Issue 
A&A
Volume 616, August 2018



Article Number  A156  
Number of page(s)  6  
Section  Numerical methods and codes  
DOI  https://doi.org/10.1051/00046361/201833206  
Published online  11 September 2018 
Sensitivity kernels for timedistance helioseismology
Efficient computation for spherically symmetric solar models
^{1}
MaxPlanckInstitut für Sonnensystemforschung, JustusvonLiebigWeg 3, 37077, Göttingen, Germany
email: fournier@mps.mpg.de
^{2}
Institut für Astrophysik, GeorgAugustUniversität Göttingen, FriedrichHundPlatz 1, 37077, Göttingen, Germany
^{3}
Magique3D, Inria Bordeaux SudOuest, Université de Pau et des Pays de l’Adour, 64013 Pau, France
Received:
9
April
2018
Accepted:
17
May
2018
Context. The interpretation of helioseismic measurements, such as wave traveltime, is based on the computation of kernels that give the sensitivity of the measurements to localized changes in the solar interior. These kernels are computed using the ray or the Born approximation. The Born approximation is preferable as it takes finitewavelength effects into account, although it can be computationally expensive.
Aims. We propose a fast algorithm to compute traveltime sensitivity kernels under the assumption that the background solar medium is spherically symmetric.
Methods. Kernels are typically expressed as products of Green’s functions that depend upon depth, latitude, and longitude. Here, we compute the spherical harmonic decomposition of the kernels and show that the integrals in latitude and longitude can be performed analytically. In particular, the integrals of the product of three associated Legendre polynomials can be computed.
Results. The computations are fast and accurate and only require the knowledge of the Green’s function where the source is at the pole. The computation time is reduced by two orders of magnitude compared to other recent computational frameworks.
Conclusions. This new method allows flexible and computationally efficient calculations of a large number of kernels, required in addressing key helioseismic problems. For example, the computation of all the kernels required for meridional flow inversion takes less than two hours on 100 cores.
Key words: Sun: helioseismology / Sun: oscillations / Sun: interior / methods: numerical
© ESO 2018
1. Introduction
The aim of timedistance helioseismology (Duvall et al. 1993) is to infer the subsurface structure of the Sun by measuring seismic wave travel times between any two points at the solar surface. The interpretation of these measurements requires understanding how waves propagate in the solar interior, i.e., solving the forward problem. Due to its simplicity, the ray approximation was initially used to invert flow velocities and soundspeed perturbations to a reference background model (Kosovichev 1996). It is still used today, for example to recover the meridional circulation (Rajaguru & Antia 2015). However, this approach is a highfrequency approximation that cannot be used to recover perturbations with sizes on the order of the local wavelength (Birch & Kosovichev 2000). Gizon & Birch (2002) derived a general framework for sensitivity kernels under the Born approximation and for random sources of excitation. Birch et al. (2004), Burston et al. (2015), and Böning et al. (2016) computed Born kernels using a normalmode summation of the eigenfunctions in a solarlike stratified background. To treat axisymmetric background media (e.g., a background that includes largescale differential rotation) and to include frequencies above the acoustic cutoff, Gizon et al. (2017) proposed solving the wave equation in frequency space by using a 2.5D finiteelement solver. All these approaches are useful, but are computationally expensive. This limits their use in the interpretation of solar data as many kernels must be computed (and averaged). In some cases it is sufficient to consider perturbations to a steady spherically symmetric reference medium. The study of meridional circulation is one such application (see, e.g., Liang et al. 2017).
In this paper, we present a way to reduce the computational time of Born sensitivity kernels in a spherically symmetric background by treating the horizontal variables (the colatitude θ and the longitude ϕ) analytically using the properties of the spherical harmonics. Here this approach is demonstrated using the scalar wave equation from Gizon et al. (2017), but it could be applied to the normalmode summation method of Böning et al. (2016) or to solving the wave equation using a highorder finitedifference scheme (Mandal et al. 2017).
2. Born sensitivity kernels
2.1. Green’s function in a spherically symmetric background
We follow the framework of Gizon et al. (2017), where the observable ψ(r, ω) at spatial location ω = (r,θ,ϕ) and frequency ω is linked to the divergence of the displacement: ψ(r,ω) = c(r) ∇ ∙ ξ(r,ω). This scalar quantity solves
where L is the spatial wave operator at frequency ω,
ρ and c are the solar density and sound speed from standard solar model S (ChristensenDalsgaard et al. 1996), γ is the attenuation, u is a background flow, and s is a stochastic source term. We assume that the sources are spatially uncorrelated and depend only on depth and frequency such that the source covariance matrix is given by
where A(r,ω) is the radial profile of the source power. The wave field ψ can be obtained using
where G is the Green’s function:
When the background is spherically symmetric (i.e., no flow and no heterogeneity), G can be written as
where r =(r,θ,ϕ), r′ = (r′,θ′,ϕ′), are the normalized spherical harmonics, , and G_{ℓ} is the Legendre component of the Green’s function
Equation (6) can be simplified by using the addition theorem (DLMF 2017, Eq. (14.18.1)) and introducing the greatcircle angle between r and r′. However, Eq. (6) is the form required in the following sections.
2.2. Crosscovariance in a spherically symmetric background
In a spherically symmetric background the expectation value of the crosscovariance between an observation point r_{1} = (r_{0},θ_{1},ϕ_{1}) and a point r = (r,θ,ϕ) is
where
and r_{0} = (r_{0},0,0) and are on the polar axis. The radius r_{0} is the observation radius, for example ∼150 km above the photosphere for SDO/HMI. In obtaining Eq. (8), we used the property that the crosscovariance depends only on the greatcircle distance between the two points. To simplify the computations we place one point on the polar axis so that the Green’s function is axisymmetric and only the mode m = 0 needs to be computed.
Using the convenient source of excitation introduced in Gizon et al. (2017), C_{ℓ}(r_{0},r,ω) is directly linked to the imaginary part of the Green’s function G_{ℓ}(r,r_{0},ω), but this assumption is not mandatory in this paper. The important assumption concerns the covariance of the sources of excitation that needs to be of the form given by Eq. (3), such that the crosscovariance depends only on depths and the greatcircle distance between the two points r_{1} and r. This assumption is common in helioseismology and is generally used in forward modeling (e,g., Kosovichev et al. 2000; Böning et al. 2016; Mandal et al. 2017).
2.3. Born flow kernels
Recovering flows in the solar interior is a major goal for local helioseismology, and so we focus here on flow kernels. The method presented here can be applied to all other types of perturbations with respect to a spherically symmetric background. The Born sensitivity kernel K = (K_{r}, K_{θ}, K_{ϕ}) connects the traveltime perturbation δτ to the vector flow u = (u_{r},u_{θ}, u_{ϕ}), such that
According to Gizon et al. (2017) we have
where W is a weighting function that relates a change in the crosscovariance to a change in traveltime (Gizon & Birch 2002) and ∇ = (∂_{r}, 1/r ∂_{θ}, 1/(r sin θ) ∂_{ϕ}) is the gradient operator with respect to the scattering location r. We note that in a spherically symmetric background, seismic reciprocity implies G(r,r′,ω) = G(r′,r,ω) for any r and r′. The reference crosscovariance also satisfies C(r′,r,ω) = C(r,r′,ω).
The expression for the kernel may differ when a different observable is chosen; however, the above integral will always involve the product of a Green’s function with the crosscovariance. One approach to obtaining the flow kernels (Böning et al. 2016; Mandal et al. 2017) is to compute the 3D Green’s function and the crosscovariance using its spherical harmonic decomposition using Eq. (6). A reference kernel is usually obtained for a fixed pair of observation points and later rotated to obtain kernels for other pairs of points. However, a fine resolution in θ and ϕ is required in order to perform this rotation accurately, which makes the computation expensive in both time and memory.
2.4. Spherical harmonic decomposition of Born flow kernels
In order to circumvent the disadvantages mentioned above (e.g., rotation) and improve accuracy, we propose a new approach based on the spherical harmonic decomposition of the kernel,
where
Decomposing G(r_{1},r,ω) and C(r_{2},r,ω) into spherical harmonics, we can obtain the spherical harmonic coefficients of each kernel.
For the u_{r} kernel, we have
where
and
The integral of three spherical harmonics over the unit sphere can be done analytically using the Gaunt formula (see, e.g., Edmonds 1960, Eq. (4.6.3))
where we have used the Wigner3j symbols (see, e.g., Edmonds 1960, p. 45). The Wigner3j symbol vanishes when , which enables us to remove the sum over m′ in the equation for K_{ur}.
It can be shown that the expression for K_{r}, K_{θ}, and K_{ϕ} can be recast in the form
where j ∈ {r, θ, ϕ and L = min(ℓ,ℓ′).
Proceeding in a similar way, for K_{r} the kernel depends on the functions f^{θ} and g^{θ} given by
The horizontal integral is
This integral I_{θ} is much more difficult to evaluate than I_{r} because of the θ derivative. In order to keep only associated Legendre polynomials in I_{θ}, we use
where the are the normalized associated Legendre polynomials. We use the convention that if m ± 1 > ℓ, so that Eq. (23) remains valid for m = ±ℓ. Integrating Eq. (22) over ϕ and using Eq. (23), I_{θ} becomes
where and
Fortunately, this integral can also be evaluated analytically. It involves a sum of products of Wigner3j symbols (see Appendix A and Dong & Lemus 2002).
The derivation of u_{ϕ} is similar to u_{θ} and requires the evaluation of the horizontal integral
Using
for m ≠ 0, we obtain
Now that we have the equations for the kernels, let us summarize the algorithm used for the resolution:
Computation and storage of the Green’s function G_{l}(r, r_{0}, ω) with the source on the polar axis, as a function of depth and harmonic degree ℓ for all frequencies;
For each greatcircle distance between r_{1} and r_{2}:
computation of the crosscovariance using Eq. (8). If the convenient source of excitation of Gizon et al. (2017) is used, the crosscovariance is directly obtained from the imaginary part of the Green’s function;
computation of the weighting function W;
computation of the functions f^{j} and g^{j};
Evaluation of the integrals I_{j} and computation of the kernel using Eq. (19).
A summary of the different terms required to compute the different components of the flow kernels using Eq. (19) is given in Table 1.
We note that the algorithm presented above could of course be used to compute sensitivity kernels for the crosscovariance amplitude using the linear definition of Nagashima et al. (2017) and the appropriate choice of W. It is also possible to obtain kernels for the crosscovariance function at a given frequency by removing the weighting function. In this case, the functions g^{j} are just the complex conjugates of f^{j}.
2.5. Numerical validation
To evaluate the flow kernels in this framework, the only ingredient to prescribe is the Green’s function as a function of the spherical harmonic degree ℓ and depth for a source located at the pole. We compute it using 1D finite elements with the solver Montjoie (Chabassier & Duruflé 2016; Fournier et al. 2017). The Green’s functions are computed with a high enough frequency resolution to resolve the modes using the mode linewidths (5671 frequencies corresponding to 4 days of observations at 60 s cadence) and ℓ_{max} = 400.
Representations of the different components of the flow kernels between two points r_{1} and r_{2} centered at 40° and separated by 42° is shown in Figs. 1 and 2. They exhibit the classical bananadoughnut shape with zero sensitivity along the ray path. Smallscale structures are visible close to the surface as we kept values of ℓ up to 400. Visually, there is no difference between the kernels computed with this new approach, the ones from Gizon et al. (2017), or the ones obtained by rotation so only one is shown here.
To allow a more quantitative approach, Fig. 3 compares kernels computed using our new approach to the approach presented by Gizon et al. (2017) where the background is axisymmetric and the Green’s function is computed for each azimuthal degree m on a 2D grid. These kernels K_{r} and K_{θ} are averaged over longitudes () where r_{1} is located at the pole and r_{2} is at a colatitude of 42° (similar to Fig. (17) in Gizon et al. 2017). The results show good agreement, validating the method presented here. We note slight differences in the structure of K_{θ} at a depth of 500 km, and attribute this to the numerics of the 2D FEM solver differing. Specifically, the 2D FEM has an inherent difficulty in computing the real part of the Green’s function close to the Dirac source location (see for details Chabassier & Duruflé 2016). In order to ensure that these small differences do not affect the interpretation of the data, we compute the travel times induced by the meridional flow model from Gizon et al. (2017). We decompose the flow in Legendre polynomial akin to Eq. (7) and compute a travel time for each according to
The bottom panel of Fig. 3 shows that this travel time as a function of is nearly indistinguishable from the travel times in Gizon et al. (2017); the differences are less than 0.5 ms.
As the computational burden of this new method depends on the maximum harmonic degree of the Green’s function ℓ_{max} and the number of required for the kernel, we illustrate the efficiency of the method on two problems of interest: kernels for meridional flow inversions and kernels for supergranulation inversions.
Fig. 1. Slices along a constant meridian of the pointtopoint 3D traveltime difference kernel for u_{r} (left) and u_{θ} (right). The 3D kernel for u_{ϕ} is zero along this slice. The kernel is computed with r_{1} and r_{2} separated by 42°, with mean latitude 40°. The green line is the ray path between the two points and the dashed black line shows the image plane of Fig. 2. 

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Fig. 2. Slices of the pointtopoint 3D traveltime difference kernel for u_{θ} (left) and u_{ϕ} (right) along the plane indicated in Fig. 1. The 3D kernel for u_{r} is mostly zero within this plane. The kernel is computed with r_{1} and r_{2} separated by 42°, with mean latitude 40°. The green cross indicates the intersection of the ray path and the image plane. The dashed black line shows the image plane of Fig. 1. 

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Fig. 3. Top and middle panels: comparison of the kernels for and 10 (blue and black, respectively) computed using the method here (solid lines) and the method of Gizon et al. (2017; dots). Bottom panel: travel times due to the radial (red) and the latitudinal (green) components of the flow for each of the kernels presented here (solid line) and those of Gizon et al. (2017; dots). 

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2.6. Computation of kernels for meridional flow
As a first test, we compute the kernels that are required to interpret meridional flow measurements. As the flow varies slowly with latitude, we can limit the number of spherical harmonic coefficients of the kernels to (see Fig. 3). As for the observations, a lowpass filter with ℓ_{max} = 300 is applied to the Green’s function. The method can be parallelized in so we use 11 cores to compute kernels up to .
For meridional flow measurements, the separation distance between the source and the receiver is generally prescribed for different values of the mean latitude (see, e.g., Liang et al. 2017). For a given separation distance, we compute 15 kernels corresponding to 15 different latitudes. We then vary the separation distance in order to probe different depths.
Table 2 shows the computational times and memory requirements of the different steps of the algorithm. The computation of the Green’s function for a source located at the pole can be done once and for all and stored as it is necessary to do it for every kernel. Therefore, the computation is very fast (3 seconds per frequency for ℓ_{max} = 300) and embarrassingly parallel in frequency so it could also be recomputed every time. The computation of the frequency integrals (f^{j} and g^{j}) consists in loading the Green’s function and the weighting function W and summing over frequencies. The reading of the Green’s function files for all frequencies takes most of the time. The spatial integrals I_{r}, I_{θ}, and I_{ϕ} can be computed once and stored for future use. However, the computational time is small compared to the full computation of the kernel, so we decide to recompute I_{j} every time as the reading time can depend upon file system I/O load. The computations are parallelized in and hence need 11~cores for each step since . The computation of the kernel for u_{r} is faster since the computation of I_{r} requires the evaluation of only two Wigner3j symbols, unlike I_{θ}. However, the major difference in computational time between K_{r} and K_{θ} comes from the sum in ℓ′ in Eq. (19). For K_{r}, the sum covers the range from to , since I_{r} = 0 for other values due to the properties of the Wigner3j. On the contrary, the sum in ℓ′ for the computation of K_{θ} must be computed for the full range from 0 to ℓ.
Even though the computational burden of K_{θ} is greater than K_{r}, the total burden remains significantly lower than for other methods (see Table 3). All the kernels required to perform a meridional flow inversion can be computed within 2 h with 100 cores, and the memory requirements do not exceed 1 GB. The approach mentioned in Sect. 2.3, where the full 3D kernel is computed and rotated to obtain different latitudes, would take 11 days using 1000 cores with very significant memory requirements. In the axisymmetric approach of Gizon et al. (2017) the computational time would be about 40 days on 1000 cores for all the same set of kernels.
The computational times presented here are for pointtopoint measurements; however, this framework can easily be extended to geometric averaging such as arctoarc measurements often performed for meridional flow measurements (e.g., Liang et al. 2017). It is only necessary to replace the product of the two spherical harmonics in Eq. (19) by a sum over all the points of the arc.
Computational time of the different steps to obtain 15 flow kernels (15 latitudes) for a given separation distance with and ℓ_{max} = 300 using 11 cores.
Comparison of the computational time and memory requirements to evaluate 225 kernels for the meridional flow.
2.7. Computation of kernels for supergranulation
Resolving smaller scale flows such as supergranules requires a high spatial resolution, and thus the Green’s function needs to include much higher harmonic degrees than for meridional flow Green’s function. For example, Duvall & Hanasoge (2013) considered measurements up to ℓ_{max} = 700, but even higher ℓ_{max} values may be required. Furthermore, supergranulation flows have maximum power around ℓ = 120, thus the kernels should at least be computed up to , or higher depending on the power distribution of the flow at large ℓ. The computational burden for these kernels is summarized in Table 4. The computation of the m = 0 component of the Green’s function now takes about 23 second per frequency, and the loading of the files to compute f^{j} around 6 min. The computation of the Wigner symbols is computationally more challenging as ℓ increases, since the number of loops scales as due to loops in ℓ, ℓ′, and m. While the computation of I_{r} is still fast, the evaluation of I_{θ} now takes 270 min on 100 cores. Computing a set of 200 kernels would take 3 days with 1000 cores, which is significantly longer than for the meridional flow kernels, but still one order of magnitude faster than the approach of Gizon et al. (2017) and with a smaller memory requirement.
Computational time of the different steps to obtain 15 flow kernels for a given separation distance with and using 100 cores.
3. Conclusions
We presented a technique that is faster than previous approaches used to compute traveltime kernels under the assumption that the background medium is spherically symmetric. This technique does not rely on the numerical computation of kernel rotations and thus the memory requirements are small. Instead, the spatial integrals are performed analytically, which also leads to higher accuracy. For example, for meridional circulation applications, the kernels can be computed one thousand times faster than with previous methods, using a tenth of the memory requirement.
Acknowledgments
The computer infrastructure was provided by the German Data Center for SDO funded by the German Aerospace Center (DLR) and by the Ministry of Science of the State of Lower Saxony, Germany.
Appendix A
Algorithm for the integral of three associated Legendre polynomials
For the sake of completeness, we summarize the algorithm of Dong & Lemus (2002) adapted to this study. The integral of three associated Legendre polynomials,
can be computed analytically in terms of sums of products of Wigner3j symbols
where the indices m_{12} = m + m′ and represent sums over the azimuthal degrees. The quantities Q_{12} and Q_{123} must be evaluated for various values of ℓ_{12} and ℓ_{123} as defined under the sums in Eq. (A.2). They depend on the Wigner3j symbols:
The value of Q_{12} (resp. Q_{123}) is nonzero only if ℓ_{12} + ℓ + ℓ′ (resp. ) is even. The last term J(ℓ_{123}, m_{123}) is the integral
which can be evaluated analytically. In this paper, we only need this value for m_{123} = ± 1. As this integral is zero for odd values of ℓ_{123}, due to the parity of the associated Legendre polynomials, we set ℓ_{123} = 2p + 1. Then, for a given m_{123}, the value of J(ℓ_{123}, m_{123}) can be evaluated recursively using
and
where n = 1, 2, ... p, together with the initial conditions
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All Tables
Computational time of the different steps to obtain 15 flow kernels (15 latitudes) for a given separation distance with and ℓ_{max} = 300 using 11 cores.
Comparison of the computational time and memory requirements to evaluate 225 kernels for the meridional flow.
Computational time of the different steps to obtain 15 flow kernels for a given separation distance with and using 100 cores.
All Figures
Fig. 1. Slices along a constant meridian of the pointtopoint 3D traveltime difference kernel for u_{r} (left) and u_{θ} (right). The 3D kernel for u_{ϕ} is zero along this slice. The kernel is computed with r_{1} and r_{2} separated by 42°, with mean latitude 40°. The green line is the ray path between the two points and the dashed black line shows the image plane of Fig. 2. 

Open with DEXTER  
In the text 
Fig. 2. Slices of the pointtopoint 3D traveltime difference kernel for u_{θ} (left) and u_{ϕ} (right) along the plane indicated in Fig. 1. The 3D kernel for u_{r} is mostly zero within this plane. The kernel is computed with r_{1} and r_{2} separated by 42°, with mean latitude 40°. The green cross indicates the intersection of the ray path and the image plane. The dashed black line shows the image plane of Fig. 1. 

Open with DEXTER  
In the text 
Fig. 3. Top and middle panels: comparison of the kernels for and 10 (blue and black, respectively) computed using the method here (solid lines) and the method of Gizon et al. (2017; dots). Bottom panel: travel times due to the radial (red) and the latitudinal (green) components of the flow for each of the kernels presented here (solid line) and those of Gizon et al. (2017; dots). 

Open with DEXTER  
In the text 
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