Aaron C. Birch

Local Helioseismology: Three-Dimensional Imaging of the Solar Interior

The Sun supports a rich spectrum of internal waves that are continuously excited by turbulent convection. The Global Oscillation Network Group (GONG) network and the SOHO/MDI (Solar and Heliospheric Observatory/Michelson Doppler Imager) space instrument provide an exceptional database of spatially resolved observations of solar oscillations, covering more than an entire sunspot cycle (11 years). Local helioseismology is a set of tools for probing the solar interior in three dimensions using measurements of wave travel times and local mode frequencies. Local helioseismology has discovered (a) near-surface vector flows associated with convection, (b) 250 m s−1 subsurface horizontal outflows around sunspots, (c) ∼50 m s−1 extended horizontal flows around active regions (converging near the surface and diverging below), (d) the effect of the Coriolis force on convective flows and active region flows, (e) the subsurface signature of the 15 m s−1 poleward meridional flow, (f) a ±5 m s−1 time-varying depth-depen...

Three-dimensional Inversion of Sound Speed below a Sunspot in the Born Approximation

We revise the inversion of acoustic travel times for the three-dimensional sound-speed structure below the solar NOAA Active Region 8243 of 1998 June. We benefit from recent progress in time-distance helioseismology that provides us with more reliable tools to infer subsurface solar properties. Among the improvements we implement here are the use of Born approximation-based travel-time sensitivity kernels that take into account finite-wavelength effects and thus are more accurate than the previously employed ray-path kernels, the inclusion of solar noise statistical properties in the inversion procedure through the noise covariance matrix, and the use of the actual variance of the noise in the temporal cross-covariances in the travel-time fitting procedure. Of these three improvements, the most significant is the application of the Born approximation to time-distance helioseismology. This puts the results of this discipline at the same level of confidence as those of global helioseismology based on inversion of normal-mode frequencies. Also, we compare inversion results based on ray-path and Born approximation kernels. We show that both approximations return a similar two-region structure for sunspots. However, the depth of inverted structures may be offset by 1 or 2 Mm, and the spatial resolution of the results is more accurately estimated with the more realistic Born sensitivity kernels. Finally, using artificial realizations of Doppler velocities of the quiet Sun, we are now able to estimate the statistical uncertainties of these inversion results.

Surface-Focused Seismic Holography of Sunspots: I. Observations

We present a comprehensive set of observations of the interaction of p-mode oscillations with sunspots using surface-focused seismic holography. Maps of travel-time shifts, relative to quiet-Sun travel times, are shown for incoming and outgoing p modes as well as their mean and difference. We compare results using phase-speed filters with results obtained with filters that isolate single p-mode ridges, and we further divide the data into multiple temporal frequency bandpasses. The f mode is removed from the data. The variations of the resulting travel-time shifts with magnetic-field strength and with the filter parameters are explored. We find that spatial averages of these shifts within sunspot umbrae, penumbrae, and surrounding plage often show strong frequency variations at fixed phase speed. In addition, we find that positive values of the mean and difference travel-time shifts appear exclusively in waves observed with phase-speed filters that are dominated by power in the low-frequency wing of the p1 ridge. We assess the ratio of incoming to outgoing p-mode power using the ridge filters and compare surface-focused holography measurements with the results of earlier published p-mode scattering measurements using Fourier – Hankel decomposition.

Linear sensitivity of helioseismic travel times to local flows

Time-distance helioseismology is a technique for measuring the time for waves to travel from one point on the solar surface to another. These wave travel times are affected by advection by subsurface flows. Inferences of plasma flows based on observed travel times depend critically on the ability to accurately model the effects of subsurface flows on time-distance measurements. We present a Born-approximation based computation of the sensitivity of time-distance travel times to weak, steady, inhomogeneous subsurface flows. Three sensitivity functions are obtained, one for each component of the 3D vector flow. We show that the depth sensitivity of travel times to horizontally uniform flows is given approximately by the kinetic energy density of the oscillation modes which contribute to the travel times. For flows with strong depth dependence, the Born approximation can give substantially different results than the ray approximation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

DIRECT MEASUREMENT OF TRAVEL-TIME KERNELS FOR HELIOSEISMOLOGY

Solar f-modes are surface gravity waves that propagate horizontally in a thin layer near the photosphere with a dispersion relation approximately that of deep water waves. At the power maximum near frequency ?/2? = 3 mHz, the wavelength of 5 Mm is large enough for various wave scattering properties to be observable. Gizon & Birch have calculated spatial kernels for scattering in the Born approximation. In this paper, using isolated small magnetic features as approximate point scatterers, a linear-response kernel has been measured. In addition, the kernel has been estimated by deconvolving the magnetograms from the travel-time maps. The observed kernel is similar to the theoretical kernel for wave damping computed by Gizon & Birch: it includes elliptical and hyperbolic features. This is the first observational evidence to suggest that it is appropriate to use the Born approximation to compute kernels (as opposed to the ray approximation). Furthermore, the observed hyperbolic features confirm that it is important to take into account scattering of the waves coming from distant source locations (as opposed to the single-source approximation). The observed kernel is due to a superposition of the direct and indirect effects of the magnetic field. A simple model that includes both monopole and dipole scattering compares favorably with the data. This new technique appears to be promising to study how seismic waves interact with magnetic flux tubes.

Sensitivity of time-distance helioseismic measurements to spatial variation of oscillation amplitudes I. Observations and a numerical model

It is well known that the observed amplitude of solar oscillations is lower in sunspots than in quiet regions of the Sun. We show that this local reduction in oscillation amplitudes, combined with the phase-speed filtering procedure in time-distance helioseismic analyses, could be a source of systematic errors in the range of 5%-40% in the measured travel-time anomalies of acoustic waves around sunspots. Removing these travel-time artifacts is important for correctly inferring the subsurface structure of sunspots. We suggest an empirical correction procedure and illustrate its usage for a small sunspot. This work uses data from SOHO MDI.

THE ADJOINT METHOD APPLIED TO TIME-DISTANCE HELIOSEISMOLOGY

For a given misfit function, a specified optimality measure of a model, its gradient describes the manner in which one may alter properties of the system to march toward a stationary point. The adjoint method, arising from partial-differential-equation-constrained optimization, describes a means of extracting derivatives of a misfit function with respect to model parameters through finite computation. It relies on the accurate calculation of wavefields that are driven by two types of sources, namely, the average wave-excitation spectrum, resulting in the forward wavefield, and differences between predictions and observations, resulting in an adjoint wavefield. All sensitivity kernels relevant to a given measurement emerge directly from the evaluation of an interaction integral involving these wavefields. The technique facilitates computation of sensitivity kernels (Frechet derivatives) relative to three-dimensional heterogeneous background models, thereby paving the way for nonlinear iterative inversions. An algorithm to perform such inversions using as many observations as desired is discussed.

Observed frequency variations of solar p-mode travel times as evidence for surface effects in sunspot seismology

Using helioseismic holography, we measure acoustic ( -mode) travel-time perturbations, observed within solar p active regions, as functions of frequency and phase speed. We find evidence for a frequency variation, at fixed phase speed, of the travel times that has not previously been reported. This variation is not expected from typical sound-speed models of sunspots, which result from the inversion of travel times and may indicate a significant contribution to the travel times from structures with vertical scales smaller than about 1 Mm near the solar surface.

SCATTERING OF ACOUSTIC WAVES BY A MAGNETIC CYLINDER: ACCURACY OF THE BORN APPROXIMATION

With the aim of studying magnetic effects in time-distance helioseismology, we use the first-order Born approximation to compute the scattering of acoustic plane waves by a magnetic cylinder embedded in a uniform medium. We show, by comparison with the exact solution, that the travel-time shifts computed in the Born approximation are everywhere valid to first order in the ratio of the magnetic to the gas pressures. We also show that for arbitrary magnetic field strength, the Born approximation is not valid in the limit where the radius of the magnetic cylinder tends to zero.

HELIOSEISMOLOGY OF PRE-EMERGING ACTIVE REGIONS. II. AVERAGE EMERGENCE PROPERTIES

We report on average subsurface properties of pre-emerging active regions as compared to areas where no active region emergence was detected. Helioseismic holography is applied to samples of the two populations (preemergence and without emergence), each sample having over 100 members, which were selected to minimize systematic bias, as described in Leka et al. We find that there are statistically significant signatures (i.e., difference in the means of more than a few standard errors) in the average subsurface flows and the apparent wave speed that precede the formation of an active region. The measurements here rule out spatially extended flows of more than about 15 m s −1 in the top 20 Mm below the photosphere over the course of the day preceding the start of visible emergence. These measurements place strong constraints on models of active region formation.