Modelling the Center-to-Limb systematic in normal-mode-coupling measurements

TL;DR

The paper tackles the challenge of inferring solar meridional circulation (MC) from global-mode cross-spectral measurements, which are severely biased by center-to-limb (C2L) variations. It develops a theoretical framework that combines axisymmetric MC coupling with a complex, phase-based C2L-bias model implemented through a modified leakage matrix, and demonstrates that neglecting C2L produces biased shallow MC inferences. Through forward modeling and preliminary inversions, the study shows that incorporating C2L improves shallow-layer MC estimates and identifies depth-dependent limitations due to declining sensitivity and signal-to-noise ratio, outlining strategies to overcome them. Finally, it presents a feasible plan for a full, gradient-based inversion using a large cross-spectral dataset (approximately $9\times10^4$ spectra) with a $140$-parameter C2L representation and a $150$-parameter MC-depth representation, leveraging high-performance computing to yield robust, time-averaged MC profiles across the solar convection zone.

Abstract

Solar meridional circulation, which manifests as poleward flow near the surface, is a relatively weak flow. While meridional circulation has been measured through various local helioseismic techniques, there is a lack of consensus about the nature of the depth profile and location of return flow, owing to its small amplitude and poor signal-to-noise ratio in observations. The measurements are strongly hampered by systematic effects, whose amplitudes are comparable to the signal induced by the flow and modelling them is therefore crucial. The removal of the center-to-limb systematic, which is the largest known feature hampering the inference of meridional flow, has been heuristically performed in helioseismic analyses, but it's effect on global modes is not fully understood or modelled. Here, we propose both a way to model the center-to-limb systematic and a method for estimation of meridional flow using global helioseismic cross-spectral analysis. We demonstrate that the systematic cannot be ignored while modelling the mode-coupling cross-spectral measurement and thus is critical for the inference of meridional circulation. We also show that inclusion of a model for the center-to-limb systematic improves shallow meridional circulation estimates from cross-spectral analysis.

Figures (23)

• Figure 1: Plot of different C2L-bias profiles based on $\sin\theta_H$ parameterization.
• Figure 2: The C2L bias, given by $\exp(if(r_\mathrm{H}))$ is expanded in terms of spherical harmonics as $\sum_s^{s_\mathrm{max}} \Lambda_s Y_{s0}$. The stars indicate real part of $\Lambda_s$ and dots indicate the imaginary part. The y-axis is in logarithmic scale. The dashed-red line shows the value of $s$ at which the spherical harmonic expansion is truncated.
• Figure 3: Error in estimating the C2L bias due to the truncation of the spherical-harmonic expansion at $s_\mathrm{max} = 7$, plotted in logarithmic scale. The true C2L-bias profile is shown as a solid-red line. The black line shows the error due to truncation of the series-expansion. In all the locations where the C2L-bias profile is non-negligible, the error due to truncation of spherical harmonic expansion is insignificant.
• Figure 4: Real part of the cross-spectrum of modes $(0, 200)$ and $(0, 202)$ at finite $t$. The panels on the left show the stacked cross-spectral measurement $S^m(\omega) = \langle \varphi^{200, m}(\omega) \varphi^{202, m+t}(\omega) \rangle$ for $m\ge 0$. The panels on the right correspond to stack-summed cross-spectra $\sum_m S^m(\omega)$.
• Figure 5: Product of leakage matrices in the expression of the cross-spectral model (Eqn. \ref{['eqn:cross-spectra-model']}). To understand the observation in Figure \ref{['fig:p4-cs-sample-nonzerot']} where the cross-spectral signal in $t=1$ channel is tiny, we show a slice of the product of leakage matrices $\tilde{L}^{\ell m}_{\ell_1 m_1} \tilde{L}^{\ell' m'}_{\ell_2 m_2}$. For illustration, we show the product for $\ell, \ell' = (200, 202)$, for different values of azimuthal difference $t=m-m'$, plotted for $m=20$. $\ell_1 = \ell, m_1=m$ and the vertical-axis represents $\Delta m = m_2 - m_1$ and the horizontal-axis represents $\Delta\ell=\ell_2 - \ell_1$. The product is small for odd-$t$.