Helioseismic Travel-Time Definitions and Sensitivity to Horizontal Flows Obtained From Simulations of Solar Convection

Since the introduction of time-distance helioseismology (Duvall et al., 1993) this method has been applied to investigate various subsurface physical properties of the quiet Sun (e.g. Kosovichev and Duvall, 1997), especially supergranulation (e.g. Duvall and Gizon, 2000;Zhao and Kosovichev, 2003;Hirzberger et al., 2007), and sunspots (e.g. Kosovichev, Duvall, and Scherrer, 2000;Zhao, Kosovichev, and Duvall, 2001;Couvidat, Birch, and Kosovichev, 2006). In these studies, the main emphasis has been on mapping both the subsurface material flows and deviations of the sound speed from a reference solar model. Travel times of wavepackets (whether from acoustic or surface gravity waves) are the basic measured quantities that are then inverted to produce estimates of the flow velocity and sound-speed perturbations. Travel times are measured from temporal cross-covariances. Over the years different definitions of these travel times have been employed (e.g., Kosovichev and Duvall, 1997;Gizon and Birch, 2002;Gizon and Birch, 2004). For a given cross-covariance, these definitions, in general, yield different travel times. It is therefore important to understand how these definitions perform in simple situations where the subsurface conditions are known. In this paper we study the sensitivity of these definitions to horizontal, steady, and uniform flows. We use vertical velocity data from a realistic numerical simulation of solar convection (Benson, Stein, and Nordlund, 2006). We introduce steady and spatially uniform horizontal flows to this vertical velocity data cube and measure the perturbations in the travel times induced by these flows. We also compare the observed relationship between travel-time differences and flow velocities with the relationship predicted by ray approximation kernels (Kosovichev and Duvall, 1997) commonly used for the inversions. We limit ourselves to the study of acoustic modes (p modes), neglecting the surface-gravity modes (f modes). The f-mode case was studied in detail by Jackiewicz et al. (2007). In Section 2 we briefly present the numerical simulation that we used for this work. In Section 3 we remind the reader of the fundamentals of the time-distance formalism. In Section 4 we present our results, and we conclude in Section 5.

We work with simulated line-of-sight velocity data [φ(r, t) where r is the horizontal position vector and t is time] obtained from a numerical three-dimensional simulation of convection in the upper solar convection zone. The numerical slab is 96 Mm×96 Mm wide and 20 Mm deep. The spatial resolution is 96 km horizontally and varies from 12 to 75 km vertically. The temporal resolution is 10 seconds and we use 8.5 hours of simulated data. This simulation was performed by Robert Stein, Ake Nordlund, Dali Georgobiani and David Benson, and is publicly available on the website sha.stanford.edu. A very similar simulation (48 Mm×48 Mm horizontally) has already been extensively analyzed by Georgobiani et al. (2007), Braun et al. (2007), and Zhao et al. (2007). The latter applied the time-distance formalism to these data to study, among other things, the validity of the inversion procedures. A detailed presentation of the code that produced these data can be found in Benson, Stein, and Nordlund (2006) and Stein and Nordlund (2000). Here we only work with the vertical velocity at the surface of the numerical simulation. The data cube we use is rebinned four times in the horizontal direction (yielding a final spatial resolution dx=384 km) and six times in time (final temporal resolution dt=60 seconds). One advantage of using this numerical simulation in place of solar data is that horizontally the boundary conditions are periodic. Therefore it is possible to simulate the impact of a steady horizontally uniform flow by simply shifting the vertical velocity maps in the horizontal direction at each time-step, using the shift theorem in the Fourier domain. Simulating an advection flow on actual solar data, e.g. on data from the SOI/MDI instrument (Scherrer et al., 1995), is possible (Jackiewicz et al., 2007) but more complicated and under normal conditions the simulated flow is nonuniform in latitude. The power spectrum of the vertical velocity φ(r, t) from the numerical simulation is very similar to the actual solar oscillations power spectrum as determined by SOI/MDI (Georgiobani et al., 2007). Therefore we expect that the results we obtain here should be fairly representative of the actual solar case.

We use the time-distance measurement procedure described in Couvidat, Birch, and Kosovichev (2006), except that we work with a center-quadrant geometry instead of a center-annulus one. We only deal with acoustic waves (p modes) and filter out the surface gravity waves (f modes) (see Jackiewicz et al., 2007, for a recent study of the travel times of f modes). The measurement of the travel times begins with the computation of cross-covariances between two observation points r 1

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