Issue 
A&A
Volume 581, September 2015



Article Number  A67  
Number of page(s)  16  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201526024  
Published online  03 September 2015 
Online material
Appendix A: Ridge filters
Parameters of the ridge filters that are used for the traveltime measurements in this paper (see text for details).
Prior to the traveltime measurements, the wavefield that is present in the Dopplergrams is filtered to select single ridges (the f modes or the p_{1} modes). The goal is to capture as much of the ridge power as possible, even if the waves are Dopplershifted 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 ω_{mode} where the ridge maximum is located as a function of wavenumber k.
The filter is constructed for each k as a plateau of width 2ω_{δ} centered around the ridge maximum ω_{mode}. The lower and upper boundaries of the plateau we call ω_{b} and ω_{c}. Next to the plateau, we add a transition region of width ω_{slope}, which consists of a raised cosine function that guides the filter from one to zero, symmetrically around ω_{mode}. The lower and upper limits of the filter we call ω_{a} and ω_{d}, respectively.
The plateau halfwidth ω_{δ} consists of the following terms (A.1)where ω_{Γ}(k) is the FWHM of the ridge (measured from the average power spectra), ω_{v}(k) = akv_{max} is the Doppler shift due to a hypothetical flow of magnitude v_{max} multiplied by a scale factor a, and ω_{const} is a constant term of small magnitude that broadens the filter predominantly at small wavenumbers.
The width of the transition region relative to the plateau width is (A.2)where j is a unitless factor.
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 ω_{mode} is a function of wavenumber, these limits can also be expressed as frequencies ω_{min} and ω_{max}.
Table A.1 lists the filter parameters we chose for the fmode and p_{1}mode ridge filters that we use throughout the paper. We note that we use the same filters for all latitudes and longitudes. For the p_{1} modes, we also list an alternative filter that we use to discuss the impact of the filter details on the traveltime measurements (see Appendix C.3).
Appendix B: Conversion of travel times into flow velocities
Pointtopoint travel times τ^{diff}(r_{1},r_{2}) are sensitive to flows in the direction of r_{1} − r_{2}. If the flow structure is known, travel times τ^{diff} 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, illposed 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 φ_{v}(r,t) that have Dopplershifted 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 Dopplershifted power spectrum; and, at each (k,ω), are independent complex Gaussian random variables with zero mean and unit variance. Employing ensures that the values φ_{v}(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 = (v_{x},0) that we add. We construct 8 h datasets φ_{v}(r,t) for v_{x} 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 v_{x} 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 v_{x} value at the solar equator.
For both methods, the 8 h datasets are ridgefiltered like the normally tracked Doppler observations (f modes and p_{1} modes). We measure travel times τ^{diff} in the x direction with the pairs of measurement points separated by 10 Mm. This distance matches the separation in the τ^{ac} measurements. The reference crosscovariance C^{ref} 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 traveltime measurements.
Fig. B.1
Pointtopoint travel times from HMI Dopplergrams with artificial velocity signal. The point separation is 10 Mm in the eastwest direction. a)f modes. b) p_{1} 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 leastsquares 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 τ^{diff} values averaged over maps and datasets are shown for HMI in Fig. B.1. For both f and p_{1} modes, the travel times from offset tracking are systematically larger than for the Dopplershifted power spectra by about 10 to 15%. In general, the traveltime magnitudes are larger for the f modes than for the p_{1} modes for the same input velocity value. The relation between input velocity v_{x} and output travel time τ^{diff} 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 p_{1} 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 fmode 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 leastsquares fit to a polynomial of degree three to the τ^{diff} measurements from Dopplershifted 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 h_{1} is used. We obtain h_{1} = −0.178 s^{2} m^{1} for the f modes and h_{1} = −0.090 s^{2} m^{1} for the p_{1} modes. For comparison, the coefficients h_{1} 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 /h_{1}. The velocities obtained from converting τ^{ac} maps we call v^{ac}.
Appendix C: Systematic errors
Appendix C.1: Centertolimb systematics
At high latitudes, the original v^{ac} and LCT ω_{z} maps for the average supergranule show strong deviations from the azimuthally symmetric peakring structures that are visible at low latitudes. Considering that the magnitude of τ^{ac} and ω_{z} is much smaller than the magnitude of τ^{oi} and div_{h} 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 v^{ac} and ω_{z} 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, τ^{diff} sensitivity kernels for measurements in the direction along the limb differ strongly from kernels for measurements in the centertolimb 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 centertolimb direction.
Fig. C.1
Circulation velocities v^{ac} of the average supergranule outflow region at solar latitude 40° derived from HMI and MDI Dopplergrams (after the correction for centertolimb systematics). The velocity maps were obtained by applying the respective conversion factors from Appendix B to the travel times τ^{ac}. The limits of the colorscale are arbitrarily set to ± 15 m s^{1}. 

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Fig. C.2
Peak v^{ac} values for p_{1} 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 p_{1} ridge filter that has been used throughout this paper. For the results in black, an alternative p_{1} 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 largescale 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 largescale and timeindependent 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 centertolimb 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 v_{r} of the average supergranule is weaker by 15 to 20% in the centertolimb 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}).
Appendix C.2: MDI instrumental systematics
Whereas for HMI the removal of geometrical centertolimb effects results in similar v^{ac} 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 fmode 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).
Appendix C.3: Selection of filter and τ^{ac} geometry parameters
We note that the v^{ac} velocity results for TD depend on the details of the ridge filter as well as the geometry parameters (Δ,n) of the τ^{ac} measurements.
To give an idea of this, we construct an alternative p_{1} ridge filter with slightly different width parameters (see Appendix A). Additionally, we select four other combinations (Δ,n) of τ^{ac} 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 p_{1} filters, we calculated v^{ac} 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 centertolimb correction since it only has a weak influence on the peak velocity magnitude at 40° latitude.
Evidently, the modified p_{1} filter results in systematically larger v^{ac} 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 phasespeed filters, Duvall & Hanasoge observed that the strength of the traveltime 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 τ^{oi} and τ^{ac} maps would be needed.
The comparison of different combinations (Δ,n) for the same filter shows that for n = 4, 6, and 8 the v^{ac} 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 v^{ac} 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 v^{ac} for n = 3.
© ESO, 2015
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