysis of lowfrequency data is required, especially for deep scatterers.
We found that the S/N in PB holography is maximum around 3.7 mHz for z_{s} = 0.7 R_{⊙} (resp. at 4.3 mHz for z_{s} = 0.9 R_{⊙}). There is no indication in our calculations that there is a benefit in using the frequencies above the acoustic cutoff (unlike predictions by Ruzmaikin & Lindsey 2003). The S/N drops to very small values above 5 mHz. One may ask if this drop is somehow compensated by the increase in spatial resolution at high frequencies. The answer is negative. Our calculations indicate that noise has a horizontal correlation length that is about half the local wavelength. Far too few independent measurements are available at high frequencies to recover an acceptable S/N by horizontal spatial averaging.
Our synthetic data do not contain a convective background. The effect of this background on S/Ns in holography should be studied. Future work should also investigate the performance of PB holography for target locations that are away from the axis of the pupil, especially for farside imaging applications.
Acknowledgments
The theoretical framework was developed by L.G. and D.F. at Mathematisches Forchungsinstitut Oberwolfach in May 2017. The numerical computations were performed by D.Y. using the Montjoie solver. D.Y. is a member of the International Max Planck Research School for Solar System Science at the University of Göttingen. We thank M. Duruflé and J. Chabassier from Inria Magique3D for the helioseismologyrelated developments of Montjoie. We also thank Chris Hanson from NYUAD for the fine tuning of the model power spectrum of solar oscillations. L.G. acknowledges support from NYUAD Institute grant G1502. The computing resources were provided in part by the German Data Center for SDO, a project funded by the German Aerospace Center (DLR).
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All Tables
Proposed propagators and associated pupils (H_{α}, A) and (H_{β}, A′) for different types of perturbations.
All Figures
Fig. 1.
Pupil geometries used to compute soundspeed or source kernels (P) and flow kernels (Qs), also see Table 2. 

In the text 
Fig. 2.
Meridional slice through the soundspeed kernel (x, r_{s}) computed in Model S at a single frequency of 3 mHz, in units of 10^{−34} kg m^{−3} s^{−3}. Both the real (left panel) and imaginary (right panel) parts of the kernel are shown. The scatterer at z_{s} = 0.7 R_{⊙} is indicated by the crosses. The observation pupil P is a polar cap of full angular size 120°. We notice the “ghost values” at the antipode. 

In the text 
Fig. 3.
Cut along the z axis through the soundspeed kernel from Fig. 2. The scatterer is at z_{s} = 0.7 R_{⊙}. 

In the text 
Fig. 4.
Left panel: image intensity 𝔼[δI_{αβ}(x)] at a single frequency of 3 mHz, as a function of position x along the zaxis. The soundspeed perturbation (see Eq. (54)) is placed at two different positions along the zaxis, z_{s} = 0.7 R_{⊙} (red) and 0.9 R_{⊙} (blue). The standard deviation of the noise is given by the black curve. We note that the jagged aspect of the curves is not due to numerical inaccuracies. Right panel: Image intensity and noise level after averaging over 101 frequencies uniformly distributed in the interval from 2.75 to 3.25 mHz. The frequency resolution is 5 μHz, implying an observation duration of T = 55.5 h. A horizontal line segment is plotted at each depth to mark half of the local wavelength. 

In the text 
Fig. 5.
Radial and horizontal widths of the frequencyaveraged soundspeed kernel ⟨𝒦^{c}⟩ as functions of scattering position. These are close to half the local wavelength at 3 mHz (dotted line). 

In the text 
Fig. 6.
S/N in PB image intensity for a 10% soundspeed perturbation over a volume V_{s}(z_{s}) placed at z_{s} along the polar axis (Eq. (54)). The results are shown at a single frequency of 3 mHz (solid curve) and after averaging over 101 frequencies in the interval from 2.75 to 3.25 mHz (dashed curve). 

In the text 
Fig. 7.
Signal, noise, and S/N as function of frequency for a soundspeed scatterer located at z_{s} = 0.7 R_{⊙}. Here we show the result for a frequency range of 2–7 mHz. The rapid changes are not due to numerical inaccuracies. The red dots indicate the values at frequency 2.4000 mHz. 

In the text 
Fig. 8.
S/N as in Fig. 7, but for a soundspeed scatterer located closer to the surface at z_{s} = 0.9 R_{⊙}. 

In the text 
Fig. 9.
Meridional slice of the soundspeed kernel with z_{s} = 0.7 R_{⊙} computed at the low frequency of 2.4000 mHz, which corresponds to the spike with a red dot in Fig. 7. This kernel displays oscillations and is not peaked as much as the 3 mHz kernel from Fig. 2. 

In the text 
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