Sub-Wavelength Resolution Imaging of the Solar Deep Interior

We derive expectations for signatures in the measured travel times of waves that interact with thermal anomalies and jets. A series of numerical experiments that involve the dynamic linear evolution of an acoustic wave field in a solar-like stratified spherical shell in the presence of fully 3D time-stationary perturbations are performed. The imprints of these interactions are observed as shifts in wave travel times, which are extracted from these data through methods of time-distance helioseismology \citep{duvall}. In situations where at least one of the spatial dimensions of the scatterer was smaller than a wavelength, oscillatory time shift signals were recovered from the analyses, pointing directly to a means of resolving sub-wavelength features. As evidence for this claim, we present analyses of simulations with spatially localized jets and sound-speed perturbations. We analyze 1 years' worth solar observations to estimate the noise level associated with the time differences. Based on theoretical estimates, Fresnel zone time shifts associated with the (possible) sharp rotation gradient at the base of the convection zone are of the order 0.01 - 0.1 s, well below the noise level that could be reached with the currently available amount of data ($\sim 0.15-0.2$ s with 10 yrs of data).Comment: Accepted, ApJ; 17 pages, 12 figure

The influence of noise sources on cross-correlation amplitudes

We use analytical examples and asymptotic forms to examine the mathematical structure and physical meaning of the seismic cross correlation measurement. We show that in general, cross correlations are not Green's functions of medium, and may be very different depending on the source distribution. The modeling of noise sources using spatial distributions as opposed to discrete collections of sources is emphasized. When stations are illuminated by spatially complex source distributions, cross correlations show arrivals at a variety of time lags, from zero to the maximum surface-wave arrival time. Here, we demonstrate the possibility of inverting for the source distribution using the energy of the full cross-correlation waveform. The interplay between the source distribution and wave attenuation in determining the functional dependence of cross correlation energies on station-pair distance is quantified. Without question, energies contain information about wave attenuation. However, the accurate interpretation of such measurements is tightly connected to the knowledge of the source distribution.Comment: 19 pages, 17 figures; Geophysical Journal Internationa

Measurements and Kernels for Source-Structure Inversions in Noise Tomography

Seismic noise cross correlations are used to image crustal structure and heterogeneity. Typically, seismic networks are only anisotropically illuminated by seismic noise, a consequence of the non-uniform distribution of sources. Here, we study the sensitivity of such a seismic network to structural heterogeneity in a 2-D setting. We compute finite-frequency cross-correlation sensitivity kernels for travel-time, waveform-energy and waveform-difference measurements. In line with expectation, wavespeed anomalies are best imaged using travel times and the source distribution using cross-correlation energies. Perturbations in attenuation and impedance are very difficult to image and reliable inferences require a high degree of certainty in the knowledge of the source distribution and wavespeed model (at least in the case of transmission tomography studied here). We perform single-step Gauss-Newton inversions for the source distribution and the wavespeed, in that order, and quantify the associated Cram\'{e}r-Rao lower bound. The inversion and uncertainty estimate are robust to errors in the source model but are sensitive to the theory used to interpret of measurements. We find that when classical source-receiver kernels are used instead of cross-correlation kernels, errors appear in the both the inversion and uncertainty estimate, systematically biasing the results. We outline a computationally tractable algorithm to account for distant sources when performing inversions.Comment: 19 pages, 12 figures, Geophysical Journal Internationa

Full Waveform Inversion for Time-Distance Helioseismology

Inferring interior properties of the Sun from photospheric measurements of the seismic wavefield constitutes the helioseismic inverse problem. Deviations in seismic measurements (such as wave travel times) from their fiducial values estimated for a given model of the solar interior imply that the model is inaccurate. Contemporary inversions in local helioseismology assume that properties of the solar interior are linearly related to measured travel-time deviations. It is widely known, however, that this assumption is invalid for sunspots and active regions, and likely for supergranular flows as well. Here, we introduce nonlinear optimization, executed iteratively, as a means of inverting for the sub-surface structure of large-amplitude perturbations. Defining the penalty functional as the $L_2$ norm of wave travel-time deviations, we compute the the total misfit gradient of this functional with respect to the relevant model parameters %(only sound speed in this case) at each iteration around the corresponding model. The model is successively improved using either steepest descent, conjugate gradient, or quasi-Newton limited-memory BFGS. Performing nonlinear iterations requires privileging pixels (such as those in the near-field of the scatterer), a practice not compliant with the standard assumption of translational invariance. Measurements for these inversions, although similar in principle to those used in time-distance helioseismology, require some retooling. For the sake of simplicity in illustrating the method, we consider a 2-D inverse problem with only a sound-speed perturbation.Comment: 24 pages, 10 figures, to appear in Ap

Propagation of seismic waves through a spatio-temporally fluctuating medium: Homogenization

Measurements of seismic wave travel times at the photosphere of the Sun have enabled inferences of its interior structure and dynamics. In interpreting these measurements, the simplifying assumption that waves propagate through a temporally stationary medium is almost universally invoked. However, the Sun is in a constant state of evolution, on a broad range of spatio-temporal scales. At the zero wavelength limit, i.e., when the wavelength is much shorter than the scale over which the medium varies, the WKBJ (ray) approximation may be applied. Here, we address the other asymptotic end of the spectrum, the infinite wavelength limit, using the technique of homogenization. We apply homogenization to scenarios where waves are propagating through rapidly varying media (spatially and temporally), and derive effective models for the media. One consequence is that a scalar sound speed becomes a tensorial wavespeed in the effective model and anisotropies can be induced depending on the nature of the perturbation. The second term in this asymptotic two-scale expansion, the so-called corrector, contains contributions due to higher-order scattering, leading to the decoherence of the wavefield. This decoherence may be causally linked to the observed wave attenuation in the Sun. Although the examples we consider here consist of periodic arrays of perturbations to the background, homogenization may be extended to ergodic and stationary random media. This method may have broad implications for the manner in which we interpret seismic measurements in the Sun and for modeling the effects of granulation on the scattering of waves and distortion of normal-mode eigenfunctions.Comment: 17 pages, 6 figures, in press, Ap