HMI Ring Diagrams Data Product Specifications Draft Proposal (revised 2008.02.13 and 2008.03.05) The fundamental scientific data products to be produced from ring-diagram analysis, in accordance with the HMI Science Plan, are full-disc and synoptic maps of horizontal velocity and sound-speed profiles from the surface to a depth of 30 Mm. In addition to these it is possible that we may be able to produce high-resolution maps to the same depths and deep-focus maps extending to 200 Mm. The distinction between full-disc and synoptic maps is not clear in the Science Plan; since the analysis normally relies on mapping onto some co-rotating surface coordinate system in any case, it may only be a matter of organization or continuity of the data. Continuous analysis of the full disc is necessary in any case to prepare the synoptic maps. Likewise the distinction between full-disc and synoptic maps on the one hand and high-resolution on the other is not made specific in the Science Plan. We regard the full-disc and synoptic data products as ones produced over the entire available grid of time and spatial coordinates at a fixed resolution, and the high-resolution products as any products of higher spatial resolution produced either irregularly or only in the vicinity of designated areas and times, such as NOAA active regions and selected comparison quiet sun regions. We propose that both the full-disc and synoptic data products for transverse velocities be sampled at a spatial scale corresponding to 2.5 deg heliographic (~30 Mm), with analysis regions of diameter 5 deg, extending out to mu = 0.986, about 80 deg from disc center. At this distance the area of foreshortened HMI pixels is comparable to that of the limit at which we are able to extract useful ring-diagram fits from MDI data. We also propose that the temporal sampling correspond to 5 degrees of Carrington rotation (1/72 of a synodic rotation, ~545 min), i.e. the time for a sampled region to rotate through its diameter. In order to invert for velocities (and other parameters) to a depth of 30 Mm, if that is even possible, it will certainly be necessary to use fits of higher-order modes only achievable with sampling areas of diameter significantly larger than 5 deg. Therefore, we intend to produce multiple full-disc datasets of inverted velocities with spatial sampling ranging from 2.5 deg to 15 or even 22.5 deg, and to combine the data from all such sets into the 2.5 deg Carrington maps, by an averaging procedure to be developed. The data for the deepest layers of such Carrington maps will of course be vastly oversampled. The data products at each grid point in the Carrington maps of sub-surface flows will be the two quantities Ux and Uy representing the zonal and meridional components of local advection relative to the rotating analysis frame, together with their formal uncertainties. The data values will be interpolated to a fixed but unequally spaced grid of depths; we propose to interpolate to a total of 16 depths between the surface and a depth of 30 Mm. The number of samples in depth of the Carrington maps should be the same as that for the full-disc data products at the largest spatial scale. Because we expect to be able to do ring diagram analysis so close to the limb with HMI resolution, the annual effect of the change in the observer heliographic latitude must be considered. It will allow us to reach, at some times of year, 5-deg diameter regions centered at latitudes as high as 85 deg in each hemisphere, and the extreme accessible longitudinal extent at each latitude will vary over the course of the year. Also, with our ability to reach to high latitudes, the longitude spacing of the analysis grid should be adjusted to retain a similar heliometric spatial sampling. We propose to have the longitude spacing in the 2.5-deg grid be 2.5 deg at latitudes between +/- 40.0 deg, 5.0 deg at latitudes between 42.5 deg and 65.0 deg in each hemisphere, 7.5 deg at latitudes between 67.5 deg and 72.5 deg in each hemisphere, and 10.0 deg at latitudes +/-75.0 and above. With these spacings, the number of possible grid points on each Carrington map, and each full-disc map at the 2.5-deg spacing, will be 3007. Of these, 2481 will always be accessible. 242 will be accessible at some times in both hemispheres (i.e. inaccesible in one hemisphere only part of the year), and 142 will be accessible in only one hemisphere part of the year. (The distinction between the latter two classes might be thought of as accessible except in "winter", and only accessible in "summer".) How finally to divide the regions of accessibility by the latitude of disc center needs to be established, but for now we propose dividing the year into quadrants with the boundaries at B0 = 0.0, +/-3.625, and +/-7.25 deg. Thus, the northern hemisphere classes, for example, would be those accessible all year, those accessible except when B0 < -3.625 (~8 months), and those only accessible when B0 >= 3.625 (~4 months). The spatial grid may be visualized with this sample for one quadrant. Points marked "A" will be included always, those marked "2" will be included in that hemisphere 8 months of the year (and both hemispheres 4 months), and those marked "1" will be included 4 months of the year. The grid is labeled by the target latitudes and longitudes in tenths of a degree. Only every other potential grid point in longitude is labeled, though all are included. lon 000 050 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 lat 850 1 1 1 1 1 1 825 1 1 1 1 1 1 1 1 800 2 1 1 1 1 1 1 1 775 2 2 2 2 1 1 1 1 750 2 2 2 2 2 1 1 1 725 A A 2 2 2 2 2 1 1 1 1 700 A A A A 2 2 2 2 1 1 1 675 A A A A A 2 2 2 1 1 1 650 A A A A A A A A A 2 2 2 2 1 1 1 625 A A A A A A A A A A A 2 2 2 1 1 600 A A A A A A A A A A A 2 2 2 1 1 575 A A A A A A A A A A A A 2 2 1 1 550 A A A A A A A A A A A A A 2 1 1 525 A A A A A A A A A A A A A 2 1 1 500 A A A A A A A A A A A A A 2 2 1 475 A A A A A A A A A A A A A 2 2 1 450 A A A A A A A A A A A A A 2 2 1 425 A A A A A A A A A A A A A A 2 1 400 A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 2 2 1 1 375 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 2 1 1 350 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 2 1 1 325 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 2 1 1 300 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 1 1 275 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 1 1 250 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 2 1 225 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 2 1 200 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 1 175 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 1 150 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 1 125 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 1 100 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 2 75 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 50 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 25 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A 00 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A For the coarser spatial grids of full-disc inversions extending to greater depths, the spacings and extents need to be adjusted accordingly, and it makes sense to sample correspondingly less frequently in time, with better frequency resolution, or at least better signal-to-noise ratios, as well. We propose a 7.5-deg grid sampled every 15 deg in Carrington rotation (~1635 min), similar to the MDI "dense-pack" mosaics but extending further from disc center, and a 15-deg grid sampled every 30 deg in Carrington rotation (~54.5 hr). It may be necessary to add a 45-deg or even larger grid to extend the depth inversions to 30 Mm if that proves feasible. For the 7.5-deg grid, approximately equal spatial sampling suggests that the longitude sampling should be 7.5 deg between +/-30.0 deg latitude, 10.0 deg at 37.5 and 45.0 deg in each hemisphere, 12.5 deg at +/-52.5 deg, 15.0 deg at +/-60.0 deg, 20.0 deg at +/-67.5 deg, and 30.0 deg at +/-75.0 deg. 0.0 7.5 7.5 7.5 15.0 7.5 22.5 7.5 30.0 7.5 37.5 10.0 45.0 10.0 52.5 12.5 60.0 15.0 67.5 20.0 75.0 30.0 Similarly the spacings for the 15-deg grid should be, in each hemisphere, 0.0 15.0 15.0 15.0 30.0 15.0 45.0 15.0 60.0 30.0 (Strictly speaking, the better spacings in longitude would be 17.5 deg at latitude 30.0 and 20.0 at latitude 45.0, but the comparatively small differences involved in oversampling here do not outweigh the advantage of uniformity.) The target quadrant grids for the 7.5-deg and 15.0-deg full-disc grids are as follows: lon 000 050 100 150 200 250 300 350 400 450 500 550 600 650 700 750 lat 750 1 675 A A 2 1 600 A A A A 2 1 525 A A A A A 2 1 450 A A A A A A A 2 375 A A A A A A A 2 300 A A A A A A A A A A 1 225 A A A A A A A A A A 1 150 A A A A A A A A A A 1 75 A A A A A A A A A A 00 A A A A A A A A A A (Total of 307, of which 261 always accessible) Note the suggested spacings of 12.5 deg in longitude at latitudes +/-52.5 deg. This is a problem because it is not am integer divisor of 360, thus the selected longitudes would not repeat in each rotation. They should be reduced to 12 deg, for a cadence of 30 samples per rotation. Also, it is not evident how often the single longitude at latitude +/-75 deg should be resampled. I suggest once every 90 deg in longitude, since tracking of all regions will presumably be done for about the duration of their disk passage. lon 000 050 100 150 200 250 300 350 400 450 500 550 600 650 700 750 lat 600 A A 1 450 A A A A 2 300 A A A A A 150 A A A A A 00 A A A A A (Total of 73, of which 65 always accessible) Because the 2.5-deg grid analyses will probably not be able to fit modes of higher order than 2, we expect about 4 depths; the 15-deg grid should have about 10 depths. For the full-sun inversions, the data at each 3-dimensional grid point, will include the best-fit inversion depth and formal uncertainty as well as the values and uncertainties for Ux and Uy. TRACKING RATE The analysis cubes will be extracted from regions tracked across their full disc passage, or at least a sufficient fraction of it to support all of the required tiles. In order to sample the "same" region from such supersets of tracking intervals, it seems necessary to track all regions at the Carrington rate. This will of course introduce "spurious" zonal flows corresponding to the mean differential rotational rate of latitude and depth. The difference between the Carrington rate and the supposed surface differential rotation rate (the Snodgrass rate currently in standard use) has a maximum value of about 263 m/s at about 50 deg latitude. (The maximum differential between the Newton & Nunn rate and Carrington rotation is about 176 m/s at about 45 deg latitude.) Since the existing ring-fitting code has been demonstrated to work even with untracked data (zonal "flows" of up to 2 km/s) this should not be a problem. It may even be arguably a convenience to reference the mean zonal flows as a function of depth and latitude to a uniform rate, just as is done in the surface functional forms of differential rotation. The duration of tracking of the original cubes, from which the analysis cubes are to be extracted, must be long enough to permit the extraction of all tiles described above. This means that the duration of tracking should extend as far as the outermost longitude tile plus the spacing. For example, for the 15-deg grid, the regions at low latidudes will be tracked for at least 150 deg in Carrington rotation (twice 60 + 15). The region at latitude +60 will be tracked for 180 degrees in rotation (twice 60 + 30) in the four months that B0 > 3.625 deg, 120 degrees the rest of the year. The only difficulty is for the regions at 75 deg latitude in the 7.5 deg grid, tracked only at times of extreme B0, for which there is no longitude spacing, as the single tracked region accounts for the full available distance to the limb. I suggest that the region be tracked for 180 degrees of rotation, with a new one initiated every 90 degrees in rotation. (This is equivalent to an effective spacing of 90 deg in longitude.) MAPPING SCALE The mapping scale in use for the standard "dense-pack" tracking with MDI was chosen to be 0.125 deg/pixel because this is close to the optical image scale of the instrument and it is a power of 2 divisor for maps of 16-degree extent (in Postel's projection). These values are somewhat flexible, since the extent of the tracked regions need only be a small amount larger than the desired apodized diameter for analysis - 15-deg in that case. For HMI the image scale will be about 3 times larger than for MDI, about 0.05 deg/pixel. (The measured image scale for HMI is 0.509 +/- 0.003 arcsec/pixel, depending on focus position. This translates to a heliographic image scale at disc center of 0.0466 deg/pixel to 0.04875 deg/pixel, depending on focus and time of year as well.) For the larger-scale tiles (15 and 30 deg), it is probably acceptable to map to a slightly degraded map scale of 0.0625 deg/pixel. This allows the tracked cubes to have areas of powers of 2 pixels (256^2 and 512^2 pixels respectively) while retaining the standard circular apodization annuli of 15/16 in the square fields, an apodization width of 0.9375. For the 5-deg tiles, 128^2 areas would allow for a map scale of 0.04 deg/pixel and map extent of 5.12 deg, corresponding to a tighter apodization width of 0.9765. A map scale of 0.45 deg/pixel would yield a map extent of 5.76 deg for an apodization width of 0.8681. Preserving the apodization width of 15/16 would require tiles of width 5-1/3 deg, which would translate to a map scale of 0.041666.. deg /pixel. An alternative is to abandon power-of-two sizes for the data cubes. We could then match an apodization width of 15/16 and a slightly degraded map scale of 0.0555 deg/pixel with areas of 96^2 pixels for example. RECOMMENDATION: Use map scale of 0.04 deg/pixel and field width of 5.12 deg. If the above recommendation is adopted, we might also consider using the same map scale of 0.04 deg for all sets of regions and adopt the uniform apodization width of 0.9765 (250/256), making the tiles for the 30-deg and 15-deg regions 15.36 and 30.72 deg in width, respectively; however, that would necessitate tile sizes of 384^2 and 768^2 rather than powers of two.