Parameters of the ridge filters that are used for the travel-time measurements in this paper (see text for details).
Prior to the travel-time measurements, the wavefield that is present in the Dopplergrams is filtered to select single ridges (the f modes or the p1 modes). The goal is to capture as much of the ridge power as possible, even if the waves are Doppler-shifted by flows. At the same time, we want to prevent power from neighboring ridges from leaking in and select as little background power as possible.
To construct the filter, we first measure the power spectra of the Dopplergrams at the equator and average over 60 days (59 days) of data in the case of HMI (MDI). After further azimuthal averaging, we identify the frequency ωmode where the ridge maximum is located as a function of wavenumber k.
The filter is constructed for each k as a plateau of width 2ωδ centered around the ridge maximum ωmode. The lower and upper boundaries of the plateau we call ωb and ωc. Next to the plateau, we add a transition region of width ωslope, which consists of a raised cosine function that guides the filter from one to zero, symmetrically around ωmode. The lower and upper limits of the filter we call ωa and ωd, respectively.
The plateau half-width ωδ consists of the following terms (A.1)where ωΓ(k) is the FWHM of the ridge (measured from the average power spectra), ωv(k) = akvmax is the Doppler shift due to a hypothetical flow of magnitude vmax multiplied by a scale factor a, and ωconst is a constant term of small magnitude that broadens the filter predominantly at small wavenumbers.
In addition, we restrict the filter to a range of wavenumbers. Above and below a k interval, the filters are set to zero. The k limits of the interval are chosen such that the ridge power is roughly twice the background power. Because ωmode is a function of wavenumber, these limits can also be expressed as frequencies ωmin and ωmax.
Table A.1 lists the filter parameters we chose for the f-mode and p1-mode ridge filters that we use throughout the paper. We note that we use the same filters for all latitudes and longitudes. For the p1 modes, we also list an alternative filter that we use to discuss the impact of the filter details on the travel-time measurements (see Appendix C.3).
Point-to-point travel times τdiff(r1,r2) are sensitive to flows in the direction of r1 − r2. If the flow structure is known, travel times τdiff can be predicted with the knowledge of sensitivity kernels. Conversely, the velocity field can be obtained from measured travel times by an inversion. Such inversions are, however, delicate, as they are, in general, ill-posed problems. A simple way to obtain rough estimates of the flow velocity while avoiding inversions is the multiplication of the travel times by a constant conversion factor. Such a conversion factor can be calculated by artificially adding the signature of a uniform flow of known magnitude and direction to Dopplergrams. The magnitude of the measured travel time divided by the input flow speed yields the conversion factor. In the following, we describe this process.
First, we create data cubes φv(r,t) that have Doppler-shifted power spectra to mimic the effect of a flow v independent of position r and time t. The data cubes are based on the noise model by Gizon & Birch (2004), so signatures from flows others than v are not present. Following the noise model, we construct in Fourier space . Here k is the horizontal wave vector; is a Doppler-shifted power spectrum; and, at each (k,ω), are independent complex Gaussian random variables with zero mean and unit variance. Employing ensures that the values φv(k,ω) are uncorrelated, which means that there is no signal from wave scattering. We use based on an average power spectrum that was measured from 60 days of HMI Dopplergrams (and 59 days of MDI Dopplergrams) at the solar equator. The quantity δω = k·v is the frequency shift due to a background flow v = (vx,0) that we add. We construct 8 h datasets φv(r,t) for vx in the range between −1000 and 1000 m s-1 in steps of 100 m s-1. For each velocity value, we compute ten realizations.
As a consistency check, we apply a second method for adding an artificial velocity signal to the HMI Dopplergram datasets. This procedure consists of tracking at an offset rate. The tracking parameters from Snodgrass (1984) are modified by a constant corresponding to a vx velocity of −100 m s-1 and 100 m s-1, respectively. The tracking and mapping procedure is as for the regular HMI observations. We produce 112 such datacubes for each vx value at the solar equator.
For both methods, the 8 h datasets are ridge-filtered like the normally tracked Doppler observations (f modes and p1 modes). We measure travel times τdiff in the x direction with the pairs of measurement points separated by 10 Mm. This distance matches the separation in the τac measurements. The reference cross-covariance Cref is taken from the regularly tracked HMI (MDI) observations averaged over 60 days (59 days) of data at the solar equator. This ensures that the artificial flow signal is captured by the travel-time measurements.
Point-to-point travel times from HMI Dopplergrams with artificial velocity signal. The point separation is 10 Mm in the east-west direction. a)f modes. b) p1 modes. The blue dots give the travel times from Dopplergram series that were constructed using the noise model by Gizon & Birch (2004). We applied a least-squares fit with a polynomial of degree three to the resulting data (black curves). The red curves show the linear term of the fit. For comparison, the black filled circles show travel times from HMI Dopplergrams that were tracked at an offset rate.
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The resulting τdiff values averaged over maps and datasets are shown for HMI in Fig. B.1. For both f and p1 modes, the travel times from offset tracking are systematically larger than for the Doppler-shifted power spectra by about 10 to 15%. In general, the travel-time magnitudes are larger for the f modes than for the p1 modes for the same input velocity value. The relation between input velocity vx and output travel time τdiff is linear only in a limited velocity range. Whereas this range spans from roughly −700 m s-1 to 700 m s-1 for the p1 modes, it only reaches from −200 m s-1 to 200 m s-1 for the f modes. For velocity magnitudes larger than 700 m s-1, the measured f-mode travel times even decrease. However, the supergranular motions that we analyze reach typical velocities of ~ 300 m s-1, which is well below that regime.
We applied a least-squares fit to a polynomial of degree three to the τdiff measurements from Doppler-shifted cubes (pink curve): (B.1)The linear term of the polynomial is shown for HMI as the red curve in Fig. B.1. For the actual conversion, only the linear coefficient h1 is used. We obtain h1 = −0.178 s2 m-1 for the f modes and h1 = −0.090 s2 m-1 for the p1 modes. For comparison, the coefficients h1 are listed for different distances in Table B.1. The table also contains the coefficients for MDI. We convert travel times into velocities by multiplying the travel times by 1 /h1. The velocities obtained from converting τac maps we call vac.
At high latitudes, the original vac and LCT ωz maps for the average supergranule show strong deviations from the azimuthally symmetric peak-ring structures that are visible at low latitudes. Considering that the magnitude of τac and ωz is much smaller than the magnitude of τoi and divh at any latitude, it is possible that even a small anisotropy in the divergent flow component of the average supergranule is picked up by the vac and ωz measurements and added to the signal from the tangential flow component that we want to measure. Such anisotropies can arise from various origins. Among them are geometrical effects that depend on the distance to the disk center.
For TD measurements, the sensitivity kernels depend on the distance to the limb. At 60° off disk center, τdiff sensitivity kernels for measurements in the direction along the limb differ strongly from kernels for measurements in the center-to-limb direction (see, e.g., Jackiewicz et al. 2007, for a discussion). Additionally, there is a gradient of the root mean square travel time in the center-to-limb direction.
Circulation velocities vac of the average supergranule outflow region at solar latitude 40° derived from HMI and MDI Dopplergrams (after the correction for center-to-limb systematics). The velocity maps were obtained by applying the respective conversion factors from Appendix B to the travel times τac. The limits of the colorscale are arbitrarily set to ± 15 m s-1.
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Peak vac values for p1 modes using different parameter combinations (Δ,n) for the average supergranule at solar latitude 40°. a) In the average outflow region. b) In the average inflow region. The blue symbols give the results for the p1 ridge filter that has been used throughout this paper. For the results in black, an alternative p1 ridge filter with slightly different parameters was used (see text for details). The error bars were computed as in Fig. 7. The annulus radii corresponding to the various combinations (Δ,n) are all within (10.0 ± 0.5) Mm.
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In the case of LCT, the shrinking Sun effect causes large-scale gradients of the horizontal velocity (of several hundred meters per second) pointing toward disk center (Lisle & Toomre 2004). This effect is presumably caused by insufficient resolution of the granules. Although HMI intensity and Doppler images have a pixel size of about 350 km at disk center, the point spread function has a FWHM of about twice that value. In Dopplergrams, the hot, bright, and broad upflows in the granule cores cause stronger blueshifts than the redshifts from the cooler, darker, and narrow downflows. Because of the insufficient resolution, the granules appear blueshifted as a whole. This blueshift adds to the blueshift of granules that move toward the observer (i.e., toward disk center), giving them a stronger signal in the Dopplergram. Lisle & Toomre argue that LCT of Dopplergrams gives more weight to these granules than to those granules that move away from the observer. However, it is not clear what causes the shrinking Sun effect in LCT of intensity images. Fortunately, the shrinking Sun effect appears to be a predominantly large-scale and time-independent effect, so it can easily be removed from LCT velocity maps by subtracting a mean image.
Another problem is the foreshortening. Far away from the disk center, the granules are not as well resolved in the center-to-limb direction as in the perpendicular horizontal direction. This introduces a dependence of the measurement sensitivity on angle. We measure at ± 60° latitude that the radial flow component vr of the average supergranule is weaker by 15 to 20% in the center-to-limb direction compared to the perpendicular direction. This corresponds to a maximum velocity difference of about 50 m s-1 for outflows and 30 m s-1 for inflows. At 40° latitude, in contrast, this difference is less than 2% (6 m s-1).
Whereas for HMI the removal of geometrical center-to-limb effects results in similar vac peak structures in the supergranule outflow regions in the whole latitude range from −60° to 60°, for MDI the peak structures appear asymmetric and distorted even after the correction. An example for f-mode TD at 40° latitude is shown in Fig. C.1. Even at disk center where geometrical effects should not play a role, there are visible systematic features (that do not appear for HMI, cf. Fig. 6). This is probably due to instrumental effects that are specific to MDI (see, e.g., Korzennik et al. 2004, for a discussion of instrumental errors in MDI).
We note that the vac velocity results for TD depend on the details of the ridge filter as well as the geometry parameters (Δ,n) of the τac measurements.
To give an idea of this, we construct an alternative p1 ridge filter with slightly different width parameters (see Appendix A). Additionally, we select four other combinations (Δ,n) of τac measurements that preserve the annulus radius R, so that R is within (10.0 ± 0.5) Mm for all the combinations (Δ,n). As we did for the standard combination (Δ = 10 Mm, n = 6), we use four different angles β for each additional combination.
For all these combinations and both the standard and modified p1 filters, we calculated vac for the average supergranule at 40° latitude. The resulting peak velocities are shown in Fig. C.2
for both inflow and outflow regions. We did not apply the center-to-limb correction since it only has a weak influence on the peak velocity magnitude at 40° latitude.
Evidently, the modified p1 filter results in systematically larger vac amplitudes. The difference with respect to the standard filter increases with decreasing Δ. For Δ = 10 Mm and n = 6, it is about 10%. This is qualitatively in line with Duvall & Hanasoge (2013). Using phase-speed filters, Duvall & Hanasoge observed that the strength of the travel-time signal from supergranulation is strongly dependent on the filter width. This shows that one should be careful when comparing absolute velocities from TD and LCT. For more reliable velocity values, an inversion of τoi and τac maps would be needed.
The comparison of different combinations (Δ,n) for the same filter shows that for n = 4, 6, and 8 the vac amplitudes are similar, so selecting the combination (Δ = 10 Mm,n = 6), as we did for most of this work, appears justified. Decreasing Δ to about 5 Mm changes the peak vac values. A possible reason is that Δ in this case becomes comparable to the wavelength of the oscillations, so it is harder to distinguish between flows in opposite directions. For small n, on the other hand, the measurement geometry deviates strongly from a circular contour. This might explain the deviations in vac for n = 3.
© ESO, 2015