Probabilistic Inversions for Time–Distance Helioseismology

Abstract

Time–distance helioseismology is a set of powerful tools to study localized features below the Sun’s surface. Inverse methods are needed to robustly interpret time–distance measurements, with many examples in the literature. However, techniques that utilize a more statistical approach to inferences, and that are broadly used in the astronomical community, are less-commonly found in helioseismology. This article aims to introduce a potentially powerful inversion scheme based on Bayesian probability theory and Monte Carlo sampling that is suitable for local helioseismology. We first describe the probabilistic method and how it is conceptually different from standard inversions used in local helioseismology. Several example calculations are carried out to compare and contrast the setup of the problems and the results that are obtained. The examples focus on two important phenomena that are currently outstanding issues in helioseismology: meridional circulation and supergranulation. Numerical models are used to compute synthetic observations, providing the added benefit of knowing the solution against which the results can be tested. For demonstration purposes, the problems are formulated in two and three dimensions, using both ray- and Born-theoretical approaches. The results seem to indicate that the probabilistic inversions not only find a better solution with much more realistic estimation of the uncertainties, but they also provide a broader view of the range of solutions possible for any given model, making the interpretation of the inversion more quantitative in nature. The probabilistic inversions are also easy to set up for a broad range of problems, and they can take advantage of software that is publicly available. Unlike the progress being made in fundamental measurement schemes in local helioseismology that image the far side of the Sun, or have detected signatures of global Rossby waves, among many others, inversions of those measurements have had significantly less success. Such statistical methods may help overcome some of these barriers to move the field forward.

Appendices

Appendix A: Supergranulation Inversion Details

This article presents a comparison of six SOLA inversions (two flow components at three target depths) to the probabilistic one for a simulated supergranule model. For completeness, details of the inversions and how they are compared are itemized here:

• i) The simulation domain is 100 Mm on a side horizontally, sampled evenly at $1/3$ Mm. It extends to −25 Mm at the bottom boundary. Cross correlations and travel times were measured in an area 50 Mm on a side at the same sampling. Each map is therefore $150\times 150$ points.

• ii) Born-sensitivity kernels are computed on the same horizontal grid as the travel times and they extend to 15 Mm below the surface using 55 points in depth. The kernels use input model power spectra that have been Fourier filtered to separately retain the first three radial orders, including the $f$-mode. Everything is computed to match the details of the travel times as discussed in Section 3.2.2.

• iii) The SOLA inversion code is the faster formulation in Fourier space that can be run in parallel (Jackiewicz et al., 2012). It includes a cross-talk parameter as in Švanda et al. (2011). Trade-off curves (L-curves) were studied to find the best regularization parameters. To obtain sensible results in terms of noise and misfit, the 3D Gaussian target functions have FWHM on the order of 10 Mm and 3 Mm in the horizontal and vertical directions, respectively. In general, the SOLA inversion results are very similar to the ones published by Dombroski et al. (2013) using an RLS inversion scheme.

• iv) The probabilistic inversion has $2\times 10^{5}$ steps. Each of the five parameters was assigned 120 walkers. With thinning every five steps, the PDF was sampled $2\times 10^{5}/(120\times 5) = 333$ times per walker, as before. More walkers were used in this inversion than the meridional example because the PDFs are more complicated, and it is suggested to use more walkers for good sampling. These values are sufficient to give good autocorrelation properties of the walker time series.

• v) The data vector ${\boldsymbol{d}}$ is composed of only a small subset (12 of the 135) of the travel-time measurement maps. To speed up the computation, each of the $150\times 150$ pixel maps was binned down to only $19\times 19$ pixels. The data vector is therefore 4332 measurements. To match the size of the data vector, the forward vector resulting from the operation $g( {\boldsymbol{m}} )$ uses kernels that are also binned down. This is noteworthy, in that the SOLA inversion uses the full maps. The probabilistic inversion is quite powerful with substantially less data. Of course, the 4332 measurements are not all completely independent, as described by the covariance matrix.

• vi) The full noise covariance matrix used in the likelihood function (Equation 17) was constructed from smaller pairs of covariances ${\mathrm{Cov}}[\delta \tau _{i},\delta \tau _{j}]$ and ordered to correctly match the travel-time data vector.

• vii) The uniform priors on the model are given in Table 1. As is common (Foreman-Mackey et al., 2013), the walkers are initialized with values in a small Gaussian ball, centered somewhere within the prior bounds. The initial guess is not critical, as the walkers soon quickly explore the full parameter space after the burn-in stage.

• viii) The entire run on a two-core desktop machine (thus minimal parallelization) was about 80 minutes. The most expensive task is the computation of the forward travel times in Equation 29, which takes about 0.2 seconds each.

Appendix B: Supergranulation Flow Inversions

A few more results are presented here to add to those in Section 3.2.3. The corner plot for the probabilistic inversion is given in Figure 9.

The flow-inversion comparison near the top of the simulation domain is shown in Figure 10. For the horizontal flows, the SOLA inversion underestimates the (smoothed) flow amplitude by about a factor of two, while the probabilistic inversion overestimates the amplitude by $15\,\%$. The profile structure is primarily determined by parameters $p_{4}$ and $p_{5}$. The peak vertical velocity is underestimated by 35% in the SOLA inversion, and it is overestimated by 60% in the probabilistic inversion. The $f$-mode is generally the most sensitive at these layers, but its sensitivity to vertical flows is weak. This affects both inversions. $v_{z}^{\mathrm{SOLA}}$ also has artifacts in its flow structure.

Figure 10

Comparison of supergranulation inversion results at the model photosphere. The left (right) panels are for the $v_{x}$ ($v_{z}$) inversion. The top rows are the flow fields for the simulation, the SOLA inversion, and the probabilistic inversion computed from the median of the PDF. Darker shading corresponds to positive velocities, and the scale is the same in each set. The bottom panels are cuts through the supergranule at $y=0$, given by the dashed line in the top panels. Twenty random samples of the PDF are drawn and computed in the model space. A few representative points from the SOLA inversion are given with uncertainties.

A comparison near 5 Mm beneath the surface is given in Figure 11. As is common in standard helioseismic inversions using typical data sets, the inferences at this depth are uninformative (Zhao et al., 2007). The vertical velocity inferred is weak and is the wrong sign, which was also noticed in Dombroski et al. (2013). The probabilistic inversion provides a good estimate of the flow structure at this depth.

Figure 11

Comparison of supergranulation inversion results near about 5 Mm beneath the model photosphere. The left (right) panels are for the $v_{x}$ ($v_{z}$) inversion. The top rows are the flow fields for the simulation, the SOLA inversion, and the probabilistic inversion computed from the median of the PDF. Darker shading corresponds to positive velocities, and the scale is the same in each set. The bottom panels are cuts through the supergranule at $y=0$, given by the dashed line in the top panels. Twenty random samples of the PDF are drawn and computed in the model space. A few representative points from the SOLA inversion are given with uncertainties.