The first operation in timedistance helioseismology is to track Doppler images at a constant angular velocity to remove the main component of solar rotation, as is done in ringdiagram analysis (see Section 4.2.1). Postel’s azimuthal equidistant projection is often used in timedistance helioseismology. The resulting data cube is Fourier transformed to obtain .
A filtering procedure is then applied to the data (Duvall Jr et al., 1997). First, frequencies below 1.5 mHz are filtered out in order to remove granulation and supergranulation noise. The data are further filtered to select parts of the wave propagation diagram.

In the case of p modes, a phase speed filter is applied to the data, and the fmode ridge is filtered out. This choice of filter is based on the fact that acoustic waves with the same horizontal phase speed travel the same horizontal distance (see Bogdan, 1997). Thus, to measure the travel time for acoustic waves propagating between two surface locations, it is appropriate to consider only those waves with the same phase speed. The choice of the phase speed depends on the travel distance. A list of Gaussian phasespeed filters,
is provided in Table 1 for various distances. Here, is the mean phase speed, is the dispersion, and is the wavenumber. The filtered signal is given by . The larger the phase speed, the deeper pmode wavepackets penetrate inside the Sun.A separate filtering procedure is applied for surface gravity waves, which are used to probe the near surface layer. In this case the filter function is 1 if and 0 otherwise. The parameter controls the width of the region around the fmode dispersion relation . A reasonable choice is . This value allows for large Doppler frequency shifts introduced by flows, and does not let the ridge through.
There is some freedom in the selection of the Fourier filters. For example one may construct filters that depend explicitly on the direction of the wavevector , and not simply on the wavenumber (see, e.g., Giles, 1999).
The basic computation in timedistance helioseismology is the temporal crosscovariance between filtered signals at two points and on the solar surface,
where is the temporal sampling and is the time duration of the observation. Times are evaluated at discrete values . Multiplication by the temporal window function is already included in the definition of , and the sum over is a short notation to mean the sum over all discrete times in the interval . The normalization factor is a correction that becomes significant only when is small. We note that in practice it is faster to compute the time convolution in the Fourier domain. The positive time lags () give information about waves moving from to , and the negative time lags about waves moving in the opposite direction.The crosscorrelation is useful as it is a phasecoherent average of inherently random oscillations. It can be seen as a solar seismogram, providing information about travel times, amplitudes, and the shape of the wave packets traveling between any two points on the solar surface. Figure 13 shows a theoretical crosscovariance as a function of distance between and , and time lag . This timedistance diagram was computed for a spherically symmetric solar model (see Sekii and Shibahashi, 2003). The first ridge corresponds to acoustic waves propagating between the two points without additional reflection from the solar surface. The next ridge corresponds to waves which arrive after one reflection from the surface, and the ridges at greater time delays result from waves arriving after multiple bounces. The backward branch associated with the second ridge corresponds to waves reflected on the farside of the Sun. In most applications, only the direct (firstbounce) travel times are measured. Figure 14 shows the pmode crosscovariance function at short distances, computed with the Fourier filters given in Table 1. In this example, crosscovariance functions were averaged over pairs of points at fixed distance to reduce noise. An example crosscovariance function for the fmode ridge is plotted in Figure 15.


The travel times of the wave packets are measured from the (firstbounce) crosscovariance function. Local inhomogeneities in the Sun will affect travel times differently depending on the type of perturbation. For example, temperature perturbations and flow perturbations have very different signatures. Given two points and on the solar surface the travel time perturbation due to a temperature anomaly is, in general, independent of the direction of propagation between and . However, a flow with a component directed along the direction will break the symmetry in travel time for waves propagating in opposite directions: Waves move faster along the flow than against it. Magnetic fields introduce a wave speed anisotropy and will have yet another travel time signature (this has not been detected yet).
Crosscovariance functions can have a large amount of realization noise due to the schochastic nature of solar oscillations, and it has proved difficult to measure travel times between two individual pixels on the solar surface. Some averaging is often required. The time over which the crosscovariances are computed is usually (this puts a limit on the temporal scale of the solar phenomena that can be studied). In order to further enhance the signaltonoise ratio, Duvall Jr et al. (1993) and Duvall Jr et al. (1997) suggested to average over points that belong to an annulus or quadrants centered at . For instance, to measure flows in the eastwest direction in the neighborhood of point , crosscovariances are averaged over two quadrant arcs and that include points a distance from :
Figure 16 shows the geometry of the averaging procedure. An example crosscovariance is shown in Figure 17 in the fmode case. Correlations at positive times () correspond to waves that propagate westward, and correlations at to waves that propagate eastward. As shown in Figure 17, crosscovariances can be further averaged along lines of constant phase within a range of distances. Likewise an average crosscorrelation is constructed from southnorth quadrants to measure flows in the meridional direction. Another average is obtained by averaging over the whole annulus, where is an annulus of radius centered at . This average can be used to measure separately the waves that propagate inward and outward from the central point.


At fixed and , the crosscovariance function oscillates around two characteristic (firstbounce) times . Centertoannulus or centertoquadrants travel times are often measured by fitting Gaussian wavelets (Duvall Jr et al., 1997; Kosovichev and Duvall Jr, 1997). This procedure distinguishes between group and phase travel times by allowing both the envelope and the phase of the wavelet to vary independently. The positivetime part of the crosscorrelation is fitted with a function of the form
where all parameters are free, and the negativetime part of the crosscorrelation is fitted separately with The times and are the socalled phase travel times. The basic observations in timedistance helioseismology are the travel time maps and , measured for each of the three averaged crosscorrelations , , and . The travel time differences are mostly sensitive to flows while the mean travel times are sensitive to wavespeed perturbations. Maps of measured travel times are shown in Figure 18: Most of the signal in these maps is due to supergranular flows (15 – 30 Mm length scales). We note that an alternative definition of travel time can be obtained by fitting a model crosscovariance function to the data (Gizon and Birch, 2002). This last definition is often used in geoseismology (see, e.g., Zhao and Jordan, 1998).

In order to maximize the potential resolution of timedistance helioseismology it is desirable to obtain travel times from crosscovariances measured with shorter and with as little spatial averaging as possible. However conventional fitting methods will fail when the crosscovariance is too noisy. A new robust definition of travel time was introduced by Gizon and Birch (2004) to measure travel times between individual pixels and as short as . According to this definition, the pointtopoint travel time for waves going from to , denoted by , and the travel time for waves going from to , denoted by , are given by
with the weight functions given by In this expression is a (smooth) reference crosscovariance function computed from a solar model and . The window function is a onesided function (zero for negative) used to separately examine the positive and negativetime parts of the crosscorrelation. The window is used to measure , and is used to measure . A standard choice is a window that isolates the firstskip branch of the crosscovariance.This definition has a number of useful properties. First, it is very robust with respect to noise. The fit reduces to a simple sum that can always be evaluated whatever the level of noise. Second, it is linear in the crosscovariance. As a consequence, averaging various travel time measurements is equivalent to measuring a travel time on the average crosscovariance. This is unlike previous definitions of travel time that involve nonlinear fitting procedures. Third, the probability density functions of and are unimodal Gaussian distributions. This means, in particular, that it makes sense to associate an error to a travel time measurement. The Born sensitivity kernels discussed in Section 4.3.5 were derived according to this definition of travel times.
In global helioseismology, it is well understood that the precision of the measurement of the pulsation frequencies is affected by realization noise resulting from the stochastic nature of the excitation of solar oscillations (see, e.g., Woodard, 1984; Duvall Jr and Harvey, 1986; Libbrecht, 1992; Schou, 1992). It is important to study these properties since the presence of noise affects the interpretation of travel time data. In particular, the correlations in the travel times must be taken into account in the inversion procedure.
An interesting approach, pioneered by Jensen et al. (2003a), consists of estimating the noise directly from the data by measuring the rms travel time within a quiet Sun region. The underlying assumptions are that the fluctuations in the travel times are dominated by noise, not by ’real’ solar signals, and that the travel times measured at different locations can be seen as different realizations of the same random process. By ’real’ solar signals we mean travel time perturbations due to inhomogeneities in the solar interior that are slowly varying over the time of the observations. Jensen et al. (2003a) studied the correlation between the centertoannulus travel times as a function of the distance between the central points, at fixed annulus radius.
Gizon and Birch (2004) derived a simple model for the full covariance matrix of the travel time measurements. This model depends only on the expectation value of the filtered power spectrum and assumes that solar oscillations are stationary and homogeneous on the solar surface. The validity of the model is confirmed through comparison with MDI measurements in a quiet Sun region. Gizon and Birch (2004) showed that the correlation length of the noise in the travel times is about half the dominant wavelength of the filtered power spectrum. The signaltonoise ratio in quietSun travel time maps increases roughly like the square root of the observation time and is maximum for a distance near half the length scale of supergranulation.
In this section we show example computations of travel time sensitivity kernels. Kernels are the functions that connect small changes in the solar model with small changes in travel times
The sum over is taken over all possible relevant types of perturbations to the model, for example local changes in sound speed, density, flows, magnetic field, or source properties. For each type of perturbation there is a corresponding kernel that depends on , the observation points and that the travel time is measured between, and a spatial location that ranges over the entirety of the solar interior. In this section we will describe the various types of approximate calculations of the that have been done. Notice that kernels can be computed for quantities other than travel times. For example, in geophysics
Dahlen and Baig (2002) computed the firstorder sensitivity of the amplitude of the observed waveform to a
small local change in sound speed. In particular it might be helpful to compute kernels for the mean
frequency and amplitude of the crosscorrelations, with the aim of using these quantities to help constrain
inversions.

Another approach to producing finitewavelength kernels is the Fresnel zone approximation (see, e.g., Sneider and Lomax, 1996; Jensen et al., 2000). The Fresnel zone approach makes a simple parametrized approximation to the kernels. In particular, in the Fresnel zone approximation, the travel time kernels, for fixed and , are written as (Jensen et al., 2001)
where is the raytheoretical travel time from to , and . The parameter is the central frequency of the wavepacket, is the corresponding period, and is related to the frequency bandwidth of the wavepacket. The amplitude of the kernel is determined by demanding that the total integral of the kernel be equal to , which is the ray theory value for the total integral of the kernel. Notice that the value of the kernel goes to infinity as tends to or . Jensen et al. (2000) showed, by comparison with direct simulation, that a two dimensional analogue of the Fresnel zone kernel (Equation (65)) was a reasonable approximation to the actual linear sensitivity of travel time to changes in the sound speed. The Rytov approximation gives a first order correction to the phase of the wavefield, instead of a first
order correction to the wavefield itself as one obtains from the Born approximation. Jensen and
Pijpers (2003) used the Rytov approximation to write travel time kernels for the effects of soundspeed
perturbations. The results were qualitatively similar to the Born approximation results of Birch and
Kosovichev (2000). No comparison has been made, in the helioseismology literature, of the ranges of
validity of the Rytov and Born approximations.


The inverse problem is to determine the perturbations to the solar model that are consistent with a particular set of observed travel times . For the inverse problem, the kernel functions and the noise covariance are in general assumed to be known.
Unlike in the ringanalysis case, where a set of one dimensional inversions are performed (see Section 4.2.3), the timedistance inversion procedure is threedimensional. So far, only RLS inversions have been implemented in timedistance helioseismology. In the RLS approach, the minimization problem reads
where we have assumed, for simplicity of notation, a diagonal covariance matrix. The are the noise estimates for each of the travel times . The regularization parameter is and is the regularization operator. Notice that we have not assumed any particular parametrization of the perturbations to the solar model . The ideal choice of regularization operator and the optimal method for choosing the regularization parameter are currently open questions.The minimization of , Equation (66), has been done using either the LSQR algorithm (Paige and Saunders, 1982), the multichannel deconvolution (MCD; Jacobsen et al., 1999), or via singular value decomposition (Hughes and Thompson, 2003). The inputs to either of these methods are the travel time kernels, which result from the linear forward problem (see Section 3.5), the regularization operator, the choice of regularization parameter, and in general also the covariance matrix of the noise.
The central question about inversions is the degree to which any particular inversion method is able to retrieve the true Sun, e.g., the actual soundspeed variations and flows in the interior, from a set of observed travel times. The degree to which any inversion method succeeds will presumably depend on a number of things: (i) the accuracy of the forward model, (ii) the noise level in the travel times, (iii) the depths and spatial scales of the real variations in the Sun, (iv) the number and type of travel times that are available as input to the inversion, and (v) the accuracy of the travel time noise covariance matrix. The two main approaches that have been taken to study inversion methods are either to invert artificial data or to invert real data for perturbations that are relatively well known from global helioseismology or from surface measurements. It is also useful to compare the results of different inversion methods applied to the same set of observed travel times.
The approach of inverting real data is appealing as it is by definition “realistic”. This approach for timedistance was pioneered by Giles (1999) who did ray theory based inversions to measure differential rotation. The results were in approximate agreement with the results of global model inversions, which validated timedistance inversions applied to slowlyvarying large scale flows. Gizon et al. (2000) inverted fmode travel times for horizontal flows in the nearsurface. Figure 22 shows a comparison between the projection of the inferred flows onto the line of sight and the directly observed lineofsight Doppler velocity. The correlation coefficient between the two is 0.7 (Gizon et al., 2000), which shows that their inversion method, an iterative deconvolution, is able to approximately retrieve the horizontal flow velocities from the travel times.


There have been a number of efforts to validate inversions by the generation and inversion of artificial data. These tests are useful as they provide an intuitive understanding of various regularization schemes, the choice of regularization parameter, the effects of limited sets of input travel times, the effects of incorrect assumptions regarding the noise covariance, the potential resolution of any particular inversion, and the crosstalk between different model parameters. A drawback to testing inversions with artificial data is that one always wonders about the realism of the artificial data. Hopefully, in the not distant future quite realistic data will be available for the purpose of testing inversion schemes.
Couvidat et al. (2004) compared inversions done using rayapproximation kernels with inversions done with Fresnelzone kernels. The results were quite similar for depths between the surface and the lower turning point of the deepest rays used in the inversion. Inversions done with the ray kernels cannot retrieve sound speed perturbations below the lower turning point of the deepest ray, while the Fresnelzone kernels extend below the depth of the deepest ray. The results of Couvidat et al. (2004), as well as those of Giles (1999), suggest that even though the ray approximation overestimates travel times for small scale perturbations this effect does not seriously corrupt the large scale features found by inversion.

Another test of inversion methods was done by Zhao and Kosovichev (2003a). The two main results of this study, regarding inversions methods, were that there is crosstalk between upflows and convergent flows, and between downflows and divergent flows, and that the MCD and LSQR methods give essentially the same results when applied to quiet sun MDI data. Figure 23 shows an example of a test of the LSQR method. The initial model is shown in the top panel. Travel times were generated using the ray approximation. The travel times were then inverted using five iterations of LSQR to obtain the flow field shown in the second panel of Figure 23. Notice that there are small vertical flows near the surface at the centers and boundaries of the supergranulations cells. After 100 iterations of LSQR, these small vertical velocity features are removed (bottom panel of Figure 23).
The first endtoend test of time distance helioseismology using artificial data was done by Jensen et al. (2003b). In this important study, artificial data were generated by numerical wave propagation through a threedimensional stratified and horizontally inhomogeneous model. Waves were generated by stochastic sources distributed over a layer 50 km below the upper boundary of the surface of the model. The artificial wave field was “observed” at the spatial resolution and temporal cadence of MDI highresolution observations. This artificial data set was then subjected to standard timedistance analysis, and the inversion was performed with the Fresnelzone kernels described by Jensen and Pijpers (2003). Figure 24 shows a comparison of the original soundspeed model and the result of the inversion of the artificial data. Notice that the inversion recovers much of the original structure. The inversion result contains noise, which is to be expected from travel times computed from finite time series. The only noise source in this example is realization noise, noise resulting from the stochastic nature of the wave excitation.
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